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+Learning to Memorize Entailment and Discourse Relations for Persona-Consistent
+Dialogues
+Ruijun Chen1, Jin Wang1*, Liang-Chih Yu2 and Xuejie Zhang1
+1School of Information Science and Engineering, Yunnan University, Yunnan, China
+2Department of Information Management, Yuan Ze University, Taiwan
+[email protected], [email protected], [email protected], [email protected]
+Abstract
+Maintaining engagement and consistency is particularly im-
+portant in dialogue systems. Existing works have improved
+the performance of dialogue systems by intentionally learn-
+ing interlocutor personas with sophisticated network struc-
+tures. One issue with this approach is that it requires more
+personal corpora with annotations. Additionally, these mod-
+els typically perform the next utterance prediction to gener-
+ate a response but neglect the discourse coherence in the en-
+tire conversation. To address these issues, this study proposes
+a method of learning to memorize entailment and discourse
+relations for persona-consistent dialogue tasks. Entailment
+text pairs in natural language inference dataset were applied
+to learn latent entailment relations as external memories by
+premise-to-hypothesis generation task. Furthermore, an in-
+ternal memory with a similar architecture was applied to the
+discourse information in the dialogue. Placing orthogonality
+restrictions on these two memory spaces ensures that the la-
+tent entailment relations remain dialogue-independent. Both
+memories collaborate to obtain entailment and discourse
+representation for the generation, allowing a deeper under-
+standing of both consistency and coherence. Experiments on
+two large public datasets, PersonaChat and DSTC7-AVSD,
+demonstrated the effectiveness of the proposed method. Both
+automatic and human evaluations indicate that the proposed
+model outperforms several strong baselines in terms of both
+persona consistency and response coherence. Our source
+code is available at https://github.com/Chenrj233/LMEDR.
+Introduction
+Traditional chit-chat models lack specificity and personal-
+ity consistency. Only when they access a sufficiently large
+dataset will they have the opportunity to generate piecemeal
+and uninformative responses in a chit-chat setting. For two
+consecutive questions with similar meanings in a two-round
+dialogue, that is, what is your job and what do you do, the
+model replied to the former: I am a lawyer, while the lat-
+ter: I am a doctor (Welleck et al. 2020). This issue arises
+because of the lack of a consistent personality as well as
+an explicit memory towards plausibility as they are typically
+trained to produce a response given only recent dialogue his-
+tory (Shum, He, and Li 2018).
+*Corresponding author
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+* I attend church regularly.
+* I love to watch scary things.
+* My favorite hobby is reading novels.
+* I work at a veterinarians office.
+* I enjoy exercising.
+* I enjoy teaching things to children.
+* I work for no money.
+* We have a boy and a girl.
+Persona (Human)
+Persona (Model)
+  Hi, what do you do for fun?
+  Hi, i usually go to yoga a few times a week. What about you?
+  Well, i like going to watch scary movies and i love reading.
+  I am actually an english teacher.
+  It is good to know there are good people out there!
+  Thank you. That's what i try to instill in my kids.
+  How many kids do you have?
+  I have 2 of my own , as well as 30 in my class.
+* I attend church regularly.
+* I love to watch scary things.
+* My favorite hobby is reading novels.
+* I work at a veterinarians office.
+* I enjoy exercising.
+* I enjoy teaching things to children.
+* I work for no money.
+* We have a boy and a girl.
+Persona (Human)
+Persona (Model)
+  Hi, what do you do for fun?
+  Hi, i usually go to yoga a few times a week. What about you?
+  Well, i like going to watch scary movies and i love reading.
+  I am actually an english teacher.
+  It is good to know there are good people out there!
+  Thank you. That's what i try to instill in my kids.
+  How many kids do you have?
+  I have 2 of my own , as well as 30 in my class.
+Entailment
+Entailment
+Incoherent
+Incoherent
+Coherent
+Coherent
+Entailment
+Incoherent
+Coherent
+Figure 1: The conceptual diagram of introducing natural lan-
+guage inference in persona-based dialogue.
+One solution that maintains consistency in a dialogue sys-
+tem is to provide a set of persona profiles that describe the
+character and then generate responses according to the per-
+sona. Persona can be defined as the composition of identity
+elements, such as profiles and background personal facts.
+The expected outcome is that dialogue models gener-
+ate a response consistent with the given persona. The Per-
+sonaChat dataset (Zhang et al. 2018), widely adopted to sup-
+port the training of persona-consistent dialogues, was man-
+ually annotated by two annotators to act as part of a prede-
+fined persona and chat naturally to know each other during
+the conversation. However, given the time and effort needed
+to annotate more persona corpora to cover all possibilities, it
+is difficult to extend the application of such persona-related
+information to the daily usage of dialogue.
+As humans, our knowledge of the concepts and the se-
+mantic relationship behind the language can allow us to re-
+arrange unstructured data so that we can understand and an-
+alyze it. Essentially, we can robustly learn novel concepts
+with minimal supervision, benefitting from the well-known
+ability of natural language inference (NLI). Figure 1 shows
+an example of introducing NLI in a persona-based dialogue.
+Given a persona as a premise, we can determine whether
+the hypothesis of the response utterance is true (entailment),
+false (contradiction), or undetermined (neutral).
+Recent studies have sought to improve the consistency
+arXiv:2301.04871v1  [cs.CL]  12 Jan 2023
+
+[SOP]
+[z]
+P1
+P|P|
+[EOP]
+P2
+...
+BART Encoder
+h[z]
+h[SOP]
+h[EOP]
+Premise
+Transformer 1
+Transformer 11
+Transformer 12
+e[z]
+e[SOP]
+e[EOP]
+...
+
+M
+z
+H1
+[SOH]
+H2
+H|H|
+H3
+...
+BART Decoder
+H1
+H2
+H3
+H4
+H|H|
+[EOP]
+Transformer 1
+Transformer 11
+Transformer 12
+e[SOH]
+...
+Entailment 
+Relation Memory
+Hypothesis
+1Pe
+2
+Pe
+2
+P
+h
+1P
+h
+| |
+Pe
+P
+| |
+P
+h
+P
+|
+| 1
+H H
+1
+H
+e
+2
+H
+e
+3
+H
+e
+| | 1
+H
+e
+H
+| |
+H
+e
+H
+...
+...
+ERM
+Figure 2: Learning to memorize the entailment relations in latent variables.
+of the dialogue system by modeling the understanding be-
+tween interlocutors (Liu et al. 2020). Song et al. (2021)
+disentangled persona-based dialogue generation into two
+subtasks—response generation and consistency understand-
+ing—and used unlikelihood training to make the decoder
+generate contradictory dialogue responses as few as possi-
+ble. However, multiple subtasks require multiple encoders,
+leading to a complex generation model structure. Nie et al.
+(2021) introduced a contradiction detection task to evaluate
+the consistency in dialogues.
+Despite continuing efforts to improve the engagement and
+consistency of dialogue systems, understanding persona-
+response consistency is still difficult. The key challenges
+are twofold: 1) Existing methods apply sophisticated struc-
+tures to learn persona consistency, which requires more an-
+notated corpora for training. However, persona-based cor-
+pora are still insufficient and difficult to collect. 2) Dialogue-
+generating models typically neglect discourse information.
+Discourse coherence is a crucial component of the effec-
+tiveness of a conversation, encompassing how utterances are
+connected and how the entire dialogue is organized to con-
+vey information to the interlocutor. Existing models usually
+perform the next utterance prediction for response genera-
+tion but ignore the dialogue discourse coherence. As indi-
+cated in Figure 1, I am actually an English teacher seems
+to be an appropriate and persona-consistent response to the
+query. However, this response is incoherent in the context of
+an entire conversation.
+To address these issues, this study proposes a method of
+learning to memorize entailment and discourse relations for
+persona-consistent dialogue tasks. We applied an encoder-
+decoder architecture from BART (Lewis et al. 2020). To ex-
+plicitly understand the consistency of personas, we designed
+an external memory to store the latent entailment relations
+between premises and the entailment hypothesis, indepen-
+dent of dialogue itself. In addition, discourse relations were
+learned and stored in internal latent memory. The latent en-
+tailment relations are ensured to be dialogue-independent
+by imposing orthogonality constraints on the two memory
+spaces. Given personas and dialogue queries, both memories
+work jointly to obtain the entailment and discourse repre-
+sentation by the BART encoder. The generation was finally
+accomplished by the BART decoder with two extra training
+objectives, which further acquired the ability to understand
+both consistency and coherence.
+Comparative experiments were conducted using the Per-
+sonaChat (Dinan et al. 2020) and DSTC7-AVSD (Alamri
+et al. 2019). Both automatic and human evaluations show
+that the proposed method generalizes well under different
+settings and outperforms several strong baselines on most
+metrics, especially persona consistency, indicating that the
+proposed method can produce better persona-consistent dia-
+logue responses.
+The remainder of this paper is organized as follows. Sec-
+tion 2 provides a brief review of the related work. Section
+3 describes the proposed model, which learns to memorize
+entailment and discourse relations by using latent variables.
+Section 4 summarizes the specific experimental setup for the
+two public dialogue datasets and the corresponding analysis
+of the results. Finally, conclusions are drawn in Section 5.
+Related Work
+Persona-based Dialogues
+Generation-based
+dialogue
+systems
+usually
+use
+the
+sequence-to-sequence (seq2seq) model (Sutskever, Vinyals,
+and Le 2014) as the backbone. After the persona is intro-
+duced into the dialogue, it is necessary to adopt an effective
+method to integrate role information into the dialogue, such
+as persona embedding (Li et al. 2016b). Subsequently, with
+the development of large-scale pre-trained language models,
+an increasing number of methods (Wolf et al. 2019; Roller
+et al. 2021; Lin et al. 2021; Zheng et al. 2020) have lever-
+aged pre-training and fine-tuning to improve persona-based
+dialogue, but the problem of dialogue consistency remains
+unsolved. Therefore, Liu et al. (2020) have attempted to
+model the understanding between interlocutors to improve
+the consistency of dialogue systems. A new perspective
+(Song et al. 2021) decomposes persona-based dialogue
+tasks into consistent understanding and dialogue generation
+significantly improves dialogue-consistent generation based
+on natural language inference.
+
+Latent Modeling
+In a dialogue scene, the factors that associate dialogue con-
+text with dialogue responses are often difficult to observe
+and explain; therefore, modeling the latent space of dialogue
+can help improve the performance of dialogue generation.
+Optimus (Li et al. 2020) combines the advantages of BERT
+(Devlin et al. 2019) and GPT-2 (Radford et al. 2020) for
+large-scale pre-training in the form of VAE (Kingma and
+Welling 2014) to model the latent variable space. PLATO
+(Bao et al. 2020) introduces discrete latent variables to
+solve the one-to-many relationship in response generation.
+DialogVED (Chen et al. 2022) introduces continuous la-
+tent variables into an enhanced encoder-decoder pre-training
+framework to improve the relevance and diversity of dia-
+logue responses. All these methods show great promise for
+modeling dialogue-related features in latent space. This pa-
+per extents the idea by additionally memorizing NLI rela-
+tions as latent dialogue-independent features.
+Learning to Memorize for Persona-consistent
+Dialogue
+The task of dialogue generation can be defined as the
+next utterance prediction, where a target response utter-
+ance R = [r1, r2, ..., r|R|] is predicted given a conversa-
+tion query Q = [q1, q2, ..., q|Q|] according to given persona
+constraints C = [c1, c2, ...c|C|]. For convenience, the sen-
+tences (R, Q, C) are mapped to the vector representation
+x = {R, Q, C}. Further, natural language inference data
+(Welleck et al. 2020; Williams, Nangia, and Bowman 2018)
+N = {P (n), H(n)}N
+n=1, which consists of the entailed text
+pairs of premise and hypothesis, was used to learn the entail-
+ment relation to preserve consistency in dialogue generation.
+Figure 2 shows the overall architecture of the proposed
+learning to memorize the entailment and discourse rela-
+tions model for a persona-consistent dialogue. The backbone
+model is based on BART (Lewis et al. 2020), which per-
+forms repeated two-stage training, i.e., learning to memorize
+and persona-consistent dialogue generation. The key insight
+of the proposed model is that it maps both the entailment
+relation and discourse information to latent spaces. Based
+on this information, an external memory module enforces
+premise-to-hypothesis generation to map the textual entailed
+pair to the Dialogue-Independent latent space, which can
+be memorized and stored in a memory structure M. Simi-
+larly, the discourse information was mapped using an inter-
+nal memory module N to learn the Dialogue-Related fea-
+tures. For generation, both entailment and discourse repre-
+sentation can be obtained from memory and enhance per-
+sona consistency in dialogue generation with additional en-
+tailment and discourse information.
+Learning to Memorize
+Entailment Relation Memory (ERM).
+ERM is an exter-
+nal memory which is used to learn and store entailment re-
+lations for persona consistency. If a given hypothesis H can
+be inferred from the premise P, the relationship of the pair
+is entailment. For persona-based dialogue, such an entail-
+ment relationship can be introduced to generate consistent
+responses.
+Given
+a
+dataset
+of
+textual
+entailed
+pairs
+N
+=
+{P (n), H(n)}N
+n=1,
+textual
+entailment
+generation
+was
+adopted to learn a latent variable z, which represents the
+latent form of entailment relations in natural language
+inference, defined as
+p(H, z |P ) = p(z |P )p(H |z , P)
+(1)
+Based on the BART encoder, we introduce a special la-
+tent token [z], a start-of-premise token [SOP], and an end-
+of-premise token [EOP] to the premise for latent entailment
+relation learning. By using the tokenizer and adding position
+embeddings, the input of the premise is transformed as
+EERM = [e[z], e[SOP], ep1, ep2, ..., ep|P |, e[EOP]]
+(2)
+We introduce a latent entailment relation memory struc-
+ture M parameterized by θ, where each element represents
+a certain latent factor, defined as
+M = [M1, ..., Mk] ∈ Rk×d
+(3)
+where k is the number of latent factors in entailment rela-
+tions, and d is the dimension of the memory element. The
+hidden state of the last layer of the BART encoder, that is,
+h[z] corresponding to e[z], was applied to learn the distribu-
+tion of the latent entailment relations z ∼ p(z |P ) by
+π = softmax(Wπh[z] + bπ)
+(4)
+where π represents the probability of each element in M.
+Then, the latent entailment representation z can be easily
+obtained from M:
+z =
+k
+�
+i=1
+πiMi
+(5)
+To memorize the latent entailment relations, we use en-
+tailment representation z from the memory M with the
+weights π, along with the premise to generate the corre-
+sponding hypothesis. The obtained entailment representa-
+tion z was added to the special start-of-hypothesis token
+[SOH] of the decoder, denote as
+ˆe[SOH] = e[SOH] + z
+(6)
+Latent memory can keep track of the entailment relation
+with the representation of the source premise by both read-
+ing and writing during generation. Notably, it can be up-
+dated by backpropagation of the premise-to-hypothesis gen-
+eration.
+The objective of the pre-training is to optimize memory
+M and model parameters ϕ by minimizing the language
+modeling loss:
+LERM = − Ez∼pθ,ϕ(z|P ) log pθ,ϕ(H |z, P)
+= − Ez∼pθ,ϕ(z|P )
+|H|
+�
+t=1
+log pθ,ϕ(Ht |z, P, H<t)
+(7)
+
+Bag of Words
+Prediction
+Orthogonal 
+Constraint
+[SOP]
+[z]
+c1
+c|C|
+[EOP]
+c2
+...
+BART Encoder
+h[z]
+Persona
+Transformer 1
+Transformer 11
+Transformer 12
+e[z]
+...
+
+M
+z
+r1
+[SOR]
+r2
+r|R|-1
+r|R|
+r3
+...
+BART Decoder
+r1
+r2
+r3
+r4
+r|R|
+[EOR]
+Transformer 1
+Transformer 11
+Transformer 12
+e[SOR]
+...
+Entailment 
+Relation Memory
+Response
+...
+...
+Response 
+Generation
+[PER]
+[z]
+c1
+[RSP]
+... [QRY] ...
+...
+Persona
+Query
+Response
+N
+zd
+Dialogue Discourse 
+Memory
+
+Multiple 
+Choice
+[QRY]
+Query
+Q
+Figure 3: Overall architecture of the proposed method for persona-consistent dialogue generation.
+Algorithm 1: Latent memory learning
+Input: A set of entailed text pair N = {P (n), H(n)}N
+n=1 and
+persona dialogue dataset x = {R, Q, C}.
+Parameter: Memory M, N parameterized by θ, φ, a pre-
+trained BART parameterized by ϕ.
+1: repeat
+2:
+Stage 1:
+3:
+Train the BART model and memory M with input N.
+4:
+Minimizing Eq. (7) and optimize θ, ϕ.
+5:
+Stage 2:
+6:
+Fixed parameter θ.
+7:
+Train the BART model and memory N with input x.
+8:
+Minimizing Eq. (19) and optimize φ, ϕ.
+9: until convergence
+Dialogue Discourse Memory (DDM).
+Discourse coher-
+ence is an important aspect of dialogue text quality. It en-
+compasses how utterances are connected, as well as how the
+entire dialogue is organized to convey information to the in-
+terlocutor (Bao et al. 2020). Similarly, we memorized the
+discourse information into an internal memory N parame-
+terized by φ, as shown in Figure 3. Let C = [c1, c2, ...c|C|]
+denote the persona of the agent, Q(1), ..., Q(m) represent m
+dialogue queries, and R(1), ..., R(m) represent the target re-
+sponses. Here, Q and R are consecutive context-response
+pairs from the same dialogue session, which are used to cap-
+ture the correct discourse relation in a dialogue. The input of
+the BART encoder is the concatenation of the persona and
+dialogue content, denoted as
+EDSM = [e[z], e[PER], C, e[QRY], Q(1), e[RSP], R(1),
+..., e[QRY], Q(m)]
+(8)
+where [QRY] and [RSP] are two special tokens that indicate
+the beginning of the query and the response, respectively.
+Latent memory N ∈ Rl×d was introduced, where l rep-
+resents the kind of latent dialogue discourse information. It
+learns the distribution of the latent dialogue discourse using
+ρ = softmax(Wρh[z] + bρ)
+(9)
+Then, the latent dialogue discourse representation was
+calculated by
+zd =
+l
+�
+j=1
+ρjNj
+(10)
+In addition, latent entailment relations should be indepen-
+dent of the dialogue context. Therefore, we propose impos-
+ing an orthogonal constraint on these two memory spaces
+to reduce the correlation between the different latent spaces.
+The orthogonal constraint can encourage the latent memory
+to learn more features and reduce redundant features. To ac-
+complish this goal, we used cosine similarity to represent the
+relationship between them, described as
+cos(M, N) =
+MN⊤
+∥M∥2∥N∥2
+(11)
+where ∥·∥2 represents L2 normalization. Once the vectors of
+different latent-memory spaces are orthogonal to each other,
+the correlation between them is the lowest. Thus, the training
+objective of the DDM can be defined to minimize the cosine
+similarity between M and N:
+LDDM =
+�
+i≤k,j≤l
+�
+MiN ⊤
+j
+∥Mi∥2∥Nj∥2
+�2
+(12)
+
+Model
+Original
+Revised
+Hits@1
+PPL
+F1
+Hits@1
+PPL
+F1
+KV Profile Memory
+54.8
+-
+14.25
+38.1
+-
+13.65
+Generative Profile Memory
+10.2
+35.01
+16.29
+9.9
+34.94
+15.71
+LIC
+17.3
+-
+17.79
+16.2
+-
+16.83
+Transfertransfo
+82.1
+17.51
+19.09
+-
+-
+-
+P2BOT
+81.9
+15.12
+19.77
+68.6
+18.89
+19.08
+Our
+89.5
+10.99
+21.99
+85.0
+13.42
+19.54
+Table 1: Automatic evaluation results of different methods for persona-based dialogue generation.
+Persona-consistent Responses Generation
+For generation, the model leverages latent memories of both
+entailment relations and dialogue discourse to preserve con-
+sistency for generating a persona-based response, denoted
+as
+p (R |Q, C ) = p(R
+��z, zd, Q, C)p(z
+��C)p(zd |Q, C)
+(13)
+We applied the persona as a premise and fed it into the
+model and obtained the latent entailment relations represen-
+tation z under the condition of persona from latent entail-
+ment memory M. The latent entailment relation represen-
+tation z and the latent dialogue discourse representation zd
+are then added to the special start-of-response token e[SOR]
+of the decoder input, denoted as
+ˆe[SOR] = e[SOR] + z + zd
+(14)
+Additionally, we use a bag-of-words loss (Zhao, Zhao,
+and Eskenazi 2017) to facilitate the latent variable,
+LBOW = − Ez∼pϕ(z|C),zd∼pφ,ϕ( zd|C,Q)
+|R|
+�
+t=1
+log pφ,ϕ(Rt
+��C, Q, z, zd)
+= − Ez∼pϕ(z|C),zd∼pφ,ϕ( zd|C,Q)
+|R|
+�
+t=1
+log
+ef(Rt)
+�
+v∈V
+ef(v)
+(15)
+where V is the whole vocabulary and f (v) represents the
+predicted probability of token v appearing in the target re-
+sponse.
+For response generation, we also trained the parameters
+of the model using a language modeling loss function:
+LLM = − Ez∼pϕ(z|C),zd∼pφ,ϕ( zd|C,Q) log pφ,ϕ(R
+��C, Q, z, zd )
+= − Ez∼pϕ(z|C),zd∼pφ,ϕ( zd|C,Q)
+|R|
+�
+t=1
+log pφ,ϕ(Rt
+��R<t, C, Q, z, zd)
+(16)
+Similar to (Wolf et al. 2019), we randomly sampled t in-
+terference responses to train the model to select the correct
+response, i.e., as a multiple-choice task. Specifically, we use
+the hidden state of the last token output by the decoder to
+predict the scores ˆyi of each candidate response and calcu-
+late the cross-entropy loss with the ground-truth label yi, de-
+noted as
+ˆy = softmax(Whheos + bh)
+(17)
+LCLS = −
+t
+�
+i=1
+ˆyi log(yi)
+(18)
+Model
+Hits@1
+PPL
+F1
+PE-Trans
+89.4
+-
+-
+ImageS2S
+-
+11.19
+21.30
+BART
+86.9
+11.85
+20.72
+Our
+89.5(90.1)
+10.99(11.00)
+21.99(21.96)
+Table 2: Comparison with pre-trained language models fine-
+tuned on PersonaChat original mode.
+Model
+Dist-1
+Dist-2
+C.Score
+LIC
+2.31
+10.71
+19.13
+P2BOT
+1.87
+10.08
+23.84
+BOB
+2.59
+13.90
+22.45
+Our
+2.47
+13.82
+25.31
+Table 3: Automatic evaluation results of different methods
+for persona consistency.
+In summary, the objective of our model is to minimize the
+loss:
+L(φ, ϕ) = LDDM + LBOW + LLM + LCLS
+(19)
+While training persona-consistent responses generation,
+we fixed the parameters of M; the specific process is de-
+scribed in Algorithm 1.
+Experiments
+Dataset
+Dialogue Dataset.
+We conducted experiments on two
+publicly available dialogue datasets to evaluate the perfor-
+mance of the proposed method.
+• ConvAI2 PersonaChat (Dinan et al. 2020) is a chi-
+tosan–chat dataset based on PersonaChat (Zhang et al.
+2018). It provides the personas of the interlocutor and is
+designed to facilitate dialogue models to generate more
+consistent and engaging responses.
+• DSTC7-AVSD (Alamri et al. 2019) provides a conver-
+sational question-answering dataset, which is similar to
+that of PersonaChat. We used this dataset to explore the
+performance of our proposed method on a contextual
+knowledge-based dialogue task. The system must gener-
+ate answers based on the given context and background
+
+Model
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+METEOR
+ROUGE-L
+CIDEr
+CMU
+0.718
+0.584
+0.478
+0.394
+0.267
+0.563
+1.094
+PLATO
+0.784
+0.637
+0.525
+0.435
+0.286
+0.596
+1.209
+ProphetNet
+0.824
+0.691
+0.582
+0.487
+0.313
+0.635
+1.382
+DialogVED
+0.822
+0.692
+0.582
+0.489
+0.312
+0.636
+1.391
+Our
+0.801
+0.680
+0.576
+0.488
+0.316
+0.631
+1.403
+Table 4: Automatic evaluation results of different methods for DSTC7-AVSD.
+Model
+Fluency
+Consistency Avg
+LIC
+3.27
+2.11
+2.69
+P2BOT
+3.51
+2.20
+2.86
+Our
+3.57
+2.31
+2.94
+Table 5: Human evaluation results.
+Hits@1
+F1
+BLEU-4
+C.Score
+BART
+86.9
+20.72
+0.01289
+21.32
+w/o ERM
+89.3
+21.70
+0.01597
+22.09
+w/o DDM
+88.4
+21.72
+0.01406
+24.66
+w/o OC
+88.8
+21.84
+0.01485
+25.01
+Our
+89.5
+21.99
+0.01561
+25.31
+Table 6: Analysis of ablation experiments on the Per-
+sonaChat original mode.
+knowledge of the conversation. We used only text infor-
+mation for the experiments.
+NLI Dataset.
+Additionally, two natural language infer-
+ence datasets were used to learn the corresponding ERM.
+• DNLI (Welleck et al. 2020) is a dialogue inference
+dataset based on PersonaChat. The dataset consisted of
+sentence pairs labeled as entailment, neutral, or contra-
+dictory.
+• MNLI (Williams, Nangia, and Bowman 2018) is a multi-
+genre natural language inference corpus and one of the
+largest corpora available for recognizing textual entail-
+ments.
+Compared Methods
+Baseline.
+Both KV profile memory and generative pro-
+file memory (Zhang et al. 2018) are the official baselines
+for PersonaChat, whereas LIC (Golovanov et al. 2020) was
+the best performing model in ConvAI2 for human evalua-
+tion, and Transfertransfo (Wolf et al. 2019) achieved the
+best performance on automatic metrics in ConvAI2. For
+DSTC7-AVSD, CMU (Sanabria, Palaskar, and Metze 2018)
+achieved the best performance for all evaluation metrics in
+the competition.
+State of the art.
+P2BOT (Liu et al. 2020) models the
+understanding between the interlocutors and obtains a new
+state-of-the-art on ConvAI2. Poly-Encoder Transformer
+(PE-Trans) (Humeau et al. 2019) was pre-trained on Reddit
+and fine-tuned on ConvAI2 to obtain the best performance
+in the response selection task. Imageseq2Seq Dodecadi-
+alogue (ImageS2S) (Shuster et al. 2020) was trained on 12
+tasks and fine-tuned on the ConvAI2 generation task. BOB
+(Song et al. 2021) disentangles persona-based dialogue gen-
+eration into consistent understanding and dialogue genera-
+tion tasks to ensure that the model has a better consistent un-
+derstanding. For DSTC7-AVSD, PLATO (Bao et al. 2020)
+was the first large-scale pre-trained dialogue language model
+that introduced a discrete variable for one-to-many relation-
+ship modeling. ProphetNets (Qi et al. 2020) proposed a
+pre-training objective for predicting multiple future tokens
+to enhance the performance of pre-trained language mod-
+els on natural language generation tasks. DialogVED (Chen
+et al. 2022) is a pre-trained model framework to enhance the
+encoder-decoder by introducing continuous latent variables
+and obtaining the state-of-the-art on the DSTC7-AVSD.
+Implementation Details
+The proposed model was initialized using BART-large.
+AdamW (Loshchilov and Hutter 2019) was applied to op-
+timize the model, with an initial learning rate of 8e-6. We
+used DNLI on PersonaChat and MNLI on DSTC7-AVSD
+for the ERM learning. The batch size was 64 for training
+stage 1, and we used a batch size of two with a gradient ac-
+cumulation of eight for training stage 2. The types of ERM
+and DDM were both set to 10 for PersonaChat and set to
+20 and 5 for DSTC7-AVSD. The proposed model trained on
+one NVIDIA RTX 3090 with PyTorch framework. For dia-
+logue generation, we used a beam search, and the maximum
+sequence length was set to 50.
+Automatic Evaluation
+Following the official automatic evaluation, we used
+Hits@1, Perplexity (PPL), and F1 for automatic evaluation
+on the PersonaChat dataset. Hits@1 is the probability that
+the golden response ranks the highest among the 20 candi-
+date responses. Perplexity was used to calculate the nega-
+tive log-likelihood of the golden response from the model.
+F1 is calculated from the precision and recall at the word
+level between the predicted and golden responses. For per-
+sona consistency, we apply Consistency Score (C.Score)
+(Madotto et al. 2020), which leverages a referee model to
+predict consistency between response and persona. Dist-1/2
+(Li et al. 2016a) is used to measure response diversity.
+The BLEU, METEOR, ROUGE-L, and CIDEr metrics
+were reported for DSTC7-AVSD automatic evaluation, sim-
+ilar to DSTC7 reviews (Alamri et al. 2019).
+
+Model
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+METEOR
+ROUGE-L
+CIDEr
+BART
+0.791
+0.668
+0.562
+0.473
+0.307
+0.619
+1.372
+w/o ERM
+0.795
+0.671
+0.565
+0.477
+0.313
+0.625
+1.395
+w/o DDM
+0.799
+0.676
+0.571
+0.482
+0.309
+0.625
+1.388
+w/o OC
+0.782
+0.660
+0.557
+0.473
+0.309
+0.620
+1.388
+Our
+0.801
+0.680
+0.576
+0.488
+0.316
+0.631
+1.403
+Table 7: Analysis of ablation experiments on the DSTC7-AVSD.
+Table 1 presents the automatic evaluation results of the
+different methods for persona-based dialogue generation. As
+indicated, the proposed model outperformed all the base-
+lines on the PersonaChat dataset, especially on the revised
+dataset. The proposed model achieved significant improve-
+ments in response selection tasks because ERM can provide
+more entailed information about the persona while DDM
+can captures the appropriate discourse information in the di-
+alogue, making it easier to distinguish the correct response
+from the candidate responses. The improvement in PPL and
+F1 also shows that ERM and DDM can further improve the
+consistency of the persona and the quality of the response.
+Table 2 compares the results of the proposed method
+with some pre-trained language models fine-tuned on the
+PersonaChat. The results of training with 14 interference
+responses are in parentheses. The proposed model also
+achieves better results compared to models trained on larger
+corpora and more tasks. Additionally, adding more interfer-
+ence responses can improve the accuracy of the model to se-
+lect the correct response. Table 3 shows the result of persona
+consistency. The proposed model still obtained the highest
+C.Score, which indicates that the responses generated by the
+proposed model perform the best on persona consistency.
+Table 4 shows the experimental results of DSTC7-
+AVSD. The proposed model achieves the best results for
+METROR and CIDEr and is close to the best model for
+the other metrics. Better performance on CIDEr shows that
+in background-based dialogue question answering, the pro-
+posed model can use ERM to capture key information in the
+background and combine the dialogue history to generate
+high-quality responses.
+Human Evaluation
+We conducted a human evaluation of the state-of-the-art
+models (LIC, P2BOT) and the proposed method. We ran-
+domly sampled 100 responses generated by these models
+from the original PersonaChat dev set for the human eval-
+uation. Both fluency and consistency were applied as cri-
+teria. Four human annotators were asked to rate fluency
+on a scale of 1 to 5 and consistency on a scale of 1 to
+3, where the Fleiss’s kappa of fluency and consistency are
+0.578 and 0.671. Here, fluency indicates the smoothness of
+responses and conversation, coherence is included, where 1
+means terrible and 5 represents very satisfying. Consistency
+represents the consistency between persona and response,
+which reflects whether the model can maintain persona con-
+sistency, where 1 means it does not match persona, 2 means
+it is irrelevant, and 3 means it is consistent with persona.
+As shown in Table 5, the results of the human evaluation
+are consistent with the automatic evaluation, and our model
+outperforms the previous best-performing model on human
+evaluation in terms of both fluency and consistency. Several
+examples of the generated responses are provided in the Ap-
+pendix to help illustrate the effectiveness of our model.
+Ablation Study
+We conducted ablation experiments on the PersonaChat and
+DSTC7-AVSD to explore the impact of each module. The
+ablation results are presented in Tables 6 and 7, respectively.
+Effect of ERM.
+After removing the ERM, the C.Score be-
+comes lower on PersonaChat, which shows that the ERM
+can make the response generated by the model more consis-
+tent with its persona and improve the persona consistency in
+the dialogue response. In dialogue question answering, ERM
+can capture the key information in the background so that the
+generated answers can achieve better results for CIDEr.
+Effect of DDM.
+The large decline in BLEU-4 indicates
+that the quality of the responses generated by the model
+deteriorates without DDM. The metric of METEOR has a
+high correlation with the results of human judgment. With-
+out DDM, METEOR drops to a large extent, validating the
+important role that DDM plays in response generation and
+DDM can effectively capture the connections of the utter-
+ances in the dialogue, thereby making the responses more
+coherent and natural.
+Effect of the Orthogonal Constraint (OC).
+Imposing or-
+thogonality constraints on the two latent spaces effectively
+reduces redundant features and makes the features captured
+by the model easier to distinguish, thereby improving the
+quality of generation and performance in response selection.
+Conclusion
+In this paper, we propose a dialogue generation method
+for learning to memorize entailment and discourse rela-
+tions with latent variables. Combining latent entailment re-
+lations and dialogue discourse relations makes generated re-
+sponses more coherent and consistent. Experiments on the
+PersonaChat dataset demonstrate the effectiveness of the
+proposed method. The results on the DSTC7-AVSD dataset
+also show that learning entailment and discourse relations
+are beneficial for dialogue question-answering generation.
+Future works will attempt to explore different latent re-
+lations in text pairs on different datasets and combine the
+discourse relations in the dialogue to make the dialogue gen-
+eration in the desired direction.
+
+Acknowledgments
+This work was supported by the National Natural Science
+Foundation of China (NSFC) under Grant Nos. 61966038
+and 62266051, the Ministry of Science and Technology, Tai-
+wan, ROC, under Grant No. MOST 111-2628-E-155-001-
+MY2 and the Postgraduate Research and Innovation Foun-
+dation of Yunnan University under Grant No.2021Z076. The
+authors would like to thank the anonymous reviewers for
+their constructive comments.
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+Appendix
+Case Study
+Tables 8 and 9 present the responses generated by the dif-
+ferent methods on PersonaChat. As indicated, the responses
+generated by the proposed method were more consistent
+with their personas and coherence on PersonaChat. In Ta-
+ble 8, the responses generated by P2BOT and BoB are in-
+consistent with other hobbies in the query, even though the
+responses are consistent with the persona. Both LIC and our
+model generate more coherent responses based on other per-
+sona. As shown in Table 9, the responses generated by LIC
+are irrelevant to the personas. P2BOT and BoB tend to ig-
+nore content from dialogue, generating repetitive responses
+that make the entire conversation incoherent. Our model can
+effectively combine dialogue content and persona to gener-
+ate appropriate responses.
+Table 10 provides the responses generated on DSTC7-
+AVSD, where the models need to generate responses based
+on the given background knowledge and dialogue content. It
+can be found that our model is able to combine dialogue con-
+tent to generate answers with key information in the context
+that is closely related to the query.
+
+Persona
+i listen to rap music.
+i produce music for artists.
+i drive a 2015 honda civic.
+my favourite food is pizza.
+Context
+Q: hi , how are you ? do you have any brothers or sisters ?
+R: no i don’t do you ?
+Q: yes , i’m 13 and i’ve an older brother .
+R: that’s nice what kind of music do you like
+Q: i do not have much time as i play soccer . you ?
+R: i am a music producer for rap artists
+Query
+cool i like rap . i hate maths though ! do you have other hobbies
+GOLD
+work takes up a lot of my time
+LIC
+i love to eat pizza .
+BoB
+i like music and i like to listen to music (Incoherent with query)
+P2BOT
+i like to listen to rap music (Incoherent with query)
+Our
+i like to drive my honda civic
+Table 8: Case analysis of response generation on the PersonaChat.
+Persona
+my dream car is a rolls royce ghost.
+i currently work for a watch dealer.
+my family does not support my career choices.
+i often shop for shoes and watches.
+i like shopping.
+Context
+Q: good evening . what are you up to ?
+R: hello , how are you ? i am well . i like to shop for shoes and watches .
+Q: i don’t shop much . prefer to relax at home with my cow .
+R: what do you do ? i’m work for a watch dealer .
+Query
+no wonder you like watches ! i’m a commercial electrician
+GOLD
+wonderful ! yeah , however , my family does not support my choice of career .
+LIC
+i love watching movies (Irrelevant to persona)
+BoB
+i work at a dealership, but i am not very good at it.
+P2BOT
+what do you do for a living ? i work for the watch . (Incoherent with context)
+Our
+my family does not support my career choices .
+Query
+why not ? family is important .
+GOLD
+i’m not sure why they do not . do you like cars ?
+LIC
+i like to watch movies (Irrelevant to persona)
+BoB
+i am not sure. i am a watch dealer. (Incoherent with context)
+P2BOT
+my family doesn ’ t support my lifestyle choices . (Incoherent with context)
+Our
+i do not know . i do have a dream car though . it is a rolls royce ghost .
+Table 9: Case analysis of response generation on the PersonaChat.
+
+Caption
+a man closes his window , then he sneezes twice before taking a drink . then
+he opens up a bag and digs through it looking for something before walking
+out of the room .
+Summary
+a man closes the window , goes to the table and goes through the items in a
+bag , takes a drink from the green cup and leaves the room .
+Context
+Q: what is the guy doing at the window?
+R: the guy is closing the window
+Q: what does he do after that?
+Baseline
+he picks up a book from the table
+PLATO
+he goes to the table and takes a drink from a green cup
+DialogVED
+he goes to the table and goes through the items in a bag before taking a
+drink
+Our
+he sneezes twice and then takes a drink
+Table 10: Case analysis of response generation on the DSTC7-AVSD.
+

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+arXiv:2301.00980v1  [gr-qc]  3 Jan 2023
+The Thermal Substructure of General Relativity
+Manfred Requardt
+Institut fuer Theoretische Physik
+Universitaet Goettingen
+Friedrich-Hund-Platz 1
+37077 Goettingen
+Germany
+(E-mail: [email protected] or
+[email protected])
+Abstract
+In a first step we will provide arguments for the understanding of
+quantum space-time (QST), that means, the microscopic substructure
+which is assumed to underly ordinary smooth classical space-time, as a
+thermal system at each (macroscopic) point x of the classical space-time
+manifold (ST). In this context we exploit among other things some recent
+findings in the foundations of quantum statistical mechanics.
+We argue that the classical metric tensor field gij(x) plays the role of an
+order parameter field, signalling the existence of a primordial phase tran-
+sition in which both space-time and (quantum) matter emerged, accom-
+panied by the spontaneous symmetry breaking (SSB) of diffeomorphism
+invariance which is induced by the existence of quantum excitations in
+QST.
+We then analyze the role the Einstein equations (EEQ) are playing
+in this context and relate them to the local thermodynamical heat influx
+into QST at point x, thus defining the Einstein tensor or Ricci tensor as
+local heat tensor. We also define local gravitational temperature, entropy
+and internal energy.
+Finally, we give a thermodynamic interpretation of the conserved Noether
+current which derives from diffeomorphism invariance, separating it into
+heat contributions and work terms. In this connection we discuss the role
+and nature of tensor and non-tensor quantities and relate our investigation
+to the socalled energy vector of Hilbert‘s classical analysis.
+
+1
+Introduction
+Without question one of the most unexpected developements in general rela-
+tivity was the realisation that there exists a close relationship between certain
+laws of classical black hole (BH) physics and the laws of thermodynamics ([1]).
+This fundamental connection was first described by Bekenstein ([2],[3]) and a
+little bit later supported by the fundamental analysis of Hawking ([4]). A nice
+overview is given for example in [5].
+That there might perhaps exist a closer and even more fundamental connec-
+tion between classical relativity and thermodynamics was, as far as we know,
+emphasized by Jacobson ([6]) and further developed in e.g. [7] and many pa-
+pers of Padmanabhan (see, for example,[8],[9] or [10]). It was shown that one
+can infer from the physics near horizons, in particular the event horizon of a
+black hole, that the Einstein equation (EEQ) must hold. One important tool
+in this context was the use of approximate Rindler-Unruh horizons and the
+corresponding results of Fulling-Unruh ([11],[12]).
+One should remark that, as far as we can see, all the many papers which do
+exist in this context emphasize the importance of horizons and the role which
+is played by the entanglement entropy of systems behind these horizons. That
+is, from this view point, entropy and more generally thermodynamics appear
+to be essentially located near these horizons and not e.g. in the bulk. These
+properties seem to be suggested, for example, by the famous entropy area law
+of black holes (BH).
+Whereas horizons apparently seem to play such a fundamental role for the
+thermal behavior of gravitation, we want to go a step further and will argue
+in this paper that in our view the situation is a different one and that both
+the Einstein equation (EEQ) and gravitation in general are of a thermal nature
+being independent of the existence of horizons and which can be inferred by
+analyzing the ordinary bulk situation in general relativity (GR). The special
+behavior near horizons is then rather a particular consequence of this more
+general scenario.
+In a first step we will provide arguments for the understanding of quantum
+space-time (QST), that is, the microscopic structure which underlies our or-
+dinary classical smooth space-time manifold ST, as a thermal system at each
+macroscopic point x ∈ ST . This will be done in the next section (some remarks
+in this direction have already been made in [13]. Early ideas can already be
+found in the work of Sakharov but without a thermal connotation (see e.g. [14]
+and [15]). We will exploit more recent findings in the foundations of modern
+quantum statistical mechanics ([16],[17],[18]). Furthermore we will introduce
+the notion of the classical space-time metric gij(x) as an order parameter field
+and classical smooth space-time as an order parameter manifold.
+By these notions we mean the following. As briefly described in [13], we as-
+sume that (Q)ST emerged as the consequence of a primordial phase transition
+PT. In this context a non-vanishing field of observables gij(x) came into exis-
+tence which was zero or rather vanishing before the phase transition happened
+in the primordial soup. That is, gij(x) and ST represent parameters which in-
+2
+
+dicate that a certain phase transition took place. Such a quantity is typically
+called an order parameter.
+We start from the following assumption or observation: Classical space-time
+ST remains to be smooth on all ordinary scales of the quantum matter regime.
+So we conjecture that the typical scale where quantum gravity effects emerge is
+the Planck scale, that is
+lp = ℏG/c3
+,
+tp = ℏG/c5 = lp/c
+(1)
+We hence assume that the typical quantum gravity degrees of freedom (DoF) live
+on the same scale. This implies that, as e.g. in hydrodynamics, there exist many
+quantum gravity DoF in the infinitesimal neighborhood of the macroscopic point
+x ∈ ST in the classical space-time manifold ST. That is, we can assume that at
+each macroscopic point x exists a microscopic subsystem in QST, consisting of
+many DoF. That these subsystems behave thermally will be proved in the next
+section.
+In a second step we will interpret the EEQ (sign convention as in [19])
+Gab := Rab − 1/2 · R · gab = κ · Tab
+,
+κ = 8πG/c4
+(2)
+in this thermodynamic context. That is, even if we assume that the global state
+of the microscopic QST is pure, the local state around some macroscopic point
+x is a thermal ensemble (density matrix) due to its intricate entanglement with
+its environment. From the findings in the papers, mentioned above, we may
+then conclude that with overwhelming probability the local states are thermal.
+As the local states around the points of the space-time manifold ST are
+thermal it seems reasonable to give also the EEQ a thermal interpretation.
+One should note that already Lorentz or Levy-Civita in 1916 or 1917 tried to
+interpret the EEQ as saying that the total energy of the universe is vanishing
+(cf. [20])
+(Tab − κ−1Gab) ≡ 0
+hence
+∂a(T a
+b − κ−1Ga
+b) ≡ 0
+∂a := ∂/∂xa
+(3)
+Einstein provided an argument against such an interpretation (see [20]) which
+seems to be sound at first glance, but one should take into account that at that
+time quantum theory was not yet fully developed (cf. e.g. our arguments, given
+in [13]).
+In this paper we will only comment on the vast field of gravitational en-
+ergy and its localization in general relativity (GR) in so far as it concerns the
+problems we want to analyze in the following.
+An interesting point of view
+is developed for example in [21], where it is claimed that gravitational energy
+should be located at places where Tab ̸= 0. However, one generally assumes
+that non-vanishing curvature has something to do with gravitational energy.
+The latter one can of course be non-zero where Tab vanishes.
+If the microscopic substructure , i.e. QST, of the classical smooth space-
+time ST is locally a thermal system, it should possess what one calls internal
+energy in thermodynamics. As in [13] we assume the existence of an interaction
+3
+
+between matter and space-time on the quantum level, that is, between QST and
+QM (quantum matter). The macroscopic ponderable matter and fields making
+up the right side of the EEQ consist of many microscopic DoF on the quantum
+scale. The same holds for the left side which describes a curvature property of
+ST.
+All this leads us to the following conjecture:
+Conjecture 1.1 We assume that
+Tab = κ−1Gab
+(4)
+is the heat influx at point x into the local thermal state of QST coming from the
+external QM system.
+Note that this concerns only the gravitational effect of matter. Thus κ−1Gab
+represents a form of heat tensor which was discussed for material systems e.g.
+in [22] or[23].
+Remark 1.2 We will later discuss in this paper the work terms in this context,
+To complete our analysis of the thermal substructure of Gravitation or QST,
+we have firstly to deal with the non-tensorial character of gravitational energy.
+Usually it holds
+∂a(√−g(T a
+b + ta
+b)) ≡ 0
+(5)
+with ta
+b the pseudo tensor of gravitational energy-momentum (brief discussions
+can be found in practically every book about GR, a particularly nice discussion
+is e.g.
+given in [24]).
+In the canonical view this pseudo tensor property is
+frequently considered to be a deficit of the theory. In section (3) we reconsider
+the situation from a perhaps new vantage point.
+It is a widespread belief that all concepts and notions which carry a physical
+meaning should be of tensorial character. It is however reasonable to analyze
+why tensorial behavior is considered to be of such a great relevance.
+If we
+have a manifold with tangent and cotangent spaces at each point, tensor fields
+can be built over tensor products of these spaces. Their importance is hence
+that they carry a geometric meaning, that is, tensors may be said to represent
+the same invariant object independent of the choice of coordinate systems. In
+this context they have the important property that they cannot be transformed
+away (transformed to zero) by an appropriate choice of coordinate system. In
+particular, their transformation properties are relatively transparent.
+On the other hand this philosphy cannot be exactly true as the wellknown
+connection coefficients or Christoffel symbols ΓI
+jk are of great geometric signif-
+icance and do not have a particularly simple transformation behavior. In our
+view, Weinberg rightly remarks that there is nothing sacred about the tensor
+transformation law ([25]). This holds the more so as physical quantities are not
+automatically elements of the multilinear algebra over tangent and/or cotangent
+space (which have essentially the meaning of velocity or momentum vectors).
+In older text books physical tensorial quantities are hence introduced as
+tuples having simply a certain transformation behavior. We therefore should
+4
+
+be prepared that there may exist physical quantities which do not transform as
+tensors. In the valuable and highly informative essay [26] there is an interesting
+remark by Einstein, when asked by the Swiss student Humm, if gravitational
+energy should be tensorial or not. He thinks not, referring for example to the
+different behavior of kinetic energy T and potential energy U under the Galilei
+group in
+∂t(T + U) = 0
+(6)
+Another argument, in our view, would be a pendulum clock with friction
+which allows us to extract energy from the gravitational field in a non-inertial
+reference system, which is not possible in a freely falling inertial system. This
+latter example suggests the following conclusion:
+Observation 1.3 The equivalence principle shows that gravitational energy can-
+not be tensorial.
+We will discuss the deeper reasons for the possibility of non-tensorial be-
+havior of various physical quantities in section (3) in a more systematic way,
+the main point being the effects of quantum theory on space-time and gravita-
+tional behavior. These quantum effects will lead to a subtle distinction between
+surface properties living over the smooth classical space-time manifold ST and
+the behavior in the underlying microscopic substructure denoted by QST. The
+former are expected to be tensorial, the latter typically not. The spontaneous
+symmetry breaking of diffeomorphism invariance will play an important role in
+this context.
+In the last section we discuss in more detail the relevance of diffeomorphism
+invariance and the corresponding conserved Noether current for our investiga-
+tion of the thermal character of the EEQ and GR in general. Wald studied the
+Noether current of diffeomorphism invariance in the framework of differential
+forms lying the main emphasis on integral representations ([27],[28],[29]). We
+will, on the other hand, rely more on the classical tensor analysis and variational
+approach, using mainly the non-integrated local point of view as it is e.g. also
+done in [8].
+But before we will do this, we will add some remarks about the quite de-
+tailed early work of Hilbert, Noether, Klein in this context (which, written in
+German, is perhaps not so widely known and which is carefully discussed in [26],
+for other aspects see also [30]). These papers dealt mainly with a proper un-
+derstanding of Hilbert’s famous socalled energy vector, which was derived from
+a quite intricate variational analysis of some invariant action functional and
+which seemed to puzzle not only the scientific community of that time quite a
+lot ([31],[32],[33],[34]. We find it remarkable that these long gone investigations
+anticipate in some sense more recent ones.
+In Hilbert’s analysis an arbitrary vector field ξi was introduced in order to
+get a conserved current from diffeomorphism invariance. It was later understood
+([34]) to be a necessary consequence of E.Noether’s famous second theorem. We
+will analyze the remarkable behavior of this conserved current, give a physical
+interpretation of the vector field ξi, which is technically the generator of the Lie
+5
+
+derivative and interpret the role of various terms occurring in the current as to
+their thermodynamic meaning and implications. This winds up to our observa-
+tion that what Hilbert used to call the energy vector is rather an expression of
+the internal energy-momentum tensor of the microscopic quantum gravitational
+substratum QST, underlying our smooth space-time manifold ST.
+2
+Quantum Space Time, QST, as a Thermal Sys-
+tem
+As we have already remarked in the introduction, thermality usually enters
+the field of (quantum) gravity or general relativity (GR) via the existence of
+horizons, most notably in the paradigmatic case of black holes (BH). We want
+to argue in this and the following sections that, in our view, classical space time
+ST is the smooth, coarse grained hull, overlying QST, which is assumed to be a
+system having quantum micro structure, that is, consists of a complex network
+of a huge number of microscopic gravitational degrees of freedom (DoF).
+Assumption 2.1 We make the simplifying assumption that we may restrict
+ourselves to an intermediate energy scale where all DoF can be treated as quan-
+tum. Furthermore, we assume the gravitational DoF to live on a scale which is
+given by the Planck units.
+From this we may infer the following. Compared to the Planck scale, the
+ordinary quantum scale on which our quantum matter is living, is many orders
+of magnitude larger. This is the reason why on the level of ordinary quantum
+theory space time can be treated as smooth and relatively invariable. On the
+other hand, we have briefly described in e.g. [13] how the microscopic gravita-
+tional DoF, living in QST cooperate so that as a consequence of a primordial
+phase transition PT the classical space time manifold ST with its points x and
+the classical metric tensorfield gab(x) do emerge, with gab(x) playing a double
+role as allowing, on the one hand, to measure distances and time intervalls and,
+on the other hand, being the carrier of the gravitational field.
+As in [13] we introduce a Hilbert space Hg, in which the global quantum
+states, ψg, of QST are living. The microscopic elementary gravitational DoF
+are represented by their corresponding local Hilbert spaces Hg
+i , i ∈ N. The local
+bases in Hg
+i are denoted by dνi
+i . Basis vectors in Hg are then given by tensor
+product states
+ψg
+I =
+�
+i
+dνi
+i , I := {ν1, ν2, · · · }
+(7)
+and a general pure quantum state in Hg is
+ψg =
+�
+I
+cIψg
+I
+(8)
+6
+
+Remark 2.2 We note that these assumptions are not really necessary prereq-
+uisites to reach the following conclusions but will represent a convenient model
+to make the derivations more concrete and precise.
+We have now to discuss what consequences can be inferred from the exis-
+tence of the nonvanishing classical metric tensor field gab(x) which lives over
+the classical space-time manifold ST and by the same token over the underlying
+microscopic substructure QST. Usually it is assumed that a quantum observable
+ˆgab(x) exists on QST with
+gab(x) =< ˆgab(x) >:= (ψg | ˆgab(x)ψg)
+(9)
+for some global state ψg which corresponds as a quantum correlate to the clas-
+sical ST.
+This, at first glance, natural assumption leads however to a number of highly
+non-trivial consequences and has now to be discussed in more detail. We as-
+sumed for example in [13] that the existence respectively non-vanishing of the
+metric tensor field gab(x) is the result of a primordial phase transition PT. In
+other words, the existence of a stable space-time distance concept is a non-trivial
+consequence of a particular physical process. Hence, following the tradition of
+e.g. quantum statistical mechanics as a paradigm, dealing with a great number
+of quantum DoF, we define:
+Definition 2.3 We call gab(x) an order parameter field and the space-time
+manifold ST an orderparameter manifold, where its non-vanishing signals the
+existence of a transition to a greater structural order in the underlying micro-
+scopic substratum (see e.g. [13] or [35],[36] and earlier references given there).
+In our context, where we have a double structure of an overlying classical
+smooth space-time manifold ST and an underlying microscopic quantum struc-
+ture QST, both being correlated in a subtle way, we have to introduce some
+particular structural elements, that is, for example, the concept of macro ob-
+servables. As far as we know, this notion was for the first time introduced by
+v.Neumann in an ingenious but perhaps little kown paper (see [39], the second
+part of [38] or [40]).
+Due to the vastly different scales of, on the one hand, ordinary quantum
+matter QM and, on the other hand, quantum space-time QST, we can assume
+that to a macroscopic point x do belong a huge number of DoF in QST or local
+Hilbert spaces Hi which live in the infinitesimal neighborhood of x. We denote
+this situation by
+Hx :=
+�
+x
+Hi ∼ [x]
+(10)
+where [x] denote the gravitational DoF in the infinitesimal neighborhood of x.
+Furthermore, to a macroscopic value, gab(x), we can choose a corresponding
+subspace [gab(x)] in �
+x Hi containing the local vectors ψg(x) with
+< ψg | ˆgab(x)ψg >= gab(x)
+(11)
+7
+
+Definition 2.4 Following the tradition in quantum statistical mechanics, we
+call [gab(x)] a phase cell and which is spanned by certain basis vectors ψg
+i (x) so
+that
+ψg(x) =
+�
+ciψg
+i (x)
+(12)
+Furthermore, in the class of global states ψg, we may select the subclass [gab]
+so that
+< ψg | ˆg(x)ψg >= gab(x)
+for all x ∈ ST
+(13)
+Remark 2.5 One should note that the various sets, [x], or [gab(x)], are not
+necessarily disjoint in QST. There may be a certain overlap.
+We now come to a central point in our analysis. Given a global state ψg
+in Hg, we can test it by sufficiently localized observables, that is, observables
+localized in an infinitesimal neighborhood of some macroscopic point x. That
+is, we may concentrate ourselves on state vectors from Hx or [gab(x)]. More
+specifically, if we have a pure global state ψg and test it with observables taken
+from B(Hx), the bounded operators operating on Hx, we can represent the
+global state ψg by a density matrix ρψ, living over Hx or [gab(x)].
+Observation 2.6 It exists a density matrix ρψ over Hx or [gab(x)] with
+< ψg | A(x)ψg >= T r(A(x) · ρψ)
+(14)
+for observables A(x) ∈ B(Hx)
+It is a relatively recent observation that much stronger results can in fact
+be derived which lead to the thermalization results, which we mentioned in the
+introduction.
+To our knowledge, early results in this direction were already
+derived by v.Neumann in [39]. More recently, these phenomena were analyzed
+in e.g. [16],[17], or [18]. In the context of decoherence by environment see also
+[40].
+Important tools in this connection are, on the one hand, the socalled
+concentration of measure phenomenon and the Levy estimates (see for example
+[16], a more systematic discussion can be found in [41]) and, on the other hand,
+the use of random (unit) vectors (that is, quantum mechanical states lying on
+high-dimensional unit spheres) and socalled typical states. We will not go into
+the complex technical details but restrict ourselves to an application of the
+results to our particular situation.
+That is, we are given the classical order parameter field gab(x), defining the
+class of microscopic quantum states ψg ∈ [gab] with
+< ψg | ˆg(x)ψg >= gab(x)
+for allx ∈ ST
+(15)
+We observed above that locally ψg can be replaced by a trace operator or density
+matrix ρψ(x).
+The results derived in the above cited papers, in particular [16], now yields
+the following in our scenario.
+8
+
+Conclusion 2.7 For a global pure state ψg ∈ [gab] it hold with very high proba-
+bility that for local observables it is equivalent to the micro canonical ensemble,
+i.e.
+Ω := d−1 · T r(1·)
+(16)
+with d the dimension of [gab] and 1 the identity operator. The local state ρψ(x)
+at some arbitrary macroscopic point is equivalent to this Ω and is called the gen-
+eralized canonical ensemble. In other words, this holds for a typical or randomly
+selected state. One should however note that Ω is a global state, ρψ(x) a local
+one.
+Remark 2.8 In case the generalized canonical ensemble can be associated with
+some form of energy, the local states have the form of the true canonical ensem-
+ble known from thermodynamics. This is the reason why the corresponding local
+state can rightly be called a generalized canonical ensemble. That is, the global
+micro canonical ensemble behaves locally as a canonical ensemble.
+3
+The Thermal Role of the Einstein Equation
+In section 2 we have argued that in an infinitesimal neighborhood of a macro-
+scopic point x ∈ ST quantum space-time (Q)ST has the characteristics of a local
+thermal system. We now want to introduce and discuss the various physical pa-
+rameters which define and characterize such a thermal system. We begin with
+a new conceptual understanding of the EEQ. Originally the EEQ are viewed
+or conceived as a dynamic or evolution equation of the space-time continuum
+under the influence of (ponderable) matter and classical fields.
+As mentioned in the introduction, much later Jacobson, Padmanabhan (and
+perhaps some other workers in the field) argued that the EEQ must hold due
+to the observed or conjectured thermal behavior near horizons, most notably,
+near the event horizon of a BH (see e.g. [6],[7],[8],[9],[10]). In the following
+we want to argue that the EEQ carry a direct thermal meaning in the bulk
+of ST, i.e., not necessarily near horizons and independent of the existence of
+horizons or singularities. This holds also for the concept of entropy, which is
+primarily understood in this context as a form of entanglement entropy, typically
+arising near horizons. In our context it will have the ordinary thermodynamical
+meaning of number of configurational alternatives.
+As we said in the introduction, already in the early days (1916,1917) e.g.
+Lorentz or Levi-Civita tried to give the EEQ a slightly different interpretation,
+which was however rejected by Einstein by an argument which was perhaps con-
+vincing at that time with quantum (field) theory still in its infancies (cf. [20]).
+Lorentz and Levi-Civita speculated that by bringing the energy-momentum ten-
+sor, occurring on the rhs of the EEQ to the lhs, we get an expression which
+vanishes identically (see equation 3 in the introduction). They then conjectured
+that this expression can then be identified with the total (vanishing) energy of
+the universe, consisting of gravitational and matter energy.
+9
+
+Einstein argued that the vanishing of the total energy of the universe would
+not prevent the material systems and the gravitational energy to annihilate
+each other. However, meanwhile we know that exactly the opposite process
+might have happened in various models of the inflationary scenario (creation
+from nothing, huge quantum fluctuations etc.). We discussed this zero energy
+universe idea in more detail in section 2 of [13].
+In [13] we introduced and described a similar but slightly different scenario.
+We argued that our universe, i.e. (quantum) matter, (Q)M, and (quantum)
+space-time, (Q)ST, emerged both as a result of a primordial phase transition
+PT from a more primordial phase, QX:
+QX →
+P T (Q)ST + (Q)M
+(17)
+While the internal energy of QST is lowered compared to the original phase
+QX as the result of an order transition by an amount ∆E, this same amount is
+transferred to QM. That is,
+E(QX) = E(QST ) + E(QM)
+(18)
+In the language of Hilbert spaces we describe the structure of our universe
+as a pure state in the tensor product
+ψ ∈ H = Hg ⊗ HM
+(19)
+with
+ψ = ψg ⊗ ψM
+(20)
+being a state where the gravitational and the matter state do exist independently
+from each other. This may be an approximation of a situation where e.g. BH’s
+are absent. The BH situation was discussed in [13].
+A thermal system should have an internal energy. In the introduction we
+formulated the following conjecture:
+Conjecture 3.1 We assume that
+Tab(x) = κ−1Gab(x)
+(21)
+is the heat influx at point x into the local thermal state of QST coming from
+the external QM system. Note that this concerns only the gravitational effect of
+matter. Thus κ−1Gab represents a form of heat tensor which was discussed for
+material systems e.g. in [22] or[23].
+That is, the EEQ is essentially a statement about the total heat influx at point
+x ∈ ST into the infinitesimal neighborhood of x in QST. We now want to
+provide arguments why we think this is indeed the case.
+In [42] we studied thermodynamics in the regime of special relativity. While
+in this field there do exist a variety of different approaches, in the approach
+we favored, i.e. with temperature being the zero component of a contravariant
+four-vector,
+T = γ · T0
+,
+γ = (1 − u2/c2)−1/2
+(22)
+10
+
+with γ the Lorentz factor, T0 the rest temperature in a comoving coordinate
+system, u the 3-velocity, T the temperature in the laboratory frame, we get
+the following interesting transformation property for the heat influx into the
+system:
+δQ = γ · δQ0
+(23)
+that is, in contrast to internal energy and work, the heat influx transforms as the
+zero component of a contravariant 4-vector. That is, it transforms covariantly
+(cf. in particular section 4.4 of [42]).
+Another paradigm is BH physics and the first law of BH-thermodynamics.
+In geometric units (c=G=1) it reads:
+δM = (8π)−1 · κδA + ΩδJ
+(24)
+with κ the surface gravity, δA the change of area, Ω angular velocity of horizon,
+δJ change of angular momentum, δM the change of mass in the center. The
+identification reads:
+δSBH = δA/4 , T = κ/2π , δM change of internal energy
+(25)
+It is perhaps surprising that heat and entropy occur at such a prominent
+place in this field while the, at first glance, more natural work contributions seem
+to be missing. We now try to explain this on a perhaps more fundamental level.
+We already briefly mentioned in the introduction that, in our view, space-time
+consists of at least two levels, a smooth surface structure, ST, with its tangential
+and cotangential structure and an underlying microscopic quantum structure,
+QST. This compound system is assumed to have emerged via a primordial phase
+transition, PT, together with a (quantum) matter component, QM.
+This phase transition was a transition to a greater order, that is, (Q)ST
+carries an extra space-time structure compared to the more primitive phase, QX,
+given, for example, by the existence (or non-vanishing) of the metric tensor field,
+gab(x), having the character of an order parameter field, and in the underlying
+QST the existence of certain phase cells, [gab(x)], containing the quantum states
+of the universe, leading to the same macroscopic metric tensor field in ST, i.e.
+gab(x). Therefore we can conclude:
+Conclusion 3.2 The primordial phase transition, PT, is accompanied by a
+spontaneous symmetry breaking, SSB, of diffeomorphism invariance (or covari-
+ance). That is, the more symmetric but less ordered phase, QX, goes over into
+the phase (Q)ST, which has lesser symmetry but higher order.
+Remark 3.3 These points were already discussed in [35] and [36].
+.
+What are the consequences of these observations for our question of non-
+covariance of e.g. gravitational energy and related questions? As we said above,
+we have a two storey structure of space-time. We argue that the smooth sur-
+face structure ST (i.e. the macroscopic part) supports observables which are
+11
+
+diffeomorphism invariant. The SSB, on the other hand, is located in the micro
+structure of the quantum regime QST. That is, we assume that the existence of
+quantum excitations, vacuum fluctuations and the emergence of the space-time
+structure [gab(x)] is responsible for the breaking of diffeomorphism invariance.
+We conjecture the following:
+Conjecture 3.4 We assume that quantities which live over the smooth macro-
+scopic manifold ST are diffeomorphism covariant, i.e. behave tensorial. On the
+other hand, quantities which rather express properties of the microscopic under-
+lying quantum structure QST we assume to behave non-tensorial in general.
+Corollary 3.5 As we already explained in the introduction, since the scale on
+which QST is living is many orders of magnitude finer than even the regime of
+ordinary quantum matter, the latter should also behave covariantly in general.
+Conclusion 3.6 As the energy-momentum tensor, Tab(x), lives essentially over
+the classical manifold ST, the heat influx, κ−1 · Gab(x), is by the same to-
+ken also tensorial or covariant. On the other hand, the part of the relativistic
+energy-momentum (pseudo) tensor, tab(x), describing the gravitational energy
+contained in QST, is non-tensorial because it describes the contributions to grav-
+itational energy which are contained in the underlying network of gravitational
+DoF. making up the microscopic structure of QST.
+4
+The Concept of Gravitational Temperature
+On general grounds we know that our local systems must have a temperature.
+From what we have said above, the local gravitational system around the point
+x ∈ ST has an internal energy U(x) and an entropy S(x).
+In section 2 we
+showed that the system at x ∈ ST can be considered as a generalized canonical
+ensemble. This entails that we can define its v.Neumann entropy
+S(x) := −T r (ρx · ln ρx) = −
+�
+i
+pi ln pi
+(26)
+with pi the probabilities of the individual states ψi(x). Therefore we can (in
+principle) calculate
+T (x) = ∂U(x)/∂S(x) with V (x) held fixed
+(27)
+(V (x) some infinitesimal volume element around x).
+We will complete this somewhat abstract definition of gravitational temper-
+ature by two different more concrete approaches in this context. Firstly, we
+discuss the Tolman-Ehrenfest approach (cf. e.g. [43],[44]) which was originally
+introduced for thermal material systems which live in ST. For convenience of
+the reader we will briefly recapitulate our own approach which we gave in [45].
+12
+
+Remark 4.1 In order that we can apply the Tolman-Ehrenfest results to purely
+gravitational systems, that is, systems not consisting of ordinary material con-
+stituents, we must at first motivate that these systems have a thermal character
+and consist of microscopic constituents. This is what we have done in this paper
+up to now.
+The Tolman-Ehrenfest result is the following:
+Observation 4.2 In thermal equilibrium in a static gravitational field we have
+for an isolated system
+T (x) ·
+�
+−g00(x) = const
+(28)
+I.e., in contrast to the non-relativistic regime (cf. sect.2), there exists in general
+a temperature gradient in a system being in thermal equilibrium in the relativistic
+regime.
+To derive this result we use the entropy maximum principle. We shall use, for
+reasons of simplicity, the weak field expansion of the gravitational field. With
+φ(x) the Newtonian gravitational potential we have
+√−g00 = (1 + 2φ/c2)1/2 ≈ 1 + φ/c2
+(29)
+Remark 4.3 Note that the gravitational potential is negative and is usually
+assumed to vanish at infinity.
+We assume an isolated macroscopic system to be in thermal equilibrium in
+such a static weak gravitational field.
+Its total entropy and internal energy
+depend on the gravitational field φ(x). We now decompose the large system into
+sufficiently small subsystems so that the respective thermodynamic variables can
+be assumed to be essentially constant in the small subsystems. As the entropy
+is an extensive quantity, we can write
+S(φ) =
+�
+i
+Si(E0
+i , Vi, Ni)
+(30)
+where E0
+i is the thermodynamical internal energy, not including the respective
+potential energy.
+Observation 4.4 It is important that in the subsystems the explicit dependence
+on the gravitational potential has vanished. The entropy in the subsystems de-
+pends only on the respective thermodynamical variables, the values of which are
+of course functions of the position of the respective subsystem in the field φ(x).
+At its maximum the total entropy is constant under infinitesimal redistri-
+bution of the internal energies E0
+i with the total energy and the remaining
+thermodynamic variables kept constant. We now envisage two neighboring sub-
+systems, denoted by (1) and (2). To be definite, we take φ(2) ≥ φ(1). We now
+transfer an infinitesimal amount of internal energy dE0
+2 from (2) to (1) (note,
+13
+
+it consists of pure heat as for example the particle numbers remain unchanged
+by assumption!). As heat has weight relativistically it gains on its way an extra
+amount of potential energy.
+Observation 4.5 By transferring dE0
+2 from (2) to (1) we gain an additional
+amount of gravitational energy
+dE0
+2 · ∆φ/c2
+,
+∆φ = φ2 − φ1
+(31)
+It is important to realize that this gravitational energy has to be transformed from
+mechanical energy into heat energy or, rather, internal energy and reinjected in
+this form into system (1) (for example by a stirring mechanism acting on system
+(1) and being propelled by the quasistatic fall of the energy dE0
+2).
+The energy balance equation now reads
+dE0
+1 = dE0
+2 + dE0
+2 · ∆φ/c2 = dE0
+2(1 + ∆φ/c2)
+(32)
+We then have (with dS1 = −dS2 in equilibrium)
+T −1
+1
+= dS1/dE0
+1 = −dS2/ − (dE0
+2(1 + ∆φ/c2)) = T −1
+2
+· (1 + ∆φ/c2)−1
+(33)
+that is
+Conclusion 4.6 It holds
+T1 = T2(1 + ∆φ/c2)
+,
+∆φ := φ2 − φ1 ≥ 0
+(34)
+Observation 4.7 The subsystem (1), having a potential energy being lower
+than (2), has a higher temperature.
+We can give the above relation another more covariant form. In the approx-
+imation we are using it holds:
+�
+1 + 2φ2/c2
+�
+1 + 2φ1/c2 = 1 + φ2/c2
+1 + φ1/c2 = 1 + (φ1 + ∆φ)/c2
+1 + φ1/c2
+= 1 + ∆φ/c2
+(35)
+Conclusion 4.8 (Covariant form) It holds
+T1 ·
+�
+−g00(1) = T2 ·
+�
+−g00(2) = const
+(36)
+The non-infinitesimal result follows by using a sequence of infinitesimal steps.
+It is perhaps useful to derive the above result in yet another, slightly differ-
+ent, way. In case φ(x) vanishes at infinity we can apply the entropy-maximum
+principle as follows. We extract the energy (bringing it to infinity)
+dE2 = dE0
+2 + dE0
+2 · φ2/c2
+(37)
+14
+
+from subsystem (2) and reinject the energy
+dE2 = dE1 = dE0
+1 + dE0
+1 · φ1/c2
+(38)
+from infinity into (1). We get
+dE0
+1 = dE0
+2 · 1 + φ2/c2
+1 + φ1/c2
+(39)
+and
+T −1
+1
+= dS1/dE0
+1 = T −1
+2
+·
+�1 + φ2/c2
+1 + φ1/c2
+�−1
+(40)
+hence
+T1 · (1 + φ1/c2) = T2 · (1 + φ2/c2)
+(41)
+As an example one may mention the Rindler/Unruh space-time. We have
+in Rindler coordinates
+g00 = −ξ2 , a = ξ−1 , T = a/2π
+(42)
+i.e.
+T √−g00 = 2π−1 = const
+(43)
+Remark 4.9 One should mention that this method to go to ∞ does make only
+sense if the thermal system extends to ∞ as well.
+In a second approach we will employ the Fulling-Unruh observation of ther-
+malisation in accelerated frames of reference (local Rindler frames were intro-
+duced by for example Padmanabhan in e.g. [9]). We will however use it in a
+slightly different way compared to Jacobson or Padmanabhan. For reasons of
+convenience we wil choose a static space-time ST and restrict ourselves to a
+sufficiently small neighborhood of some arbitrary point x ∈ ST .
+We choose a local inertial frame (LIF) at x with Lorentz-orthonormal coor-
+dinates (X, T ) so that x has the coordinates (0, 0). We assume that the LIF
+moves along a certain geodesic through x. We place a thermometer system at
+the point x which is gauged so that it yields the result zero in a LIF. However, we
+assume that the thermometer is in principle sensitive against the thermal exci-
+tations of the underlying microscopical gravitational system. The thermometer
+now experiences the gravitational field gab(x) at the macroscopic point x ∈ ST .
+Now, relative to the LIF (moving on a geodesic through x) and employing
+the inertial coordinates (X, T ) an observer at the point x (and the thermometer)
+experiences an acceleration a(X, T ) (the detailed numerical calculations, which
+are a little bit intricate, can be found in [22] exercise in section 9.6). One should
+however note that everything we have said holds only locally. But locally, near
+the origin, we can switch from inertial coordinates (X, T ) to socalled Rindler
+coordinates (xR, tR).
+T = xR · sinh (κtR) , X = xR · cosh(κtR)
+(44)
+15
+
+with κ the corresponding proper acceleration.
+In full Rindler space, i.e.
+the right wedge WR, we know from the work
+of Fulling ([11]) and Unruh ([12]) that the thermometer will detect thermal
+Rindler modes. In our case the situation is only locally, i.e. in a neighbor-
+hood of x or (X, T ) = (0, 0), Rindler like. Therefore we cannot expect to have
+such fully developed Rindler modes. However we know already that ST at x
+or (X, T ) = (0, 0) behaves microscopically as a thermal system, that is, as a
+socalled generalized canonical ensemble, hence carrying a distribution of local
+thermal excitations. Therefore we make the conjecture:
+Conjecture 4.10 The observer at x or (X, T ) = (0, 0) will detect a certain
+thermal excitation spectrum, consisting of certain quasi particles (which may
+be approximations of corresponding Rindler modes), leading to a corresponding
+local temperature.
+5
+Diffeomorphism Invariance and its Conserved
+Noether Current
+In section 3 we discussed the consequences of the possible SSB of diffeomorphism
+invariance due to the emergence of quantum DoF in the underlying micro struc-
+ture of ST, i.e.
+QST as a result of the primordial phase transition PT. We
+argued that we may have two classes of observables, the one class of covari-
+ant observables which live above the macroscopic space-time manifold ST and
+another class, which we assume to consist of quantities which rather describe
+properties of the underlying micro structure within QST. We argued that the
+elements of this latter class need not behave covariantly under geometric trans-
+formations of the surface structure ST. A prominent example in this context is
+the concept of gravitational energy. Another important role will be played by
+the work terms which occur in the complete conserved Noether current we will
+derive below. Their covariance comes about by exploiting the vector field ξi(x)
+which enters via the Lie derivative, which is a covariant operation.
+But befor we enter into the technical details, we want to recapitulate what
+we said in the introduction concerning the work of Hilbert, Noether, Klein etc.
+([31],[32],[33],[34]). We were surprised to see that Hilbert essentially got after a
+long and intricate calculation a conservation law for what he considered to be
+the gravitational energy vector. This energy vector contained an arbitrary vec-
+tor field ξi, the deeper role of which was not really understood and appreciated
+at that time, while technically it is a consequence of E.Noether’s second theo-
+rem. In our view Hilbert already performed calculations which were much later
+repeated in a similar context, being apparently unaware of the earlier results.
+Below we will explain the physical role of the arbitrary vector field ξi while
+its mathematical role is clear, it is simply the vector field which generates the
+diffeomorphism group and occurs in the Lie derivative. In traditional physicist’s
+notation:
+xi → xi = xi + ǫξi(x)
+(45)
+16
+
+with ǫ infinitesimal. These intensive investigations and discussions performed
+in the Hilbert group were carefully studied in two beautiful essays by Rowe and
+Sauer ([26],[30]).
+We give now a brief description of the sequence of necessary steps which
+lead to the form of conserved Noether current deriving from the assumption of
+diffeomorphism invariance. We will perform most of these steps in an appendix.
+Our main motivation for this is a consequence of the observation that in many
+representations of this stuff for example important and nonvanishing boundary
+terms are frequently dropped which then results in only partial results.
+Remark 5.1 If for example certain boundary terms are dropped one gets only
+the contracted Bianchi identity instead of the full conserved current. This is in
+our view dangerous because in contrast to ordinary variations diffeomorphisms
+typically need not vanish at infinity. Furthermore, we will show that variations,
+having a local support, lead to results which differ in some important respects.
+In the following we restrict ourselves to a variation of the Hilbert-Einstein
+action
+S[gab] =
+�
+R(gab) · √−g d4x
+(46)
+with R the scalar curvature
+R := Ra
+a with Rab the Ricci tensor Rac := Rb
+abc
+(47)
+the rhs being the Riemann curvature tensor. g is the determinant of the metric
+tensor. R is a scalar and √−g d4x, the canonical volume element, is an invariant
+under coordinate transformations or diffeomorphisms.
+Hence S is invariant
+under diffeomorphisms
+φλ : x → x(λ) or infinitesimal: x → ¯x = x + dλ ξ(x)
+(48)
+where the vector field ξ(x) induces the flow of the diffeomorphism group φλ.
+If T (x) is some arbitrary tensor field, we construct an λ-dependent tensor
+field in the following way. We shift back the tensor at x(λ) to the point x
+Tλ(x) := φ∗
+−λ(T (xλ))
+(49)
+with φ∗
+−λ = (φ∗
+λ)−1 the map, induced by φλ : x → x(λ). Now all these Tλ(x)
+are defined at the same point x and we can take the derivative with respect to
+λ at λ = 0. We get
+0 = d S[gab]/d λ =
+�
+d/dλ L(gab(x; λ) d4 x
+(50)
+where in the following the derivative is taken always at λ = 0 and L is the scalar
+density R(gab) · √−g.
+Remark 5.2 Technical details can be found, for example, in [46], Appendix C.
+17
+
+Observation 5.3 The derivative with respect to λ at λ = 0 is nothing but the
+Lie derivative, Lξ, of the tensor T or more general geometric objects. As with
+ordinary derivatives, the Lie derivative obeys the Leibniz rule.
+Definition 5.4 In the following we abreviate
+Lξ(T )(x) := δT (x)
+(51)
+that is, in particular
+d/dλ gab(x; λ) = Lξ gab(x) = δgab(x)
+(52)
+It is an important observation that, as the Lie derivative is defined for general
+differentiable manifolds, it is independent of the concept of covariant derivative.
+It can hence be expressed with the help of, on the one hand, partial derivatives,
+on the other hand, expressed by means of an arbitrary covariant derivative
+operator. For example with the help of the covariant derivative, induced by
+gab(x):
+Lξ gab = ∇aξb + ∇bξa = ξc∂cgab + gcb∂aξc + gacξc
+(53)
+While the Lie derivative for tensor fields is canonically given via φ∗
+−λ(T (xλ))
+as
+Lξ(T )(x) := lim
+λ→0 (φ∗
+−λ(T (xλ)) − T (x))/λ
+(54)
+one has to say some words in the case of e.g. densities.
+Remark 5.5 A scalar density like L(x) becomes an invariant if multiplied by
+the volume element d4x. If we have a scalar density at point x(λ), its translate
+back to x has to be again a scalar density so that e.g. L d4(xλ) remains invariant.
+What we have said in the remark, allows us to fix the necessary transformation
+properties and define the Lie derivative of a scalar density.
+Lemma 5.6 We have
+Lξ(√−g · R) = −√−g∇a(R · ξa) = −∂a(R · ξa · √−g)
+(55)
+We are now going to describe the strategy which will lead to the derivation
+of a conserved current, following from diffeomorphism invariance.
+We shall
+however not follow the perhaps more obvious strategy to exploit the above
+integral expression for S[gab] but will use a local approach as it is for example
+done in [47] in the case of the ordinary energy-momentum tensor conservation
+law following from 4-translation invariance in quantum field theory and also
+used in [8]. Our approach will consist of essentially two steps.
+i) We take the Lagrange density ([gab(x)] which is a lengthy expression in
+gab(x) and its first and second partial derivatives and, using the Leibniz rule
+and that partial derivatives and Lie derivative do commute, represent its Lie
+derivative as a long expression consisting ultimately terms containing the Lie
+18
+
+derivative of the basic building blocks gab(x) which we derived above. Note that
+we use the abbreviation
+d/dλ gab(x; λ) = Lξ(gab(x) =: δgab(x)
+(56)
+Diffeomorphism invariance enters in the way that L[gab] does not explicitly
+depend on the coordinates but only via the field gab and its derivatives. That
+is, this approach exploits the structural form of the Lagrange density.
+ii) In a second step we simply directly calculate the Lie derivative of the
+density R√−g as described in formula (55). We then equate the numerically
+identical expressions and bring them on the same side, thus getting a vanishing
+expression which can be written as a conserved current.
+Now, using formula 74 in the appendix and the contracted Bianch identity,
+we get for the variation of L[gab]:
+δL[gab] = √−g · ∇a(va + 2Gabξb)
+(57)
+with
+va = ∇b(∇aξb + ∇bξa) − gcd∇a(∇cξd + ∇dξc)
+(58)
+(some technical remarks:
+∇a := gab∇b
+and
+∇bgac = 0 = ∇bgac
+(59)
+for the Levi-Civita connection).
+Now using in the second step the direct Lie derivative of the Lagrange density
+R · √−g, which we already wrote down above (see formula 55), we arrive at the
+conserved Noether current.
+Theorem 5.7 The Lie derivative of R·√−g, derived in the two ways, described
+above, yields the conserved Noether current
+0 = ∇a(Rξa + 2Gabξb + va)
+(60)
+with
+va = ∇b(∇aξb + ∇bξa) − gcd∇a(∇cξd + ∇dξc)
+(61)
+We can rewrite this formula to get a slightly different result. With
+2Gabξb = 2Rabξb − Rgabξb
+,
+gabξb = ξa
+(62)
+we get
+Corollary 5.8 A variant of the above result is
+0 = ∇aja
+,
+ja = (2Rabξb + va)
+(63)
+19
+
+6
+The Thermal Meaning of the Conserved Noether
+Current
+We want now to come back to the interpretation of the conserved Noether cur-
+rent we have derived above. We mentioned in the introduction that Hilbert and
+his colleagues had great difficulties to understand the role the arbitrary vector
+field ξi(x) is playing in this expression. As we have an energy-momentum ten-
+sor Tab, playing a fundamental role in the theory, the occurrence of a conserved
+energy vector (as Hilbert liked to call it) like our ji(x) was puzzling.
+In this context we remind the reader what we said in section 3 concerning
+the role of tensorial or covariant quantities compared to non-tensorial or non-
+covariant quantities, the former referring to properties of the smooth classical
+surface structure ST, the latter ones referring to the underlying quantum me-
+chanical micro structure QST. If we assume the vector field ξi(x) is chosen to be
+time like, we can associate it with the orbits of observers or measuring devices.
+Then expressions like
+Tab(x)ξb(x)
+or
+Gab(x)ξb(x)
+(64)
+represent (energy) flows as observed or measured by the respective moving ob-
+servers. That is, they have an objective (geometric) quality. In this sense they
+should have a covariant tensorial character and therefore can create a covariant
+conserved current.
+Observation 6.1 The (time like) vector fields ξi(x) can be assumed to be the
+orbits of observers, floating through space-time.
+In section 3 we argued that
+Gab(x) = κ · Tab(x)
+(65)
+is a statement about gravitational heat energy influx at macroscopic space-
+time point x. Correspondingly the first part of the conserved Noether current
+describes the flow of heat energy contributing to the internal energy of the
+gravitational system. We now analyze the physical meaning of the second part
+vi(x).
+va = ∇b(∇aξb + ∇bξa) − gcd∇a(∇cξd + ∇dξc)
+(66)
+If ξi(x) is a Killing vector field, that is, if it induces a symmetry of the metric
+tensor,
+Lξgab(x) = 0
+or
+∇aξb + ∇bξa = 0
+(67)
+and using
+∇bξa = ∇b(gacξc) , ∇aξb = gac∇cξb , ∇bgac = 0
+(68)
+we get:
+Theorem 6.2 If ξi(x) is a Killing vector field we have vi(x) = 0.
+20
+
+Since in the preceding sections we argued that QST is a thermal system,
+possessing at each macroscopic space-time point x the local state functions
+internal energy and entropy as well as a notion of local heat influx given by
+Tab(x) or Gab(x), it suggests itself to regard the vector field vi(x), which contains
+mainly contributions built from the metric, gab(x), and the vector field ξi(x) as
+work terms.
+Conjecture 6.3 We assume that the vector field vi(x) contains the work terms
+of our gravitational system QST. It contains essentially the effects of 4-volume
+changes (compressions and decompressions).
+Observation 6.4 This interpretation is supported by the above result that vi(x) =
+0 for Killing vector fields inducing geometric symmetries, Lξgab = 0, i.e., es-
+sentially no 4-volume changes.
+In the introduction we mentioned the work of Sakharov. Induced Gravity is
+assumed to result from the deformation of the structure of vacuum fluctuations
+by curvature. Our above working philosophy is a related one. Locally com-
+pressing or decompressing QST affects the local level structure of the thermal
+system at point x and thus is a kind of work, done at the system.
+We would like to make some remarks concerning a point which irritates or
+irritated many researchers. It is frequently argued that mere coordinate trans-
+formations can completely alter the numerical values of quantities like energy or
+work, even make them vanish in case we are dealing with non-tensor quantities.
+We dealt already with such problems in [40] in the context of special relativity
+(SR), in particular if volume changes due to Lorentz contraction have to be
+included in thermodynamic work terms.
+Some researchers have the attitude to consider Lorentz contraction as not
+being real, whatever that actually means. We think Pauli in [20] sect.5 made
+this point particularly clear.
+He argues that the atomic physics, underlying
+the contraction of a measuring rod is complicated but has to obey Lorentz
+covariance. Therefore Lorentz contraction, in his view, is, on the one hand, an
+objective process, but, as it is at the same time a result of Lorentz symmetry, it
+can as well be explained with the help of the general Lorentz invariance of SR.
+The same is true, in our view, in the case of curvature effects on the micro
+structure of QST. Furthermore, we can choose at each space-time point x a local
+geodesic coordinate system in which SR does hold, thus establishing the close
+relatedness of SR and GR. What concerns pure coordinate transformations,
+there exist basically two possibilities. On the one hand, they may be related to
+concrete changes of reference systems to which our above remarks do apply. On
+the other hand, they may not be implementable by concrete reference systems.
+In that case we may assume that transformation behavior should be regarded
+as a consequence following from consistency reasons.
+21
+
+7
+Appendix: The Conserved Noether Current
+We begin with the calculation of the variation of R[gab] · √−g, that is, reducing
+δ(R[gab] · √−g) to an expression which contains only terms like δgab, remem-
+bering that δ denotes the Lie derivative or d/dλ at λ = 0. Furthermore we use
+the above expression for the Lie derivative
+Lξ gab = ∇aξb + ∇bξa = ξc∂cgab + gcb∂aξc + gacξc
+(69)
+As R is a relative complex expression in the variables gab and its first and second
+partial derivatives, the calculations are, as is often the case in this context,
+lengthy and a little bit intricate.
+We have (see [46] p.453)
+δ(R · √−g) = √−g(δRab)gab + √−gRabδgab + Rδ(√−g)
+(70)
+with Rab the Ricci tensor. Furthermore ([46] p.185)
+gabδRab = ∇ava
+va = ∇b(δgab) − gcd∇a(δgcd)
+(71)
+and (see [46] p.453, [20] section 23)
+δ(√−g) = −1/2 · √−ggabδgab = +1/2 · √−ggabδgab
+(72)
+(0 = δ(gabgab) = gabδgab + δgabgab)
+(73)
+Conclusion 7.1 The above formulas yield
+δL(gab) = (Rab − 1/2Rgab)δgab√−g + ∇ava
+√−g
+(74)
+Observation 7.2 Gab = (Rab − 1/2Rgab) is called Einstein tensor with Gab =
+8π · Tab. It fulfills the contracted Bianchi identity ∇aGab = 0 which can also
+be derived from diffeomorphism invariance if one neglects boundary terms (see
+[48] p.138).
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+

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@@ -0,0 +1,2940 @@
+arXiv:2301.11959v1  [math.OC]  27 Jan 2023
+Approximation of Optimal Feedback Controls for Stochastic
+Reaction-Diffusion Equations
+Wilhelm Stannat and Alexander Vogler
+Technische Universit¨at Berlin, Berlin, Germany
+January 31, 2023
+Abstract
+We approximate optimal feedback controls for stochastic reaction-diffusion equations
+by using a general type of approximation that allows us to reduce the control problem to
+an optimization over deterministic controls. Similar to [SW20] we derive necessary opti-
+mality conditions and prove the existence of an optimal control for the reduced problem.
+Furthermore we derive explicit convergence rates and numerically investigate a gradient
+descent algorithm for the approximation of the optimal feedback control using a radial
+basis approximation as a particular example.
+1
+Introduction
+The main objective of this paper is to provide approximation results for optimal control prob-
+lems of stochastic partial differential equations. For fixed time horizon T > 0 and bounded
+domain Λ ⊂ R we consider the controlled stochastic partial differential equation (SPDE)
+�
+dug
+t = [Aug
+t + F(ug
+t) + gt]dt + BdWt,
+t ∈ [0, T]
+ug
+0 = u0,
+(1)
+where W : [0, T] × Ω → L2(Λ) denotes a L2(Λ) valued cylindrical Q-Wiener process and
+A : D(A) ⊂ L2(Λ) → L2(Λ) is a densely defined self adjoint, positive definite linear operator
+with compact inverse (for example the negative Dirichlet-Laplace). For a ONS of eigenvectors
+(en)n in L2(Λ) with corresponding eigenvalues (λn)n≥1 the domain of A is characterized as
+x ∈ D(A) := {x =
+∞
+�
+n=1
+xnen ∈ L2(Λ) : ∥x∥2
+2 :=
+∞
+�
+n=1
+λnx2
+n < ∞}.
+The objective of the control problem is to minimize the cost functional
+J(g) = (J1(ug) + J2(g)) ,
+over the set of admissible controls
+A = L2([0, T] × Ω, (Ft)t∈[0,T]; U),
+Keywords and phrases — stochastic optimal control, reaction-diffusion equations, adjoint calculus, varia-
+tional methods, artificial neural networks
+Mail: • [email protected] • [email protected]
+1
+
+where U is a closed and convex subset of L2(Λ) and
+J1(u) := E
+�� T
+0
+�
+Λ
+l(t, x, ut(x))dxdt +
+�
+Λ
+m(x, uT (x))dx
+�
+,
+J2(g) := E
+�� T
+0
+∥gt∥2
+L2(Λ)dt
+�
+.
+The mathematical theory of these problems is by now well understood (see e.g. [CDP18],
+[DM13], [FGS17b], [FZ20], [FHT13], [FO16], [FHT18], [LZ15], [LZ18], [SW21b], [SW22],
+[Wes22]), however the efficient numerical approximation of these control problems still faces
+serious difficulties due to the computational complexity of classical approaches that require ei-
+ther to approximate an infinite dimensional Hamilton-Jacobi-Bellman equation or a backward
+SPDE.
+In the last couple of years there is a rising interest in more efficient methods for the
+numerical approximation of optimal control problems ([BEJ19], [CL22] [DKK21], [DP16],
+[DMPV19],[EHJ17], [GKM18], [KK18], [NR21], [OSS22], [SW21a]).
+In particular applica-
+tions of machine learning algorithms to the approximation of backward stochastic differential
+equations and optimal control problems have drawn a lot of attention recently([EHJ17],[CL21],
+[CL22]). However the literature in this topic for SPDE’s is still very sparse. One main diffi-
+culty in the case of optimal control of SPDE’s is that the solution to the state equation (1)
+takes values in an infinite dimensional space. Assuming the optimal control ˆg is of markovian
+feedback type, i.e. ˆgt = ˆG(t, u ˆG
+t ), the function ˆG still depends through the solution on some
+infinite dimensional object, which makes it much more difficult to approximate using neural
+networks.
+In our approach we consider finitely based approximations for the optimal feedback func-
+tion ˆG instead and show that it is indeed sufficient if one restricts the optimization to some
+ansatz space of admissible controls that can approximate those finitely based approximations,
+in order to reach the optimal cost. Based on this observation we can construct explicit exam-
+ples of ansatz spaces that allow an efficient numerical approximation of the control problem.
+Furthermore we derive explicit convergence rates under additional assumptions on the regu-
+larity of ˆG and provide some numerical illustrations of our findings.
+The rest of this paper is organized as follows: In section 2 we provide standing assumptions
+and state our main results. In Section 2.2 we will discuss sufficient assumptions under which
+the control problem can be reduced to a feedback control problem. In section 3 we will give a
+prove for our first main theorem, which is reminiscent of a universal approximation result. In
+section 4 we will prove our second main result, which provides convergence rates. Section 5 we
+will provide explicit examples of ansatz spaces where the assumptions of our main theorems
+are satisfied.
+Numerical illustrations are provided in the companion manuscript [SVW23].
+2
+Standing Assumptions and Main Results
+2.1
+Standing Assumptions
+Let Λ ⊂ Rd be a bounded domain and H = L2(Λ). Furthermore let A : D(A) ⊂ H → H be
+a self-adjoint, negative definite linear operator such that A has a compact inverse. It follows
+that A has a discrete spectrum and there exists orthonormal basis (en)n≥1 of H consisting
+of eigenfunctions of A with corresponding eigenvalues (−λn)n≥1 arranged in increasing order
+0 ≤ λ1 ≤ λ2 ≤ . . ..
+2
+
+For r ∈ R we can then define the fractional operator (−A)
+r
+2 : D((−A)
+r
+2 ) → L2(Λ) by
+(−A)
+r
+2u :=
+∞
+�
+n=1
+λ
+r
+2nunen,
+for all
+u ∈ D((−A)
+r
+2) := {u =
+∞
+�
+n=1
+unen ∈ H : ∥u∥2
+r :=
+∞
+�
+n=1
+λr
+nu2
+n < ∞}.
+To simplify notations let Hr := D((−A)
+r
+2 ) and note that ∥u∥r = ∥(−A)
+r
+2u∥H defines a
+seminorm on Hr. By L0
+2 we denote the space of Hilbert-Schmidt operators Φ : H → H with
+norm
+∥Φ∥2
+L0
+2 :=
+∞
+�
+n=1
+∥Φen∥2
+H
+and for Φ taking values in Hr we define ∥Φ∥L0
+2,r := ∥A
+r
+2 Φ∥L0
+2.
+Remark 1. If A = ∆, where ∆ denotes the Laplace operator with Dirichlet boundary condi-
+tions, it is well known that H1 = H1
+0(Λ) and H2 = H2(Λ) ∩ H1
+0(Λ). In the case of Neumann
+boundary conditions one can consider the operator (A − αI) for some α > 0 and f(u) + αu
+in the non-linearity instead.
+We impose the following assumptions on the coefficients of the controlled spde (1):
+Assumption 2.
+H1. We assume that F is of Nemytskii type, i.e. F(u)(x) = f(u(x)) for
+some f : R → R continuously differentiable and Lipschitz continuous
+|f(u1) − f(u2)| ≤ C|u1 − u2|
+∀u1, u2 ∈ R,
+for some constant C > 0.
+H2. The dispersion operator B√Q is Hilbert-Schmidt.
+H3. For any t ∈ [0, T] and x ∈ R the functions l(t, x, u) : [0, T] × R × R → R+ and m(x, u) :
+R × R → R+ are differentiable with respect to the variable u.
+H4. For any t ∈ [0, T] and x ∈ R the functions l(t, x, u) and m(x, u) are locally Lipschitz
+continuous w.r.t. u such that
+|l(t, x, u1) − m(t, x, u2)| ≤ C(1 + |u1| + |u2|)|u1 − u2|,
+|m(x, u1) − m(x, u2)| ≤ C(1 + |u1| + |u2|)|u1 − u2|
+∀u1, u2 ∈ R,
+for some constant C > 0.
+H5. The initial condition u0 : Ω → H is F0- measurable with
+E
+�
+∥u0∥p
+H
+�
+< ∞,
+for all p ≥ 2.
+3
+
+2.2
+Reduction to Feedback Controls
+As already mentioned in the introduction, we will work in a setting where there exists an
+optimal control ˆg ∈ A of feedback type, i.e.
+inf
+g∈A J(g) = J(ˆg)
+and
+ˆgt = ˆG(t, u
+ˆG
+t ),
+for some feedback map ˆG : [0, T] × H → H, such that the equation
+�
+du ˆG
+t = [∆u ˆG
+t + f(u ˆG
+t ) + ˆG(t, u ˆG
+t )]dt + BdWt,
+t ∈ [0, T]
+u ˆG
+0 = u0,
+(2)
+has a unique probabilistic strong solution. Furthermore we will assume that the feedback map
+ˆG is jointly continuous and satisfies a linear growth condition
+∥ ˆG(t, u)∥H ≤ C(1 + ∥u∥H).
+The simplest example where these assumptions are matched is the case of a linear quadratic
+control problem, where F is linear and ˜l(t, x, u) = ∥u∥2
+H, m(t, u) = ∥u∥2
+H. In this case, the
+optimal control g∗ is indeed of feedback form, given by
+g∗
+t = P(t)ug∗
+t
+(3)
+where P : [0, T] → L(H) is the solution of the Riccati equation [Tud90]
+�
+∂tP(t) + P(t)∆ + ∆P(t) − Id + P 2(t) = 0, t ∈ [0, T]
+P(T) = −Id.
+(4)
+We will give a short explanation how one can construct such an optimal feedback function
+ˆG in a more general situation. To this end we consider the Hamilton-Jacobi-Bellman equation
+
+
+
+
+
+∂tV + 1
+2tr(BQB∗D2V ) + ⟨DV, Au + F(u)⟩H +
+�
+Λ l(t, x, u(x)) dx
++ infG∈U{⟨Dv, G⟩ + ∥G∥2
+H} = 0,
+u ∈ H, t ∈ [0, T]
+VT (u) =
+�
+Λ m(x, u(x)) dx, u ∈ H.
+(5)
+We assume that the HJB equation (5) has a unique mild solution V : [0, T] × H → R in the
+sense of ([FGS17b], Def. 4.70), such that
+V ∈ C0,1([0, T] × H, R).
+If in addition the function γ(p) := arginfg∈U{⟨p, g⟩ + ∥g∥2
+H} is continuous and equation (2)
+has a unique strong solution for ˆG(t, u) = γ(DV (t, u)), then an optimal control of (SCP) is
+given by
+ˆgt = ˆG(t, u
+ˆG
+t ).
+For the existence, regularity of a solution to the HJB equation (5) and examples we refer to
+[FGS17b], in particular section 4.8.1 and section 6.11. A sufficient condition for (2) to have
+a unique strong solution is that ˆG is Lipschitz continuous in u, see e.g. [LR15]. This is in
+particular the case if the solution V of the HJB equation (5) has a bounded second derivative
+in u, see Theorem 4.155 and Remark 4.202 in [FG´S17a]. Note that ˆG(t, u) is in particular
+bounded if U is a bounded subset of H.
+In the following we will consider the feedback control problem
+4
+
+Problem 3 (FCP). Minimize
+J(G) := J1(uG) + J2(G(·, uG
+· )),
+subject to the spde
+�
+duG
+t
+= [AuG
+t + F(uG
+t ) + G(t, uG
+t )]dt + BdWt,
+t ∈ [0, T]
+uG
+0
+= u0,
+(6)
+over the set of admissible feedback controls
+Uad := {G ∈ C([0, T] × H, U)| equation (6) has a unique strong solution and G(t, uG
+t ) ∈ A}.
+Remark 4. It is obvious that under the above assumptions we have
+inf
+g∈A J(g) =
+inf
+G∈Uad J(G).
+2.3
+Approximation Results
+For the approximation of the feedback function ˆG, let (Sh)h∈(0,1] be a family of finite dimen-
+sional subspaces of H1 satisfying
+∥Phu − u∥H → 0, h → 0,
+where Ph : H → Sh denotes the L2-orthogonal projection onto Sh. We will also need the
+orthogonal projection Rh : H1 → Sh onto Sh w.r.t. the inner product
+a(u, v) := ⟨(−A)
+1
+2u, (−A)
+1
+2 v⟩H.
+We then define for any map G : [0, T]×H → H the corresponding finitely based approximation
+of order h ∈ (0, 1] by
+Gh : [0, T] × H → Sh, Gh(t, u) := PhG(t, Phu).
+(7)
+Here finitely-based means that Gh(t, u) depends on finitely many coordinates of u only. In a
+first step we can then approximate our feedback function ˆG by the finitely based vector fields
+( ˆGh)h∈(0,1].
+2.3.1
+Uniform Approximation Result
+Definition 5. Let G : [0, T] × H → H. We say that a subset U ⊂ Uad satisfies the uniform
+approximation property with respect to G, if there exists a family (Gh,m)h∈(0,1],m∈N ⊂ U that
+satisfies the following linear growth condition uniformly in m
+∥Gh,m(t, u)∥H ≤ Ch(1 + ∥u∥H),
+(8)
+for some constant Ch > 0, such that for any R > 0
+lim
+m→∞
+sup
+(t,u)∈[0,T]×BH(0,R)
+∥Gh(t, u) − Gh,m(t, u)∥2
+H = 0,
+(9)
+where
+BH(0, R) := {u ∈ H|∥u∥H ≤ R}
+(10)
+and Gh is given as in (7).
+5
+
+Theorem 6. Assume that Assumption 2 is in force. Let U ⊆ Uad satisfy the uniform approx-
+imation property with respect to ˆG, then
+inf
+g∈A J(g) = inf
+G∈U J(G).
+Remark 7. Theorem 6 tells us, if any finitely based approximation of ˆG can be approximated
+on bounded sets by elements of U, then we reach the optimal cost by minimizing over U.
+2.3.2
+Convergence Rates
+In order to formulate our second result we need to impose stronger assumptions on our control
+problem.
+Assumption 8.
+S1. The dispersion operator B√Q takes values in H1 and is Hilbert-Schmidt,
+i.e.
+∥B
+�
+Q∥L0
+2,1 < ∞
+S2. The initial condition u0 : Ω → H1 is F0- measurable with
+E
+�
+∥u0∥p
+H1
+�
+< ∞,
+for all p ≥ 2.
+S3. The admissible controls take values in H1, in particular we assume that
+U ⊆ H1.
+Furthermore, the optimal feedback ˆG(t, u) : [0, T] × H → U is Lipschitz continuous in u
+with Lipschitz constant L independent of t, i.e.
+∥ ˆG(t, u1) − ˆG(t, u2)∥H ≤ L∥u1 − u2∥H
+∀t ∈ [0, T], u1, u2 ∈ H
+Unfortunately the existence of a solution to the HJB equation (5) with Lipschitz regularity
+is difficult to check. However Assumption 8 is satisfied in the linear quadratic case (see Section
+2.2), i.e. if f is linear and l is a quadratic function, for example
+l(t, x, u) = |u|2.
+Since also the numerical approximation of the state equation will contribute to the con-
+vergence rates, we will also specify additional assumptions on our given finite element approx-
+imation:
+Assumption 9.
+R1. For all u ∈ Hs, s ∈ {1, 2}, h ∈ (0, 1] it holds
+∥Rhu − u∥H ≤ Chs∥u∥Hs
+and
+∥Phu∥1 ≤ C∥u∥H1 for all u ∈ H1, h ∈ (0, 1].
+R2. The Ritz-projection coincides with the L2-orthogonal projection on H1, i.e.
+Ph|H1 = Rh
+6
+
+Remark 10. By the best approximation property of the orthogonal L2-projection we get for
+any u ∈ Hs, with s ∈ {1, 2}
+∥Phu − u∥H ≤ ∥Rhu − u∥H ≤ Chs∥u∥Hs.
+Assumption R2 is satisfied for example when A = ∆, where ∆ denotes the Laplace operator
+with Dirichlet boundary conditions on the unit interval (0, 1) ⊂ R, and the finite dimensional
+subspaces Sh are given by
+Sh = span{ek, k = 1, ..., N},
+for h = λ
+− 1
+2
+N+1, N ∈ N, where ek(·) =
+√
+2 sin(kπ·) are the orthonormal eigenfunctions of A
+with corresponding eigenvalues λk = k2π2 (see [Kru1403]).
+We now introduce the following approximating control problem.
+Problem 11 (FESCP). Let h ∈ (0, 1]. Minimize
+Jh(G) =
+�
+J1(uG,h) + J2(G(·, uG,h))
+�
+,
+over the set
+UL
+ad := {G ∈ C([0, T] × H, U)|G(t, u) is Lipschitz continuous in u, uniformly in t},
+subject to the discretized SPDE
+�
+duG,h
+t
+= [AhuG,h
+t
++ PhF(uG,h
+t
+) + PhG(t, uG,h
+t
+)]dt + PhBdWt,
+t ∈ [0, T]
+uG,h
+0
+= Phu ∈ Sh,
+(11)
+where Ahuh for uh ∈ Sh is defined as the unique element in Sh with
+a(uh, vh) = ⟨Ahuh, vh⟩H, for all vh ∈ Sh.
+The following lemma is easily shown by standard arguments.
+Lemma 12. For any G ∈ U L
+ad there exists a unique strong solution uG,h : [0, T] × Ω → Sh to
+(11) and for any p ≥ 2 it holds
+sup
+h∈(0,1]
+�
+sup
+t∈[0,T]
+E
+�
+∥uG,h
+t
+∥p
+H +
+�� T
+0
+∥uG,h
+t
+∥2
+H1dt
+�p/2��
+< ∞.
+Definition 13. Let G : [0, T] × H → H and h ∈ (0, 1]. We say that a sequence of subsets
+(Uh,m)m∈N ⊂ Uad satisfies the uniform Lipschitz approximation property of order h with
+respect to G, if there exists a sequence of Lipschitz continuous controls (Gh,m)m∈N with
+Lipschitz constants independent of m, such that Gh,m ∈ Uh,m, and for all R > 0
+ǫh,R
+m
+:=
+sup
+(t,u)∈[0,T]×BH(0,R)
+∥Ph(Gh(t, u) − Gh,m(t, u))∥2
+H
+(12)
+→ 0,
+(13)
+as m → ∞.
+Our two main theorems of this section can then be formulated as follows:
+7
+
+Theorem 14. Let h ∈ (0, 1]. We assume that the assumptions 2, Assumption 8 and Assump-
+tion 9 R1 are in force.
+Let (Uh,m)m∈N ⊂ Uad satisfies the uniform Lipschitz approximation property of order h
+with respect to ˆG, then it holds
+inf
+G∈Uh,m Jh(G) − inf
+g∈A J(g) ≤ C
+�
+1 + E
+�� T
+0
+∥ ˆG(t, u
+ˆG
+t )∥2
+H1 dt
+�1/2�
+h + Ch
+�
+ǫh,R
+m
++ 1
+R
+�1/2
+,
+for all R > 0, some uniform constant C > 0 and some constant Ch > 0 that is independent
+of m.
+Under additional convexity assumptions on the coefficients we can prove a lower bound
+on the approximating costs. To this end consider the following modified assumptions
+Assumption 15. H1’. The function f : R → R is linear.
+H2’. For any t ∈ [0, T], x ∈ Λ and all u1, u2 ∈ R it holds
+l(t, x, u2) − l(t, x, u1) − ∂ul(t, x, u1)(u2 − u1) ≥ 0
+m(x, u2) − m(x, u1) − ∂um(x, u1)(u2 − u1) ≥ 0.
+Theorem 16. In addition to the assumptions of Theorem 14 we assume that Assumption 9
+and Assumption 15 are in force. Then for any U ⊆ Uad it holds
+inf
+g∈A J(g) − inf
+G∈U Jh(G) ≤ Ch,
+for some constant C > 0, which is independent of h.
+3
+Proof of Theorem 6
+To simplify notations we denote V = H1, Pn := P 1
+n , Gn := ˆG
+1
+n and Gn,m(t, u) := G
+1
+n ,m(t, u).
+Furthermore let
+un := uGn
+be the unique strong solution to equation (6) w.r.t. Gn and
+un,m := uGn,m
+be the unique strong solution to equation (6) w.r.t. Gn,m. Due to the linear growth assump-
+tions on ˆG and Gh,m, one can obtain the following a priori estimates by standard arguments.
+Lemma 17. For any p ≥ 2 it holds
+sup
+n∈N
+E
+�
+sup
+t∈[0,T]
+∥un
+t ∥p
+H +
+�� T
+0
+∥un
+t ∥2
+V dt
+�p/2�
+≤ C(1 + ∥u0∥p
+H).
+Furthermore for any p ≥ 2 and fixed n ∈ N we have
+sup
+m∈N
+E
+�
+sup
+t∈[0,T]
+∥un,m
+t
+∥p
+H +
+�� T
+0
+∥un,m
+t
+∥2
+V dt
+�p/2�
+≤ Cn(1 + ∥u0∥p
+H).
+The following tightness result is a standard consequence of the a-priori bound given in
+Lemma 17, for further detail see [FG95].
+Lemma 18. The sequence (un)n∈N is tight in C([0, T], V ′) ∩ L2([0, T], H) and w.r.t.
+the
+weak topology in L2([0, T], V ) and for any fixed n ∈ N the sequence (un,m)m∈N is tight in
+C([0, T], V ′) ∩ L2([0, T], H) and w.r.t. the weak topology in L2([0, T], V ).
+8
+
+3.1
+Finitely Based Approximation of the Optimal Cost
+In the first part we will prove that the optimal cost can be approximated by the cost of the
+finitely based vector-fields gn
+t := Gn(t, un
+t ). In particular we will show that for
+J(gn) → J(ˆg) = inf
+g∈A J(g)
+(14)
+along a subsequence. In the following we set
+Γ := (L2([0, T], H) ∩ C([0, T], V ′)) × C([0, T], U1),
+where U1 is a Hilbert space such that the embedding U ⊂ U1 is Hilbert-Schmidt. Now by
+Prohorov’s theorem and the Skorohod representation theorem, there exist a probability space
+(˜Ω, ˜F, ˜P) and a sequence of Γ-valued random variables (˜un, ˜W n)n and (˜u, ˜W), such that
+1. ˜W n = ˜W for any n ∈ N, ˜P-a.s.
+2. L(˜un, ˜W n) = L(un, W)
+3. ˜P-a.s.,
+∥˜un − ˜u∥L2([0,T],H) + ∥˜un − ˜u∥C([0,T],V ′) → 0.
+It is not difficult to see that (˜un)n∈N satisfies for any p ≥ 2
+sup
+n∈N
+˜E
+�
+sup
+t∈[0,T]
+∥˜un
+t ∥p
+H
+�
++ sup
+n∈N
+˜E
+��� T
+0
+∥˜un
+t ∥2
+V dt
+��p/2
+< ∞.
+Furthermore, since ∥˜un − ˜u∥L2([0,T],H) → 0, we have by using Fatou’s lemma
+˜E
+�
+sup
+t∈[0,T]
+∥˜ut∥p
+H +
+�� T
+0
+∥˜ut∥2
+V dt
+�p/2�
+≤ lim inf
+n→∞
+˜E
+�
+sup
+t∈[0,T]
+∥˜un
+t ∥p
+H +
+�� T
+0
+∥˜un
+t ∥2
+V dt
+�p/2�
+< ∞.
+This implies in particular, that ˜u ∈ L2 �
+˜Ω, ˜P, L2([0, T], V )
+�
+and ˜un → ˜u weakly.
+We will now prove that ˜u is a weak solution to (6). Recall that
+⟨Au, v⟩ = −⟨(−A)
+1
+2 u, (−A)
+1
+2v⟩,
+so that ∥Au∥V ′ ≤ ∥u∥V , which implies that
+˜E
+�� T
+0
+∥A˜un
+t ∥2
+V ′dt
+�
+< ∞.
+For any v ∈ L2 �
+˜Ω, ˜P, L2([0, T], V )
+�
+it follows that
+˜E
+�� T
+0
+⟨A˜un
+t , vt⟩ dt
+�
+= ˜E
+�� T
+0
+⟨˜un
+t , Avt⟩ dt
+�
+→ ˜E
+�� T
+0
+⟨˜ut, vt⟩ dt
+�
+= ˜E
+�� T
+0
+⟨A˜ut, vt⟩ dt
+�
+which implies that A˜un → A˜u weakly in L2 �
+˜Ω, ˜P, L2([0, T], V ′)
+�
+.
+It remains to investigate the nonlinear drift coefficients.
+9
+
+Lemma 19. There exists a subsequence (˜unk)k∈N such that
+� T
+0
+∥Gnk(t, ˜unk
+t ) − ˆG(t, ˜ut)∥2
+Hdt → 0
+˜P-a.s. and in Lp(˜Ω, ˜P)
+for all p ≥ 2 and
+� T
+0
+∥F(˜unk
+t ) − F(˜ut)∥2
+Hdt → 0
+˜P-a.s. and in Lp(˜Ω, ˜P)
+for all p ≥ 2. Furthermore, the process ˜u is a H-valued continuous process that satisfies
+⟨˜ut, v⟩ = ⟨u0 +
+� t
+0
+A˜utdt +
+� t
+0
+F(˜ut)dt +
+� t
+0
+ˆG(t, ˜ut)dt + B ˜Wt, v⟩,
+t ∈ [0, T],
+for all v ∈ V , ˜P-a.s. as a V ′-valued process.
+Proof. We just need to show the first statement of the lemma, the second part follows by
+standard arguments. We have that
+∥Gn(t, ˜un
+t ) − ˆG(t, ˜ut)∥H = ∥Pn ˆG(t, Pn˜un
+t ) − ˆG(t, ˜ut)∥H
+≤ ∥Pn ˆG(t, Pn˜un
+t ) − Pn ˆG(t, Pn˜ut)∥H + ∥Pn ˆG(t, Pn˜ut) − Pn ˆG(t, ˜ut)∥H
++ ∥Pn ˆG(t, ˜ut) − ˆG(t, ˜ut)∥H
+≤ ∥ ˆG(t, Pn˜un
+t ) − ˆG(t, Pn˜ut)∥H + ∥ ˆG(t, Pn˜ut) − ˆG(t, ˜ut)∥H
++ ∥Pn ˆG(t, ˜ut) − ˆG(t, ˜ut)∥H.
+(15)
+Since ˜un
+t → ˜ut in L2([0, T], H) it follows, passing to some subsequence again denoted with
+(˜un), that ˜un
+t → ˜ut in H ˜P ⊗ dt-a.e. . Now (15), ∥Pnv − v∥H → 0 for all v ∈ H and continuity
+of ˆG imply that
+∥Gn(t, ˜un
+t ) − ˆG(t, ˜ut)∥H → 0
+˜P ⊗ dt − a.e.
+For convergence of
+� T
+0 ∥ ˆGn(t, ˜un
+t ) − ˆG(t, ˜ut)∥2
+Hdt in Lp(˜Ω, ˜P) for all p ≥ 2, it now suffices to
+show
+sup
+n
+˜E
+�� T
+0
+∥Gn(t, ˜un
+t ) − ˆG(t, ˜ut)∥p
+Hdt
+�
+< ∞
+(16)
+for all p ≥ 2, since the latter implies uniform integrability of ∥Gn(t, ˜un
+t )− ˆG(t, ˜ut)∥p
+H, and thus
+� T
+0
+∥Gnk(t, ˜unk
+t ) − ˆG(t, ˜ut)∥2
+Hdt → 0
+˜P-a.s. and in Lp(˜Ω, ˜P)
+for all p ≥ 2.
+17 implies that
+˜E
+�� T
+0
+∥Gn(t, ˜un
+t ) − ˆG(t, ˜ut)∥p
+Hdt
+�
+≤ 2p ˜E
+�� T
+0
+∥Pn ˆG(t, ˜un
+t )∥p
+H + ∥ ˆG(t, ˜ut)∥p
+Hdt
+�
+≤ 2pC ˜E
+�� T
+0
+2 + ∥Pn˜un
+t ∥p
+H + ∥˜ut∥p
+Hdt
+�
+≤ 2p+1C T
+�
+1 + ˜E
+�
+sup
+t∈[0,T]
+∥˜un
+t ∥p
+H
+��
+< ∞
+10
+
+for all p ≥ 2.
+The convergence
+� T
+0
+∥F(˜unk
+t ) − F(˜ut)∥2
+H dt → 0
+˜P-a.s. and in Lp(˜Ω, ˜P) for all p ≥ 2 follows by similar (even simpler) calculations, due to the
+Lipschitz continuity of f.
+Lemma 20. There exists a subsequence (˜unk)k∈N such that for any t ∈ [0, T]
+∥˜unk
+t
+− ˜ut∥H → 0,
+˜P-a.s. and in Lp(˜Ω, ˜P).
+Proof. We have for any t ∈ [0, T]
+∥˜un
+t − ˜ut∥2
+H = 2
+� t
+0
+⟨˜un
+s − ˜us, A(˜un
+s − ˜us)⟩ds +
+� t
+0
+⟨˜un
+s − ˜us, F(˜un
+s ) − F(˜us)⟩Hds
++
+� t
+0
+⟨˜un
+s − ˜us, Gn(s, ˜un
+s ) − ˆG(s, ˜us)⟩Hds
+≤ −2
+� t
+0
+∥˜un
+s − ˜us∥2
+V ds + 4
+� t
+0
+∥˜un
+s − ˜us∥2
+Hds
++
+� t
+0
+∥F(˜un
+s ) − F(˜us)∥2
+Hds +
+� t
+0
+∥Gn(s, ˜un
+s ) − ˆG(s, ˜us)∥2
+Hds.
+Using the Lipschitz continuity of f and Gronwall inequality, we end up with
+∥˜un
+t − ˜ut∥2
+H ≤ CT
+� T
+0
+∥Gn(s, ˜un
+s ) − ˆG(s, ˜us)∥2
+Hds.
+Now by Lemma 19 there exists a subsequence, again denoted by (˜un
+t )n, such that
+� T
+0
+∥Gn(s, ˜un
+s ) − ˆG(s, ˜us)∥2
+Hds → 0,
+˜P-a.s. and in Lp(˜Ω, ˜P). This finishes the proof.
+Now for a generalized probability space µ := (Ωµ, Fµ, (Fµ
+t )t, Pµ, W µ) we define for any
+g ∈ L2(Ωµ × [0, T], H)
+Jµ(g) := Jµ
+1 (ug) + Jµ
+2 (g),
+where
+Jµ
+1 (u) := Eµ
+�� T
+0
+�
+Λ
+l(t, x, ut(x))dxdt +
+�
+Λ
+m(x, uT (x))dx
+�
+,
+Jµ
+2 (g) := Eµ
+�� T
+0
+∥gt∥2
+Hdt
+�
+,
+and ug is the solution to the spde
+�
+dug
+t
+= [Aug
+t + F(ug
+t) + gt]dt + BdW µ
+t ,
+t ∈ [0, T]
+ug
+0
+= u0.
+(17)
+To show (14) we need the following lemma.
+11
+
+Lemma 21. Let µ be a generalized probability space and u1, u2 ∈ L2(Ωµ × [0, T], H) and
+g1, g2 ∈ A, such that
+∥g1∥L2(Ωµ×[0,T],H) + ∥g2∥L2(Ωµ×[0,T],H) ≤ κ,
+and
+∥u1∥L2(Ωµ×[0,T],H) + ∥u2∥L2(Ωµ×[0,T],H) ≤ κ,
+then it holds
+|J1(u1) − J1(u2)| ≤ C(κ)
+�
+∥u1 − u2∥L2(Ωµ×[0,T],H) + ∥u1(T, ·) − u2(T, ·)∥L2(Ωµ,H)
+�
+and
+|J2(g1) − J2(g2)| ≤ C(κ)∥g1 − g2∥L2(Ωµ×[0,T],H),
+for some constant C(κ) > 0.
+Proof. Using the assumptions on l and m we get
+|J1(u1) − J1(u2)|
+≤ Eµ
+�� T
+0
+�
+Λ
+|l(t, x, u1(t, x)) − l(t, x, u2(t, x))|dxdt
+�
++ Eµ
+��
+Λ
+|m(x, u1(T, x)) − m(x, u2(T, x))|dx
+�
+≤ L Eµ
+�� T
+0
+�
+Λ
+(1 + |u1(t, x)| + |u2(t, x)|) |u1(t, x) − u2(t, x)|dxdt
+�
++ L Eµ
+��
+Λ
+(1 + |u1(T, x)| + |u2(T, x)|) |u1(T, x) − u2(T, x)|dx
+�
+.
+On the other hand we have
+|J2(g1) − J2(g2)|
+≤ Eµ
+�� T
+0
+��∥g1(t)∥2
+H − ∥g2(t)∥2
+H
+�� dt
+�
+≤ Eµ
+�� T
+0
+�
+Λ
+(1 + |g1(t, x)| + |g2(t, x)|)|g1(t, x) − g2(t, x)|dxdt
+�
+.
+An application of the Cauchy-Schwarz inequality now yields the result.
+Lemma 22. There exists a subsequence (˜unk)k∈N, such that
+Jµ(Gnk(·, unk
+· )) → inf
+g∈A J(g)
+Proof. By Lemma 19 and Lemma 20, there exists a subsequence (˜unk)k∈N, such that ∥˜unk −
+˜u∥L2([0,T],H) + ∥˜unk
+T
+− ˜uT ∥H → 0, ˜P-almost surely and in Lp(˜Ω, ˜P), and L(˜u|˜P) = L(u|P),
+L(˜un|˜P) = L(un|P). Therefore by the a-priori estimates on ˜un, ˜u and the previous lemma, we
+have that for ˜µ := (˜Ω, ˜F, ( ˜Ft)t, ˜P), µ := (Ω, F, (Ft)t, P) it holds
+Jµ(Gnk(·, unk
+· )) = J ˜µ(Gnk(·, ˜unk
+· )) → J ˜µ( ˆG(·, ˜u·)) = Jµ( ˆG(·, u
+ˆG
+· )) = inf
+g∈A J(g)
+12
+
+3.2
+Approximation of the Finitely Based Minimizing Sequence
+We turn to the the second part of the proof for Theorem 6. We will show that for any n ∈ N
+we have
+J(gn,m) → J(gn),
+along a subsequence, where gn,m
+t
+:= Gn,m(t, un,m
+t
+). The proof is quite similar to the previous
+proof. The only difference is the argument for the convergence result for (˜un,m), due to the
+different assumption on the approximating sequence Gn,m, m ≥ 1.
+First note that due to the tightness of the sequence (un,m)m∈N stated in Lemma 18 there
+exist a probability space (˜Ω, ˜F, ˜P) and a sequence of Γ-valued random variables (˜un,m, ˜W m)m
+and (˜un, ˜W), such that
+1. ˜W m = ˜W for any m ∈ N, ˜P-a.s.
+2. L(˜un,m, ˜W m) = L(un,m, W)
+3. ˜P-a.s.,
+∥˜un,m − ˜un∥L2([0,T],H) + ∥˜un,m − ˜un∥C([0,T],V ′) → 0.
+Again, for any p ≥ 2, it holds that
+sup
+m∈N
+˜E
+�
+sup
+t∈[0,T]
+∥˜un,m
+t
+∥p
+H +
+�� T
+0
+∥˜un,m
+t
+∥2
+V dt
+�p/2�
+< ∞,
+(18)
+and therefore also
+˜E
+�
+sup
+t∈[0,T]
+∥˜un
+t ∥p
+H +
+�� T
+0
+∥˜un
+t ∥2
+V dt
+�p/2�
+< ∞.
+Again, we can conclude from this that ˜un ∈ L2 �
+˜Ω, ˜P, L2([0, T], V )
+�
+and ˜un,m → ˜un weakly.
+To identify ˜un as a weak solution to (6) we again have that A˜un,m → A˜un weakly in
+L2 �
+˜Ω, ˜P, L2([0, T], V ′)
+�
+, so that it remains to investigate the nonlinear drift coefficients.
+Lemma 23. Fix n. Then there exists a subsequence (˜un,mk
+·
+)k sucht that
+lim
+m→∞
+˜E
+�� T
+0
+∥Gn,mk(t, ˜un,mk
+t
+) − Gn(t, ˜un
+t )∥2
+Hdt
+�
+= 0
+˜P-a.s. and in Lp(˜Ω, ˜P)
+for all p ≥ 2 and
+lim
+m→∞
+˜E
+�� T
+0
+∥F(˜un,mK
+t
+) − F(˜un
+t )∥2
+Hdt
+�
+= 0
+˜P-a.s. and in Lp(˜Ω, ˜P)
+for all p ≥ 2. Furthermore the process ˜un is a H-valued continuous process that satisfies
+⟨˜un
+t , v⟩ = ⟨un
+0 +
+� t
+0
+A˜un
+t dt +
+� t
+0
+F(˜un
+t )dt +
+� t
+0
+Gn(t, ˜un
+t )dt + B ˜Wt, v⟩,
+t ∈ [0, T],
+for all v ∈ V , ˜P-a.s. as a V ′-valued process.
+13
+
+Proof. Again, we only need to show the first statement of the theorem, the second part follows
+again by standard arguments. This time we can estimate
+∥Gn,m(t, ˜un,m
+t
+) − Gn(t, ˜un
+t )∥H ≤ ∥Gn,m(t, ˜un,m
+t
+) − Gn(t, ˜un,m
+t
+)∥H + ∥Gn(t, ˜un,m
+t
+) − Gn(t, ˜un
+t )∥H
+= Im
+1 + Im
+2 .
+(19)
+Passing to some subsequence (still denoted by (˜un,m)), we have that
+∥Pn˜un,m
+t
+− Pn˜un
+t ∥H ≤ ∥˜un,m
+t
+− ˜un
+t ∥H → 0
+˜P ⊗ dt-.a.e.
+and continuity of ˆG then implies that
+Im
+2 ≤ ∥ ˆG(·, Pn˜un,m) − ˆG(·, Pn˜un)∥L2([0,T],H) → 0
+˜P ⊗ dt-.a.e.
+For the proof of convergence of Im
+1
+to 0 let R := supt∈[0,T] ∥˜un
+t ∥H < ∞ ˜P-a.s. Then for
+˜ω ∈ ˜Ω and t ∈ [0, T] with R(˜ω) < ∞ and ∥˜un,m
+t
+(˜ω) − ˜un
+t (˜ω)∥H → 0, the assumptions on Gn,m
+now imply that
+∥Gn,m(t, ˜un,m
+t
+(˜ω)) − Gn(t, ˜un,m
+t
+(˜ω))∥H → 0.
+Since this is true for ˜P ⊗ dt-a.e. (˜ω, t) ∈ ˜Ω × [0, T] we conclude that
+I1
+m → 0
+˜P ⊗ dt-a.s.
+The convergence of
+� T
+0 ∥Gn,m(t, ˜un,m
+t
+) − Gn(t, ˜un
+t )∥2
+Hdt in Lp(˜Ω, ˜P) for all p ≥ 2, now follows
+similar to the correpsonding argument in Lemma 19 from the uniform intergrability, implied
+by the bound
+sup
+m
+˜E
+�� T
+0
+∥Gn,m(t, ˜un,m
+t
+) − Gn(t, ˜un
+t )∥p
+Hdt
+�
+< ∞
+(20)
+for all p ≥ 2. (20) can be proven exactly in the same way as the proof of (16), using the
+uniform linear growth condition on Gn,m w.r.t. m.
+Finally, the convergence
+� T
+0
+∥F(˜un,m
+t
+) − F(˜un
+t )∥2
+H dt → 0
+˜P-a.s. and in Lp(˜Ω, ˜P) for all p ≥ 2 follows by similar (even simpler) calculations, due to the
+Lipschitz continuity of f.
+Proof. (of Theorem 6) Since ∥˜un,m − ˜un∥L2([0,T],H) → 0, ˜P-almost sure and L(˜un|˜P) =
+L(un|˜P) and L(˜un,m|˜P) = L(un,m|˜P) we have by Lemma 21 for ˜µ := (˜Ω, ˜F, ( ˜Ft)t, ˜P), µ :=
+(Ω, F, (Ft)t, P)
+Jµ(Gn,m(·, un,m
+·
+)) = J ˜µ(Gn,m(·, ˜un,m
+·
+)) → J ˜µ(Gn(·, ˜un
+· )) = Jµ(Gn(·, un
+· ))
+along a subsequence. Now together with the result from Section 3.1 we can find a sequence
+(G
+n)n∈N ⊆ U, such that
+Jµ(G
+n(·, uG
+n
+·
+)) → Jµ( ˆG(·, u
+ˆG
+· )) = inf
+g∈A J(g).
+14
+
+Therefore we obtain
+inf
+G∈U J(G) ≤ J(G
+n)
+= Jµ(G
+n(·, uG
+n
+·
+)) → inf
+g∈A J(g),
+hence
+inf
+G∈U J(G) ≤ inf
+g∈A J(g).
+Since G
+n(·, uG
+n
+·
+) ∈ A for all n ∈ N we get
+inf
+G∈U J(G) = inf
+g∈A J(g).
+4
+Proof of Theorem 14, 16
+4.1
+Finite Element Approximation
+We start with upper and lower bound estimates for the optimal cost of the finite element
+discretiation (FESCP) of the control problem (SCP).
+4.1.1
+Upper Bound
+The main theorem concerning the upper bound of the approximating cost functional Jh is
+the following:
+Theorem 24. Let h ∈ (0, 1]. Assume that Assumption 2 is in force, then it holds
+inf
+g∈A J(g) ≥ Jh( ˆG) − ǫ1(h),
+where
+ǫ1(h) = C(T)
+�
+1 + E
+�� T
+0
+∥ ˆG(t, u
+ˆG
+t )∥2
+H1 dt
+�1/2�
+h,
+for some C(T) > 0, independent of h. This shows in particular that
+inf
+g∈A J(g) ≥
+inf
+G∈UL
+ad
+Jh(G) − ǫ1(h),
+since ˆG is an admissible feedback control.
+We will now use the rest of this subsubsection to prove Theorem 24. First we quantify
+the difference between the cost of a control and its finite element approximation in terms of
+the parameter h.
+Lemma 25. Let G ∈ U L
+ad and uG be the unique solution to (6) and uG,h be the unique solution
+to (11), for h ∈ (0, 1]. Then it holds
+|J1(uG) − J1(uG,h)| ≤ C(T)h
+15
+
+and
+|J2(G(·, uG
+· )) − J2(PhG(·, uG,h
+·
+))| ≤ C(T)hE
+�� T
+0
+∥G(t, uG
+t )∥2
+H1 dt
+�1/2
+,
+for some constant C(T) > 0 which only depends on the Lipschitz constant of G, but is inde-
+pendent of h.
+Proof. By Lemma 12 it holds
+∥uG,h∥L2(Ω×[0,T],H) ≤ C,
+for some constant C > 0 which may depend on the Lipschitz constant of G, but is independent
+of h. Using ∥PhG(t, uG,h
+t
+)∥H ≤ ∥G(t, uG,h
+t
+)∥H and the fact that G(t, u) is at most of linear
+growth w.r.t. u, due to the Lipschitz property, we get
+E
+�� T
+0
+∥PhG(t, uG,h
+t
+)∥2
+H dt
+�
+≤ CE
+�� T
+0
+1 + ∥uG,h
+t
+∥2
+Hdt
+�
+≤ C
+for some constant C > 0 that depends only on the Lipschitz constant of G. Now we can apply
+Lemma 21 to obtain
+|J1(uG) − J1(uG,h)|2 ≤ CE
+�� T
+0
+∥uG
+t − uG,h
+t
+∥2
+H
+�
++ CE
+�
+∥uG
+T − uG,h
+T
+∥2
+H
+�
+.
+Using Lemma 21 and the Lipschitz continuity of G again,
+|J2(G(·, uG
+· )) − J2(PhG(·, uG,h
+·
+))|2 ≤ C E
+�� T
+0
+∥G(t, uG
+t ) − PhG(t, uG,h
+t
+)∥2
+Hdt
+�
+≤ C E
+�� T
+0
+∥Ph(G(t, uG
+t ) − G(t, uG,h
+t
+))∥2
+Hdt
+�
++ C E
+�� T
+0
+∥G(t, uG
+t ) − PhG(t, uG
+t )∥2
+Hdt
+�
+≤ C E
+�� T
+0
+∥uG
+t − uG,h
+t
+∥2
+Hdt
+�
++ C E
+�� T
+0
+∥G(t, uG
+t ) − PhG(t, uG
+t )∥2
+Hdt
+�
+(21)
+where the constant C may differ from line to line, but will be independent of h.
+Since
+G(t, uG
+t ) ∈ H1, we get from Remark 10
+∥G(t, uG
+t ) − PhG(t, uG
+t )∥2
+H ≤ h2∥G(t, uG
+t )∥2
+H1
+and from [Kru1403] we get
+E
+�� T
+0
+∥uG
+t − uG,h
+t
+∥2
+H
+�
+≤ C(T)h2
+for some constant C(T) independent of h. Inserting both estimates into (21) yields the desired
+result.
+16
+
+Proof of Theorem 24.
+Lemma 25 implies that
+inf
+g∈A J(g) = J1(u
+ˆG) + J2( ˆG(·, u
+ˆG
+· ))
+≥ J1(u
+ˆG,h) + J2(Ph ˆG(·, u
+ˆG,h
+·
+)) − C(T)
+�
+1 + E
+�� T
+0
+∥ ˆG(t, u
+ˆG
+t )∥2
+H1 dt
+��
+h
+= Jh( ˆG) − C(T)
+�
+1 + E
+�� T
+0
+∥ ˆG(t, u
+ˆG
+t )∥2
+H1dt
+��
+h.
+4.1.2
+Lower Bound
+Under the additional convexity assumptions specified in Assumption 15 we can also prove
+the lower bound. The main result concerning the lower bound of the approximating cost
+functional Jh is the following:
+Theorem 26. Let h ∈ (0, 1]. In addition to the assumptions of Theorem 24 we assume that
+the Assumption 15 and Assumption 9 are in force. Then for any G ∈ Uad
+Jh(G) + Ch ≥ J( ˆG),
+for some constant C > 0 independent of h and G. This shows in particular that
+inf
+g∈A J(g) ≤
+inf
+G∈Uad Jh(G) + Ch.
+The proof of the lower bound uses the necessary optimality condition for the control
+problem (SCP). To this end let H(t, ·) : H × U × H → R
+H(t, u, g, p) =
+�
+Λ
+l(t, x, u(x)) dx + ∥g∥2
+H + ⟨p, F(u) + g⟩H
+denote the reduced Hamiltonian associated with (SCP) and let L(t, u)(x) = l(t, x, u(x)),
+(t, x) ∈ [0, T] × Λ, u ∈ H, be the Nemytskii operator induced by l(t, x, u) and respectively
+M(u)(x) = m(x, u(x)) the Nemytskii operator induced by m(x, u). By Assumption H1’ and
+H2’ the Hamiltonian is convex in the variables u, g, i.e.
+H(t, u2, g2, p) − H(t, u1, g1, p) − ∂uH(t, u1, g1, p)(u2 − u1) − ∂gH(t, u1, g1, p)(g2 − g1) ≥ 0,
+for any t ∈ [0, T], u1, u2 ∈ H1, g1, g2 ∈ A and p ∈ H.
+Theorem 7.2 in [SW21b] now proves a stochastic maximum principle for the case where
+the running cost l(t, x, u) does not depend on t. A straightforward generalization to the time
+dependent case now yields
+Theorem 27. There exist adapted processes (q, p) with
+p ∈ L2([0, T] × Ω, H1) ∩ L2(Ω, C([0, T], H))
+and
+q ∈ L2([0, T] × Ω, L0
+2)
+satisfying
+�
+dpt
+= −
+�
+Apt + DF(uˆg
+t)pt + DL(t, uˆg
+t)
+�
+dt + qtdWt,
+t ∈ [0, T]
+pT
+= DM(uˆg
+T ),
+(22)
+17
+
+such that
+H(t, uˆg, v, pt) ≥ H(t, uˆg, ˆgt, pt)
+for all v ∈ U and almost every (t, ω) ∈ [0, T] × Ω. In particular it holds
+⟨∂gH(t, uˆg, ˆg, pt), v − ˆgt⟩H ≥ 0,
+for all v ∈ U and almost every (t, ω) ∈ [0, T] × Ω.
+Proof. (of Theorem 26) We first rewrite
+Jh(G) − J( ˆG) = J1(uG,h) − J1(u
+ˆG) + J2(PhG(·, uG,h
+·
+)) − J2( ˆG(·, u
+ˆG
+· ))
+= E
+�� T
+0
+�
+Λ
+l(t, x, uG,h
+t
+(x)) − l(t, x, u
+ˆG
+t (x))dx dt
+�
++ E
+��
+Λ
+m(x, uG,h
+T
+(x)) − m(x, u
+ˆG
+T (x))dx
+�
++ E
+�� T
+0
+∥PhG(t, uG,h
+t
+)∥2
+H − ∥ ˆG(t, u
+ˆG
+t )∥2
+Hdt
+�
+.
+Let (p, q) be the solution to the adjoint equation w.r.t. ˆgt = ˆG(t, u ˆG
+t ) (22), then we can write
+Jh(G) − J( ˆG) = T1 + T2,
+where
+T1 = E
+�
+⟨uG,h
+T
+− u
+ˆG
+T , pT ⟩H
+�
++ E
+�� T
+0
+�
+Λ
+l(t, x, uG,h
+t
+(x)) − l(t, x, u
+ˆG
+t (x))dx dt
+�
++ E
+�� T
+0
+∥PhG(t, uG,h
+t
+)∥2
+H − ∥ ˆG(t, u
+ˆG
+t )∥2
+Hdt
+�
+T2 = E
+��
+Λ
+m(x, uG,h
+T
+(x)) − m(x, u
+ˆG
+T (x))dx
+�
+− E
+�
+⟨uG,h
+T
+− u
+ˆG
+t , DM(u
+ˆG
+T )⟩H
+�
+.
+We start by estimating the term T2. By the convexity of m in u, we get
+m(x, u′) − m(x, u) − ∂um(x, u)(u′ − u) ≥ 0
+for every x ∈ Λ, u′, u ∈ R, hence
+T2 ≥ 0.
+For the other term T1 we consider the equation for uG,h
+t
+− u ˆG
+t :
+
+
+
+
+
+d(uG,h
+t
+− u ˆG
+t ) = [Au ˆG
+t − AhuG,h
+t
++ (PhF(uG,h
+t
+) − F(u ˆG
+t )) + (PhG(t, uG,h
+t
+) − ˆG(t, u ˆG
+t ))]dt
++(Ph − I)BdWt,
+t ∈ [0, T]
+(uG,h
+0
+− u ˆG
+0 ) = (Ph − I)u0.
+(23)
+By Itˆo’s formula we get
+d⟨uG,h
+t
+− u
+ˆG
+t , pt⟩H = ⟨uG,h
+t
+− u
+ˆG
+t , dpt⟩H + ⟨pt, d(uG,h
+t
+− u
+ˆG
+t )⟩H + d⟨uG,h − u
+ˆG, p⟩t.
+18
+
+Recall that Ph|H1 = Rh by R2 (Assumption 9). This now implies that
+⟨AhuG,h
+t
+, pt⟩H = ⟨AhuG,h
+t
+, Phpt⟩H = ⟨AhuG,h
+t
+, Rhpt⟩H
+= ⟨A
+1
+2 uG,h
+t
+, A
+1
+2Rhpt⟩H = ⟨A
+1
+2uG,h
+t
+, A
+1
+2 pt⟩H
+= ⟨uG,h
+t
+, Apt⟩H,
+so that
+⟨Au
+ˆG
+t − AhuG,h
+t
+, pt⟩H + ⟨uG,h
+t
+− u
+ˆG
+t , Apt⟩H = ⟨Au
+ˆG
+t , pt⟩H − ⟨u
+ˆG
+t , Apt⟩H = 0.
+Now
+d⟨uG,h
+t
+− u
+ˆG
+t , pt⟩H = −⟨uG,h
+t
+− u
+ˆG
+t , DF(u
+ˆG
+t )pt + DL(t, u
+ˆG
+t )⟩Hdt
++ ⟨pt, (PhF(uG,h
+t
+) − F(u
+ˆG
+t )) + (PhG(t, uG,h
+t
+) − ˆG(t, u
+ˆG
+t ))⟩Hdt
++ ⟨q∗
+t (uG,h
+t
+− u
+ˆG
+t ), dWt⟩H
++ ⟨(Ph − I)∗pt, BdWt⟩H
++ ⟨qt
+�
+Q, (Ph − I)B
+�
+Q⟩L0
+2dt.
+Taking expectation and invoking the definition of the Hamiltonian, we arrive at
+T1 = E
+�� T
+0
+H(t, u
+ˆG
+t , PhG(t, uG,h
+t
+), pt) − H(t, u
+ˆG
+t , ˆG(t, u
+ˆG
+t ), pt)dt
+�
++ E
+�� T
+0
+⟨pt, PhF(uG,h
+t
+) − F(u
+ˆG
+t ) − DF(u
+ˆG
+t )(uG,h
+t
+− u
+ˆG
+t )⟩Hdt
+�
++ E
+�� T
+0
+�
+Λ
+l(t, x, uG,h
+t
+(x)) − l(t, x, u
+ˆG
+t (x)) − ∂ul(t, x, u
+ˆG
+t (x))(uG,h
+t
+− u
+ˆG
+t )(x)dx dt
+�
++
+� T
+0
+⟨qt
+�
+Q, (Ph − I)B
+�
+Q⟩L0
+2dt.
+The maximum principle and convexity of l in u now imply that
+T1 ≥ E
+�� T
+0
+⟨pt, PhF(uG,h
+t
+) − F(u
+ˆG
+t ) − DF(u
+ˆG
+t )(uG,h
+t
+− u
+ˆG
+t )⟩Hdt
+�
++
+� T
+0
+⟨qt
+�
+Q, (Ph − I)B
+�
+Q⟩L0
+2dt.
+Since f is linear, hence PhF(uG,h
+t
+) = F(PhuG,h
+t
+) = F(uG,h
+t
+), the first term on the right hand
+side vanishes
+E
+�� T
+0
+⟨pt, PhF(uG,h
+t
+) − F(u
+ˆG
+t ) − DF(u
+ˆG
+t )(uG,h
+t
+− u
+ˆG
+t )⟩Hdt
+�
+= 0
+and Assumption 9 implies that the second term on the right hand side is of order h, since
+⟨qt
+�
+Q, (Ph − I)B
+�
+Q⟩L0
+2 ≤ ∥qt
+�
+Q∥L0
+2∥(Ph − I)B
+�
+Q∥L0
+2
+≤ Ch∥qt
+�
+Q∥L0
+2∥B
+�
+Q∥L0
+2,1.
+We thus obtain that T1 ≥ −Ch, which together with T2 ≥ 0, implies the assertion.
+19
+
+4.2
+Approximation of the Optimal Feedback Control
+In the whole section we fix h ∈ (0, 1]. However, every result of this section remains true in
+the limit h ↓ 0, i.e. if we consider the optimal control problem (FCP) instead of the finite
+element approximation (FESCP). The proofs can be easily adapted to this situation.
+Lemma 28. Under the Assumption 2, for any G1, G2 ∈ U L
+ad there exists a constant C > 0
+depending only on T and the Lipschitz constant of G2, such that
+E
+�
+sup
+r∈[0,T]
+∥uG1,h
+r
+− uG2,h
+r
+∥2
+H +
+� T
+0
+∥uG1,h
+r
+− uG2,h
+r
+∥2
+H1dr
+�
+≤ CE
+�� T
+0
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥2
+Hdr
+�
+,
+and
+E
+�
+sup
+r∈[0,T]
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG2,h
+r
+))∥2
+H
+�
+≤ CE
+�
+sup
+r∈[0,T]
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥2
+H
+�
+.
+Proof. For any r ∈ [0, T], the Lipschitz continuity of G2 implies that
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG2,h
+r
+))∥H
+≤ ∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥H + ∥Ph(G2(r, uG1,h
+r
+) − G2(r, uG2,h
+r
+))∥H
+≤ ∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥H + C∥uG1,h
+r
+− uG2,h
+r
+∥H,
+(24)
+where C is the Lipschitz constant of G2. Now by Itˆo’s formula we get for any r ∈ [0, T]
+∥uG1,h
+r
+− uG2,h
+r
+∥2
+H
+= −2
+� r
+0
+∥uG1,h
+s
+− uG2,h
+s
+∥2
+1ds + 2
+� r
+0
+⟨Ph(F(uG1,h
+s
+) − F(uG2,h
+s
+)), uG1,h
+s
+− uG2,h
+s
+⟩Hds
++ 2
+� t
+0
+⟨Ph(G1(s, uG1,h
+s
+) − G2(s, uG2,h
+2
+)), uG1,h
+s
+− uG2,h
+s
+⟩Hds
+≤ −2
+� r
+0
+∥uG1,h
+s
+− uG2,h
+s
+∥2
+1ds + 2
+� r
+0
+∥F(uG1,h
+s
+) − F(uG2,h
+s
+))∥H∥uG1,h
+s
+− uG2,h
+s
+∥Hds
++ 2
+� r
+0
+∥Ph(G1(s, uG1,h
+s
+) − G2(s, uG2,h
+2
+))∥H∥uG1,h
+s
+− uG2,h
+s
+∥Hds.
+Using Young’s inequality and the Lipschitz continuity of f we obtain by the previous consid-
+erations
+sup
+r∈[0,t]
+∥uG1,h
+r
+− uG2,h
+r
+∥2
+H +
+� t
+0
+∥uG1,h
+r
+− uG2,h
+r
+∥2
+1dr
+≤ C
+� t
+0
+sup
+r∈[0,s]
+∥uG1,h
+r
+− uG2,h
+r
+∥2
+Hds + ∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥2
+Hdr,
+where the constant C depends on the Lipschitz constants of G2 and of f. Gronwalls inequality
+now implies that
+sup
+r∈[0,T]
+∥uG1,h
+r
+− uG2,h
+r
+∥2
+H +
+� T
+0
+∥uG1,h
+r
+− uG2,h
+r
+∥2
+1dr
+≤ C
+� T
+0
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥2
+Hdr.
+(25)
+20
+
+which yields the first inequality taking expectations.
+For the proof of the second inequality we first take the supremum in (24)
+sup
+r∈[0,T]
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG2,h
+r
+))∥H
+≤ sup
+r∈[0,T]
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥H + C sup
+r∈[0,T]
+∥uG1,h
+r
+− uG2,h
+r
+∥H,
+and inserting (25) yields that
+sup
+r∈[0,T]
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG2,h
+r
+))∥H
+≤ sup
+r∈[0,T]
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥H + C
+� T
+0
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥2
+Hdr
+≤ C sup
+r∈[0,T]
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥H.
+Taking squares and expectations gives the second inequality.
+Lemma 29. Under the Assumptions 2, for any G1, G2 ∈ U L
+ad there exists a constant C > 0
+depending only on T and the Lipschitz constants of G1 and G2, such that for all R > 0 it
+holds
+|Jh(G1) − Jh(G2)|
+≤ CE
+�
+sup
+r∈[0,T]
+∥Ph(G1(r, uG1,h
+r
+) − G2(r, uG1,h
+r
+))∥2
+H
+�1/2
+.
+Proof. Lemma 12 implies that
+E
+�
+sup
+t∈[0,T]
+∥PhGi(t, uGi,h
+t
+)∥2
+H + sup
+t∈[0,T]
+∥uGi,h
+t
+∥2
+H
+�
+≤ C,
+for i = 1, 2, for some C > 0 that depends only on the Lipschitz constants of G1 and G2 and
+T, but is independent of h. Therefore a simple application of Lemma 21 yields
+|Jh(G1) − Jh(G2)|2
+≤ C
+�
+E
+�
+sup
+t∈[0,T]
+∥uG1,h
+t
+− uG2,h
+t
+∥2
+H
+�
++ E
+�
+sup
+t∈[0,T]
+∥Ph(G1(t, uG1,h
+t
+) − G2(t, uG2,h
+t
+))∥2
+H
+��
+,
+for some constant C > 0 depending only on T and the Lipschitz constants of G1 and G2. Now
+Lemma 28 yields the desired result.
+We are now in the position to prove our main results.
+Proof of Theorem 14.
+Let R > 0 and
+BR :=
+�
+sup
+t∈[0,T]
+∥u
+ˆG,h
+t
+∥H < R
+�
+.
+Then
+P (Bc
+R) ≤ 1
+R2 E
+�
+sup
+t∈[0,T]
+∥u
+ˆG,h
+t
+∥2
+H
+�
+≤ C 1
+R2
+21
+
+for some constant C that is independent of h by Lemma 12. This implies, again using Lemma
+12, that for any m ∈ N
+E
+�
+sup
+t∈[0,T]
+∥Ph(Gh,m(t, u
+ˆG,h
+t
+) − ˆG(t, u
+ˆG,h
+t
+))∥2
+H
+�
+= E
+�
+�
+1BR + 1Bc
+R
+�
+sup
+t∈[0,T]
+∥Ph(Gh,m(t, u
+ˆG,h
+t
+) − ˆG(t, u
+ˆG,h
+t
+))∥2
+H
+�
+≤ E
+�
+1BR sup
+t∈[0,T]
+∥Ph(Gh,m(t, u
+ˆG,h
+t
+) − ˆG(t, u
+ˆG,h
+t
+))∥2
+H
+�
++ P (Bc
+R)1/2 E
+�
+sup
+t∈[0,T]
+∥Ph(Gh,m(t, u
+ˆG,h
+t
+) − ˆG(t, u
+ˆG,h
+t
+))∥4
+H
+�1/2
+≤ ǫh,R
+m
++ Ch
+1
+R
+for some constant Ch independent of m. Lemma 29 now yields
+|Jh(Gh,m) − Jh( ˆG)| ≤
+�
+ǫh,R
+m
++ Ch
+1
+R.
+Using Theorem 24 we get
+inf
+g∈A J(g) ≥ Jh( ˆG) − Cǫ1(h)
+≥ Jh(Gh,m) − Cǫ1(h) −
+�
+ǫh,R
+m
++ Ch
+1
+R
+≥
+inf
+G∈Uh,m Jh(G) − Cǫ1(h) −
+�
+ǫh,R
+m
++ Ch
+1
+R,
+where
+ǫ1(h) = C
+�
+1 + E
+�� T
+0
+∥ ˆG(t, u
+ˆG
+t )∥2
+H1 dt
+�1/2�
+h
+and constants C, Ch > 0, where the constant C does not depend on h.
+The second main result is just a simple consequence of Theorem 26.
+Proof of Theorem 16.
+By Theorem 26 we get
+inf
+g∈A J(g) ≤
+inf
+G∈Uad
+Jh(G) + Ch
+for some constant C independent of h, and since U ⊆ Uad we obtain
+inf
+g∈A J(g) ≤ inf
+G∈U Jh(G) + Ch.
+Remark 30. We can also derive an approximation error estimate with respect to the non-
+discretized cost functional J. As mentioned in the beginning of this section, it is not difficult
+to show that for G1, G2 ∈ U L
+ad
+|J(G1) − J(G2)| ≤ CE
+�
+sup
+r∈[0,T]
+∥G1(r, uG1
+r ) − G2(r, uG1
+r )∥2
+H
+�1/2
+.
+(26)
+22
+
+In particular we have for any G ∈ U L
+ad and ˆGh(t, u) := Ph ˆG(t, Phu),h ∈ (0, 1]
+|J(G) − J( ˆG)|
+≤ |J(G) − J( ˆGh)| + |J( ˆGh) − J( ˆG)|
+≤ CE
+�
+sup
+t∈[0,T]
+∥G(t, u) − Ph ˆG(t, Phu
+ˆGh)∥2
+H
+�1/2
++ |J( ˆGh) − J( ˆG)|.
+Due to the Lipschitz continuity of ˆG and our assumptions on the orthogonal projections Ph
+we have
+∥ ˆG(t, u
+ˆG
+t ) − ˆGh(t, u
+ˆG
+t )∥H ≤ ∥ ˆG(t, u
+ˆG
+t ) − Ph ˆG(t, u
+ˆG
+t )∥H
++ ∥Ph ˆG(t, u
+ˆG
+t ) − Ph ˆG(t, Phu
+ˆG
+t )∥H
+≤ h∥ ˆG(t, u
+ˆG
+t )∥H1 + C∥u
+ˆG
+t − Phu
+ˆG
+t ∥H
+≤ h∥ ˆG(t, u
+ˆG
+t )∥H1 + Ch∥u
+ˆG
+t ∥H1.
+Using the result from Lemma 28 for h ↓ 0, we get
+E
+�
+sup
+t∈[0,T]
+∥u
+ˆG
+t − u
+ˆGh
+t
+∥H
+�
+≤ Ch(E
+�� T
+0
+∥ ˆG(t, u
+ˆG
+t )∥H1dt
+�
++ E
+�� T
+0
+∥u
+ˆG
+t ∥H1dt
+�
+)
+and one can show in a similar fashion that
+E
+�� T
+0
+∥ ˆG(t, u
+ˆG
+t ) − ˆGh(t, u
+ˆGh
+t
+)∥Hdt
+�
+≤ Ch(E
+�� T
+0
+∥ ˆG(t, u
+ˆG
+t )∥H1dt
+�
++ E
+�� T
+0
+∥u
+ˆG
+t ∥H1dt
+�
+).
+Lemma 21 now implies
+|J( ˆGh) − J( ˆG)| ≤ Ch,
+for some constant C > 0 which depends only on the Lipschitz constant of ˆG. Therefore we
+obtain
+|J(G) − J( ˆG)| ≤ CE
+�
+sup
+t∈[0,T]
+∥G(t, u) − Ph ˆG(t, Phu
+ˆGh)∥2
+H
+�1/2
++ Ch,
+for some constant C that only depends on the Lipschitz constant of ˆG. Now one can easily
+show an analog result of Theorem 14 for the non-discretized cost functional J.
+5
+Examples
+In this section we will provide some examples to demonstrate the scope of our two main
+theorems. In the following we consider U = BH(0, R), for some R > 0. Furthermore we assume
+that the solution to the HJB equation (5) has bounded second derivatives on [0, T − ǫ] × H,
+for all ǫ ∈ ]0, T[. Sufficient conditions for this assumption to be true are given by ([FGS17b],
+Theorem 4.1.55). Then by ([FGS17b], Theorem 4.201) there exists a unique optimal control
+given in feedback form by
+ˆgt = ˆG(t, u
+ˆG
+t ),
+23
+
+for some bounded feedback ˆG : [0, T] × H → H which is Lipschitz continuous in u, and where
+u ˆG is the unique strong solution to equation (2).
+In the first example regarding Theorem 6 we will construct an ansatz space of approx-
+imating controls U that satisfies the uniform approximation property with respect to the
+optimal feedback ˆG by using artificial neural networks with one layer. In the second example
+regarding Theorem 14 we will construct a sequence of ansatz spaces (Uh,m)m∈N that satisfies
+the Lipschitz approximation property with respect to the optimal feedback ˆG using Gaussian
+radial basis neural networks and specify explicit approximation rates.
+5.1
+Universal Approximation by Neural Networks
+Regarding Theorem 6, we show that the set
+U :=
+�
+G
+�
+t,
+∞
+�
+i=1
+uiei
+�
+(x) =
+n
+�
+i=1
+ψi(t, ηl(u1, . . . , un))ei(x)
+����∥G(t, u)∥H ≤ R, where ψ ∈ N n, n, l ∈ N
+�
+satisfies every assumption of Theorem 6, where
+N n :=
+∞
+�
+k=1
+N n
+k
+(27)
+and
+N n
+k :=
+�
+ψ(x) = Cθ(Ax + b)
+�� A ∈ Rk×n, b ∈ Rk, C ∈ Rn×k�
+denotes the set of all 1-layer artificial neural networks from Rn to Rn with k neurons, for a
+given non-polynomial, Lipschitz continuous activator function θ.
+To this end we recall the following classical universal approximation result by [Pin99]:
+Theorem 31. Let θ ∈ C(R), then we define for u = (u1, . . . , ud) ∈ Rd
+θ(u)i := θ(ui).
+(28)
+If θ is not polynomial, then for any n, m ∈ N, compact set K ⊂ Rn, h ∈ C(K, Rm) and ǫ > 0,
+there exists k ∈ N, A ∈ Rk×n, b ∈ Rk, C ∈ Rm×k, such that
+sup
+u∈K
+|h(u) − ψ(u)| < ǫ,
+(29)
+where ψ is the 1-layer artificial neural network
+ψ(u) := Cθ(Au + b).
+(30)
+Recall the finitely based approximation (7)
+ˆGh : [0, T] × H → Sh
+ˆGh(t, u) := Ph ˆG(t, Phu).
+Let Nn be the dimension of S 1
+n , e1, . . . , eNn be an orthonormal basis of S 1
+n and define
+gn : [0, T] × RNn → RNn
+gn
+i (t, u) := ⟨ ˆG(t,
+Nn
+�
+j=1
+ujej), ei⟩,
+i = 1, . . . , Nn.
+(31)
+24
+
+Since gn is continuous, there exist for any m ∈ N a 1-layer artificial neural network ψn,m :
+[0, T] × RNn → RNn, such that
+sup
+(t,u)∈[0,T]×BRNn (0,m)
+|gn(t, u) − ψn,m(t, u)| < R
+m.
+(32)
+If we consider for any m ∈ N the continuous function ηm : RNn → RNn
+ηm(x) =
+�
+x
+|x| ≤ m
+m x
+|x|
+|x| > m
+and define
+˜ψn,m(t, u) := ψn,m(t, η(u)),
+then clearly ˜ψn,m = ψn,m on [0, T] × BRNn(0, m) and for any (t, u) ∈ [0, T] × BRNn(0, m)
+|ψn,m(t, u)| ≤ |gn(t, u)| + |ψn,m(t, u) − gn(t, u)|
+≤ R + R
+m ≤
+�
+1 + 1
+m
+�
+R,
+since
+|gn(t, u)| = ∥P 1
+n
+ˆG(t,
+Nn
+�
+j=1
+ujej)∥H ≤ ∥ ˆG(t,
+Nn
+�
+j=1
+ujej)∥H ≤ R.
+Furthermore, on [0, T] × BRNn(0, m)c we have
+| ˜ψn,m(t, u)| = |ψn,m(t, m u
+|u|)|
+≤ |gn(t, m u
+|u|)| + |ψn,m(t, m u
+|u|) − gn(t, m u
+|u|)|
+≤
+�
+1 + 1
+m
+�
+R.
+Now let
+Gn,m(t, u) :=
+�
+1 − 1
+m
+� Nn
+�
+i=1
+˜ψn,m
+i
+(t, ⟨u, e1⟩H, . . . , ⟨u, en⟩H))ei.
+(33)
+Then
+∥Gn,m(t, u)∥ =
+�
+1 − 1
+m
+�
+| ˜ψn,m (t, ⟨u, e1⟩H, . . . , ⟨u, eNn⟩H) |
+≤
+�
+1 − 1
+m
+� �
+1 + 1
+m
+�
+R ≤ R.
+Since the elements of N n
+k are Lipschitz continuous, so is Gn,m, in particular Gn,m ∈ U is an
+admissible control. Furthermore, (Gn,m)m∈N approximates Gn. Indeed, for any R > 0 and
+any ǫ > 0 there exists an M ∈ N, such that BH(0, R) ⊂ BH(0, m) and 2R
+m < ǫ for every
+25
+
+m ≥ M. Therefore, we have for any m ≥ M
+sup
+(t,u)∈[0,T]×BH(0,R)
+∥Gn,m(t, u) − Gn(t, u)∥
+≤
+sup
+(t,u)∈[0,T]×BH(0,m)
+∥Gn,m(t, u) − Gn(t, u)∥
+≤
+sup
+(t,u)∈[0,T]×BRNn (0,m)
+|
+�
+1 − 1
+m
+�
+ψn,m(t, u) − gn(t, u)|
+≤
+sup
+(t,u)∈[0,T]×BRNn (0,m)
+�
+1 − 1
+m
+�
+|ψn,m(t, u) − gn(t, u)| + 1
+m|gn(t, u)|
+< 2R
+m < ǫ.
+5.2
+Interpolation by radial basis functions
+In our second example we will interpolate the finitely based approximations by Gaussian
+radial basis function neural networks and define a suitable ansatz space of controls for the
+approximation that satisfies a uniform Lipschitz approximation property with respect to the
+optimal feedback ˆG. The main idea is to determine the optimal feedback at some mayor states
+(t1, x1), ..., (tN , xN) and interpolate afterwards. For some κ > 0 let
+Φ(x, y) := exp(−κ|x − y|2)
+denote the Gaussian kernel. Furthermore we define for O ⊆ Rd
+FΦ(O) = span{Φ(·, y) | y ∈ O}
+∥·∥Φ,
+where ∥ · ∥Φ denotes the norm associated with the scalar product
+⟨
+N
+�
+i=1
+αiΦ(·, xj),
+M
+�
+j=1
+βjΦ(·, yj)⟩Φ :=
+N
+�
+i=1
+M
+�
+j=1
+αiβjΦ(xi, yj).
+Now we define the ’point evaluation’ map
+R : FΦ(O) → C(O), R(f)(x) := ⟨f, Φ(·, x)⟩Φ
+to identify abstract elements of FΦ(O) with functions and introduce the corresponding native
+Hilbert space of Φ by
+NΦ(O) := R(FΦ(O)),
+with the inner product
+⟨f1, f2⟩NΦ(O) := ⟨R−1f1, R−1f2⟩Φ.
+If O = Rd, the native space is given by
+NΦ(Rd) = {f ∈ L2(Rd) ∩ C(Rd) | ˆf/ˆΦ1/2 ∈ L2(Rd)},
+with the inner product
+⟨f1, f2⟩NΦ(Rd) = (2π)−d/2⟨ ˆf1/ˆΦ1/2, ˆf2/ˆΦ1/2⟩L2(Rd),
+26
+
+where ˆf denotes the analytic Fourier transform of f.
+For a given function g ∈ NΦ(O) we will consider approximations of the type
+sg,X(x) :=
+K
+�
+k=1
+αkΦ(x, xk),
+(34)
+for a discrete set X = {x1, ..., xK} ⊆ O and some α1, ..., αK ∈ R, such that
+sg,X(xk) = g(xk),
+(35)
+for k = 1, ..., K.
+Lemma 32. Let g ∈ NΦ(Rd), X = {x1, ..., xK} ⊆ O ⊆ Rd and sg,X be of the form (34),
+satisfying (35) for g|O ∈ NΦ(O).
+Then the natural extension of sg,X to Rd is Lipschitz
+continuous with Lipschitz constant given by 2κ∥g∥2
+NΦ(Rd).
+Proof. We first observe that for any x ∈ Rd, using R(sg,X) = sg,X and R(Φ(·, x)) = Φ(·x),
+sg,X(x) = ⟨sg,X, Φ(·, x)⟩Φ = ⟨sg,X, Φ(·, x)⟩NΦ(Rd)
+and therefore for any i = 1, ..., d
+∂isg,X(x) = ⟨sg,X, ∂iΦ(·, x)⟩NΦ(Rd).
+Using the Cauchy-Schwarz inequality we get
+|∂isg,X(x)|2 ≤ ∥sg,X∥2
+Φ∥∂iΦ(·, x)∥2
+NΦ(Rd).
+Now
+∥∂iΦ(·, x)∥2
+NΦ(Rd) = ∂1
+i ∂2
+i Φ(y, y) = 2κ.
+Furthermore we have
+∥sg,X∥2
+Φ ≤ ∥g∥2
+NΦ(O) ≤ ∥g∥2
+NΦ(Rd).
+This concludes the proof of this lemma.
+In the following we define for X = {x1, . . . , xK} ⊆ O
+hX,O := sup
+x∈O
+min
+1≤j≤K |x − xj|2.
+Then we consider the following two results from [Wen04]:
+Lemma 33. Let O ⊆ BR(x0), for some R > 0 and X = {x1, ..., xK} ⊆ O be quasi uniform
+(q.u.) with respect to cqu > 0, i.e.
+qX ≤ hX,O ≤ cquqX,
+where qX := 1
+2 mini̸=j |xi − xj|2. Then it holds
+hX,O ≤ 2RcquK−1/d.
+27
+
+Theorem 34. Let f ∈ NΦ(O) and sf,X denote its interpolant based on the quasi uniform set
+X = {x1, ..., xK} ⊆ O. Then there exists a constant C > 0, such that
+sup
+x∈O
+|f(x) − sf,X(x)| ≤ Clhl
+X,O∥f∥NΦ(O),
+for any l ∈ N.
+We now apply the above result to the approximation of the optimal feedback ˆG with
+Gaussian radial basis networks. Therefore we again recall the finitely based approximation
+(7)
+ˆGh : [0, T] × H → Sh
+ˆGh(t, u) := Ph ˆG(t, Phu).
+Let Nn be the dimension of S 1
+n , e1, . . . , eNn be an orthonormal basis of S 1
+n and define
+gn : [0, T] × RNn → RNn
+gn
+i (t, u) := ⟨ ˆG(t,
+Nn
+�
+j=1
+ujej), ei⟩,
+i = 1, . . . , Nn.
+(36)
+In the following we assume that the functions gn
+i , i = 1, . . . Nn are elements of NΦ([0, T]×RNn).
+Then
+Un,m := {G(t,
+∞
+�
+i=1
+uiei)(x) =
+Nh
+�
+i=1
+si(t, (u1, ..., uNn))ei(x)|si ∈ N m,n}
+where
+N m,n := {s(t, x) =
+m
+�
+k=1
+αkΦ((t, x), (tk, xk))|((t1, x1), ..., (tm, xm)) ∈ BRNn(0, m1/2Nn) × [0, T] q.u.},
+satisfies the uniform Lipschitz approximation property with respect to ˆG of order 1
+n. Indeed,
+let l ∈ N, then for any R > 0 there exists an M ∈ N, such that BRNn(0, R) ⊆ BRNn(0, m1/(2Nn))
+for all m ≥ M. Furthermore by Theorem 34 and Lemma 33 there exists for all m ≥ M an
+element Gn,m ∈ Un,m,
+Gn,m(t, u) =
+Nn
+�
+i=1
+sn
+i (t, (u1, ..., uNn))ei,
+for some sn(t, u) := (sn
+1(t, u), ..., sn
+Nn(t, u)) , such that sn
+i satisfies (35) for gn
+i , such that
+sup
+(t,u)∈[0,T]×BH(0,R)
+∥Gn,m(t, u) − ˆGn(t, u)∥H
+≤
+sup
+(t,u)∈[0,T]×BRNn (0,R)
+|sn(t, u) − gn(t, u)|
+≤
+sup
+(t,u)∈[0,T]×BRNn (0,m1/(2n))
+|sn(t, u) − gn(t, u)|
+≤ Clcqum−l/(2Nn)
+28
+
+for all m ≥ M. Due to Lemma 32 any Gm,n has Lipschitz constant independent of m and
+since
+ǫn,R
+m
+=
+sup
+(t,u)∈[0,T]×BH(0,R)
+∥Gn,m(t, u) − ˆGn(t, u)∥H
+≤ Clcqum−l/(2Nn) → 0,
+(Um,n)m∈N satisfies the Lipschitz approximation property with respect to ˆG of order 1
+n.
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+

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+Sequential Fair Resource Allocation under a Markov Decision Process Framework
+Parisa Hassanzadeh1 , Eleonora Kreaˇci´c1 , Sihan Zeng2 , Yuchen Xiao1 and Sumitra Ganesh1
+1J.P. Morgan AI Research, 2Georgia Institute of Technology
+{parisa.hassanzadeh, eleonora.kreacic, yuchen.xiao, sumitra.ganesh}@jpmorgan.com,
+[email protected]@jpmorgan.com
+Abstract
+We study the sequential decision-making problem
+of allocating a limited resource to agents that re-
+veal their stochastic demands on arrival over a fi-
+nite horizon.
+Our goal is to design fair alloca-
+tion algorithms that exhaust the available resource
+budget. This is challenging in sequential settings
+where information on future demands is not avail-
+able at the time of decision-making. We formu-
+late the problem as a discrete time Markov deci-
+sion process (MDP). We propose a new algorithm,
+SAFFE, that makes fair allocations with respect to
+the entire demands revealed over the horizon by ac-
+counting for expected future demands at each ar-
+rival time. The algorithm introduces regularization
+which enables the prioritization of current revealed
+demands over future potential demands depending
+on the uncertainty in agents’ future demands. Us-
+ing the MDP formulation, we show that SAFFE
+optimizes allocations based on an upper bound on
+the Nash Social Welfare fairness objective, and we
+bound its gap to optimality with the use of concen-
+tration bounds on total future demands. Using syn-
+thetic and real data, we compare the performance
+of SAFFE against existing approaches and a re-
+inforcement learning policy trained on the MDP.
+We show that SAFFE leads to more fair and effi-
+cient allocations and achieves close-to-optimal per-
+formance in settings with dense arrivals.
+1
+Introduction
+The problem of multi-agent resource allocation arises in nu-
+merous disciplines including economics [Cohen et al., 1965],
+power system optimization [Yi et al., 2016; Nair et al.,
+2018], and cloud computing [Balasubramanian et al., 2007;
+Abid et al., 2020]. It comes in various flavors depending on
+the nature of resources, the agents’ behavior, the timing of al-
+location decisions, and the overall objective of the allocation
+process. A common element shared by various settings is the
+existence of a limited resource that needs to be distributed to
+various agents based on their reported requests.
+There are various objectives that one might wish to opti-
+mize in the context of resource allocation. Efficiency max-
+imization only aims at reducing the leftover resource, while
+other objectives combine efficiency with various notions of
+fairness. Among different fairness metrics, the Nash Social
+Welfare (NSW) is a well-known objective defined as the ge-
+ometric mean of the agents’ “satisfaction” with their allo-
+cation. In the offline setting where we observe all requests
+prior to making an allocation decision, the NSW enjoys a
+few favorable properties including being Pareto-efficient (for
+any other allocation there would be at least one agent who is
+worse off compared to the current one), envy-free (no agent
+prefers another agent’s allocation), and proportionally fair
+(every agent gets their fair share of the resource), and thus
+exhausts the total budget under a limited resource [Varian,
+1973].
+In this work, we study the allocation of divisible resources
+to truthful agents with the aim of optimizing the NSW. We
+consider the much more challenging online (or sequential)
+setting where the agents place requests in a sequential manner
+and allocation decisions have to be made instantaneously. As
+the decision cannot be modified once it is made, every allo-
+cation affects all future allocations which leads to a compro-
+mise between fairness and efficiency. Intuitively, to achieve
+fairness we need to reserve the resource for anticipated future
+agents, which can lead to wasted resources if the expected
+future agents do not arrive.
+Online allocation algorithms are commonly evaluated in
+comparison to their offline (or hindsight) counterpart [Gal-
+lego et al., 2015; Sinclair et al., 2020; Sinclair et al., 2022].
+However, there is no standard notion of fairness in the sequen-
+tial setting, as it may be impossible to satisfy all of the favor-
+able fair properties simultaneously[Sinclair et al., 2022]. We
+aim to fill this gap on the formulation and to design efficient
+and reliable algorithms for the online allocation problem. Our
+contributions are as follows.
+Main Contributions
+• We formulate a notion of fair allocations in sequential set-
+tings where agents can submit multiple demands over the
+horizon.
+• We formulate the problem as a Markov decision process
+(MDP), and design the reward function so that its cumu-
+lative rewards matches our fairness objective. This offers
+us a tool to quantitatively evaluate the performance of al-
+location policies.
+arXiv:2301.03758v1  [cs.LG]  10 Jan 2023
+
+• We introduce SAFFE, a novel intuitive sequential algo-
+rithm that makes close-to-optimal allocations and is easy
+to implement. We further introduce SAFFE-D, character-
+ized by a tunable parameter that balances efficiency and
+fairness. The parameter can be tuned with the use of con-
+centration bounds around expected future demands.
+• We provide theoretical upper bounds on the sub-
+optimality gap using concentration bounds around the ex-
+pected demand, and illustrate the close-to-optimal perfor-
+mance of SAFFE-D in different settings using numerical
+simulations and real data.
+2
+Problem Formulation
+We consider a supplier that has a divisible resource with a
+limited budget size B, and N agents that arrive sequentially
+over T time steps requesting (possibly fractional) units of the
+resource. At time t ∈ {1, . . . , T}, agent i ∈ {1, . . . , N}
+arrives and reveals its demand Xt
+i ∈ R sampled from dis-
+tribution PXt
+i |X1
+i ,...,Xt−1
+i
+.
+We assume that each agent has
+at least one demand over the horizon T. The supplier ob-
+serves demands Xt = (Xt
+1, . . . , Xt
+N), and makes an allo-
+cation At = (At
+1, . . . , At
+N), where At
+i ∈ R≥0 denotes the
+resource amount allocated to agent i. We assume that the al-
+locations are immediately taken by the agents and removed
+from the inventory. We also assume that the setting is re-
+peated, i.e., after T time steps, a new budget is drawn and the
+next allocation round of T time steps starts.
+Agent i has a utility function u(Ai, Xi), representing its
+satisfaction with the allocation Ai = (A1
+i , . . . , AT
+i ) given
+its (latent) demands. The utility function is a non-decreasing
+non-negative concave function. In this work, we consider the
+following utility function, where an agent’s utility linearly in-
+creases with its total allocation up to its total request
+u(Ai, Xi) =
+T
+�
+t=1
+min
+�
+At
+i, Xt
+i
+�
+.
+(1)
+An agent with the utility function in (1) only values an allo-
+cation in the time step it requested the resource, and not in
+earlier or later steps, which is suitable in settings where the
+allocation process is time sensitive and the supplier is not able
+to delay the decision-making outside of the current time step.
+If all agent demands X1, . . . , XN are known to the supplier
+at the time of decision-making, they can be used for deter-
+mining the allocations. This setting is often referred to as
+offline. However, in the online or sequential setting, the agent
+demands are gradually revealed over time, such that the sup-
+plier only learns about Xt
+1, . . . , Xt
+N at time t.
+Notation
+(x)+ = max{x, 0}. For vectors X and Y, we
+use X ≥ Y to denote Xi ≥ Yi for each i. 0 denotes a zero
+vector. N(µ, σ2) denotes the Normal distribution with mean
+µ and variance σ2.
+2.1
+Fairness in Allocation Problems
+The supplier aims to allocate the resource efficiently, i.e.,
+with minimal leftover at the end of horizon T, and in a
+fair manner maximizing the NSW. The NSW is defined as
+�N
+i=1 u(Ai, Xi)wi, where wi ∈ R+ reflects the weight as-
+signed to agent i by the supplier. The NSW is a balanced
+compromise between the utilitarian welfare objective, which
+maximizes the utility of a larger number of agents, and the
+egalitarian welfare objective, which maximizes the utility of
+the worst-off agent. Since
+arg max
+Ai
+N
+�
+i=1
+u(Ai, Xi)wi = arg max
+Ai
+N
+�
+i=1
+wi log u(Ai, Xi),
+(2)
+the logarithm of NSW is often used as the objective function
+in allocation problems, such as in the Eisenberg-Gale pro-
+gram [Eisenberg and Gale, 1959]. We refer to this as the
+log-NSW objective.
+In the sequential setting, it is not guaranteed that there
+always exists an allocation that is simultaneously Pareto-
+efficient, envy-free, and proportional [Sinclair et al., 2020,
+Lemma 2.3]. Motivated by the properties of the NSW ob-
+jective in the offline setting, and the fact that our repeated
+setting can be viewed as multiple draws of the offline setting,
+we propose a modified NSW objective for the sequential set-
+ting. Our goal in the sequential setting is to find an allocation
+that maximizes the log-NSW in expectation. Specifically,
+arg max
+Ai
+N
+�
+i=1
+wiEXi
+�
+log u(Ai, Xi)
+�
+.
+(3)
+With the goal of measuring the performance of an allo-
+cation in the sequential setting, [Sinclair et al., 2020] intro-
+duces approximate fairness metrics ∆Amax and ∆Amean. Let
+Aonline
+i
+denote the allocation vector of agent i given by an
+online algorithm subject to latent demands, and let Ahindsight
+i
+denote the allocation vector in hindsight after all demands
+are revealed, defined as (14) in Sec. 4. The expected dif-
+ference between the overall allocations for each agents can
+be used to measure the fairness of the online algorithm. Let
+∆Ai :=
+��� �T
+t=1 At,hindsight
+i
+− �T
+t=1 At,online
+i
+���. Then,
+∆Amax = E
+�
+max
+i
+∆Ai
+�
+, ∆Amean = 1
+N
+N
+�
+i=1
+E[∆Ai], (4)
+are concerned with the worst-off agent and average agent in
+terms of cumulative allocations in hindsight, respectively1.
+While our optimization objective is not to minimize ∆Amax
+or ∆Amean, we use them as metrics to evaluate the fairness of
+online allocation algorithms in the experiments in Sec. 7.
+3
+Markov Decision Process Formulation
+Determining the optimal allocations under the NSW objec-
+tive is a sequential decision-making problem as the alloca-
+tion in one step affects the budget and allocations in future
+1∆A is defined for the additive utility function in (1), and may
+need to be redefined for alternative utilities.
+
+steps. We formulate the problem as a finite-horizon total-
+reward Markov Decision Process (MDP) modeled as a tu-
+ple < {St, At, Pt, Rt}t=1,...,T >.
+St denotes the under-
+lying time-dependent state space, At is the action space,
+Pt : St × At × St+1 → R≥0 describes the state transi-
+tion dynamics conditioned upon the previous state and ac-
+tion, Rt : St × At → R≥0 is a non-negative reward func-
+tion, and T is the horizon over which the resource is allo-
+cated. The goal of the supplier is to find an allocation policy
+π = {π1, . . . , πT | πt : St → At} mapping the state to
+an action, in order to maximize the expected sum of rewards
+E[�T
+t=1 Rt(st, πt(st))]. Next, we describe the components
+of the MDP in details.
+State Space
+The state space St is time-dependent, and the
+state size increases with time step t since the state captures
+the past demand and allocation information. Specifically, the
+state at step t is defined as st = (X1:t, A1:t−1, Bt), where
+X1:t := (X1, . . . , Xt), A1:t := (A1, . . . , At), and
+Bt =
+�
+Bt−1 − �N
+i=1 At−1
+i
+t ≥ 1
+B
+t = 1
+(5)
+Action Space
+The actions space is state and time-
+dependent. For any st = (X1:t, A1:t−1, Bt) ∈ St, we have
+At = {At ∈ RN :
+N
+�
+i=1
+At
+i ≤ Bt}
+(6)
+The state and action space are both continuous and therefore
+infinitely large. However, At is a compact polytope for any
+st ∈ St, and St is compact if the requests are bounded.
+State Transition Function
+Given state s ∈ St and action
+a ∈ At, the system transitions to the next state s′ ∈ St+1
+with probability
+P(s, a, s′) = Prob(st+1 = s′ | st = s, at = a),
+(7)
+where Xt+1
+i
+∼ PXt+1
+i
+|X1
+i ,...,Xt
+i .
+Reward Function
+The reward at time step t ∈ {1, . . . , T},
+is defined as follows
+Rt(st, πt(st)) =
+N
+�
+i=1
+1{Xt
+i > 0} . wi(U t
+i − U t−1
+i
+),
+(8)
+where
+U t
+i = log
+�
+t
+�
+τ=1
+min{Aτ
+i , Xτ
+i } + ϵ
+�
+, t ∈ {1, . . . , T}, (9)
+U 0
+i = 0, and ϵ is a small value added for to ensure values are
+within the domain of the log function2. The indicator function
+1{ �
+Xt
+i > 0} ensures that we only account for the agents with
+a demand at time t. Then, the expected sum of rewards over
+the entire horizon T is equivalent to the expected log-NSW
+objective defined in (3) for ϵ → 0.
+2For ϵ ≪ 1, this will lead to a maximum error of ϵ for x ≥ 1.
+At time t, with state st ∈ St and action (allocation)
+At ∈ At, the optimal Q-values satisfy the Bellman optimal-
+ity equation [Bellman, 1966]:
+Qt(st, At) = Rt(st, At) + E
+�
+max
+At+1∈At+1
+Qt+1(st+1, At+1)
+�
+.
+(10)
+We denote the optimal policy corresponding to Eq. (10) by
+π⋆
+t . Then, the optimal allocation is the solution to
+arg max
+At Qt(st, At).
+(11)
+We show in Appendix A that an optimal policy exists for
+the MDP, and it can be derived by recursively solving (11)
+backward in time.
+This quickly becomes computationally
+intractable, which motivates us to find alternative solutions
+that trade-off sub-optimality for computational efficiency. In
+Sec. 5, we introduce a heuristic algorithm that uses estimates
+of future demands to determine the allocations, and we dis-
+cuss a learning-based approach in Sec. 7.5.
+4
+Offline (Hindsight) Setting
+If the supplier could postpone the decision-making till the end
+of the horizon T, it would have perfect knowledge of all de-
+mands X1, . . . , XT . Let �
+Xi = �T
+τ=1 Xτ
+i denote the total
+demands of agent i. Then, the supplier solves the following
+convex program, referred to as the Eisenberg-Gale program
+[Eisenberg, 1961], to maximize the NSW objective in (3) for
+allocations �A = ( �A1, . . . , �AN),
+max
+�
+A≥0
+N
+�
+i=1
+wi log
+�
+u( �Ai, �
+Xi)
+�
+s.t.
+N
+�
+i=1
+�Ai ≤ B.
+(12)
+While the optimal solution to (12) may not be achievable in
+the sequential setting, it provides an upper bound on the log-
+NSW achieved by any online algorithm, and serves as a base-
+line for comparison. With the utility function in Eq. (1), allo-
+cating resources beyond the agent’s request does not increase
+the utility. Therefore, solving (12) is equivalent to the follow-
+ing
+max
+�
+A
+N
+�
+i=1
+wi log( �Ai)
+s.t. 0 ≤ �Ai ≤ �
+Xi, ∀i
+N
+�
+i=1
+�Ai ≤ B.
+(13)
+Then, any distribution of agent i’s allocation �Ai across the T
+time steps that satisfies the demand constraint at each step,
+would be an optimal allocation in hindsight Ahindsight
+i
+, i.e.,
+Ahindsight
+i
+∈
+�
+(A1
+i , . . . , AT
+i ) : 0 ≤ At
+i ≤ Xt
+i, ∀t,
+T
+�
+τ=1
+At
+i = �Ai
+�
+.
+(14)
+
+The optimal solution to (13) takes the form
+�A⋆
+i
+=
+min{ �
+Xi, µ}, where µ is a function of budget B and demands
+�X such that �N
+i=1 �Ai = B. The solution can be efficiently
+derived by the water-filling algorithm in Algorithm 1, and the
+threshold µ can be interpreted as the water-level shown in
+Fig. 1a.
+(a)
+(b)
+Figure 1: Water-filling solution for equal unit weights: Each bar
+shows the demands or allocations of an agent. (a) In hindsight, the
+resource is allocated using water-level µ determined such that the
+total allocations are equal to the budget size. (b) In the sequential
+setting, at time t, the cumulative prior allocations and total expected
+future demands are accounted for when determining water-level µt.
+Algorithm 1 Water-filling algorithm with weights for solving
+(13)
+1: Input: number of agents N, resource budget B, demand
+vector �X ∈ RN, weight vector w ∈ RN
+2: Output: allocation vector �A ∈ RN
+3: Find an ordered index set {i1, . . . , iN} by sorting the
+agents such that
+�
+Xi1
+wi1 ≤ · · · ≤
+�
+XiN
+wiN
+4: αj =
+wij
+wij +···+wiN for j = 1, . . . , N
+5: j ← 1
+6: while j ≤ N and B > 0 do
+7:
+if B ≤ �
+Xij/αj then
+8:
+�Aik = αkB for k = j, . . . , N
+9:
+break
+10:
+else
+11:
+�Aij = �
+Xij
+12:
+B ← B − �Aij
+13:
+j ← j + 1
+14:
+end if
+15: end while
+5
+Sequential Setting: Heuristic Algorithm
+In this section, we propose a simple and computationally effi-
+cient heuristic algorithm, which we name Sequential Alloca-
+tions with Fairness based on Future Estimates (SAFFE), as
+well as its variant SAFFE-Discounted. This online algorithm
+uses Algorithm 1 as a sub-routine to compute the allocations
+at each step t.
+5.1
+SAFFE Algorithm
+SAFFE is based on the simple principle of substituting unob-
+served future requests with their expectations. By doing so,
+we convert the sequential decision-making problem into solv-
+ing the offline problem in each time step. With the algorithm
+formally stated in Algorithm 2, at each time step t we use the
+expected future demands to determine the total resources we
+expect to allocate to each agent by time T. This allows us to
+reserve a portion of the available budget for future demands.
+Specifically, at t = 1, . . . , T, we solve the following problem
+max
+Ct
+N
+�
+i=1
+1{Y t
+i > 0} . wi log( �At
+i + Ct
+i)
+s.t. 0 ≤ Ct
+i ≤ Y t
+i , ∀i,
+N
+�
+i=1
+Ct
+i ≤ Bt,
+(15)
+where
+Y t
+i = Xt
+i + E
+�
+T
+�
+τ=t+1
+Xτ
+i
+�
+,
+i = 1, . . . , N
+(16)
+The indicator function 1{Y t
+i > 0} ensures that we do not
+consider absent agents, i.e., those with no current or expected
+future demands3. Ct
+i denotes the total allocation for agent i
+over the period t, . . . , T, if the future demands would arrive
+exactly as their expectations. In other words, Ct
+i consists of
+the allocation in the current time step and the reserved allo-
+cations for the future. Then, the current allocation At
+i is a
+fraction of Ct
+i computed as
+At
+i = Ct
+i
+Xt
+i
+Y t
+i
+.
+(17)
+Similar to the hindsight problem (13), we can efficiently solve
+(15) using a variant of the water-filling algorithm given in
+Algorithm 3 in Appendix B. As illustrated in Fig. 1b, at time
+step t, the water-level µt is calculated while accounting for
+all previous allocations to the agents and their total expected
+future demands.
+Algorithm 2 SAFFE Algorithm
+1: Input: number of agents N, resource budget B, demand
+vectors X1, . . . XT ∈ RN, weight vector w ∈ RN, de-
+mand distributions PX1, . . . , PXN
+2: Output: allocation vectors A1, . . . , AT ∈ RN
+3: for For t = 1, . . . , T do
+4:
+Y t
+i = Xt
+i + E[�T
+τ=t+1 Xτ
+i ] for i = 1, . . . , N
+5:
+�At
+i = �t
+τ=1 Aτ
+i for i = 1, . . . , N
+6:
+Ct ← Algorithm 3 with inputs (N, B, Yt, w, �At) in
+App. B
+7:
+At
+i = Ct
+iXt
+i/Y t
+i for i = 1, . . . , N
+8:
+B ← B − �N
+i=1 At
+i
+9: end for
+5.2
+SAFFE-Discounted Algorithm
+While SAFFE is simple and easy to interpret, it is sub-
+optimal. In fact, SAFFE maximizes an upper bound of the
+3With a slight abuse of notation, in the objective function, we
+assume 0 × −∞ = 0 to ensure that an agent without demand is not
+included, and gets no allocation.
+
+x;
+Ai|A2|A3|A4|A5YicIc?
+μt
+A!optimal Q-values of the MDP defined in Sec. 3 as ϵ → 0.
+This implies that SAFFE overestimates the expected reward
+gain from the future and reserves the budget excessively for
+future demands. To correct the over-reservation, we propose
+SAFFE-Discounted (SAFFE-D), which penalizes uncertain
+future requests by their standard deviations. At every step
+t, the algorithm computes
+Y t
+i = Xt
+i +
+T
+�
+τ=t+1
+�
+E[Xτ
+i ] − λ std(Xτ
+i )
+�+
+(18)
+for some regularization parameter λ ≥ 0 and solves (15).
+As in SAFFE, the current allocation At
+i is split proportion-
+ally from Ct
+i according to (17). The regularizer λ is a hyper-
+parameter that can be tuned to provide the optimal trade-off
+between consuming too much of the budget at the current step
+and reserving too much for the future. We show in Sec. 6,
+the uncertainty in future demands reduces as we approach
+T, and we expect better performance with a decreasing time-
+dependent function such as λ(t) =
+√
+T − t λ for some λ > 0.
+Alternatively, λ(t) can be learned as a function of time.
+Remark 1. In this paper, we assume access to historical
+data from which expected future demands can be estimated,
+and we mainly focus on the decision-making aspect for al-
+locations. These estimates are directly used by SAFFE and
+SAFFE-D. We empirically study how sensitive the algorithms
+are to estimation errors in Sec. 7.3.
+6
+On the Sub-Optimality of SAFFE-D
+In order to determine how sub-optimal SAFFE-D is, we
+upper bound the performance gap between SAFFE-D and
+the hindsight solution, in terms of ∆Amax introduced in
+(4). Let Aoracle denote the allocations derived using SAFFE-
+D in a setting where an oracle has perfect knowledge of
+the sequence of incoming demands, i.e., when SAFFE-D
+is used with no stochasticity.
+We refer to this setup as
+SAFEE-Oracle. We upper bound ∆Amax, the distance be-
+tween ASAFFE-D and Ahindsight, in two steps: by bounding the
+distance between SAFFE-D and SAFFE-Oracle (Sec. 6.1)
+and between SAFFE-Oracle and hindsight (Sec. 6.2). Fi-
+nally, Theorem 4 upper bounds the distance ∆Amax between
+SAFFE-D and hindsight in terms of the sum of these two dis-
+tances. Our arguments rely on concentration inequalities and
+are in spirit similar to the idea of guardrails in [Sinclair et al.,
+2022].
+6.1
+SAFFE-Discounted vs SAFFE-Oracle
+Under mild assumptions on the distribution of demands, we
+can quantify the distance between SAFFE-D and SAFFE-
+Oracle allocations using concentration inequalities on the de-
+viation of future demands from their expected value. Let us
+define
+Y
+t
+i = Xt
+i + E
+�
+T
+�
+τ=t+1
+Xτ
+i
+�
++
+√
+T − t
+√ξ
+std(Xτ
+i )
+(19)
+Yt
+i = Xt
+i + E
+�
+T
+�
+τ=t+1
+Xτ
+i
+�
+−
+√
+T − t
+√ξ
+std(Xτ
+i )
+(20)
+For simplicity assume that agent i’s demands are i.i.d (this
+assumption can be relaxed see Appendix C). Then, for ξ > 0,
+with probability at least 1−ξ, based on Chebyshev’s inequal-
+ity we have
+Yt
+i ≤ Xt
+i +
+T
+�
+τ=t+1
+Xτ
+i ≤ Y
+t
+i
+(21)
+We further assume that all agents have equal std(Xτ
+i ), but
+their expectations might differ (the assumption simplifies the
+presentation, and is not crucial. See Appendix C). We first
+present the worst-case scenario bound, where we allow highly
+unbalanced future demands for different agents, i.e. there is
+an agent k such that Yt
+k ≥ Y
+t
+j for all other agents j ̸= k.
+Theorem 1 (Unbalanced demands bound). Let At,SAFFE-D
+i
+and At,oracle
+i
+denote allocations by SAFFE-D for λ(t) =
+�
+T −t
+ξ
+and SAFFE-Oracle, respectively. Then, for all agents
+i we have
+���At,SAFFE-D
+i
+− At,oracle
+i
+��� ≤
+�
+�
+�
+�
+�
+�
+�
+2N
+�
+(T −t)
+ξ
+std(Xt
+i)
+if Bt ≤
+N
+�
+i=1
+Y
+t
+i,
+4
+�
+(T −t)
+ξ
+std(Xt
+i)
+if Bt ≥
+N
+�
+i=1
+Y
+t
+i
+with probability at least 1 − ξ.
+Proof. The detailed proof is in Appendix C. Intuitively, the
+discrepancy scales with the number of agents, since if all ag-
+nets j ̸= k submit demands according to their upper bound
+Y
+t
+j, then the water-level moves so that their SAFFE-Oracle
+allocations increase compared to SAFFE-Oracle with Yt
+j,
+which happens on the account of agent j who now gets less.
+Finally, using (21) completes the proof, as we can translate
+the discrepancy between SAFFE-Oracle with Yt
+i and Y
+t
+i to
+that between SAFFE-D and SAFFE-Oracle with Y t
+i .
+In order to move beyond the worst-case scenario of highly
+unbalanced request distributions, we now assume that all
+agents have the same Y
+t
+i and Yt
+i, e.g. if their demands are
+from the same distribution, the discrepancy between alloca-
+tions scales better.
+Theorem 2 (Balanced demands bound). If for any two agents
+i and j we have the same bounds in (21), i.e. Y
+t
+i = Y
+t
+j and
+Yt
+i = Yt
+j, then for all agents i, with probability at least 1 − ξ,
+we have
+���At,SAFFE-D
+i
+− At,oracle
+i
+��� ≤
+�
+�
+�
+�
+�
+�
+�
+0
+if Bt ≤
+N
+�
+i=1
+Yt
+i,
+4
+�
+(T −t)
+ξ
+std(Xt
+i)
+if Bt ≥
+N
+�
+i=1
+Yt
+i
+Proof. See Appendix C for the detailed proof, which follows
+a similar argument as the proof of Theorem 1.
+
+6.2
+SAFFE-Oracle Matches Hindsight
+Now that we have upper bounded the discrepancy between
+the allocations of SAFFE-D and SAFFE-Oracle, our next step
+is to show that SAFFE-Oracle achieves the optimal alloca-
+tions in hindsight.
+Theorem 3. SAFFE-Oracle achieves optimal allocations in
+hindsight, i.e. for all i we have �T
+t=1 At,oracle
+i
+= �
+Ai, where
+�
+Ai is the solution to the Eisenber-Gale program (12).
+Proof. See Appendix D for a detailed proof, which relies on
+the following observations: 1) For t = 1, total allocations for
+the current step and the reserved allocations for the future C1
+i ,
+equals the solution �
+Ai of (12), and 2) SAFFE-Oracle fully
+distributes the total reserved allocations for future by the end
+of the horizon T.
+6.3
+SAFFE-Discounted vs Hindsight
+We can now bound the gap between SAFFE-D and hindsight.
+Theorem 4 (Gap between SAFFE-D and hindsight). In the
+unbalanced demands setting, for the gap between SAFFE-D
+and hindsight measured by ∆Amax introduced in (4) we have
+∆Amax ≤ N T 3/2
+√ξ std(Xt
+i).
+(22)
+In the balanced demands setting, we have
+∆Amax ≤ 2T 3/2
+√ξ
+std(Xt
+i).
+(23)
+Proof. Firstly, let us note that Theorem 3 guarantees that
+∆Amax = E
+�
+max
+i
+�����
+T
+�
+t=1
+At,SAFFE-D −
+T
+�
+t=1
+At,oracle
+�����
+�
+,
+as SAFFE-Oracle achieves the hindsight solution. It remains
+to employ Theorems 1 and 2 together with the triangle in-
+equality in order to obtain an upper bound on the right hand
+side.
+In the case of unbalanced demands, this results in
+∆Amax ≤ N T 3/2
+√ξ std(Xt
+i). In the case of balanced demands,
+we have ∆Amax ≤ 2T 3/2
+√ξ
+std(Xt
+i)
+Remark 2. As ∆Amean ≤ ∆Amax, Theorem 4 also provides
+an upper bound on ∆Amean.
+7
+Experimental Results
+In this section, we use synthetic data to evaluate SAFFE-
+D under different settings and to compare its performance
+against baseline sequential algorithms in terms of fairness
+metrics and budget allocation efficiency. In the experiments,
+we study: 1) how the fairness of SAFFE-D allocations is af-
+fected as the budget size, number of agents and time horizon
+vary (Sec. 7.1), 2) whether the algorithm favors agents de-
+pending on their arrival times or demand sizes (Sec. 7.2), 3)
+how sensitive the algorithm is to future demand estimation
+errors (Sec. 7.3), 4) how the discounting in SAFFE-D im-
+proves fairness (Sec. 7.4), and 5) how SAFFE-D compares to
+allocation policies learned using reinforcement learning (RL)
+(Sec. 7.5). Finally, in Sec. 7.6, we evaluate SAFFE-D on real
+data.
+Evaluation Metrics
+We consider the following metrics to
+compare the fairness and efficiency of sequential allocation
+algorithms:
+• Log-NSW: Expected log-NSW in hindsight as in Eq. (3).
+Since the value of log-NSW is not directly comparable
+across different settings, we use its normalized distance
+to the hindsight log-NSW when comparing algorithms,
+denoted by ∆Log-NSW.
+• Utilization (%): The fraction of available budget dis-
+tributed to the agents over the horizon. Due to the stochas-
+ticity in experiments described next, we may have an
+over-abundance of initial budget that exceeds the total de-
+mands over the horizon. Therefore, we define this met-
+ric by only considering the required fraction of available
+budget B as
+�N
+i=1
+�T
+t=1 At
+i
+min{B, �N
+i=1
+�T
+t=1 Xt
+i}
+× 100
+• ∆Amean and ∆Amax: The average and maximum nor-
+malized deviation of per-agent cumulative allocations
+compared to hindsight allocations as defined in Eq. (4).
+For better scaling in visualizations, we make a slight
+change by normalizing these metrics with respect to hind-
+sight as
+∆Ai
+�T
+t=1 At,hindsight
+i
+. Since these metrics measure dis-
+tance, an algorithm with lower ∆A is considered more
+fair in terms of this metric.
+Allocation Algorithm Baselines
+We compare our algo-
+rithms SAFFE and SAFFE-D with the following baselines:
+• Hindsight: As discussed in Sec. 4, the solution to (13)
+represents a baseline for evaluating sequential algorithms
+since its Log-NSW provides an upper bound for other al-
+gorithms.
+• HOPE-Online: Motivated by the mobile food-bank al-
+location problem, this algorithm was proposed in [Sin-
+clair et al., 2020] for a setting where N agent demands
+are sequentially revealed over T = N. As we explain in
+Appendix E, HOPE-Online coincides with SAFFE in the
+special case where each agent makes only one request.
+• Guarded-HOPE: It was proposed in [Sinclair et al.,
+2022] to achieve the correct trade-off between fairness
+(in terms of envy-freeness) and efficiency with an input
+parameter LT derived based on concentration arguments.
+The algorithm is designed for a setting of one request per
+horizon for an individual. We modify Guarded-HOPE
+(Algorithm 4 in Appendix F) to be applicable to our set-
+ting, and to provide meaningful confidence bounds for the
+multiple demand case. As in [Sinclair et al., 2022], we
+use LT = T −1/2 and LT = T −1/3.
+Demand Processes
+We investigate various settings by con-
+sidering the following demand processes, for which we are
+able to analytically derive the future demand means and stan-
+dard deviations:
+• Symmetric Setting (Homogeneous Bernoulli arrivals
+with Normal demands). At time t, agent i makes a re-
+quest Xi ∼ N(µi, σ2
+i ) with probability p, independently
+
+from other agents. The distribution parameters are such
+that µi ∼ Uniform(10, 100), and σi = µi/5. We study
+regimes where p = c
+T , 1 ≤ c ≤ T, is the average number
+of arrivals per agent.
+• Non-symmetric
+Arrivals
+Setting
+(Inhomogeneous
+Bernoulli arrivals with Normal demands).
+In this
+setup, we assume that the likelihood of arrivals over
+the horizon varies across agents, which we use to
+evaluate whether the algorithms favor agents based on
+their arrival times. We implement this using Bernoulli
+arrivals with a time-varying parameter p. We consider
+three groups. Ask-Early: 1/3rd of the agents are more
+likely to frequently visit earlier in the horizon such
+that pt
+i ∝ (T − t), Ask-Late: 1/3rd of the agents are
+more likely to frequently visit later in the horizon with
+pt
+i ∝ t, and Uniform: the remaining agents do not have
+a preference in terms of time with constant pt
+i.
+The
+Bernoulli parameters of the three groups are normalized
+to have equal expected number of visits.
+• Non-symmetric
+Demands
+Setting
+(Homogeneous
+Bernoulli arrivals and varying Normal demands).
+We
+consider homogeneous arrivals but with time-varying
+demand sizes across agents. We have three groups with
+agents that make requests with the same probability p
+over T. More-Early: 1/3rd of the agents have Normal
+demands with mean µt
+i ∝ (T − t), More-Late: 1/3rd
+of the agents have µt
+i ∝ t, and Uniform: the remaining
+agents have constant µt
+i. The parameters are normalized
+to have the same total demands across the groups.
+Standard deviation σi is assumed to be time-independent
+for all agents.
+Real Data
+We evaluate SAFFE-D using a real dataset4 con-
+taining five years of store-item sales data for 10 stores. We
+assume that each store (agent) requests its total daily sales
+from a warehouse (supplier) with limited inventory, and the
+warehouse aims to be fair when supplying the stores on a
+weekly basis, i.e., when T = 7. We use the first three years
+to estimate the expected demands and standard deviations for
+each weekday, and we use the remaining two years to evalu-
+ate SAFFE-D for fair daily allocations.
+We report the results on synthetic data as an average
+over 200 experiment realizations with one standard deviation.
+For SAFFE-D we use λ(t) = λ
+√
+T − t, where the hyper-
+parameter λ is optimized with respect to Log-NSW. We ex-
+press the budget size B as the fraction of total expected de-
+mands such that a budget of 0.1 means enough budget to meet
+only 10% of the expected demands.
+7.1
+Scaling System Parameters
+In Fig. 2a, we compare SAFEE-D with the baselines as the
+supplier’s budget increases from 0.1 to 1. We consider the
+Symmetric setting with N = 50 agents and 2 expected ar-
+rivals over horizon T = 40. We observe that SAFFE-D out-
+performs all other approaches in terms of both ∆Log-NSW
+and resource utilization across all budgets. The advantage of
+4https://www.kaggle.com/c/demand-forecasting-kernels-only
+SAFFE-D in comparison to other algorithms is that it pre-
+vents over-reserving for the future, and thus achieves higher
+utilization and lower ∆Log-NSW. In contrast, SAFFE does
+not impose the regularization term which potentially results
+in overestimating future demands.
+Hope-Guardrail is de-
+signed to find a balance between utilization and envy, which
+does not necessarily mean balance in terms of Log-NSW. Ad-
+ditional experiments showing how the performance of the al-
+gorithms scale with the number of agents N, time horizon
+T and the per-agent expected arrivals are provided in Ap-
+pendix G.1. The results suggest that SAFFE-D, although sub-
+optimal, performs close to Hindsight.
+7.2
+Non-Symmetric Arrivals or Demands
+We investigate whether SAFFE-D allocates to agents differ-
+ently based on their arrival or demand patterns. We consider
+N = 50 agents with 2 expected arrivals over the horizon
+T = 40, and Normal demands with µi = 50, and budget size
+0.5. Under the Non-symmetric Arrivals setting, we compare
+the agents’ allocations for each algorithm with respect to the
+Hindsight allocations in Fig. 3. In this setting, agents have
+different likelihoods of arriving over the horizon. We ob-
+serve that despite being optimized for the log-NSW objective,
+SAFFE-D outperforms all other algorithms on average, and is
+more uniform across the three groups compared to SAFFE
+and Guarded-HOPE which favor the Ask-Late agents.
+In
+terms of the worst-case agent, SAFFE outperforms all other
+methods and has similar ∆Amax across all three groups. The
+algorithms are compared under the Non-symmetric Demands
+setting in Appendix G.2.
+Table 1: Improvement of SAFFE-D with choice of λ.
+Log-NSW (↑)
+Utilization % (↑)
+∆Amean (↓)
+∆Amax (↓)
+Hindsight
+236.57±10.21
+100.0±0.0
+−
+−
+SAFFE-D (λ(t))
+235.91±10.26
+99.45±1.5
+0.05±0.02
+0.45±0.23
+SAFFE-D (λ)
+234.12±10.67
+97.86±3.5
+0.09±0.03
+0.65±0.22
+SAFFE (λ = 0)
+232.89±10.91
+95.77±4.5
+0.11±0.03
+0.69±0.21
+7.3
+Sensitivity to Estimation Errors
+We investigate how sensitive the algorithms are to estimation
+errors by scaling each agent’s mean estimate at each step as
+�µt
+i = (1 + δ)µt
+i, where δ denotes the noise level. Fig. 2b
+shows the change in performance as we vary the noise level
+δ from −0.5 to 0.5, for the Symmetric setting with N = 50
+agents, 2 expected per-agent arrivals, T = 40, and budget
+0.5. We observe that SAFFE-D is robust to mean estimation
+noise while all other algorithms are impacted. This results
+from the discounting term in SAFFE-D, for which the hyper-
+parameter λ can be tuned to the estimation errors. However,
+when over-estimating the future demands, SAFFE and other
+methods will be initially more conservative which can lead to
+lower utilization and less fair allocations due to the leftover
+budget. When under-estimating the expected demands, they
+use the budget more greedily earlier on and deplete the budget
+sooner, resulting in less allocations to agents arriving later
+(especially in Gaurded-HOPE).
+7.4
+Choice of λ for SAFFE-D
+The discounting term in SAFFE-D uses the confidence in es-
+timates to improve the performance of SAFFE by balancing
+
+(a) Different budget size B.
+(b) Sensitivity to estimation errors.
+Figure 2: Symmetric Setting with N = 50 and T = 40. (a) SAFFE-D performs close to Hindsight achieving high utilization and low
+∆Log-NSW, ∆Amean and ∆Amax, (b) SAFFE-D is robust to expected demands estimation errors.
+Figure 3: Non-symmetric Arrivals Setting: SAFFE-D outperforms
+in the average case across groups with different arrival patterns and
+SAFFE is more uniform in the worst case.
+the budget between allocating now vs reserving for future ar-
+rivals. We compare the performance of SAFFE-D in terms of
+the discounting function in Table 1 for constant or decreasing
+λ over the horizon. In both cases, the parameter is tuned to
+maximize Log-NSW. As expected from Sec. 5.2, the perfor-
+mance of SAFFE-D improves for λ(t) decreasing over time.
+This is because the uncertainty in expected future demands
+reduces and the supplier can be less conservative when mak-
+ing allocation decisions.
+7.5
+Reinforcement Learning Results
+We investigate learning an allocation strategy using RL under
+the Symmetric setting. We consider N = 10 agents, T = 10
+and varying budget size sampled from Uniform(0.4, 0.8).
+We use the Soft Actor-Critic (SAC) [Haarnoja et al.,
+2018a; Achiam, 2018] method with automatic entropy tun-
+ing [Haarnoja et al., 2018b] to learn a policy (See Ap-
+pendix H). The results averaged over 200 experiment roll-
+outs for five random seeds are reported in Table 2, for two
+cases where we expect to have 2 and 4 per-agent arrivals.
+We observe that while the RL policy is not able to match the
+hindsight performance, it outperforms SAFFE-D and other
+approaches in terms of Log-NSW under sparse (2 per-agent)
+arrivals. With denser (4 per-agent) arrivals, the RL policy per-
+forms slightly worse than SAFFE-D while still outperforming
+others. These observations are in line with other experiments
+illustrating the close-to-optimal performance of SAFFE-D es-
+pecially in settings with more arrivals.
+7.6
+Real Data
+In contrast to the demand processes considered in previ-
+ous experiments, in this case, we need to estimate the ex-
+pected demands and their standard deviations. The efficiency
+and fairness metrics are reported in Table 3 for budget size
+0.5.
+We observe that SAFFE-D is optimal, and SAFFE
+and HOPE-Online perform very close to Hindsight in terms
+of Utilization and Log-NSW. As shown in Fig. 4c in Ap-
+pendix G.1, we expect that SAFFE-D achieves (close to)
+hindsight performance for dense agent arrivals as with the
+daily arrivals in this setting. In order to investigate less dense
+scenarios, we impose sparsity in the arrivals by erasing each
+arrival with probability p. As observed in Table 3, SAFFE-D
+is no longer optimal but outperforms all other methods. Ad-
+ditional results are provided in Appendix G.3.
+8
+Related Work
+Offline resource allocation is a classic problem that has
+been extensively studied for single and multiple resources,
+various utility functions, and resource types.
+In the non-
+strategic setup where agents are assumed to truthfully report
+their demands and to take the entirety of their allocations,
+several heuristics-based and optimization-based approaches
+have been proposed achieving different levels of optimal-
+ity guarantees and computational efficiency [Bretthauer and
+Shetty, 1995; Katoh and Ibaraki, 1998; Lee and Lee, 2005;
+Patriksson, 2008]. This problem finds important applications
+in computer and vehicle network management, where exist-
+ing works take advantage of the problem structure [Wong et
+al., 2004; Zhao et al., 2015; Wang et al., 2018; Gatzianas et
+al., 2021; Xu et al., 2021].
+
+Hindsight
+SAFFE-D
+SAFFE
+HOPE-Online
+Guarded-HOPE (L↑ = T-1/2)
+Guarded-HOPE (LT = T-1/3)
+100
+0.14 -
+0.5 
+4.0
+98
+3.5 
+0.12 
+(%)
+96
+0.4
+-NSW
+0.10 
+3.0
+Utilization
+0.3
+2.5
+92
+△Log-I
+0.08
+2.0
+0.06
+0.2
+1.5
+88
+0.04
+1.0 
+86
+0.0
+0.1 
+0.5 
+84
+0.00
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+Budget Fraction
+Budget Fraction
+Budget Fraction
+Budget Fraction100 
+1.6 
+00
+0.35 
+95
+0.25 
+0.30 
+1.4
+(%) 
+ 06
+Utilization 
+0.20 -
+1.2
+85 
+0.25 
+80 
+0.20
+75
+0.10
+0.15 -
+0.8 
+70 -
+0.05 -
+0.10
+0.6
+65 
+0.00
+-0.4
+-0.2
+0.0
+0.2 
+-0.4
+-0.2 
+0
+0.2 
+0.4
+0.4
+-0.2 
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
++
+Noise Level §
+Noise Level §
+Noise Level §
+Noise Level §0.40
+All
+0.35
+Ask-Early
+1.0
+0.30
+Uniform
+Ask-Late
+0.8
+mean
+0.25
+max
+0.20
+0.6
+0.15
+0.4
+0.10
+0.2
+0.05
+0.00
+0.0
+SAFFE-D
+SAFFE
+HOPE-Online
+SAFFE-D
+SAFFE
+Guarded-HOPE
+HOPE-Online
+Guarded-HOPE
+(L = T~1/3)
+= T1/3 )
+Algorithm
+AlgorithmTable 2: SAFFE-D vs RL allocation policy in the Symmetric setting for N = 10 agents with different per-agent arrivals.
+2 Per-agent Arrivals (Sparse)
+4 Per-agent Arrivals (Dense)
+Log-NSW (↑)
+Utilization % (↑)
+∆Amean (↓)
+∆Amax (↓)
+Log-NSW (↑)
+Utilization % (↑)
+∆Amean (↓)
+∆Amax (↓)
+Hindsight
+35.74±1.23
+100.0±0.0
+−
+−
+47.37±2.20
+100.0±0.0
+−
+−
+RL Policy (SAC)
+35.11±1.20
+95.63±0.49
+0.13 ± 0.01
+0.43 ± 0.01
+47.00±2.20
+98.30±0.45
+0.12±0.01
+0.34±0.02
+SAFFE-D
+35.01±1.28
+99.54±0.42
+0.15±0.05
+0.51± 0.16
+47.12± 2.22
+99.82±0.36
+0.11±0.04
+0.32±0.09
+SAFFE
+34.77±1.19
+92.74±0.41
+0.16±0.02
+0.53±0.03
+46.87±2.15
+97.14±0.43
+0.12±0.02
+0.35±0.03
+Table 3: Experiments on real data for daily per-agent demands (p = 1), and more sparse demand arrivals (p = 0.5).
+p = 1
+p = 0.5
+Log-NSW (↑)
+Utilization % (↑)
+∆Amean (↓)
+∆Amax (↓)
+Log-NSW (↑)
+Utilization % (↑)
+∆Amean (↓)
+∆Amax (↓)
+Hindsight
+42.72±0.00
+100.0±0.0
+−
+−
+35.24±0.94
+100.0±0.0
+−
+−
+SAFFE-D
+42.72±0.00
+100.0±0.0
+0
+0
+35.16±0.94
+100.0±0.0
+0.06±0.04
+0.17±0.14
+SAFFE
+42.72±0.02
+99.98±0.15
+0.01±0.01
+0.01±0.03
+34.82±1.10
+99.16±2.94
+0.17±0.08
+0.37±0.16
+HOPE-Online
+42.71±0.02
+99.98±0.15
+0.10±0.04
+0.04±0.01
+34.44±1.10
+99.10±3.03
+0.27±0.06
+0.60±0.15
+Guarded-HOPE (LT = T −1/2)
+42.48±0.30
+97.93±2.80
+0.07±0.02
+0.16±0.05
+33.55±3.67
+95.93±6.24
+0.27±0.06
+0.62±0.15
+Guarded-HOPE (LT = T −1/3)
+42.55±0.21
+98.64±1.94
+0.07±0.02
+0.15±0.05
+33.59±3.67
+96.54±5.61
+0.28±0.07
+0.63±0.15
+Other settings involve agents that can misreport their true
+demands to receive more favorable allocations, where allo-
+cation mechanisms need to be designed to incentivize truth-
+ful behavior [Guo et al., 2009; Cole et al., 2013; Sinha and
+Anastasopoulos, 2015; Abebe et al., 2020; Tan et al., 2020].
+In particular, [Cole et al., 2013] achieves this by strategically
+withholding a portion of the resources at the cost of compro-
+mised efficiency. [Tan et al., 2020] collects payments from
+agents when they place requests, and [Guo et al., 2009] stud-
+ies the repeated allocation of a single resource among the
+same agents and collects an artificial payment which intu-
+itively captures the credibility of the agents in the past.
+In the online setting, a number of works [Tesauro and
+others, 2005; Asadpour et al., 2020; Gallego et al., 2015;
+Wang et al., 2017; Sinclair et al., 2020; Sinclair et al., 2022]
+consider real-time allocation decision-making, among which
+[Sinclair et al., 2020; Sinclair et al., 2022] are closest to
+our setup. They study allocating multiple divisible resources
+among truthful agents that are characterized by a finite num-
+ber of types. Motivated by the food-bank allocation problem,
+they consider one request per agent. In our work, we study
+the more general scenario where the agents’ requests may ar-
+rive multiple times and the number of types need not be finite.
+Our goal is to be fair to the agents considering all their allo-
+cations over the horizon.
+9
+Conclusions
+This paper studies the problem of fair resource allocation
+in the sequential setting, and mathematically formulates the
+NSW fairness objective under the MDP framework. With the
+goal to maximize the return in the MDP, we propose SAFFE-
+D, a new intuitive algorithm that enjoys a worst-case theoret-
+ical sub-optimality bound. We show that SAFFE-D signifi-
+cantly outperforms existing approaches through extensive ex-
+periments, and effectively balances how much of the budget
+should be used at each step versus preserved for the future.
+Empirical results over several generated demand processes
+demonstrate the superiority of SAFFE-D under different bud-
+get sizes, number of agents, and time horizon lengths, espe-
+cially in settings with dense arrivals. The uncertainty-based
+discount in SAFFE-D also improves the robustness of the al-
+locations to errors in future demand estimations. Future work
+includes exploring learning-based allocation policies, learn-
+ing the optimal λ(t) in SAFFE-D, and extensions to settings
+with strategic agents.
+Disclaimer
+This paper was prepared for informational pur-
+poses in part by the Artificial Intelligence Research group of
+JPMorgan Chase & Co˙and its affiliates (“JP Morgan”), and
+is not a product of the Research Department of JP Morgan.
+JP Morgan makes no representation and warranty whatsoever
+and disclaims all liability, for the completeness, accuracy or
+reliability of the information contained herein. This docu-
+ment is not intended as investment research or investment ad-
+vice, or a recommendation, offer or solicitation for the pur-
+chase or sale of any security, financial instrument, financial
+product or service, or to be used in any way for evaluating the
+merits of participating in any transaction, and shall not con-
+stitute a solicitation under any jurisdiction or to any person,
+if such solicitation under such jurisdiction or to such person
+would be unlawful.
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+Sequential Fair Resource Allocation under a Markov Decision
+Process Framework
+Supplementary Materials
+A
+Existence of Optimal Policy
+The state space of the MDP is continuous, the action space
+is continuous and state-dependent, and the reward is bounded
+and continuous due to the constant ϵ in Eq. (9). As a result of
+[Furukawa, 1972][Theorem 4.2], a stationary optimal policy
+π⋆ = (π⋆
+1, . . . , π⋆
+T ) is known to exist.
+If we restrict the action space to allocations that satisfy
+At
+i ≤ Xt
+i for all i and t, the reward at step t (8) becomes
+Rt(st, At) =
+N
+�
+i=1
+1{Xt
+i > 0} . wi
+�
+log( �At−1
+i
++ At
+i + ϵ) − log( �At−1
+i
++ ϵ)
+�
+,
+(24)
+where �At
+i := �t
+τ=1 Aτ
+i denotes the cumulative allocated re-
+sources to agent i till time t. At time step T, π⋆
+T maximizes
+QT (sT , AT ) = RT (sT , AT ) which does not depend on fu-
+ture allocations. The optimal allocation AT ⋆ can be directly
+computed from QT since there is no uncertainty about future
+demands. Then, the optimal policy for all t can be derived by
+recursively solving (11) backward in time. This quickly be-
+comes computationally intractable, which motivates us to find
+alternative solutions that trade-off sub-optimality for compu-
+tational efficiency.
+B
+Water-Filling Algorithm with Past
+Allocations
+Algorithm 3 extends the water-filling algorithm presented in
+Algorithm 1 for agents with different weights to a setting
+where each agent may have past allocations denoted by A0.
+This algorithm is used as a base algorithm in SAFFE (Al-
+gorithm 2) to compute the allocations during each time step
+based on past allocations, and current and expected future de-
+mands. In line 3, the agents are ordered according to their
+demands, past allocations and weights.
+This ordering de-
+termines which agent is used to compute the water-level µ.
+For each selected agent ij, the condition in line 6 determines
+whether there is enough budget to fully satisfy agent ij’s de-
+mand Xij. If there is enough budget, agent ij receives its
+full request (line 10) and the supplier moves on to the next
+agent in order. Otherwise, the available budget is fully di-
+vided among the remaining agents (line 8) according to a
+water-level µ computed in line 7. The water-level accounts
+for the agent’s past allocations as well as their weights.
+C
+SAFFE-D vs SAFFE-Oracle
+Lemma 1 (Chebyshev’s inequality). For a random variable
+Y with finite expectation and a finite non-zero variance, we
+Algorithm 3 Water-Filling Algorithm with Past Allocations
+1: Input: number of agents N, resource budget B, demand
+vector X ∈ RN, weight vector w ∈ RN, past allocations
+A0 ∈ RN
+2: Output: allocation vector A ∈ RN
+3: Find an ordered index set {i1, . . . , iN} by sorting the
+agents such that
+Xi1+A0
+i1
+wi1
+≤ · · · ≤
+XiN +A0
+iN
+wiN
+4: j ← 1
+5: while j ≤ N and B > 0 do
+6:
+if B ≤ �N
+k=j
+� wik
+wij (Xij + A0
+ij) − A0
+ik
+�+
+then
+7:
+Solve �N
+k=j(
+wik
+wij µ − A0
+ij)+ = B for µ
+8:
+Aik = (
+wik
+wij µ − A0
+ij)+ for k = j, . . . , N
+9:
+break
+10:
+else
+11:
+Aij = Xij
+12:
+B ← B − Aij
+13:
+j ← j + 1
+14:
+end if
+15: end while
+have
+P
+�
+|Y − E(Y )| ≥
+1
+√ξ
+�
+Var(Y )
+�
+≤ ξ
+In particular, under the assumption from Section 6.1 that
+agent i’s requests are i.i.d, Chebyshev’s inequality gives us
+that Eq. (21) holds with probability 1 − ξ. Note however that
+the assumption is not crucial, and in a more general case we
+would have
+Yt
+i = E[
+T
+�
+τ=t+1
+Xτ
+i ] − 1
+√ξ
+�
+�
+�
+�Var
+�
+T
+�
+τ=t+1
+Xτ
+i
+�
+and
+Y
+t
+i = E[
+T
+�
+τ=t+1
+Xτ
+i ] + 1
+√ξ
+�
+�
+�
+�Var
+�
+T
+�
+τ=t+1
+Xτ
+i
+�
+.
+See also Remark 3 on how the assumption on equal standard
+deviations std(Xτ
+i ) = std(Xτ
+j ) for any two agents i and j is
+not crucial for the analysis.
+We proceed as follows.
+First, we state the result for
+SAFFE-Oracle in the unbalanced demands regime, i.e. when
+there is an agent k such that Yt
+k ≥ Y
+t
+j for all j ̸= k. Then, we
+
+state the result for the case of balanced demands, i.e. when
+Yt
+i = Yt
+j and Y
+t
+i = Y
+t
+j for any two i and j. Finally, we
+present the proof for both results.
+Lemma 2 (Unbalanced demands SAFFE-Oracle). Let A
+t
+i
+and At
+i denote the allocations of SAFFE-Oracle with Y t
+i =
+Y
+t
+i and Y t
+i = Yt
+i, respectively. Then for all agents i we have
+|A
+t
+i − At
+i| ≤
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+2N
+�
+(T −t)
+ξ
+std(Xτ
+i )
+if Bt ≤
+N
+�
+i=1
+Yt
+i,
+2N
+�
+(T −t)
+ξ
+std(Xτ
+i )
+if
+N
+�
+i=1
+Yt
+i ≤ Bt ≤
+N
+�
+i=1
+Y
+t
+i,
+4
+�
+(T −t)
+ξ
+std(Xτ
+i )
+if Bt ≥
+N
+�
+i=1
+Y
+t
+i
+(25)
+Lemma 3 (Balanced demands SAFFE-Oracle). Let A
+t
+i and
+At
+i denote the allocations of SAFFE-Oracle with Y t
+i = Y
+t
+i
+and Y t
+i = Yt
+i, respectively. Then we have
+|A
+t
+i − At
+i| ≤
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+if Bt ≤
+N
+�
+i=1
+Yt
+i,
+4
+�
+(T −t)
+ξ
+std(Xτ
+i )
+if
+N
+�
+i=1
+Yt
+i ≤ Bt ≤
+N
+�
+i=1
+Y
+t
+i,
+4
+�
+(T −t)
+ξ
+std(Xτ
+i )
+if Bt ≥
+N
+�
+i=1
+Y
+t
+i
+(26)
+Proof of Lemma 2 and Lemma 3.
+Case Bt
+≤
+�N
+n=1 Yt
+i.
+Agents whose lower bound
+demands are fully filled by the lower bound solution with
+Y t
+i
+= Yt
+i might increase the water-level when submitting
+larger demands, which will cause other agents to have smaller
+C
+t
+j in comparison to Ct
+j. In the worst-case scenario, where
+agent i’s lower bound demand is greater than all other agents’
+upper bound demands, the difference Ct
+i − C
+t
+i can be at most
+�
+j̸=i
+�
+Y
+t
+i − Yt
+i
+�
+,
+i.e., agent i has to compensate for the excess demands of
+all other agents.
+If std(Xτ
+i ) is equal for all agents, the
+difference in the worst-case scenario is at most 2(N −
+1)
+�
+(T −t)
+ξ
+std(Xτ
+i ), which leads to a difference between
+At
+i − A
+t
+i of at most 2N
+�
+(T −t)
+ξ
+std(Xτ
+i ) as a consequence
+of Lemma 4.
+On the other hand if instead of the worst-case scenario
+we assume that Y
+t
+i and Yt
+i are the same for all agents (e.g.
+same demand distributions), then the water-level would not
+move in SAFFE using Y t
+i = Y
+t
+i in comparison to SAFFE
+using Y t
+i = Yt
+i, and thus there would be no change in the
+allocations resulting from the two problems.
+Case �N
+n=1 Yt
+i ≤ Bt ≤ �N
+n=1 Y
+t
+i.
+There is enough
+budget to satisfy all agents’ demands under the lower bound
+constraint but not eoug budget under the upper bound one.
+Compared to Ct
+i, in the worst-case scenario, C
+t
+i can decrease
+by at most
+�
+j̸=i
+Y
+t
+i −
+�
+j̸=i
+Yt
+i
+where agent i’s lower bound demand is larger than any other
+agents’ upper bound demands, and therefore, the water-filling
+algorithm will satisfy the upper bound demands of all other
+agents on the account of agent i. If all agents have equal
+std(Xτ
+i ), in the worst-case scenario, the difference is again at
+most 2(N − 1)
+�
+(T −t)
+ξ
+std(Xτ
+i ). Following Lemma 4, this
+leads to a difference of at most 2N
+�
+(T −t)
+ξ
+std(Xτ
+i ) between
+At
+i and A
+t
+i.
+On the other hand, if we assume that �
+j̸=i Y
+t
+i and
+�
+j̸=i Yt
+i are the same for all agents, then all agents will re-
+ceive an equal share of the surplus, i.e. the difference between
+Ct
+i and C
+t
+i becomes
+1
+N
+�
+�Bt −
+�
+j̸=i
+Yt
+i
+�
+� .
+Given that Bt ≤ �N
+n=1 Y
+t
+i, and due to (21), this is at most
+2
+�
+(T −t)
+ξ
+std(Xτ
+i ). As a consequence of Lemma 4, this leads
+to difference of at most 4
+�
+(T −t)
+ξ
+std(Xτ
+i ) between At
+i and
+A
+t
+i.
+Case �N
+n=1 Y
+t
+i ≤ Bt.
+There is enough budget to sat-
+isfy all agents’ demands even under the upper bound, so the
+difference between C
+t
+i and Ct
+i is purely driven by the differ-
+ence between lower and upper bound demands, i.e. it equals
+Y
+t
+i − Yt
+i which is 2
+�
+(T −t)
+ξ
+std(Xτ
+n). Following Lemma 4,
+this leads to a difference of at most 4
+�
+(T −t)
+ξ
+std(Xt′
+i ) be-
+tween At
+i and A
+t
+i.
+Lemma 4. If |C
+t
+i − Ct
+i| ≤ D, then we have
+|A
+t
+i − At
+i| ≤ D + 2
+�
+(T − t)
+ξ
+std(Xτ
+i ).
+Proof. Firstly, let us note that
+Xt
+iC
+t
+iYt
+i − Xt
+iCt
+iY
+t
+i = Xt
+iYt
+i(C
+t
+i − Ct
+i) + Xt
+iCt
+iYt
+i
+− Xt
+iCt
+iYt
+i − Xt
+iCt
+i
+�
+Y
+t
+i − Yt
+i
+�
+
+and hence we have
+���Xt
+iC
+t
+iYt
+i − Xt
+iCt
+iY
+t
+i
+��� ≤ Xt
+iYt
+i
+���C
+t
+i − Ct
+i
+��� + Xt
+iCt
+i
+���Y
+t
+i − Yt
+i
+���
+≤ Xt
+iYt
+i · D + 2Xt
+iCt
+i
+�
+T − t
+ξ
+std(Xτ
+i )
+≤ Xt
+iYt
+i
+�
+D + 2
+�
+T − t
+ξ
+std(Xτ
+i )
+�
+where in the last inequality we use Ct
+i ≤ Yt
+i.
+Thus
+|A
+t
+i − At
+i| =
+�����
+Xt
+iC
+t
+i
+Y
+t
+i
+− Xt
+iCt
+i
+Yt
+i
+�����
+=
+�����
+Xt
+iC
+t
+iYt
+i − Xt
+iCt
+iY
+t
+i
+Y
+t
+iYt
+i
+�����
+≤
+(D + 2
+�
+(T −t)
+ξ
+std(Xτ
+i ))Xt
+i
+Y
+t
+i
+≤ D + 2
+�
+(T − t)
+ξ
+std(Xτ
+i )
+Remark 3. The equal standard deviation assumption that for
+any two agents i and j, we have std(Xτ
+i ) = std(Xτ
+j ) is
+used in the proofs only to simplify �
+j̸=i std(Xτ
+j ) = (N −
+1) std(Xτ
+j ). Thus, the assumption can be removed.
+D
+SAFFE-Oracle Matches hindsight
+Lemma 5. At t=1, the solution C1,oracle
+i
+to the SAFFE-Oracle
+problem coincides with the solution ˜Ai to the Eisenber-Gale
+program (12) i.e. C1,oracle
+i
+= �
+Ai.
+Proof. At t = 1, SAFFE-Oracle solves the following prob-
+lem
+max
+{C1
+i }
+�
+i:�T
+t=1 Xt
+i >0
+log(C1
+i )
+s.t. 0 ≤ C1
+i ≤
+T
+�
+t=1
+Xt
+i,
+N
+�
+n=1
+C1
+i ≤ B.
+This is equivalent to problem (12) with Ai = C1
+i , and hence,
+we have C1,oracle
+i
+= ˜Ai.
+Lemma 6. With SAFFE-Oracle, for any agent i we have
+C1,oracle
+i
+=
+T
+�
+t=1
+At,oracle
+i
+i.e. its future allocation from the first round is allocated in
+full by the end of the T th round.
+Proof. First, we prove that for t = 2, . . . , T we have
+t−1
+�
+τ=1
+Aτ,oracle
+i
++ Ct,oracle
+i
+=
+t−2
+�
+τ=1
+Aτ,oracle
+i
++ Ct−1,oracle
+i
+i.e. the past plus future allocation from the previous step is
+preserved in the following step. Recall that in each step we
+are solving
+max
+{Ct
+i }
+�
+i:�T
+τ=t Xτn>0
+log(
+t−1
+�
+τ=1
+Aτ,oracle
+i
++ Ct
+i)
+(27)
+s.t. 0 ≤ Ct
+i ≤
+T
+�
+τ=t
+Xτ
+i ,
+N
+�
+n=1
+Ct
+i ≤ Bt.
+where the sequence of future demands is known to the oracle.
+It is easy to see that
+t−1
+�
+τ=1
+Aτ,oracle
+i
++ Ct,oracle
+i
+≤
+t−2
+�
+τ=1
+Aτ,oracle
+i
++ Ct−1,oracle
+i
+because the constraints in step t are tighter. If this didn’t
+hold, the previous round solution Ct−1,oracle
+i
+would not be
+the maximizing solution in step t − 1. It remains to note
+that �t−1
+τ=1 Aτ,oracle
+i
++ Ct,oracle
+i
+can achieve its upper bound,
+because compared to the previous step t − 1, the remaining
+budget was reduced by exactly
+N
+�
+n=1
+�t−1
+�
+τ=1
+Aτ,oracle
+i
+−
+t−2
+�
+τ=1
+Aτ,oracle
+i
+�
+.
+Therefore,
+Ct,oracle
+i
+= Ct−1,oracle
+i
+−
+�t−1
+�
+τ=1
+Aτ,oracle
+i
+−
+t−2
+�
+τ=1
+Aτ,oracle
+i
+�
+satisifes the constraints.
+This means that Ct,oracle
+i
+=
+Ct−1,oracle
+i
+− At−1,oracle
+i
+, i.e., the current plus future alloca-
+tion in step t equals its counterpart in step t − 1 minus the
+allocation in step t − 1. In particular, we get
+C2,oracle
+i
+= C1,oracle
+i
+− A1,oracle
+i
+(28)
+= C1,oracle
+n
+−
+X1
+i
+�T
+τ=1 Xτ
+i
+C1,oracle
+i
+(29)
+=
+�
+1 −
+X1
+n
+�T
+τ=1 Xτn
+�
+C1,oracle
+i
+,
+(30)
+C3,oracle
+i
+= C2,oracle
+i
+− A2,oracle
+i
+(31)
+=
+�
+1 −
+X2
+n
+�T
+τ=2 Xτn
+�
+C2,oracle
+i
+(32)
+=
+�
+1 −
+X2
+n
+�T
+τ=2 Xτn
+� �
+1 −
+X1
+n
+�T
+τ=1 Xτn
+�
+C1,oracle
+i
+(33)
+
+and in general we have
+Ct,oracle
+i
+=
+�
+1 −
+Xt−1
+i
+�T
+τ=t−1 Xτ
+i
+�
+· · ·
+�
+1 −
+X1
+i
+�T
+τ=1 Xτ
+i
+�
+C1,oracle
+i
+(34)
+and
+At,oracle
+n
+=
+�
+1 −
+Xt−1
+i
+�T
+τ=t−1 Xτ
+i
+�
+· · ·
+�
+1 −
+X1
+i
+�T
+τ=1 Xτ
+i
+�
+Xt
+i
+�T
+τ=t Xτ
+i
+C1,oracle
+i
+(35)
+Proof of Theorem 3. As a consequence of Lemmas 5 and 6,
+we have ˜At
+i = �T
+t=1 At,oracle
+i
+for every i. This completes the
+proof.
+E
+SAFFE as a generalization of HOPE-Online
+The heuristic algorithm HOPE-Online in [Sinclair et al.,
+2020] coincides with SAFFE under a simplified demand pro-
+cess. HOPE-Online is designed for a setting where the sup-
+plier visits a set of agents sequentially in order. SAFFE gen-
+eralizes the allocation algorithm to a setup where agents can
+arrive simultaneously and several times over the horizon T.
+Similar to SAFFE-D which improves SAFFE using the un-
+certainty of future demand estimates, Guarded-HOPE pro-
+posed in [Sinclair et al., 2022] improves HOPE-Online and
+achieves the correct trade-off between a measure of fairness
+(envy-freeness) and resource leftover by learning a “lower
+guardrail” on the optimal solution in hindsight. We highlight
+that Guarded-HOPE is designed for the setting where agents
+are not making repeated demands over the horizon i.e. each
+individual agent has a request at most once during the hori-
+zon. In that sense, our SAFFE and SAFFE-D represent gen-
+eralizations as we allow for agents with multiple demands. In
+our analysis of the optimality of SAFFE-D, we rely on con-
+centration inequalities for deviation of future demands from
+their expected values, which is in spirit similar to the opti-
+mality analysis of the guardrail approach in [Sinclair et al.,
+2022].
+F
+Extension of Guarded-HOPE to Our
+Setting
+Since Guarded-HOPE was designed for a setting where in
+each time-step a number of individuals of the same type ar-
+rive, we slightly modify it to be applicable to our setting in Al-
+gorithm 4. The key components of the algorithm are the upper
+and lower guardrails defined in lines 7 and 10. Specifically,
+X, X ∈ RN are high-confidence lower and upper bounds on
+the future demands, and A, A ∈ RN, which [Sinclair et al.,
+2020] refers to as upper and lower guardrails, respectively,
+are the optimal hindsight allocations under X, X. When the
+demands Xt are revealed in time step t, the condition in line
+13 first checks if the budget is insufficient to even allow an
+allocation according to the lower guardrail. If so, the budget
+is allocated right away and none is reserved. The condition
+in line 15 checks if the current demands can be allocated ac-
+cording to the upper guardrail assuming that the remaining
+budget is still enough to allow for allocations according to
+the lower guardrail for anticipated future demands. If so, the
+the upper guardrail is allocated to the agents with demands at
+the current step. Otherwise, the lower guardrail is allocated.
+Algorithm 4 Guarded-HOPE (Modified Compared to [Sin-
+clair et al., 2022])
+1: Input: number of agents N, resource budget B, de-
+mand vectors X1, . . . XT ∈ RN, demand distributions
+PX1, . . . , PXN , bound on envy LT
+2: Output: allocation vectors A1, . . . , AT ∈ RN
+3: Define
+confidence
+bound
+CONFi
+=
+�
+std(X1
+i )E[X1
+i ](T − 1)
+4: EXi := �T
+τ=1 E[Xτ
+i ]
+5: ci = Lt(1 + CONFt
+i/EXi) − CONFt
+i/EXi for all i =
+1, . . . , N
+6: Solve A for X := EX(1 − c) using Algorithm 1
+7: Ai ← Ai/Xi
+8: γi = CONFt
+i/EXi
+9: Solve A for X := EX(1 + γ) using Algorithm 1
+10: Ai ← Ai/Xi
+11: for For t = 1, . . . , T do
+12:
+Define
+confidence
+bound
+CONFt
+i
+=
+�
+std(Xt
+i)E[Xt
+i](T − t)
+13:
+if B ≤ �N
+i=1 Xt
+iAi then
+14:
+At
+i = 1{Xt
+i > 0} × B/
+� �N
+i=1 1{Xt
+i > 0}
+�
+for
+i = 1, . . . , N
+15:
+else if B ≥ �N
+i=1 Xt
+iAi + �N
+i=1 Ai
+�
+E[�T
+τ=t Xτ
+i ] +
+CONFt
+i
+�
+then
+16:
+At
+i = 1{Xt
+i > 0} × Xt
+iAi for i = 1, . . . , N
+17:
+else
+18:
+At
+i = 1{Xt
+i > 0} × Xt
+iAi for i = 1, . . . , N
+19:
+end if
+20:
+B ← B − �N
+i=1 At
+i
+21: end for
+G
+Additional Experiments
+G.1
+Scaling System Parameters
+SAFFE-D is compared to the baselines in terms of utilization
+and fairness metrics in Fig. 4, as different system parame-
+ters vary. Unless otherwise stated, the settings have N = 50
+agents, budget size 0.5, time horizon T = 40, and there are
+2 expected arrivals per-agent. In all experiments, we observe
+that SAFFE-D is more efficient and more fair compared to
+other methods, i.e., it achieves higher utilization, and lower
+∆Log-NSW, ∆Amean and ∆Amax. In Fig. 4a, we observe
+that as the number of agents increases over the same horizon,
+the algorithms initially achieve higher utilization and lower
+∆Log-NSW, which eventually levels out. However, in terms
+
+of ∆Amax fairness which measures the allocation difference
+with respect to hindsight for the worst agent, SAFEE-D is
+the least affected across different number of agents, while all
+other algorithms become worse. Fig. 4b shows the metrics
+for varying horizon T while having the same number of ex-
+pected arrivals, i.e., when the arrivals are more spread out
+across time. In this setting, utilization and fairness metrics
+seem relatively unaffected across all methods.
+In Fig. 4c, we compare the algorithms as the number of
+arrivals over the horizon increases. We observe that SAFFE-
+D is able to use all the budget and match the fair hindsight
+allocations when it is expected to have more than 5 arrivals
+per-agent over T = 40. SAFFE is able to match the perfor-
+mance of SAFFE-D in terms of utilization and ∆Log-NSW
+as the arrivals become denser. As discussed in Appendix E,
+HOPE-Online and Guarded-HOPE are designed for settings
+with a single per-agent arrival. As expected, when there are
+several arrivals per agents, they are not able to match SAFFE-
+D or SAFFE since they do not account for an agent’s past
+allocations when distributing the budget.
+G.2
+Non-Symmetric Demands
+Fig. 5 shows how SAFFE-D compares to other baselines in
+terms of allocations using the Non-symmetric Demands Set-
+ting, where some agents request larger demands earlier, while
+others have larger requests later or have no preference. We
+observe that SAFFE-D and SAFFE achieve lower ∆Amean,
+and have low variability across the groups on average. When
+considering the worst-case agent, SAFFE-D is less fair to
+agents that have larger demands earlier in the horizon, as it
+reserves the budget early on accounting for future expected
+demands. However, Uniform and More-Late agents receive
+allocations closer to hindsight compared to other algorithms.
+G.3
+Real Data
+In order to study how SAFFE-D performs on real data with
+sparser arrivals over the horizon, we enforce less per-agent
+requests by erasing each arrival with probability p.
+Since
+T = 7, setting p = 2
+7 corresponds to two weekly demands
+per store. As observed from Fig. 6, SAFFE-D outperforms
+the other methods in terms of efficiency and fairness. We re-
+mark that while with the uniform random erasures, we have
+imposed Bernoulli arrivals similar to the demand processes
+described in Sec. 7, the results presented here are based on
+the real demand sizes and reflect using imperfect estimates
+that are computed based on real data. However, further ex-
+periments on real datasets are needed to compare the perfor-
+mance of SAFFE-D under more general arrival processes that
+are correlated across time.
+H
+Training of Reinforcement Learning Policy
+We
+implement
+a
+variation
+of
+the
+Soft
+Actor-Critic
+method [Haarnoja et al., 2018a] with clipped double-Q learn-
+ing [Achiam, 2018] as well as automatic entropy [Haarnoja
+et al., 2018b] tuning.
+We train the SAC policy for 100K
+episodes for 5 random seeds, and then evaluate the 5 check-
+points under another 200 rollouts in order to compare with
+other baselines. We report the average performance with one
+standard deviation over the four metrics of Log-NSW, Uti-
+lization, ∆Amean and ∆Amax in Table 2.
+The policy network architecture consists of three Fully
+Connected (FC) layers followed by one output layer, where
+each FC layer has 256 neurons with ReLU activation func-
+tions.
+Since the MDP state is time dependent, to pre-
+vent the input state vector size from growing over the time
+horizon T during training, we represent the state vector as
+⟨ �Xt, �At, Bt, t⟩, where
+�
+Xt
+i =
+t
+�
+τ=1
+Xτ
+i ,
+�At
+i =
+t
+�
+τ=1
+Aτ
+i .
+We guarantee the step-wise budget constraint �N
+i=1 At
+i ≤
+Bt, by designing the output layer to have two heads: one head
+with a Sigmoid activation function, which outputs a scalar de-
+termining the step-wise budget utilization ratio ut ∈ [0, 1];
+and another head with a Softmax activation function, which
+outputs an allocation ratio vector, zt ∈ RN : �N
+i=1 zt
+i = 1
+over the agents. The final allocation for each agent is deter-
+mined by At
+i = zt
+iutBt.
+We perform hyper-parameter tuning using a grid search
+over a set of candidates. The results in Table 2 are achieved
+by using the hyper-parameters summarized in Table 4.
+Table 4: SAC Hyper-parameters used in experiments.
+Parameter
+Sparse
+Dense
+(2 arrivals)
+(4 arrivals)
+Learning rate
+3 · 10−4
+3 · 10−4
+Replay-buffer size
+8 · 105
+106
+Batch size
+512
+512
+Target smoothing coefficient(τ)
+0.005
+0.005
+Update interval (step)
+5
+1
+Update after (step)
+105
+5 · 105
+Uniform-random action selection (step)
+105
+5 · 105
+I
+Multiple-Resource Setting
+The setting in this paper can be easily extended to the sup-
+plier having M divisible resources, where resource j
+∈
+{1, . . . , M} has a limited budget of size Bj, and B =
+(B1, . . . , BM). At time t ∈ {1, . . . , T}, agent i’s demands
+i denoted by Xt
+i = (Xt
+i,1, . . . , Xt
+i,M) ∈ RM, and the sup-
+plier’s allocation is At
+i = (At
+i,1, . . . , At
+i,M) ∈ RM.
+For
+multiple resources, Θi = (X1
+i,j, . . . , XT
+i,j, Vi), where Vi =
+(vi,1, . . . , vi,M) denotes the per-unit value that agent i asso-
+ciates to each resource,
+u(Ai, Θi) =
+T
+�
+t=1
+M
+�
+j=1
+vi,j min
+�
+At
+i,j, Xt
+i,j
+�
+(36)
+While we present the paper in terms of a single resource, all
+algorithms and proofs can be extend to the multiple-resource
+setting. In this case, the optimal allocation in hindsight or the
+sequential setting will not have closed-form solutions as in
+the water-filling algorithm, but can be computed using convex
+optimization techniques.
+
+(a) Different number of agents N.
+(b) Different horizons T.
+(c) Different per-agent expected arrivals.
+Figure 4: Symmetric Setting : SAFFE-D performs close to Hindsight and outperforms other methods by achieving higher utilization and
+lower ∆Log-NSW, ∆Amean and ∆Amax.
+Figure 5: Non-symmetric Demands Setting: SAFFE-D allocates
+uniformly across agents with different arrival patterns on average,
+while SAFFE outperforms in the worst case.
+
+Hindsight
+SAFFE-D
+SAFFE
+HOPE-Online
+Guarded-HOPE (Lt = T-1/2)
+Guarded-HOPE (LT = T-1/3)
+100.0
+0.08 
+2.25 
+97.5
+0.30
+2.00 
+(%)
+95.0 
+0.06
+0.25
+1.75 
+Utilization
+92.5
+ISN-
+max
+1.50 -
+ALog-i
+0.20
+90.0 
+0.04
+1.25
+87.5
+0.15
+1.00
+85.0
+0.02 
+0.10
+0.75 
+82.5
+0.50
+0.00 -
+25
+50
+75
+100
+125
+150
+175
+200
+25
+50
+75
+100
+125
+150
+175
+200
+25
+50
+75
+100
+125
+150
+175
+200
+25
+50
+75
+100
+125
+150
+175
+200
+Number of Agents
+Number of Agents
+Number of Agents
+Number of AgentsHindsight
+SAFFE-D
+SAFFE
+HOPE-Online
+Guarded-HOPE (Lt = T-1/2)
+Guarded-HOPE (LT = T-1/3)
+100
+1.6 
+0.06
+- 86 
+00
+(%)
+1.4 
+96
+0.05
+MSN-
+t6
+0.25
+Utilization
+0.04
+1.2
+92
+△Log-I
+0.20
+K
+0.03
+1.0
+0.02
+0.15 -
+0.8 -
+88
+0.01
+0.10 
+0.6 -
+84
+25
+50
+75
+100
+125
+150
+175
+25
+50
+75
+100
+125
+150
+175
+25
+50
+75
+100
+125
+150
+175
+25
+50
+75 
+100
+125
+150
+175
+Horizon T
+Horizon T
+Horizon T
+Horizon THindsight
+SAFFE-D
+SAFFE
+HOPE-Online
+Guarded-HOPE (Lt = T-1/2)
+Guarded-HOPE (LT = T-1/3)
+1.75-
+100.0
+0.30 
+0.08
+97.5
+1.50 
+0.25 
+(%)
+95.0
+0.06 -
+1.25
+0.20 
+Utilization
+92.5
+max
+1.00 
+ 0'06
+0.15
+0.75
+87.5
+0.10 
+0.50
+85.0
+0.02 
+0.05 
+82.5
+0.25 
+80.0
+0.00 
+0.00
+0.00 -
+10
+15
+20
+25
+30
+35
+40
+0
+5
+10
+15
+20
+25
+30
+35
+40
+10
+15
+20
+25
+30
+35
+10
+15
+20
+25
+30
+35
+40
+Expected Per-Agent Arriavls
+Expected Per-Agent Arriavls
+Expected Per-Agent Arriavls
+Expected Per-Agent Arriavls0.40
+1.6
+All
+0.35
+More-Early
+1.4
+Uniform
+0.30
+1.2
+More-Late
+mean
+0.25
+xew
+1.0
+0.20
+0.8
+0.15
+0.6
+0.10
+0.4
+0.05
+0.2 
+0.00
+0.0
+SAFFE-D
+SAFFE
+HOPE-Online
+Guarded-HOPE
+SAFFE-D
+SAFFE
+HOPE-Online
+Guarded-HOPE
+(L = T~1/3)
+Algorithm
+AlgorithmFigure 6: Experiments on real data for different budgets when p = 2
+7. SAFFE-D performs close to Hindsight achieving high utilization and
+low ∆Log-NSW, ∆Amean and ∆Amax.
+
+Hindsight
+SAFFE-D
+SAFFE
+HOPE-Online
+Guarded-HOPE (Lt = T-1/2)
+Guarded-HOPE (Lt = T-1/3)
+100
+0.4 
+0.45 -
+1.2 -
+0.40 
+(%) oez!n
+86
+-NSW
+EO
+0.35
+1.0 
+mean
+- 96
+-00
+max
+0.8
+0.25
+ t6
+0.20
+0.6 -
+0.1
+0.15 
+92 
+0.10 
+0.0
+0.2
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+Budget Fraction
+Budget Fraction
+Budget Fraction
+Budget Fraction

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+Nuclear level densities and γ−ray strength functions in 120,124Sn isotopes: impact of
+Porter-Thomas fluctuations
+M. Markova,1, ∗ A. C. Larsen,1, † P. von Neumann-Cosel,2 S. Bassauer,2 A. G¨orgen,1
+M. Guttormsen,1 F. L. Bello Garrote,1 H. C. Berg,1 M. M. Bjørøen,1 T. K. Eriksen,1 D. Gjestvang,1
+J. Isaak,2 M. Mbabane,1 W. Paulsen,1 L. G. Pedersen,1 N. I. J. Pettersen,1 A. Richter,2
+E. Sahin,1 P. Scholz,3, 4 S. Siem,1 G. M. Tveten,1 V. M. Valsdottir,1 and M. Wiedeking5, 6
+1Department of Physics, University of Oslo, N-0316 Oslo, Norway
+2Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, D-64289 Darmstadt, Germany
+3Institut f¨ur Kernphysik, Universit¨at zu K¨oln, D-50937 K¨oln, Germany
+4Department of Physics, University of Notre Dame, Indiana 46556-5670, USA
+5Deptartment of Subatomic Physics to SSC Laboratory,
+iThemba LABS, Somerset West 7129, South Africa
+6School of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa
+(Dated: January 16, 2023)
+Nuclear level densities (NLDs) and γ-ray strength functions (GSFs) of 120,124Sn have been ex-
+tracted with the Oslo method from proton-γ coincidences in the (p, p′γ) reaction. The functional
+forms of the GSFs and NLDs have been further constrained with the Shape method by studying
+primary γ-transitions to the ground and first excited states. The NLDs demonstrate good agreement
+with the NLDs of 116,118,122Sn isotopes measured previously. Moreover, the extracted partial NLD
+of 1− levels in 124Sn is shown to be in fair agreement with those deduced from spectra of relativistic
+Coulomb excitation in forward-angle inelastic proton scattering.
+The experimental NLDs have been applied to estimate the magnitude of the Porter-Thomas
+(PT) fluctuations. Within the PT fluctuations, we conclude that the GSFs for both isotopes can
+be considered to be independent of initial and final excitation energies, in accordance with the
+generalized Brink-Axel hypothesis. Particularly large fluctuations observed in the Shape-method
+GSFs present a considerable contribution to the uncertainty of the method, and may be one of the
+reasons for deviations from the Oslo-method strength at low γ-ray energies and low values of the
+NLD (below ≈ 1 · 103 − 2 · 103 MeV−1).
+I.
+INTRODUCTION
+Numerous experimental and theoretical efforts have
+been dedicated to the study of γ−decay processes in
+atomic nuclei. The decay properties of excited nuclei are
+not only pivotal for the basic nuclear physics research,
+but also are the core ingredients for large-scale calcula-
+tions of nucleosynthesis and element abundances in the
+universe [8, 24].
+While gradually moving from the lowest to higher ex-
+citation energies of a nucleus, the spacing between indi-
+vidual excited states becomes smaller, and the sensitiv-
+ity of experimental techniques might be no longer suffi-
+cient to resolve them separately. Here, the nucleus enters
+the quasi-continuum regime and the concept of the nu-
+clear level density (NLD), i.e.
+the number of nuclear
+states per excitation energy unit, becomes an indispens-
+able tool for a statistical description of nuclei. By anal-
+ogy, the γ-ray strength function (GSF), or the average,
+reduced γ-transition probability, becomes more suitable
+to describe the numerous γ-transitions. The statistical
+model as formulated by Hauser and Feshbach [37] with
+ingredients such as the NLD and GSF, provides the main
+∗ [email protected]
+† [email protected]
+framework for modelling nuclear reactions and calculat-
+ing their cross-sections for astrophysical purposes (e.g.
+[8]), the design of nuclear reactors [18], and the transmu-
+tation of nuclear waste [61].
+Among all experimental techniques used for the extrac-
+tion of GSFs [25], the Oslo method has been widely used
+to obtain the dipole strength below the neutron threshold
+by studying the γ-decay of residual nuclei formed in light-
+ion induced reactions [33, 34, 63]. The main advantage
+of the method is a simultaneous extraction of the NLD
+and GSF from primary γ−decay spectra at excitation
+energies below the neutron separation energy Sn. The
+GSFs for many nuclei obtained by employing different
+experimental techniques have previously been reported
+to provide a rather good agreement in absolute values
+and/or general shapes with the Oslo method strengths
+[13, 51, 70]. A few cases of large discrepancies have also
+been reported (e.g. the comparison of the Oslo and (γ, γ′)
+data for 89Y and 139La presented in Ref. [25]).
+A large fraction of theoretical and experimental tech-
+niques focusing on calculating or measuring the GSF,
+including the Oslo method, rely on the validity of the
+generalized Brink-Axel (gBA) hypothesis [10, 14]. In its
+most general form, the hypothesis states that the GSF
+is independent of excitation energies, spins and parities
+of initial and final states and depends solely on the γ-
+ray energy of involved transitions.
+This is often used
+as a necessary approximation and simplification in many
+arXiv:2301.05319v1  [nucl-ex]  12 Jan 2023
+
+2
+methods, and allows to set a direct correspondence be-
+tween strengths extracted from the γ-decay and photo-
+excitation processes [14, 50]. Even though this hypothesis
+is experimentally established at high energies, i.e., in the
+vicinity of the giant dipole resonance, its validity below
+the neutron threshold still triggers quite some debate.
+For example, several theoretical studies suggest the need
+of a modification of the hypothesis [15, 38, 40, 44, 54],
+while experimental studies range from claims of a viola-
+tion [7, 42, 43, 56] to a validity [16, 32, 50, 51, 64]. The
+question regarding the validity is a rather difficult one,
+partially depending on what degree of violation is consid-
+ered acceptable in different experimental and theoretical
+applications.
+A crucial point to be considered when addressing the
+applicability of the gBA hypothesis is the presence of
+fluctuations of partial radiative widths, or the so-called
+Porter-Thomas (PT) fluctuations [59]. The partial radia-
+tive widths are proportional to the corresponding reduced
+transition strengths (B(XL) values where X is the elec-
+tromagnetic character and L the angular momentum of
+the γ ray). At sufficiently high excitation energies and
+high NLD values, the nuclear wave functions are quite
+complex with many components. In this region, accord-
+ing to random-matrix theory [69], the partial widths fol-
+low a χ2
+ν behavior with ν = 1 degree of freedom, while
+the total widths are more narrowly distributed with the
+variance inversely proportional to the number of inde-
+pendently contributing partial widths.
+Such a variation of partial widths is directly reflected
+in the variation of the GSF, which may mask the ex-
+citation energy independence of the strength, and thus
+a test of the gBA hypothesis might become especially
+difficult. Indeed, for relatively light nuclei, e.g.
+64,65Ni
+[16] and 46Ti [31], the NLDs are rather low, and tests of
+the gBA hypothesis are limited. On the other hand, the
+238Np nucleus [32] with a particularly high NLD, makes
+a perfect case for studying the GSF as a function of ini-
+tial and final excitation energies, as fluctuations of the
+strength are strongly suppressed. The Sn isotopes inves-
+tigated here present an intermediate case for studying to
+what degree the PT fluctuations are expected to distort
+excitation energy dependence of the GSF.
+Moving away from the valley of stability opens up
+new perspectives for studying exotic, neutron-rich nuclei,
+with applications to heavy-element nucleosynthesis [48],
+using for example the β-Oslo method [65] and the Oslo
+method in inverse kinematics [41]. However, this leads
+to additional complications, such as the lack of neutron-
+resonance data for normalizing the NLD and GSF from
+the Oslo-method data. Moreover, some of the light-ion
+induced reactions may lead to a population of a limited
+spin range, which might introduce additional assump-
+tions and uncertainties when extracting the shapes of the
+NLD and GSF.
+A novel technique, the Shape method [71], has recently
+been proposed to mend this problem. Applied to the pri-
+mary γ-transitions to several low-lying discrete states at
+consecutive excitation energy bins, it allows for an inde-
+pendent determination of the shape of the GSF. Thus,
+the shape of the strength and the interlinked slope of the
+NLD extracted with the Oslo method can be addition-
+ally constrained by the Shape method. However, as the
+latter is using data on direct decays to low-lying discrete
+states only, the PT fluctuations of the involved partial
+widths are expected to be significantly larger than for
+the Oslo-method GSF.
+In this work, the potential role of PT fluctuations in
+establishing the validity of the gBA hypothesis as well as
+the application of the Shape method are addressed for
+120Sn and 124Sn. Both the Oslo method and the Shape
+method have been applied to the same data sets. Exper-
+imental NLDs have been used to estimate fluctuations of
+the strengths for different specific initial and final excita-
+tion energies and compared with previous Oslo-method
+NLDs for even-even isotopes [4, 66, 67]. In Sec. II the
+details of the experimental procedure, the application of
+the Oslo method (Subsec. II A) and the Shape method
+(Subsec. II B) are presented. Section III focuses on the
+NLDs for 120,124Sn and the comparison with other exper-
+imental and theoretical results. In Sec. IV the procedure
+of estimating fluctuations of the strengths is presented
+together with the Shape method results, and the study
+of fluctuations and GSFs as functions of initial and fi-
+nal excitation energies. Finally, the main conclusions are
+summarised in Sec. V.
+II.
+DETAILS OF THE EXPERIMENT AND
+DATA ANALYSIS
+Experiments on both 120Sn and 124Sn were performed
+in February 2019 at the Oslo Cyclotron Laboratory
+(OCL). The isotopes were studied through the inelastic
+scattering reactions 120,124Sn(p, p′γ) with a proton beam
+of energy 16 MeV and intensity I ≈ 3 − 4 nA provided
+by the MC-35 Scanditronix cyclotron. Both targets used
+in the experiment were self-supporting with thicknesses
+and enrichments of 2.0 mg/cm2, 99.6% for 120Sn and 0.47
+mg/cm2, 95.3% for 124Sn, respectively. The 120Sn target
+was placed in the beam for approximately 24 hours, while
+the whole run on 124Sn lasted approximately 17 hours. A
+self-supporting 28Si target (natural Si, 92.2% 28Si) with
+thickness of 4 mg/cm2 was placed in the same proton
+beam for ≈ 1.5 hours at the end of the experiment for
+the energy calibration of the γ detectors.
+The experimental setup at the OCL comprises of
+the target chamber surrounded by 30 cylindrical large-
+volume LaBr3(Ce) detectors (Oslo SCintillator ARray,
+OSCAR for short) [41, 74], and 64 Si particle ∆E − E
+telescopes (SiRi) [30]. The LaBr3(Ce) scintillator detec-
+tors with �3.5′′×8′′ crystals were mounted on a trun-
+cated icosahedron frame with all front-ends fixed at a
+distance of 16.3 cm from the center of the target cham-
+ber, thus covering ≈ 57% of the total solid angle. The
+full-energy peak efficiency and energy resolution of OS-
+
+3
+2
+4
+6
+8
+10
+12
+14
+E (MeV)
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+ E (MeV)
+D
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+p
+d
+t
+Sn g.s.
+124
++
+Sn 2
+124
+Sn g.s.
+122
+FIG. 1.
+Experimental E-∆E spectrum measured for the
+124Sn isotope. The proton channel used for the data analysis
+is marked with the red solid line. The ground and first excited
+states of 124Sn in the proton channel and the ground state of
+122Sn in the triton channel, used for the calibration of the
+particle telescope, are marked with yellow circles.
+CAR have been measured to be ≈ 20% and ≈ 2.7%,
+respectively, at Eγ = 662 keV for the 137Cs calibration
+source placed at the same distance from the front-ends
+of the detectors.
+The (p, p′γ) reaction on 120,124Sn was one of the first
+in the series of experiments performed with OSCAR, in-
+stalled in 2018 at the OCL. As compared to the pre-
+viously used array CACTUS, consisting of 28 5′′×5′′
+NaI(Tl) detectors [28], OSCAR provides greatly im-
+proved timing and γ−energy resolution.
+All the scin-
+tillator crystals in the OSCAR array are coupled to
+Hamamatsu R10233-100 photomultiplier tubes with ac-
+tive voltage dividers (LABRVD) [60].
+In these experiments, the SiRi particle-telescope array
+was placed in backward angles with respect to the beam
+direction, covering a rather narrow range of angles from
+126◦ to 140◦ and making up ≈ 6% of the total solid-
+angle coverage. SiRi consists of eight trapezoidal-shaped
+∆E −E telescopes with a thick E-detector and a thinner
+∆E-detector with thicknesses of 1550 µm and 130 µm,
+respectively. Each of the eight ∆E detectors is segmented
+into eight curved pads, amounting to 2◦ of particle scat-
+tering angle per pad, yielding an angular resolution of
+2◦. For the 120,124Sn(p, p′γ) experiment, the full width
+at half maximum (FWHM) for SiRi was estimated to be
+≈ 100 − 120 keV from a Gaussian fit to the elastically
+scattered protons. All particle-γ coincidences in the ex-
+periment were recorded using XIA digital electronics [72].
+SiRi enables the exploitation of the ∆E − E technique
+to differentiate between the various observed reaction
+channels, as shown in Fig. 1.
+The elastic peak in the
+proton channel and the ground-state peak in the triton
+channel, combined with the known energy deposition in
+each of the 64 ∆E − E pads, were used to perform a
+linear calibration of the SiRi detectors for both targets.
+The kinematics of the reactions were used to convert the
+proton energies deposited in the SiRi detectors into the
+corresponding excitation energies of the target nucleus.
+As previously shown for �3.5′′×8′′ LaBr3(Ce) detec-
+tors coupled to the same type of photomultiplier and volt-
+age divider, the energy response of the detector remains
+rather linear up to ≈ 17 − 18 MeV [22]. However, to ac-
+count for minor non-linearity effects, a quadratic calibra-
+tion was applied to all 30 OSCAR detectors. Prominent
+γ transitions in 28Si ranging from 1.78 to 7.93 MeV were
+used for this purpose. Further, by applying graphical en-
+ergy (see Fig. 1) and timing cuts on the studied proton
+channel, putting gates on the prompt timing peak and
+subtracting background for particle and γ detection in
+SiRi and OSCAR, a so-called raw coincidence matrix was
+constructed for both studied nuclei. The raw matrices are
+shown in Figs. 2(a) and (d) for 120Sn and 124Sn, respec-
+tively. Consecutive diagonals indicate direct transitions
+to the ground and first excited states. For excitation en-
+ergies between 7 and 9 MeV, peaks that are due to minor
+12C and 16O contaminants in the targets are observed.
+At further stages of the analysis these peaks were re-
+moved1 to minimize the effect of these contaminants and
+any related artefacts on the final results. Approximately
+5.3×107 and 1.3×107 p-γ events in the excitation-energy
+range up to the neutron separation energy were collected
+for 120Sn and 124Sn, respectively.
+The γ spectra for each excitation-energy bin of the
+coincidence matrices were further unfolded according to
+the procedure outlined in [34], using the most recent re-
+sponse function of the OSCAR detectors [76] simulated
+with the Geant4 simulation tool [3, 5, 6]. This proce-
+dure has been applied to a large number of Oslo-type
+data published throughout the past two decades, and
+has been repeatedly shown to provide valuable results.
+A great advantage of the method is the preservation of
+statistical fluctuations of the raw coincidence spectrum
+into the unfolded one by using the so-called Compton
+subtraction method [34]. This technique strongly sup-
+presses additional, artificial fluctuations. The unfolded
+matrices for 120Sn and 124Sn are shown in Figs. 2(b) and
+(e).
+The main objective of the analysis is to extract the sta-
+tistical nuclear properties, namely the NLD and GSF, by
+exploiting their proportionality to the decay probability
+1 The contaminants were removed after unfolding of the γ spectra.
+A narrow graphical gate is put on each Gaussian-like contami-
+nant peak in the unfolded matrix, and the parts of the spectra
+within the gate are obtained by interpolating the neighbouring
+regions of the matrix.
+
+4
+0
+2
+4
+6
+8
+10
+ (MeV)
+γ
+E
+0
+2
+4
+6
+8
+10
+ (MeV)
+x
+E
+Sn
+120
+(a)
+n
+S
+Raw coincidence matrix
+0
+2
+4
+6
+8
+10
+ (MeV)
+γ
+E
+0
+2
+4
+6
+8
+10
+ (MeV)
+x
+E
+Sn
+120
+n
+S
+(b)
+Unfolded matrix
+0
+2
+4
+6
+8
+10
+ (MeV)
+γ
+E
+ (MeV)
+x
+E
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+Number of counts
+Sn
+120
+n
+S
+(c)
+1
+2
+3
+Primary matrix
+0
+2
+4
+6
+8
+10
+ (MeV)
+γ
+E
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+ (MeV)
+x
+E
+Sn
+124
+(d)
+n
+S
+0
+2
+4
+6
+8
+10
+ (MeV)
+γ
+E
+0
+2
+4
+6
+8
+10
+ (MeV)
+x
+E
+Sn
+124
+(e)
+n
+S
+0
+2
+4
+6
+8
+10
+ (MeV)
+γ
+E
+0
+2
+4
+6
+8
+10
+ (MeV)
+x
+E
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+Number of counts
+Sn
+124
+n
+S
+(f)
+1
+2
+3
+FIG. 2.
+Experimental raw p − γ coincidence ((a) and (d)), unfolded ((b) and (e)) and primary ((c) and (f)) matrices for
+120,124Sn obtained in the (p, p′γ) experiments. Yellow dashed lines indicate the neutron separation energies. Red and green
+dashed lines in panels (c) and (f) confine transitions to the ground (region 1) and the first excited Jπ = 2+ (region 2) states.
+Blue solid lines (region 3) indicate the areas of the primary matrices used further in the Oslo method. Bin sizes are 64 keV×64
+keV and 80 keV×80 keV for 120Sn and 124Sn, respectively. Blue arrows mark the sequence of the analysis steps.
+at each specific excitation energy and γ energy. Infor-
+mation regarding this decay probability can be obtained
+by isolating the first γ rays in a cascade emitted by the
+nucleus at a certain excitation energy, i.e.
+primary γ
+rays originating directly from the nucleus decaying from
+this excited state, or the so-called first-generation γ rays.
+The unfolded matrix contains all possible generations of
+γ rays emitted in every cascade from all excitation ener-
+gies up to the neutron separation energy. The γ-ray spec-
+tra for each excitation-energy bin in the unfolded matrix
+are expected to contain the same γ rays as in the lower-
+lying bins, in addition to the γ rays originating from the
+excited states confined by this energy bin. This fact is
+the key for the iterative subtraction technique, the so-
+called first-generation method, applied to both unfolded
+matrices for 120Sn and 124Sn. This technique relies on
+the assumption that γ decay is independent of whether
+states were populated directly in a reaction or via decays
+from higher-lying states. The details of the procedure are
+outlined in [33]. The primary matrices obtained after 23
+iterations for both nuclei are shown in Figs. 2(c) and (f).
+At this stage, two alternative methods can be used
+in order to extract the GSF from the primary matrix,
+namely the Oslo method and the Shape method. The
+former is a well-developed procedure primarily used to
+extract nuclear properties from the OCL data and it has
+been in use for more than two decades (see e.g. [47]).
+In addition to the GSF, it provides the simultaneous ex-
+traction of the NLD, which are the main characteristics of
+interest in this article. The latter procedure, the Shape
+method, has been recently presented and published in
+Ref. [71]. The two methods are expected to complement
+each other and a combined analysis yields an improved
+normalisation of the GSF and, therefore, the NLD. All
+details of these procedures applied to the 120,124Sn iso-
+topes are provided in the subsequent sections.
+A.
+Analysis with the Oslo method
+As already mentioned, the primary matrix is propor-
+tional to the decay probability from a set of initial excited
+states i within a chosen bin Ei to final states f confined
+within a bin Ef of the same size with γ rays of energy
+Eγ = Ei − Ef. The first step of the Oslo-type of anal-
+ysis is the decomposition of the primary matrix into the
+density of final states ρf and the γ-ray transmission co-
+efficient Ti→f:
+P(Eγ, Ei) ∝ Ti→f · ρf.
+(1)
+
+5
+Here, Ti→f, the transmission coefficient, is a function
+of γ-ray energy depending on both the initial and final
+state.
+The thorough derivations of this decomposition
+using Fermi’s golden rule and the Hauser–Feshbach the-
+ory of statistical reactions as starting points can be found
+in Refs. [52] and [53], correspondingly. This relation is
+expected to hold for relatively high excitation energies
+below the neutron threshold, corresponding to the com-
+pound states and their decay [47]. This energy range es-
+sentially coincides with the range of applicability of the
+first generation method.
+This form of dependence on Ei, Ef and Eγ, however,
+does not allow a simultaneous extraction of the transmis-
+sion coefficient and NLD. To enable such an extraction,
+the gBA hypothesis is adopted as one of the central as-
+sumptions in the Oslo method [10, 14]. As mentioned
+previously, the gBA hypothesis suggests an independence
+of the GSF (and, therefore, the transmission coefficient)
+of spins, parities and energies of initial and final states,
+leading to a dependence on γ-ray energy only. This signif-
+icantly simplifies the form of the relation given in Eq. (1):
+Ti→f → T (Eγ) and ρf = ρ(Ef) = ρ(Ei − Eγ).
+In earlier applications of the Oslo method, the gBA hy-
+pothesis has been found to be adequate for the relatively
+low-temperature regimes studied (T ≈ 0.7 − 1.5 MeV)
+[31]. However, as the Oslo method relies on the gBA hy-
+pothesis, it cannot be used alone to test its validity. To
+investigate the validity of the hypothesis, either compar-
+isons of independent experimental methods [50] or addi-
+tional tests suggested in, e.g., [16, 32] are required. This
+matter is of particular importance and will be discussed
+in more detail in Sec. IV.
+The next step of the Oslo method includes an itera-
+tive χ2-minimization procedure between the experimen-
+tal first-generation matrix P(Eγ, Ei) normalized to unity
+for each Ei bin and the theoretical Pth(Eγ, Ei) given by
+the following expression [63]:
+Pth(Eγ, Ei) =
+T (Eγ)ρ(Ei − Eγ)
+�Ei
+Eγ=Emin
+γ
+T (Eγ)ρ(Ei − Eγ)
+.
+(2)
+This χ2 fit of the transmission coefficient and NLD nor-
+mally gives a very good agreement with the experimental
+matrix P(Eγ, Ei) when applied to the statistical region
+of excitation energies.
+The step-by-step description of
+the minimization procedure is provided in Ref. [63]. To
+ensure the applicability of the statistical assumptions,
+minimum excitation energies of Emin
+i
+= 4.5 MeV for
+120Sn and 5.0 MeV for 124Sn were chosen.
+Sufficient
+statistics at higher energies allows us to set Emax
+i
+to the
+neutron separation energy for each isotope, Sn = 9.1
+and 8.5 MeV for 120Sn and 124Sn, respectively. To ex-
+clude regions where counts have been over-subtracted in
+the first-generation procedure, minimum γ-ray energies
+Emin
+γ
+= 1.3 and 1.6 MeV were set accordingly for 120Sn
+and 124Sn. The resulting areas where the Oslo method
+was applied in this work are marked by the blue lines in
+Figs. 2(c) and (f).
+The global χ2 fit yields only functional forms of the
+transmission coefficient T (Eγ) and NLD ρ(Ei − Eγ). It
+can be shown mathematically that one can construct an
+infinite set of T (Eγ) and ρ(Ei − Eγ) combinations cor-
+responding to the obtained fit and given by the forms
+[63]:
+˜ρ(Ei − Eγ) =Aρ(Ei − Eγ) exp(α(Ei − Eγ)),
+˜T (Eγ) =BT (Eγ) exp(αEγ),
+(3)
+where ρ and T are two fixed solutions, A and B are the
+scaling parameters, and α is the slope parameter shared
+by both the transmission coefficient and NLD. For each
+studied nucleus this ambiguity must be removed via de-
+termining unique normalization parameters A, B and α
+from external experimental data. If available, low-lying
+discrete states and neutron-resonance data are the main
+input parameters, combined with models for the spin dis-
+tribution and for extrapolations where there is a lack of
+experimental data.
+The first step of the normalization procedure is to de-
+termine the unique NLD solution ρ(Ei−Eγ). The param-
+eters A and α can be constrained by fitting the NLD to
+low-lying discrete states [1] in the excitation-energy range
+where the level scheme can be considered complete. At
+the neutron separation energy, the NLD can be normal-
+ized to the total NLD calculated from neutron-resonance
+spacings [55]. These data also provide the average, to-
+tal radiative width ⟨Γγ⟩ used to determine the scaling
+parameter B for the transmission coefficient. All details
+of the normalization procedure for 120Sn and 124Sn have
+been presented in the Supplemental Material of Ref. [50].
+However, some minor changes were introduced in this
+work to improve the normalization and the estimated un-
+certainties. We would like to stress that these changes do
+not affect the results presented in Ref. [50] in any signif-
+icant way, and do not undermine any of the presented
+conclusions. To avoid any confusion regarding the nor-
+malization parameters, we provide the updated and com-
+plete description of this procedure in the following.
+The most recent compilation of the discrete states [1]
+was used to anchor the NLD for 120,124Sn at low excita-
+tion energies. As compared to the compilation from 2003
+used in the previous analysis, some changes in the num-
+ber and the excitation energies of low-lying states appear
+and give a slightly different slope of the NLD. The anchor
+point at the neutron separation energy, ρ(Sn), is usually
+extracted from the neutron resonance spacing D0 for s-
+wave neutrons or D1 for p-wave neutrons. As 123Sn is an
+unstable target nucleus (T1/2 = 129.2 d [1]), no neutron
+resonance data are available, and we used other means
+to estimate ρ(Sn) and ⟨Γγ⟩ for 124Sn.
+The normalization procedure for
+120Sn is rather
+straightforward, in accordance with the steps outlined
+in Ref. [47], due to the available s-wave neutron capture
+data. The target spin of 119Sn is Iπ
+t = 1/2+, thus spins
+0+ and 1+ of the compound nucleus 120Sn are populated
+in s-wave capture. Assuming that both positive and neg-
+ative parities contribute equally to ρ(Sn), the average
+
+6
+5
+6
+7
+8
+9
+10
+11
+20
+40
+60
+80
+100
+120
+140
+160
+ (meV)
+〉 
+γ
+Γ
+〈
+Sn
+124
+Estimated value for 
+Even Sn isotopes, exp. value
+Odd Sn isotopes, exp. value
+ fit
+2
+χ
+Sn
+113
+Sn
+115
+Sn
+117
+Sn
+119
+Sn
+121
+Sn
+123
+Sn
+125
+Sn
+114
+Sn
+116
+Sn
+118
+Sn
+120
+Sn
+124
+(a)
+5
+6
+7
+8
+9
+10
+11
+ (MeV)
+n
+S
+4
+10
+5
+10
+6
+10
+)
+-1
+) (MeV
+n
+(S
+ρ
+Sn
+113
+Sn
+115
+Sn
+117
+Sn
+119
+Sn
+121
+Sn
+123
+Sn
+125
+Sn
+114
+Sn
+116
+Sn
+118
+Sn
+120
+Sn
+124
+(b)
+Sn
+124
+Estimated value for 
+Sn, shifted
+124
+Value for 
+Even isotopes, exp. value
+Even isotopes, shifted
+Odd isotopes, exp. value
+ fit
+2
+χ
+Sn
+120
+FIG. 3.
+(a) Experimental systematics for the average total
+radiative width for Sn isotopes. (b) Experimental systematics
+for the NLD at the neutron separation energy. The estimated
+values of ⟨Γγ⟩ and ρ(Sn) for 124Sn are marked with stars, the
+experimental ⟨Γγ⟩ values are taken from [55], the level den-
+sities are obtained from the D0 values given in [55]. Arrows
+mark ρ(Sn) values shifted by the neutron pair-gap values for
+the χ2 fit.
+s-wave neutron resonance spacing D0 can be written as
+[47]:
+1
+D0
+= 1
+2 [ρ(Sn, It + 1/2) + ρ(Sn, It − 1/2)] .
+(4)
+A transformation of the partial NLD for specific spins
+into the total NLD can be performed by adopting the
+back-shifted Fermi gas model (BSFG) for the NLD
+ρ(Ex, J) = ρ(Ex)·g(Ex, J) (Ex here stands for the excita-
+tion energy variable) with the spin distribution function
+given by [20, 23]:
+g(Ex, J) ≃ 2J + 1
+2σ2
+exp
+�
+−(J + 1/2)2
+2σ2
+�
+,
+(5)
+where σ is the spin-cutoff parameter. Given this distri-
+bution function, Eq. (5) can be rewritten for the total
+NLD at the neutron separation energy as a function of
+the experimental resonance spacing D0 (taken from Ref.
+[55]) and the target nucleus spin [47]:
+ρ(Sn) = 2σ2
+D0
+1
+(It + 1) exp
+�
+− (It+1)2
+2σ2
+�
++ It exp
+�
+− I2
+t
+2σ2
+�.
+(6)
+Note that the spin-cutoff parameter is an excitation-
+energy dependent function. The form of the spin-cutoff
+parameter at Sn of Ref. [68] was chosen for 120,124Sn
+σ2(Sn) = 0.0146A5/3 1 +
+�
+1 + 4a(Sn − E1)
+2a
+.
+(7)
+Here, a and E1 are the level-density parameter and the
+back-shift parameter for the BSFG model taken from
+Ref. [68].
+In the Oslo method, the measured level densities do
+not reach up to Ex = Sn due to the non-zero minimum
+γ-ray energy limit in the extraction of ρ(Ei−Eγ). To use
+the ρ(Sn) value as an anchor point for the normalization,
+the experimental Oslo data were extrapolated using the
+constant temperature (CT) level density model [21, 23,
+68]:
+ρCT (Ex) =
+1
+TCT
+exp
+�Ex − E0
+TCT
+�
+,
+(8)
+characterized by temperature (TCT ) and shift energy
+(E0) parameters. Earlier Oslo-method analyses exploited
+the BSFG model as an alternative for the interpolation
+procedure [47], however, the choice between these two al-
+ternatives is defined by the fit quality in each particular
+case (see Sec.III).
+As
+the
+experimental
+information
+on
+the
+s-wave
+neutron-resonance spacing is available for 120Sn, Eq. (6)
+was used directly to transform the D0 value into ρ(Sn).
+For 124Sn, this value was estimated from the systematics
+for even-even and even-odd Sn isotopes in the following
+way. The ρ(Sn) values were estimated for each Sn iso-
+tope with available neutron-resonance spacing D0 using
+Eq. (6). The resulting systematics for the ρ(Sn) values
+are shown in the lower panel of Fig. 3. The values of
+ρ(Sn) for even-even isotopes were shifted by the corre-
+sponding values of the neutron pairing gaps calculated
+from the AME 2003 mass evaluation [9] using Eq. (1) of
+Ref. [19]. Finally, the value of ρ(Sn) for 124Sn was cal-
+culated from a log-linear fit through the data points for
+even-odd and shifted even-even isotopes as shown by the
+red dashed line in Fig. 3(b).
+The second step after constraining the A and α param-
+eters for the NLD is to normalize the transmission coeffi-
+cient (and thus the GSF). As the slope α is already deter-
+mined by the NLD normalization, the scaling parameter
+B is the only parameter that remains to be estimated.
+The starting point for normalizing the γ-transmission co-
+efficient is the following relation [46]:
+⟨Γ(Ex, J, π)⟩ =
+1
+2πρ(Ex, J, π)
+�
+XL
+�
+Jf ,πf
+� Ex
+Eγ=0
+dEγ×
+× TXL(Eγ)ρ(Ex − Eγ, J, π),
+(9)
+
+7
+where ⟨Γ(Ex, J, π)⟩ is the average radiative width for
+states with spin J, parity π at excitation energy Ex, and
+X and L indicate the electromagnetic character and mul-
+tipolarity, respectively. The GSF, fXL(Eγ), is connected
+to the transmission coefficient by [2]:
+TXL(Eγ) = 2πE(2L+1)
+γ
+fXL(Eγ).
+(10)
+At high excitation energies, there is experimental ev-
+idence the dipole radiation is dominant (L = 1) (e.g.,
+Ref. [46]).
+The Oslo-type of experiments and analysis
+does not allow for distinguishing between different types
+of radiation, and, thus, the strength extracted with the
+Oslo method is presented by the total contribution of
+both electric and magnetic types of dipole transitions,
+E1 and M1.
+Insertion into Eq. (9) links the experimental dipole
+GSF f(Eγ) to the value of the total average radiative
+width ⟨Γγ⟩ obtained from s-wave neutron capture [55].
+For a target nucleus with ground state spin It and parity
+πt, Eq. (9) can be rewritten as
+⟨Γγ⟩ = ⟨Γ(Sn, It ± 1/2, πt)⟩ =
+1
+2ρ(Sn, It ± 1/2, πt)×
+×
+� Sn
+Eγ=0
+dEγE3
+γf(Eγ)ρ(Sn − Eγ)×
+×
+1
+�
+J=−1
+g(Sn − Eγ, It ± 1/2 + J).
+(11)
+Here, we adopt again the assumption on an equal contri-
+bution of states with positive and negative parities, and
+apply the spin distribution function of Eq. (5). It can be
+easily seen that the 1/ρ(Sn, It ± 1/2, πt) term equals the
+D0 value. For the spin-cutoff parameter dependence on
+the excitation energy, we follow the procedure outlined
+in Ref. [17]:
+σ2(Ex) = σ2
+d + Ex − Ed
+Sn − Ed
+[σ2(Sn) − σ2
+d],
+(12)
+where σd is estimated from the discrete lower-lying states
+at Ex ≈ Ed [1] (see Table I).
+In the case of 120Sn, the average total radiative width
+⟨Γγ⟩ was estimated as an average of three s-wave neu-
+tron resonances with energies in the range of ≈ 455-
+828 eV [55]. The remaining two resonances presented in
+[55] were excluded due to either being possibly of p-wave
+nature, or having a significantly lower value as compared
+to values for confirmed s-wave resonances found in the
+neighbouring Sn isotopes. In the case of 124Sn, we per-
+formed a linear fit through all values of ⟨Γγ⟩ available for
+other Sn isotopes as shown in Fig. 3(a) to estimate the
+⟨Γγ⟩ value for 124Sn.
+Ideally, the fit of the NLD to the low-lying discrete
+levels and the ρ(Sn) value are sufficient to constrain the
+slope parameter α for the GSF and NLD. However, the
+latter can be influenced by the range of experimentally
+0
+2
+4
+6
+8
+1
+−
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+)
+-1
+) (MeV
+x
+ (E
+ρ
+Level density 
+ Oslo data
+ Oslo data, red. 
+ Known descrete levels 
+) from neutron res. data 
+n
+(S
+ρ
+ 
+ CT model 
+Sn
+120
+(a)
+0
+2
+4
+6
+8
+ (MeV)
+x
+Excitation energy E
+1
+−
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+)
+-1
+) (MeV
+x
+ (E
+ρ
+Level density 
+ Oslo data 
+ Oslo data, red. 
+ Known descrete levels 
+) from syst. 
+n
+(S
+ρ
+ 
+)  from syst., red. 
+n
+(S
+ρ
+ 
+ CT model
+Sn
+124
+(b)
+FIG. 4.
+Experimental nuclear level densities for 120Sn (a) and
+124Sn (b). The NLDs at Sn are marked with crosses, discrete
+states are shown as shaded areas. For the 124Sn isotope both
+the total and reduced NLDs are shown. The first two vertical
+arrows at lower Ex energies on each figure constrain the lower
+excitation energy fit region, while the last two arrows at higher
+Ex energies mark the lower and upper limits for the higher
+excitation energy fit region.
+populated spins, which might be narrower than the in-
+trinsic2 spin distribution. This issue was previously dis-
+cussed in Refs. [73, 75]. An analysis of the observed tran-
+sitions in the unfolded matrices below Ei ≈ 4 − 5 MeV
+and their relative intensities can aid to reveal the popu-
+lated spins of the 120,124Sn nuclei populated in the (p, p′γ)
+reaction. However, this method has a large uncertainty
+in the determination of the exact maximum spin popu-
+lated in the reaction. Alternatively, one can make use
+of the new Shape method [71] to obtain the NLD slope
+that corresponds to the experimental spin range. This
+is of particular importance for 124Sn with no available
+neutron-resonance parameters.
+The application of the
+Shape method will be discussed in detail in Sec. II B.
+From the Shape method we obtained a reduction factor
+β for ρ(Sn), representing a certain fraction of the total
+2 All existing spins possible for a given nucleus at a given excitation
+energy.
+
+8
+TABLE I. Parameters used for the normalization of the nuclear LD and GSF for 120,124Sn.
+Nucleus
+Sn
+D0
+a
+E1
+Ed
+σd
+σ(Sn)
+ρ(Sn)
+T
+E0
+β
+⟨Γγ⟩
+(MeV)
+(eV)
+(MeV−1)
+(MeV)
+(MeV)
+(105 MeV −1)
+(MeV)
+(MeV)
+(meV)
+120Sn
+9.105
+95(14)
+13.92
+1.12
+2.53(4)
+3.4(5)
+5.82
+3.66(54)
+0.72+1
+−2
+0.19+9
+−4
+0.70
+121(25)b
+124Sn
+8.489
+–
+12.92
+1.03
+2.77(3)
+3.3(5)
+6.00
+1.38(30)a
+0.75+2
+−2
+-0.11+11
+−6
+0.20
+82(19)a
+a From systematics.
+b Modified with respect to the value pubished in Ref. [55].
+spin distribution from Eq. (5), corresponding to the re-
+duced spin range from J = 0 to a certain maximum spin.
+This was done by requesting optimally matching slopes
+of the Oslo method and the Shape method GSFs above
+Eγ ≈ 5 MeV. A rather strong reduction of the level den-
+sity in 124Sn at the neutron separation energy might re-
+flect some maximum limit of the experimental spin range.
+However, it is important to note that using experimental
+systematics of the ρ(Sn) and ⟨Γγ⟩ might have large un-
+certainties. In the case of 124Sn, it is quite probable that
+such a large reduction factor is needed due to, e.g., an
+overestimated ρ(Sn) from the χ2 fit of the systematics.
+The simultaneous use of the Oslo and Shape methods
+can therefore significantly reduce systematic uncertain-
+ties for the slopes of extracted strengths and level densi-
+ties. All parameters used in the normalization procedure
+for 120,124Sn are listed in Table I. The resulting NLDs
+for 120Sn and 124Sn with their estimated error bands are
+shown in Fig. 4.
+We note that the errors in Table I and the resulting
+error bands for the NLD and the GSF presented in sec-
+tions III and IV combine statistical and systematic com-
+ponents.
+The latter includes uncertainties introduced
+by the unfolding and the first-generation procedures for
+both 120,124Sn isotopes. These are propagated through
+the Oslo method according to the procedure outlined
+in Ref. [63].
+In addition, systematic uncertainties due
+to the normalization parameters are included. For the
+120Sn isotope, the experimental uncertainty (1 standard
+deviation) of the D0 value was propagated to estimate
+the error for the NLD at the neutron separation energy.
+The experimental uncertainties of the radiative widths
+in 120Sn [55] were used to estimate the error of the aver-
+age, total radiative width ⟨Γγ⟩, contributing to the un-
+certainty of the scaling factor B. In the case of the 124Sn
+isotope, the errors of the ρ(Sn) and ⟨Γγ⟩ were calculated
+from the uncertainties of the χ2 fit parameters and prop-
+agated into the total uncertainties of the NLD and GSF.
+In the previously published result on 124Sn [50], a 50%
+uncertainty of ρ(Sn) was assumed to account for presum-
+ably underestimated errors from the χ2 fit. However, the
+excellent agreement within the estimated error bands of
+the slopes of the GSFs obtained with the Oslo and Shape
+method allows us to apply a more modest error band as
+presented in this work. All errors of the normalization
+parameters described above are summarised in Table I.
+B.
+Analysis with the Shape method
+Quite often, nuclei with no available neutron resonance
+data and/or a restricted experimental spin range are en-
+countered. One possible way to overcome this is the use
+of isotopic systematics comprising of nuclei with stable
+neighbouring A − 1 isotopes as applied in the present
+case for 124Sn.
+However, this is often not possible in
+other isotopic chains due to the lack of data (e.g. 127Sb
+[57]).
+Moreover, the question on whether systematics
+from neighbouring isotopic chains can be used for a given
+nucleus, and to what extent one can rely on these sys-
+tematics, is still open. Hence, an alternative way to con-
+strain the normalization parameters is required.
+The
+novel Shape method [71] provides a way to determine
+the slope parameter α for the NLD and the GSF without
+making use of neutron resonance data.
+The starting point for the method is extracting ex-
+perimental intensities of first generation γ transitions to
+specific final states with spins and parities Jπ at final
+excitation energies Ef, represented by diagonals in the
+primary matrix. The intensities (related to the branch-
+ing ratios) of these γ transitions are proportional to the
+number of counts ND in the diagonals.
+The selection
+of which diagonals are to be used depends on a partic-
+ular nucleus, the spacing between the final states, and
+whether the resolution is sufficient to distinguish between
+different diagonals. The main concept behind the Shape
+method is that the intensities of the γ transitions are
+proportional to the partial widths and hence to the GSF.
+By taking intensities of transitions in successive excita-
+tion energy bins, the functional form of the GSF can be
+obtained.
+In the case of 120,124Sn, the only two diagonals clearly
+seen in the primary matrices are the ground state di-
+agonal D1 and the diagonal corresponding to the first
+excited state D2 (marked accordingly as regions 1 and 2
+in Fig. 2(c) and (f)). For given initial excitation-energy
+bins Ei (horizontal line) they define the direct decay to
+the final excitation energy Ef at the ground state with
+Jπ = 0+ and the first excited state with Jπ = 2+ with
+γ-ray energies Eγ = Ei − Ef.
+The Shape method adopts the same form of the spin
+distribution, given by Eq. (5), as used in the Oslo
+method, and assumes γ transitions to be of predomi-
+nantly dipole nature (this has been confirmed by mea-
+suring angular distributions). According to Eq. (13) in
+
+9
+Ref. [71], the number of counts in a chosen diagonal
+ND corresponding to the final energy Ef is proportional
+to the population cross-section of initial states Ei with
+Ji = Jf −1, Jf, Jf +1, spin distribution function g(Ei, Ji)
+and the partial γ-decay width. For the case of 120,124Sn
+with the ground and first excited state diagonals D1 and
+D2, the following relations can be written
+f(Eγ1) ∝
+ND1
+E3
+γ1 · g(Ei, 1)
+f(Eγ2) ∝
+ND2
+E3
+γ2 · [g(Ei, 1) + g(Ei, 2) + g(Ei, 3)].
+(13)
+By varying Ei, one obtains corresponding pairs of values
+f(Eγ = Ei) and f(Eγ = Ei − Ex(2+)). As Eqs. (13)
+only give the proportionality with the GSF, these pairs
+are not normalized in absolute value.
+Firstly, the consecutive pairs of values are normalized
+internally, as shown and described in Fig. 2 of Ref. [71],
+to reconstruct the functional shape of the GSF. Thus,
+one can extract two GSFs, corresponding to decays to
+the ground state and decays to the first excited state.
+Secondly, the general shape of both GSF must be scaled
+to match any available strength below the neutron sepa-
+ration energy, i.e. normalizing to external experimental
+data. This is the main limitation of the method, as it
+provides only a slope, or a shape of the strength, but not
+the absolute GSF, and therefore requires some additional
+experimental information. For the 120,124Sn isotopes, the
+GSFs extracted from relativistic Coulomb excitation in
+forward-angle inelastic proton scattering below the neu-
+tron separation energy [12] were used to scale the GSF
+points obtained for both diagonals separately [50].
+The upper excitation energy limit for the application
+of the Shape method can, in principle, be extended to
+Sn, whilst the definition of the lower limit is rather ar-
+bitrary.
+The applicability of Eqs. (13) is restricted to
+the statistical excitation energy region where the spin
+distribution function g(Ex, J) can be trusted. There is
+no clear criterion for the minimum level density which
+can be considered high enough to assume this is fulfilled.
+In this work, we require that the level density must be
+at least 10 levels per excitation energy bin for the spin
+distribution g(Ex, J) to be applied.
+III.
+NUCLEAR LEVEL DENSITIES
+The experimental NLDs of 120,124Sn displayed in Fig. 4
+follow nicely the discrete low-lying states up to ≈ 3 MeV
+for 120Sn and ≈ 2.7 MeV for 124Sn.
+At higher ener-
+gies, the NLDs increase rapidly and reach an exponen-
+tial, constant-temperature behavior. This suggests that
+the level schemes used for the normalization of the NLDs
+can be considered complete up to ≈ 3 and 2.7 MeV for
+120Sn and 124Sn, respectively. The energy resolution is
+sufficient to distinguish the ground state and the first ex-
+cited states, presented by two bumps at 0 and ≈ 1.1−1.2
+MeV for both nuclei. The presence of the data points
+between the ground and first excited states can be ex-
+plained by the finite excitation energy resolution of order
+100 keV and presence of the leftover counts between the
+diagonals in the primary matrices after the background
+subtraction procedure. At higher excitation energies, the
+experimental points are following the CT model predic-
+tion, starting from ≈ 4 MeV. The normalization of the
+NLDs was found to be rather insensitive to the exact
+choice of the two upper normalization limits (the two ar-
+rows at higher excitation energies in Fig. 4), due to the
+smooth behaviour of the NLDs at higher excitation en-
+ergies.
+In
+Fig.
+5
+we
+show
+a
+comparison
+of
+the
+total
+NLDs for Sn isotopes extracted with the Oslo method,
+including
+the
+present
+results
+for
+120,124Sn.
+The
+116,117,118,119,121,122Sn isotopes were previously studied
+with a 38-MeV beam of 3He using the (3He, αγ) and
+(3He, 3He γ) reaction channels and reported in Refs. [4,
+66, 67]. The slopes of the NLDs for 120,124Sn are quite
+similar to each other (T = 0.72 and 0.75 MeV, see Ta-
+ble I) and those of other even-mass isotopes. All NLDs of
+even-mass nuclei agree quite well within the estimated er-
+ror bands below the neutron separation energy. However,
+it is important to note some differences in the normaliza-
+tion procedures in the newest analysis of 120,124Sn and
+the older analyses of even-mass isotopes. Firstly, all pre-
+vious analyses exploited the BSFG for the extrapolation
+of the highest experimental NLD points to the ρ(Sn) val-
+ues instead of the CT model. As was previously shown in
+Ref. [27] and confirmed for 120,124Sn, the CT model re-
+sults in a better χ2 fit value. For example, between ≈ 4.8
+and 6.8 MeV in 124Sn, the reduced χ2 value is a factor
+of 4 smaller for the CT model than for the BSFG model.
+This factor becomes larger and might exceed 10 if lower
+excitation energy points above ≈ 3 MeV are included.
+Secondly, the different form of the spin-cutoff parame-
+ter taken from Ref. [23] was used in the older analyses.
+The immediate consequence of this choice is slightly less
+steep slopes of the NLDs if the CT extrapolation is used.
+However, in combination with the BSFG extrapolation
+model, the resulting slopes of the NLDs in 116,118,122Sn
+are expected to be close to those obtained for 120,124Sn,
+as can also be observed in Fig. 5.
+In general, the NLDs of odd-mass Sn isotopes are by
+a factor of 7-8 higher than for the even-mass isotopes,
+primarily due to the unpaired valence neutron [29]. As
+compared to other even-mass isotopes, 120,124Sn demon-
+strate essentially the same features, such as the well-
+defined bumps at the ground and the first excited state
+and a step-like structure right below 3 MeV excitation
+energy.
+Earlier studies exploiting microscopic calcula-
+tions based on the seniority model link the latter fea-
+ture to breaking of consecutive nucleon Cooper pairs [62].
+Due to the closed proton shell, Z = 50, the break-
+ing of proton Cooper pairs is suppressed until higher
+excitation energies are reached.
+Thus, these step-like
+structures are likely to be correlated with the break-
+
+10
+0
+2
+4
+6
+8
+ (MeV)
+x
+Excitation energy E
+1
+−
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+)
+-1
+) (MeV
+x
+ (E
+ρ
+Level density 
+Sn, Agvaanluvsan et al. (2009) 
+116
+ 
+Sn, Agvaanluvsan et al. (2009) 
+117
+ 
+Sn, Toft et al. (2010) 
+118
+ 
+Sn, Toft et al. (2010) 
+119
+ 
+Sn, present work
+120
+ 
+Sn, Toft et al. (2011) 
+121
+ 
+Sn, Toft et al. (2011) 
+122
+ 
+Sn, present work 
+124
+ 
+FIG. 5.
+Experimental total nuclear level densities for 116Sn
+[4], 117Sn [4], 118Sn [66], 118Sn [66], 120Sn, 121Sn [67], 122Sn
+[67], 124Sn.
+ing of neutron pairs at energies exceeding 2∆n = 2.6
+and 2.5 MeV [19] for 120Sn and 124Sn, respectively. For
+higher excitation energies, where a continuous “melting”
+of Cooper pairs sets in, the NLDs follow a smooth trend
+with no distinctive structures, as previously observed for
+116,118,122Sn [4, 66, 67].
+The inelastic proton scattering data [12], used for the
+absolute normalization of the Shape method GSFs, can
+also provide information on the partial NLD. The NLD
+of 1− states in 124Sn was extracted for the excitation-
+energy range ≈ 4.5 − 14.5 MeV by means of the fluctu-
+ation analysis [36], applying procedures analog to those
+used in Refs. [13, 58]. All details of the extraction proce-
+dure can be found in Ref. [11]. To compare with the Oslo
+data, we apply the spin distribution function in Eq. (5)
+to the total NLD of 124Sn to reduce it to the density
+of J = 1 levels for excitation energies above ≈ 3.2 MeV,
+where this function can be assumed to be applicable. Fur-
+ther, applying the assumption on equal contribution of
+positive- and negative-parity states [45, 49], the density
+of J = 1− states was obtained. In contrast to the previ-
+ously published results on 96Mo [51] and 208Pb [13], there
+is in fact a region of overlap between the two data sets, as
+shown in Fig. 6. The Oslo data, as well as the CT model
+used in the normalization procedure (blue dashed line),
+lie within, but closer to the lower edge of the error band
+for the inelastic proton scattering data up to ≈ 10.5 MeV.
+This provides support of the spin-cutoff model adopted
+in the Oslo-method normalization. A model predicting a
+higher spin-cutoff value than presented in Table I would
+imply a wider spin distribution and, therefore, a signifi-
+cantly lower fraction of J = 1 states leading to a larger
+2
+4
+6
+8
+10
+12
+14
+16
+ (MeV)
+x
+Excitation energy E
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+)
+-1
+) (MeV
+x
+ (E
+ρ
+Level density 
+ Oslo data
+ (p,p') data
+) from syst. 
+n
+(S
+ρ
+ 
+ Composite formula, G&C (1965)
+ BSFG model fit
+ CT model fit
+ Goriely et al. (2008)
+ Hilaire et al. (2013)
+FIG. 6.
+Experimental nuclear level densities for 1± states
+for 124Sn obtained with the Oslo method (blue data points)
+and the (p, p′) data [12] (orange data points).
+The pre-
+diction of the CT model used for the normalization of the
+Oslo method data is shown by the dashed blue line.
+A
+fit with the BSFG through all data and with the compos-
+ite formula [23] are shown by the dashed magenta and solid
+cyan lines.
+Predictions of the microscopic Hartree-Fock-
+Bogoliubov+combinatorial method [26] and Hartree-Fock-
+Bogolyubov+Gogny force calculations [39] are marked by the
+dashed light and dark grey lines respectively.
+discrepancy between the Oslo and the (p, p′) data in the
+overlapping area. Thus, we can conclude that the spin-
+cutoff estimate provided by Eq. (7) is reasonable, and
+probably lies closer to the upper limit in the range of
+acceptable spin-cutoff values that would make the two
+experimental NLDs agree with each other.
+The constant temperature regime, characterized by the
+pair-breaking process, continues at least up to the neu-
+tron separation energy or higher, where the temperature
+begins to rise and the Fermi gas behaviour of nucleons
+sets in.
+As shown in Fig. 6, the CT model begins to
+deviate quite drastically from the (p, p′) data at higher
+excitation energies, well above the Sn value.
+For this
+reason, the BSFG model is expected to provide a more
+accurate description of the NLD at high excitation en-
+ergies, although it is not an appropriate model at lower
+excitation energies. The global fit of all data with the
+BSFG model only indeed fails to reproduce the regime of
+increasing nuclear temperature between ≈ 6.5−14 MeV,
+especially in the vicinity of the neutron separation energy
+and slightly above. The composite NLD formula, intro-
+duced by Gilbert and Cameron in Ref. [23] (denoted as
+G&C), combines the CT model at lower excitation ener-
+gies and the BSFG model at higher energies, and appears
+to be more suitable for the simultaneous description of
+
+11
+the Oslo and (p, p′) data. From the result of the fit with
+the composite NLD formula, the constant temperature
+regime holds up to ≈ 8.5 MeV, i.e. in the vicinity of the
+neutron separation energy. Even though this formula re-
+produces the general trend and performs better than the
+BSFG, it is still not able to completely describe the NLD
+above the neutron separation limit.
+Microscopic spin- and parity-dependent NLD calcula-
+tions based on the Hartree-Fock-Bogoliubov plus com-
+binatorial method [26] deviate from both the Oslo and
+the (p, p′) data throughout the whole energy range
+(from 3.2 to 14 MeV), being higher by a factor of ≈4-
+5 on average.
+On the other hand, the NLD calcu-
+lated within the temperature-dependent Hartree-Fock-
+Bogolyubov approach with the Gogny force [39] follows
+the (p, p′) data and the composite formula prediction
+nicely from ≈ 6.5 MeV excitation energy and above,
+while still being about a factor of 3 higher than the Oslo-
+method NLD. For the case of the total NLD, this devia-
+tion might reach up to two orders of magnitude. We con-
+clude that although microscopic models are appealing, as
+they should in principle grasp the underlying physics in
+contrast to simple analytical formulae, they are at this
+point not able to describe experimental data well enough
+over a wide excitation-energy range.
+IV.
+PORTER-THOMAS FLUCTUATIONS AND
+γ−RAY STRENGTH FUNCTIONS
+The experimental GSFs extracted with the Oslo
+method result from averaging γ transitions over relatively
+wide excitation-energy windows, ≈ 4.6 for 120Sn and 3.5
+MeV for 124Sn (region 3 in Fig. 2(c) and (f)). Therefore,
+any variations of the strength due to PT fluctuations are
+expected to be strongly suppressed, lying well within the
+estimated error bands. As such, PT fluctuations play a
+minor role and have little influence on the overall shapes
+of the GSFs. However, to test the gBA hypothesis, it is
+necessary to investigate how the GSF varies as a function
+of excitation energy (and also, in principle, spin and par-
+ity of the initial and final states). Then, a complication
+arises, because the action of narrowing down the aver-
+aging interval to study the GSF for different specific ini-
+tial and final excitation energies will inevitably introduce
+larger uncertainties due to increased PT fluctuations of
+the partial radiative widths.
+Oslo-method data have previously been used to study
+the shapes of the GSFs as functions of initial and final
+excitation energies to address the question on the validity
+of the gBA hypothesis [16, 31, 32, 35]. With the excep-
+tion of Ref. [16], which presents a detailed discussion and
+estimates of the PT fluctuations for the case of 64,65Ni,
+the role of these fluctuations are approached mostly in a
+qualitative way. Due to the particularly high density of
+initial and accessible final states in 238Np, studied in Ref.
+[32], reaching up to ≈ 4.3·106 states at Sn = 5.488 MeV,
+the PT fluctuations are expected to be negligible for the
+comparison of individual GSFs for different individual
+initial and final excitation energies with the Oslo-method
+strength.
+An excellent agreement of all strengths was
+found, and this indeed serves as a strong argument for
+the validity of the gBA hypothesis [32]. Such a compari-
+son, however, is much more difficult in the case of lighter
+nuclei such as 46Ti [31], 64,65Ni [16], and 92Zr [71]. For
+example, the density of levels at Sn = 9.658 MeV in 64Ni
+is only ≈ 2.6·103 MeV−1, and variations on the strengths
+for specific excitation energies might reach some tens of
+percent of the absolute value [16]. In this regard, the nu-
+clei studied in this work present an intermediate case be-
+tween the heavy 238Np and relatively light 64,65Ni nuclei,
+with the total NLDs of ≈ 2.5 · 105 MeV−1 at Sn = 9.104
+MeV for 120Sn and ≈ 8.8·104 MeV−1 at Sn = 8.489 MeV
+for 124Sn.
+To study the variation in the GSFs of 120,124Sn, we
+follow the procedure outlined in Refs. [16, 35], assuming
+that the fluctuations of the GSF follow a χ2
+ν distribution
+with the number of degrees of freedom corresponding to
+the number of γ-ray transitions n(Eγ) at a given tran-
+sition energy Eγ. Relative fluctuations of the GSF are
+given by the ratio between the deviation σP T and average
+µ, or r = σP T /µ =
+�
+2/ν, of the χ2
+ν distribution [59].
+The number of transitions (i.e., the number of partial
+widths, or primary transitions) n can be calculated for
+each Eγ for specific initial and final excitation energies,
+allowing to study how the fluctuations evolve with γ-ray
+and excitation energy. We adopt the following relation
+from Refs. [16, 35] to estimate the number of transitions
+n(Eγ, Ei):
+n(Eγ, Ei) =∆E2 �
+Jπ
+1
+�
+L=−1
+�
+π′
+ρ(Ei, J, π)×
+× ρ(Ei − Eγ, J + L, π′),
+(14)
+where we consider dipole transitions only, and ∆E is the
+excitation energy bin width. By substituting Ei with Ef
+and Ei − Eγ with Ef + Eγ, it is also possible to obtain
+the number of transitions as a function of Eγ and final
+excitation energy.
+We limit ourselves to two types of cases in estimating
+the GSF fluctuations. Firstly, we study the case when the
+initial Ei and final Ef excitation energies both lie within
+the quasi-continuum region, for which the spin distribu-
+tion of Eq. (5) is considered applicable. This allows to
+apply this distribution to account for the spin dependence
+of the NLDs in Eq. (14). Further, it is assumed again an
+equal contribution of positive- and negative-parity states
+within the quasi-continuum.
+We also require a mini-
+mum level density of 10 levels per bin, corresponding
+to Ef ≈ 3.2 MeV in 120Sn and Ef ≈ 3.0 MeV in 124Sn.
+Note that this is a rather crude estimate that should be
+taken with some caution. However, since we want to ob-
+tain an approximate magnitude of the fluctuations, small
+deviations from the spin distribution formula are not ex-
+pected to impact the results. Secondly, we consider ini-
+tial excited states within the quasi-continuum and final
+
+12
+0
+1
+2
+3
+4
+5
+6
+ (MeV)
+γ
+E
+4
+−
+10
+3
+−
+10
+2
+−
+10
+ )
+γ
+r(E
+ = 9.09 MeV
+i
+ E
+ = 8.19 MeV
+i
+ E
+ = 7.17 MeV
+i
+ E
+ = 6.14 MeV
+i
+ E
+ = 5.12 MeV
+i
+ E
+Sn
+120
+(a)
+0
+1
+2
+3
+4
+5
+6
+ (MeV)
+γ
+E
+4
+−
+10
+3
+−
+10
+2
+−
+10
+ )
+γ
+r(E
+ = 8.48 MeV
+i
+ E
+ = 7.36 MeV
+i
+ E
+ = 6.40 MeV
+i
+ E
+ = 5.44 MeV
+i
+ E
+Sn
+124
+(b)
+FIG. 7.
+Relative fluctuations of the GSF r(Eγ, Ei) for dif-
+ferent initial excitation energies for (a) 120Sn and (b) 124Sn.
+All initial Ei and final energies Ei − Eγ lie within the quasi-
+continuum region. The excitation and γ-ray energy bins are
+128 keV for 120Sn and 160 keV for 124Sn.
+states with known parities and spins within the discrete
+region. Here, the level density at the final excitation en-
+ergy can be calculated directly using the known states
+from Ref. [1].
+Figure
+7
+shows
+the
+relative
+GSF
+fluctuations
+r(Eγ, Ei) =
+�
+2/n(Eγ, Ei) as functions of Eγ for transi-
+tions from different initial excitation-energy bins within
+the quasi-continuum for 120Sn and 124Sn. The data are
+shown for Ef ≥ 3.2 MeV for 120Sn and Ef ≥ 3.0 MeV
+for 124Sn, so that the final excitation energies of the in-
+cluded transitions lie within the quasi-continuum. The
+experimental level densities were used for the calculation.
+Similar to the results for 64,65Ni [16], the fluctuations in-
+crease exponentially with γ-ray energy for a given Ei, as
+well as from the lowest to the highest initial excitation
+energy at a given Eγ. This behaviour can easily be ex-
+plained by the decreasing number of possible transitions
+for consecutively lower excitation energies, given the ex-
+ponentially decreasing density of accessible levels.
+The magnitudes of the fluctuations in both nuclei are
+quite similar due to the similar values of the total NLDs,
+and all minor differences stem primarily from a slight
+difference in the bin width. At the neutron separation
+0
+2
+4
+6
+8
+ (MeV)
+γ
+E
+3
+−
+10
+2
+−
+10
+1
+−
+10
+1
+10
+2
+10
+ )
+γ
+r(E
+ g.s.)
++
+ = 0 MeV (0
+f
+ E
+)
++
+ = 1.17 MeV (2
+f
+ E
+)
+-
+, 7
++
+, 4
++
+, 2
+-
+ = 2.43 MeV (3
+f
+ E
+ = 3.33 MeV
+f
+ E
+ = 4.35 MeV
+f
+ E
+ = 5.38 MeV
+f
+ E
+ = 6.40 MeV
+f
+ E
+Sn
+120
+(a)
+0
+2
+4
+6
+8
+ (MeV)
+γ
+E
+3
+−
+10
+2
+−
+10
+1
+−
+10
+1
+10
+2
+10
+ )
+γ
+r(E
+ g.s.)
++
+ = 0 MeV (0
+f
+ E
+)
++
+ = 1.13 MeV (2
+f
+ E
+)
++
+, 4
+-
+, 5
++
+ = 2.24 MeV (0
+f
+ E
+ = 3.20 MeV
+f
+ E
+ = 4.16 MeV
+f
+ E
+ = 5.12 MeV
+f
+ E
+ = 6.08 MeV
+f
+ E
+Sn
+124
+(b)
+FIG. 8.
+Relative fluctuations of the GSF r(Eγ, Ef) for dif-
+ferent final excitation energies for (a) 120Sn and (b) 124Sn.
+All initial energies Ei − Eγ lie within the quasi-continuum
+region. The same applies to the different final energies Ef
+represented by blue lines. The red dashed line corresponds to
+the ground state as the final state, the green one corresponds
+to the first excited 2+ state as the final state, and the yellow
+one corresponds to several discrete final low-lying states. The
+excitation and γ-ray energy bins are 128 keV for 120Sn and
+160 keV for 124Sn.
+energy, fluctuations in both nuclei range from ≈ 10−4 to
+4 − 5 · 10−3 %, while for the lower excitation energy they
+reach up to ≈ 3 − 6%. Fluctuations of these orders of
+magnitude are indeed expected for the relatively heavy
+120,124Sn nuclei. For example, based on the NLD of 64Ni
+[16] and 120Sn, the number of transitions at Ei ≈ 7.7
+MeV at Eγ ≈ 2.3 MeV in 120Sn is roughly by a factor
+of 1000 larger than in 64Ni, which indeed yields larger
+fluctuations in 64Ni by approximately a factor of 30.
+The relative GSF fluctuations calculated from the tran-
+sitions to specific final excitation energies demonstrate an
+opposite trend, exponentially decreasing with γ-ray en-
+ergies, as shown in Fig. 8. These trends are displayed
+with an approximately equal spacing for several final ex-
+citation energy bins within the quasi-continuum, as well
+as the bins containing the ground state, the first excited
+state, and several known low-lying excited states. In con-
+trast to the lowest initial excitation energies, fluctuations
+at final excitation energies below Ef ≈ 3 MeV reach up
+
+13
+to tens of percent and might become a considerable con-
+tribution to the total uncertainty of the GSF.
+The estimates of the PT fluctuations can be fur-
+ther put into the context of testing the gBA hypothesis
+for 120,124Sn.
+By analogy with the 238Np results from
+Ref. [32], the experimental data obtained for 120,124Sn
+can be readily used to test whether the transmission co-
+efficients, and, therefore, the GSFs, are dependent on the
+initial and final excitation energies. Equation (1) can be
+rewritten in the form [32]:
+P(Eγ, Ei)N(Ei) = T (Eγ) · ρ(Ei − Eγ),
+(15)
+where we introduce an additional energy-dependent fac-
+tor N(Ei) given by:
+N(Ei) =
+� Ei
+0
+T (Eγ) · ρ(Ei − Eγ)dEγ
+� Ei
+0
+P(Eγ, Ei)dEγ
+.
+(16)
+Here, we make use of the transmission coefficient ex-
+tracted from the Oslo method, and hence averaged over
+a wide range of excitation energies. We can deduce the
+transmission coefficient as a function of excitation energy
+and γ energy through
+T (Eγ, Ei) = P(Eγ, Ei)N(Ei)
+ρ(Ei − Eγ)
+.
+(17)
+A similar relation can be obtained for the final excitation
+energy by substituting Ei with Ef + Eγ.
+The GSFs for several initial excitation energies in the
+case of 120Sn were previously published in Ref. [50], where
+they were compared with the strength extracted with the
+Oslo method. In this work, we present the comparison
+of the individual GSFs for different initial and final ex-
+citation energy bins for both 120Sn and 124Sn with the
+corresponding Oslo-method results. Individual strengths
+are shown together with the error band due to the sta-
+tistical uncertainty propagated through the method, de-
+noted by statistical for short. As the Oslo-method GSF
+is an averaged strength with heavily suppressed PT fluc-
+tuations, it is shown with the total error band as well
+as additional error bars, denoting the expected PT fluc-
+tuations, or rather expected deviations of the individual
+strengths due to PT fluctuations. The latter is essential
+to assess whether there is an agreement or not between
+the strengths extracted for various excitation-energy bins
+and the Oslo-method strengths.
+The results for 120Sn at four initial excitation energies
+are shown in the upper row of Fig. 9.
+The dark grey
+shaded areas indicate regions of potential infinite fluctu-
+ations due to the expected zero values of the NLD at the
+final excitation energies in the energy gaps between the
+first few discrete states. As can be seen from Fig. 4, the
+experimental NLD has small non-zero values between the
+ground state and the first and second excited states at
+≈ 1.171 and 1.875 MeV due to the experimental resolu-
+tion and the presence of some residual counts in the raw
+matrix after the background subtraction. The analysis
+applied to each individual excitation energy Ei gener-
+ates a continuous data set for the GSF from the high-
+est possible gamma-ray energy at Eγ = Ei downward
+to gamma-ray energies below 2 MeV shown for 120Sn in
+Fig. 9(a)-(d). The GSF values in the dark grey region at
+higher gamma energies belong to hypothetical primary
+gamma-ray transitions in the energy range between the
+ground state and 1.171 MeV, while the dark grey region
+at lower energies belongs to decays into the energy range
+from 1.171 MeV to 1.875 MeV. However, it should be
+mentioned that direct gamma decays to those final exci-
+tation energy regions are physically not possible and that
+the corresponding data points are artifacts of the contin-
+uous analysis. It is, however, interesting to observe that
+the PT fluctuation analysis reveals those regions by un-
+usually large PT fluctuations
+In case of fixed initial excitation energies, light grey
+shaded areas correspond to energy bins where the fluc-
+tuations can not be estimated either due to to Eγ > Ei
+or unambiguous spins of some final excited states. In the
+latter case it is no longer possible to define what spins of
+initial states within the quasi-continuum yielding dipole
+transitions must be included to the sum in Eq. 14 . For
+the rest of the experimental points, the fluctuations were
+estimated and shown in Fig. 9 as vertical error bars. The
+values of these errors exceed or are of the same magni-
+tude as the statistical uncertainties for high Eγ for all
+of the presented cases. For the highest initial excitation
+energies in Fig. 9, Ei = 7.74 and 7.10 MeV, they become
+increasingly suppressed as compared to the statistical er-
+rors, by roughly a factor of 10 at Eγ ≈ 4.5 MeV, gradu-
+ally increasing to ≈ 102 toward Eγ ≈ 1 MeV. For lower
+initial excitation energies, this factors are of order 1 and
+10. Except for the strong deviations in the areas with ex-
+pected large fluctuations (dark grey areas), all strengths
+are in fairly good agreement with the Oslo-method result
+within its error band.
+Similar results with an excitation energy bin width of
+160 keV are shown for 124Sn in the upper row of Fig. 10.
+Since the range of populated spins might be limited in
+this case, using the total NLD provides a lower estimate
+of the PT fluctuations, and they might be slightly larger
+than shown in the figure.
+By analogy with the case
+of 120Sn, the GSFs for different initial excitation ener-
+gies are in rather good agreement with the Oslo-method
+strength within the shown error bands and areas of ex-
+pected finite PT fluctuations. These results for both the
+120,124Sn isotopes bring further support to the GSF be-
+ing independent on the initial excitation energy, in ac-
+cordance with the gBA hypothesis.
+At lower excitation energies, the uncertainty due to PT
+fluctuations is expected to gradually outweigh the statis-
+tical error bar. This effect becomes especially apparent
+for the GSFs extracted for specific final excitation-energy
+bins. The GSF for the ground state and the first excited
+state at 1.171 MeV in 120Sn are demonstrated in compari-
+son with the Oslo-method GSF in Figs. 9(e) and (f). The
+data are shown for Ef + Eγ ≥ 3.2 MeV. The area below
+
+14
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+ = 5.82 MeV
+i
+ E
+ PT fluctuations
+ Oslo method
+(a)
+)
+-3
+GSF (MeV
+ = 6.46 MeV
+i
+ E
+(b)
+8
+−
+10
+7
+−
+10
+6
+−
+10
+ = 7.10 MeV
+i
+ E
+(c)
+8
+−
+10
+7
+−
+10
+6
+−
+10
+ = 7.74 MeV
+i
+ E
+(d)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+(e)
+ = 0 MeV, g.s.
+f
+ E
+ PT fluctuations
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+(f)
+ = 1.17 MeV
+f
+ E
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+(g)
+ = 2.50 MeV
+f
+ E
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+Sn
+120
+(h)
+ = 3.26 MeV
+f
+ E
+FIG. 9.
+GSFs for 120Sn at initial excitation energies (a) 5.82 MeV, (b) 6.46 MeV, (c) 7.10 MeV, (d) 7.74 MeV and final
+excitation energies (e) ground state, (f) first excited state, (g) 2.50 MeV, (h) 3.26 MeV compared to the Oslo method strength
+(blue shaded band). For each strength the statistical error band is shown together with the error due to the PT fluctuations.
+Dark grey regions correspond to the areas of expected infinite PT fluctuations, light grey area marks energies for which the
+fluctuations of the strength were not determined. The γ-ray and excitation energy bin widths are both 128 keV.
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+ = 5.52 MeV
+i
+ E
+ PT fluctuations
+ Oslo method
+(a)
+)
+-3
+GSF (MeV
+ = 6.16 MeV
+i
+ E
+(b)
+8
+−
+10
+7
+−
+10
+6
+−
+10
+ = 6.96 MeV
+i
+ E
+(c)
+8
+−
+10
+7
+−
+10
+6
+−
+10
+ = 7.76 MeV
+i
+ E
+(d)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+(e)
+ = 0 MeV, g.s.
+f
+ E
+ PT fluctuations
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+(f)
+ = 1.13 MeV
+f
+ E
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+(g)
+ = 2.80 MeV
+f
+ E
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+Sn
+124
+(h)
+ = 3.44 MeV
+f
+ E
+FIG. 10.
+GSFs for 124Sn at initial excitation energies (a) 5.52 MeV, (b) 6.16 MeV, (c) 6.96 MeV, (d) 7.76 MeV and final
+excitation energies (e) ground state, (f) first excited state, (g) 2.80 MeV, (h) 3.44 MeV compared to the Oslo method strength
+(blue shaded band). For each strength the statistical error band is shown together with the error due to the PT fluctuations.
+Dark grey regions correspond to the areas of expected infinite PT fluctuations, light grey area marks energies for which the
+fluctuations of the strength were not determined. The γ-ray and excitation energy bin widths are both 160 keV.
+this energy and the area corresponding to Ef + Eγ > Sn
+are shaded. The fluctuations of the ground-state strength
+are large below ≈ 5 MeV, where they reach ≈ 40% of the
+absolute value. Between Eγ ≈ 3.3 and 5 MeV, the fluc-
+tuations of the strength are ≈ 60% on average and reach
+up to 90% toward the lowest γ energy. The latter case
+corresponds to only 1-3 possible dipole transitions at this
+Eγ. Applying the χ2
+ν distribution for fluctuations of so
+
+15
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+Oslo method
+-rays to the ground state
+γ
+Shape method, 
+Porter-Thomas fluctuations
+(a)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+-rays to the first excited state
+γ
+Shape method, 
+Porter-Thomas fluctuations
+Sn
+120
+(b)
+FIG. 11.
+Shape-method GSFs of 120Sn for γ−rays feeding
+the ground state (a) and the first excited state (b) compared
+to the Oslo method result (blue band). The Shape method re-
+sults are shown together with the statistical error propagated
+through the method, shown as a band (significantly smaller
+in width than the size of the data points), and the error bars
+due to the PT fluctuations. The Oslo method GSF is shown
+with the total (stat.+syst.) error band.
+few transition widths is not justified as it is valid solely
+in the statistical regime. Thus, the estimation procedure
+should be taken with great care when r(Eγ) approaches
+values of 1.
+Below ≈ 5 MeV, some strong deviations of the ground
+state strength from the Oslo-method result are observed.
+Besides the strong PT fluctuations at these γ-ray ener-
+gies, there might be some quadrupole transitions that
+cause methodical problems in this region. As the extrac-
+tion of the GSF relies on dipole radiation being dom-
+inant, quadrupole transitions from numerous low-lying
+2+ states to the ground state could distort the strength
+as the factor of E5
+γ should be used instead of E3
+γ.
+At
+higher γ-ray energies, the ground-state strength repro-
+duces the slope of the Oslo method strength, lying well
+within the Oslo-method error band. Similar effects can
+be seen for the 124Sn (in Fig. 10(e), Ef +Eγ ≥ 3.0 MeV),
+where the fluctuations were again estimated with the to-
+tal NLD and, therefore, should be considered lower-limit
+estimates.
+High PT fluctuations of 10-60% are observed also for
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+Oslo method
+-rays to the ground state
+γ
+Shape method, 
+Porter-Thomas fluctuations
+(a)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+ (MeV)
+γ
+E
+8
+−
+10
+7
+−
+10
+6
+−
+10
+)
+-3
+GSF (MeV
+-rays to the first excited state
+γ
+Shape method, 
+Porter-Thomas fluctuations
+Sn
+124
+(b)
+FIG. 12.
+Same as Fig. 11, but for 124Sn.
+the GSF to the first excited states in both isotopes, as
+shown in Fig. 9(f) and 10(f).
+For both nuclei these
+strengths reproduce the slopes of the Oslo method GSF
+in the region between 5 and 6.5 MeV quite well. For the
+higher final excitation energies, the fluctuations of the
+strengths are at most by one order of magnitude larger
+than the statistical uncertainties at low γ−ray energies,
+whilst at higher γ−ray energies they are by one order of
+magnitude smaller. For these strengths it is challenging
+to argue for an exact agreement with the Oslo method
+result.
+If taking a general agreement of the strengths
+within the error bars as a criterion, it can be possible to
+claim an overall independence of the strengths of final
+excitation energy for 120,124Sn.
+As the PT fluctuations become more significant at
+lower final excitation energies, they are expected to make
+a considerable contribution to the total error band of the
+Shape-method results. In figures 11 and 12, the GSFs
+for γ rays feeding the ground state and the first excited
+2+ state are shown for 120Sn and 124Sn, respectively, to-
+gether with the corresponding Oslo-method strengths.
+To test what a reasonable minimum excitation-energy
+limit would be for the application of the Shape method,
+we choose Ei = 4 MeV in both nuclei as a starting point.
+The Shape method results are presented with their statis-
+tical uncertainties, propagated through the unfolding and
+the first generation method. The Oslo-method strength
+
+16
+is shown with the total error band and the expected vari-
+ations of the corresponding ground-state or first-excited
+state strengths due to the PT fluctuations. Both of these
+strengths for 120Sn follow the shape of the Oslo-method
+strength quite well from the neutron separation energy
+and down to ≈ 5.5 − 6 MeV. Here, they start deviating
+gradually for lower γ-ray energies. In 124Sn, the agree-
+ment between the GSFs is quite good from Eγ ≈ 5 MeV
+and higher.
+Remarkably, the ground-state strengths and the first-
+excited state strengths for 120,124Sn demonstrate quite
+significant enhancements between 3 and 5 MeV, which
+can not be attributed to any real features of the strength.
+Moreover, there are no noticeable structures on the diag-
+onals at 4 < Ei < 5 MeV that might have induced these
+features. No similar effect was previously reported for
+even-even isotopes [71]. The appearance of these bumps
+might partly arise from the failure of the internal normal-
+isation technique at relatively low γ-ray energies where
+large fluctuations of the strengths are observed. The fluc-
+tuations of the ground-state strength in 120,124Sn range
+from ≈30 to 70% below 5.5 MeV, and from ≈15-35%
+below 4.3 MeV for the GSF corresponding to the first
+excited state. Since the pairs of data points for the two
+diagonals at each excitation energy are normalized in-
+ternally to each other (see Ref. [71]), large variations of
+the strengths could lead to an erratic internal normal-
+ization at relatively low γ-ray energies. When reaching
+densities of 1·103-2·103 levels per MeV, the distorting ef-
+fect due to the PT fluctuations becomes smaller, and
+the Shape-method results follow nicely the Oslo-method
+strength in both cases. This potential problem should be
+considered in future studies performed with the Shape
+method. When approaching the neutron separation en-
+ergies in 120,124Sn, fluctuations of the strengths do not
+exceed a few percent, which is comparable to the sta-
+tistical error bands shown in Figs. 11 and 12, whilst for
+the rest of the energy range, the PT fluctuations make a
+noticeable contribution to the uncertainties.
+Additional explanations for the smooth bump-like
+structures observed in the GSF might come from the fail-
+ure of some basic assumptions in the Shape method such
+as a symmetric parity distribution of the initial nuclear
+levels, pure dipole transitions of the involved γ-ray de-
+cays, and a spin-independent excitation probability in
+the (p, p′γ) reaction at 16 MeV. The lower the excitation
+energy, the less the assumption of a symmetric parity dis-
+tribution might be justified, especially in the magic Sn
+isotopes, so this may lead to deviations when using the
+Shape method at excitation energies below 5-6 MeV. Fur-
+thermore, similar to the discussion of the Oslo method,
+potential contributions of quadrupole transitions can dis-
+tort the analysis procedure due to the different energy
+factor of E5
+γ as compared to E3
+γ for dipole transitions.
+In particular, the excited 2+ states will most likely de-
+cay (on average) preferably to the first 2+ instead to the
+ground state. Within the Shape method, this can lead
+to the fact that the value of the GSF for the ground
+state γ-decay is (on average) smaller than for the decay
+into the first 2+ state. Thus the value pair in the Shape
+method has an increasing course towards low gamma en-
+ergies due to fto 2+[Ei − Ex(2+)] > fto g.s.[Ei − Ex(g.s.)]
+and might explain the increasing bump-like trend of the
+GSF. It remains an open question as to why the devia-
+tion of the strengths is systematically upward (always an
+increase) and whether the PT fluctuations, asymmetric
+parity distributions or the specific decay behavior of 2+
+states at low excitation energies are the main cause of
+the observed deviation
+V.
+CONCLUSIONS
+The nuclear level densities and γ-ray strength func-
+tions of 120,124Sn were extracted using the Oslo method,
+and the slopes of the strengths were additionally con-
+strained with the Shape method. The NLDs were found
+to be in good agreement with previously deduced NLDs
+for 116,118,122Sn, with slight deviations primarily due to
+some differences in the normalization procedures. The
+Oslo-method NLD for 1− states in 124Sn is in fairly
+good agreement within the estimated error bands with
+the result obtained from the fluctuation analysis of high-
+resolution inelastic proton scattering spectra above 6
+MeV. Given the model-independence of the (p, p′) result,
+this agreement supports the choice of the spin distribu-
+tion function and the spin-cutoff parameter employed in
+the Oslo method. The combined results covering excita-
+tion energies up to 14 MeV clearly demonstrate the tran-
+sition between the constant temperature and the Fermi
+gas regimes at ≈ 7 MeV.
+The experimental NLDs were used to estimate the role
+of the Porter-Thomas fluctuations in assessing the gen-
+eralised Brink-Axel hypothesis below the neutron sepa-
+ration energy in 120,124Sn, as well as the applicability of
+the Shape method. Most of the deviations of the GSFs
+for different initial and final excitation energies from
+the Oslo-method strength can be explained by strong
+PT fluctuations due to very few γ transitions. For the
+ground-state and the first-excited state strengths, this
+effect is especially apparent, with the PT fluctuations
+reaching up to 90-100% at low γ-ray energies. Despite
+some local discrepancies, the individual GSFs are in over-
+all good agreement with the Oslo-method strength within
+the error bands, suggesting an independence of initial
+and final excitation energies in support of the general-
+ized Brink-Axel hypothesis within uncertainties of the
+Oslo method.
+Strong PT fluctuations were found to play a notice-
+able role in the extraction of the GSFs with the Shape
+method, as they might contribute to considerable devia-
+tions from the Oslo-method result at low γ-ray energies.
+The reliability of the Shape method applied to 120,124Sn
+is under question for values of the NLDs below 1·103-
+2·103 levels per MeV, but quite satisfactory above this
+limit in both nuclei. Further investigations are needed
+
+17
+to understand why the Shape method seemingly leads to
+an overestimate of the low-energy strength in the region
+where the PT fluctuations are large.
+ACKNOWLEDGMENTS
+The authors express their thanks to J. C. M¨uller,
+P. A. Sobas, and J. C. Wikne at the Oslo Cyclotron
+Laboratory for operating the cyclotron and providing
+excellent experimental conditions. A. Zilges is sincerely
+thanked for stimulating discussions and for providing the
+120,124Sn targets. This work was supported in part by
+the National Science Foundation under Grant No. OISE-
+1927130 (IReNA), by the Deutsche Forschungsgemein-
+schaft (DFG, German Research Foundation) under Grant
+No. SFB 1245 (project ID 279384907), by the Norwegian
+Research Council Grant 263030, and by the National Re-
+search Foundation of South Africa (Grant No. 118846).
+A. C. L. gratefully acknowledges funding by the Euro-
+pean Research Council through ERC-STG-2014 under
+Grant Agreement No. 637686, and from the Research
+Council of Norway, project number 316116.
+J. I. ac-
+knowledges the support by the State of Hesse within the
+Research Cluster ELEMENTS (Project ID 500/10.006)
+and within the LOEWE program “Nuclear Photonics”
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+

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+arXiv:2301.01958v1  [math.CO]  5 Jan 2023
+Interaction graphs of isomorphic automata networks I:
+complete digraph and minimum in-degree
+Florian Bridoux∗, K´evin Perrot†, Aymeric Picard Marchetto‡, Adrien Richard§,
+January 6, 2023
+Abstract
+An automata network with n components over a finite alphabet Q of size q is a discrete
+dynamical system described by the successive iterations of a function f : Qn → Qn. In most
+applications, the main parameter is the interaction graph of f: the digraph with vertex set
+[n] that contains an arc from j to i if fi depends on input j. What can be said on the set
+G(f) of the interaction graphs of the automata networks isomorphic to f? It seems that this
+simple question has never been studied. Here, we report some basic facts. First, we prove
+that if n ≥ 5 or q ≥ 3 and f is neither the identity nor constant, then G(f) always contains
+the complete digraph Kn, with n2 arcs. Then, we prove that G(f) always contains a digraph
+whose minimum in-degree is bounded as a function of q. Hence, if n is large with respect to
+q, then G(f) cannot only contain Kn. However, we prove that G(f) can contain only dense
+digraphs, with at least ⌊n2/4⌋ arcs.
+1
+Introduction
+An automata network with n components over a finite alphabet Q is a function
+f : Qn → Qn,
+x = (x1, . . . , xn) �→ f(x) = (f1(x), . . . , fn(x)).
+The components fi : Qn → Q, are usually called the local transition functions of the network,
+while f is referred as the global transition function (but here we identify this function as the
+network). Automata networks are also called finite dynamical systems, and in the binary case,
+when Q = {0, 1}, the term Boolean networks is applied.
+The dynamics described by f is explicitly represented by the digraph Γ(f) with vertex set
+Qn and an arc from x to f(x) for every x ∈ Qn. Two automata networks f, h : Qn → Qn are
+isomorphic, f ∼ h in notation, if Γ(f) and Γ(h) are isomorphic in the usual sense; equivalently,
+there is permutation π of Qn such that π ◦f = h◦π. An equivalence class of ∼ then corresponds
+to an unlabeled digraph with |Q|n vertices in which each vertex has out-degree exactly one.
+Automata networks have many applications.
+In particular, they are omnipresent in the
+modeling of neural and gene networks (see [5] for a review). The “network” terminology comes
+from the fact that the interaction graph of f is often considered as the main parameter: it is
+the digraph G(f) with vertex set [n] = {1, . . . , n} and such that, for all i, j ∈ [n], there is an arc
+∗Universit´e Cˆote d’Azur, CNRS, I3S, Sophia Antipolis, France. ([email protected])
+†Aix-Marseille Universit´e, Universit´e de Toulon, CNRS, LIS, Marseille, France. ([email protected])
+‡Universit´e Cˆote d’Azur, CNRS, I3S, Sophia Antipolis, France. ([email protected])
+§Universit´e Cˆote d’Azur, CNRS, I3S, Sophia Antipolis, France. ([email protected])
+1
+
+x
+f(x)
+000
+000
+001
+100
+010
+000
+011
+100
+100
+000
+101
+100
+110
+000
+111
+100
+Γ(f)
+000
+100
+010
+110
+001 101 011 111
+f1(x) = x3
+f2(x) = 0
+f3(x) = 0
+G(f)
+1
+2
+3
+x
+h(x)
+000
+000
+001
+111
+010
+111
+011
+111
+100
+111
+101
+000
+110
+000
+111
+000
+Γ(h)
+000
+111
+101
+110
+100 010 001 011
+h1(x) = x1 + (x2 ∨ x3)
+h2(x) = x1 + (x2 ∨ x3)
+h3(x) = x1 + (x2 ∨ x3)
+G(h)
+1
+2
+3
+Figure 1: Two isomorphic automata networks on {0, 1}3 with different interaction graphs.
+from j to i if fi depends on input j, that is, if there exist x, y ∈ Qn which only differ in xj ̸= yj
+such that fi(x) ̸= fi(y).
+In many applications, as modeling of gene networks, the interaction graph is often well ap-
+proximated while the actual dynamics is not [7, 5]. One is thus faced with the following question:
+what can be said on the dynamics described by f from G(f) only? There are many results in
+this direction; see [4] for a review. In most cases, the studied dynamical properties are invariant
+by isomorphism: number of fixed points, number of images, number of periodic configurations,
+number and size of limit cycles, transient length and so on. However, the interaction graph is
+not invariant by isomorphism: even if f and h are isomorphic, their interaction graphs can be
+very different, as shown by Figure 1. This variaton can give some limit to the central question
+stated above and we think it deserves some study.
+Hence, we propose a systematic study of this variation, considering the set G(f) of interaction
+graphs of automata networks isomorphic to f:
+G(f) = {G(h) | h : Qn → Qn, h ∼ f}.
+If G ∈ G(f) we say that f can be produced by G. For instance, if f is constant, we write this
+f = cst, then G(f) contains a single digraph, the digraph on [n] without arcs, and if f is the
+identity, we write this f = id, then G(f) also contains a single digraph, the digraph on [n] with n
+loops (cycles of length one) and no other arcs. So the identity and constant automata networks
+can be produced by a single digraph.
+Even if the systematic study of G(f) seems to us a new perspective, some classical results
+can be stated in this setting. For instance, let us say that a digraph is a companion digraph if
+it can be obtained from a directed path with n vertices by adding between 0 and n arcs, all
+pointing to the first vertex of the path. If f is a linear network with n components over the
+field Q = Fp for some prime p, that is if f(x) = Mx for some n × n matrices M over Fp, then
+2
+
+considering the Frobenius normal form of M, we obtain that f is isomorphic to a linear network
+h whose interaction graph G is a disjoint union of companion digraphs [2]. Consequently, any
+linear network f can be produced by a digraph G ∈ G(f) which is a disjoint union of companion
+digraphs, and which is thus very sparse: G has at most 2n − 1 arcs.
+The results we give on G(f) in this paper differ in that they do not rely on any hypothesis
+made on f: we work with the whole set F(n, q) of automata networks with n components over
+an alphabet of size q. Our first result shows that, excepted few exceptions, f can always be
+produced by the complete digraph on [n], that is the digraph on [n] with n2 arcs, denoted Kn.
+Theorem 1. Let n, q ≥ 2 with n ≥ 5 or q ≥ 3. If f ∈ F(n, q) and f ̸= cst, id, then Kn ∈ G(f).
+This indeed gives some limitations of what can be deduced on f from G(f): if G(f) = Kn
+nothing can be said on the dynamics of f up to isomorphism except that f has at least two
+images and at most qn − 1 fixed points. As explained above, if f = cst or f = id then f can
+only be produced by one digraph: the empty digraph and the digraph with n loops, respectively.
+The previous theorem says that if f ̸= cst, id can only be produced by one digraph, then this
+digraph is necessarily Kn. But is there at least one such automata network f, whose dynamics
+is completely specific to the complete digraph? For n = 2 and q ≥ 3 we exhibit such f, and we
+show that it does not exist when n is large enough with respect to q.
+We obtain these results by studying the minimum in-degree of the digraphs in G(f). We first
+prove, using a difficult result in additive number theory, that G(f) always contains a digraph
+whose minimum in-degree is bounded by a constant cq that only depends on q, and which is 5 for
+q = 2. Hence, any automata network f is isomorphic to an automata network h which contains
+a “simple” local transition function hi, in that hi only depends on at most cq components.
+Conversely, we prove that, for some f ∈ F(n, q), every digraph in G(f) has minimum in-degree
+at least 2. To state these results compactly, for n, q ≥ 2 we denote by δ−(n, q) the minimum
+number d such that, for any f ∈ F(n, q), some digraph in G(f) has minimum in-degree at most
+d.
+Theorem 2. For all n, q ≥ 2 with n ≥ 3 or q ≥ 3 we have
+2 ≤ δ−(n, 2) ≤ 5
+and
+2 ≤ δ−(n, q) ≤ (2 + o(1))q
+where o(1) tends to 0 as q tends to infinity, and does not depend on n.
+Hence, for any q ≥ 3, since δ−(2, q) ≥ 2 there is f ∈ F(2, q) such that every digraph G(f)
+has minimum in-degree at least 2, and thus G(f) = {K2}. So there are indeed some automata
+networks in F(2, q) which are only produced by K2. On the other hand, for n large enough
+with respect to q, we have δ−(n, q) < n, so for any f ∈ F(n, q) there is a digraph G ∈ G(f)
+with minimum in-degree at most n − 1, which is thus distinct from Kn: no automata network
+in F(n, q) is only produced by Kn. That some automata networks in F(n, q) are only produced
+by Kn when q is large enough with respect to n is open.
+Even if for n large with respect to q there are no automata networks only produced by Kn,
+we may ask if there are automata networks which are only produced by dense digraphs. We
+answer this positively (and thus the fact, mentioned above, that a linear network can always be
+produced by a sparse digraph is specific to the linear case).
+Theorem 3. For all n, q ≥ 2, there exists f ∈ F(n, q) such that every digraph in G(f) has at
+least ⌊n2/4⌋ arcs.
+The previous results shed light on the case |G(f)| = 1. We may take the converse direction
+and focus on automata networks f for which |G(f)| is large. Certainly |G(f)| ≤ 2n2 and the
+3
+
+inequality is strict, since otherwise G(f) contains the empty digraph, so that f = cst and thus
+|G(f)| = 1. So |G(f)| ≤ 2n2 − 1. In a subsequent paper we will prove, in particular, that
+this bound can be reached. Hence, while Kn is in some sense universal in that any automata
+network f ̸= cst, id with n components can be produced by Kn, there are also universal automata
+networks f with n components in that f can be produced by all the digraphs on [n] except the
+empty one.
+The paper is organized as follows. Theorems 1, 2 and 3 are proved in Sections 2, 3 and 4
+respectively. Concluding remarks are given in Section 5.
+Notations and terminologies
+n and q are always integers greater than 1. We set [n] =
+{1, . . . , n} and �q� = {0, 1, . . . , q − 1}. Elements of �q�n are called configurations, and those of
+[n] are called components. We denote by F(n, q) the set of functions f : �q�n → �q�n; so we
+always assume, without loss, that Q = �q�. In the following, the members of F(n, q) are referred
+as functions instead of automata networks. We denote by 0 (resp. 1) the configurations x such
+that xi = 0 (resp. 1) for all i ∈ [n]. For k ∈ �q�, we denote kei the configuration x such that
+xi = k and xj = 0 for all j ̸= i. We write ei instead of 1ei. The (Hamming) weight of x ∈ �q�n
+is the number of i ∈ [n] with xi ̸= 0. For x, y ∈ �q�n, the sum x + y is applied component-wise
+modulo q. Hence, x and x + ei differ only in component i. So the interaction graph G(f) of
+f ∈ F(n, q) has an arc from j to i if and only if there is x ∈ �q�n such that fi(x) ̸= fi(x + ej). A
+fixed point is a configuration x such that f(x) = x. A limit cycle of f is a cycle of Γ(f). Hence,
+fixed points correspond to limit cycles of length one. An independent set of f is an independent
+set of Γ(f), equivalently, it is a set A ⊆ �q�n such that f(A) ∩ A = ∅.
+2
+Complete digraph
+In this section, we will prove Theorem 1 which states that Kn ∈ G(f) for every f ∈ F(n, q) with
+f ̸= cst, id and n ≥ 5 or q ≥ 3. For that, we exhibit several sufficient conditions for Kn ∈ G(f),
+and we then prove that at least one of these sufficient conditions must be satisfied. As the proof
+for q ≥ 3 is much simpler, we will start with it. We will then treat the case n ≥ 5 independently.
+2.1
+Case q ≥ 3
+We say that f ∈ F(n, q) contains a complete-pattern if there is x ∈ �q�n and a subset A ⊆ �q�n
+of size n such that x, f(x) /∈ A ∪ f(A) and either x ̸= f(x) or A ∩ f(A) = ∅; in that case we say
+that the complete-pattern is rooted in (x, A).
+We prove, in Lemma 1 below, that, for q ≥ 3 and n ≥ 2, we have Kn ∈ G(f) whenever f
+has a complete-pattern. We then prove, in Lemma 2, that if f ̸= cst, id then f must contain a
+complete-pattern. Together, this proves Theorem 1 for q ≥ 3.
+Lemma 1. Let f ∈ F(n, q) with q ≥ 3 and n ≥ 2. If f has a complete-pattern then Kn ∈ G(f).
+Proof. Suppose that f has a complete-pattern rooted in (x, A). We have three cases:
+• Case 1: x ̸= f(x) and f(A) ∩ A ̸= ∅. Let π be any permutation of �q�n such that:
+– π(x) = 0,
+– π(f(x)) = 1,
+– π(A) = {2e1, . . . , 2en},
+– π(f(A) \ A) ⊆ {0, 2}n \ {0, 2e1, . . . , 2en}.
+4
+
+See Figure 2 for an illustration. Note that since f(A) ∩ A ̸= ∅ the size of f(A) \ A is at
+most n − 1, and since n − 1 ≤ 2n − (n + 1) the configurations in f(A) \ A can indeed have
+distinct images by π inside {0, 2}n \ {0, 2e1, . . . , 2en}. Let h = π ◦ f ◦ π−1 and let us prove
+that G(h) = Kn. For all i ∈ [n], the configurations 2ei and 0 differ only in component i,
+while h(2ei) ∈ {0, 2}n and h(0) = 1 differ in every component. Hence, G(h) has an arc
+from i to every component. Thus, G(h) = Kn and since h ∼ f we have Kn ∈ G(f).
+• Case 2: x ̸= f(x) and f(A) ∩ A = ∅. Let π be any permutation of �q�n such that:
+– π(x) = 0,
+– π(f(x)) = 1,
+– π(A) = {e1, . . . , en},
+– π(f(A)) ⊆ {2e1, . . . , 2en}.
+See Figure 3 for an illustration. Let h = π ◦ f ◦ π−1 and let us prove that G(h) = Kn. For
+all i ∈ [n], the configurations ei and 0 differ only in component i, while h(ei) ∈ {0, 2}n and
+h(0) = 1 differ in every component. Hence, G(h) has an arc from i to every component.
+Thus, G(h) = Kn and since h ∼ f we have Kn ∈ G(f).
+• Case 3: x = f(x) and f(A) ∩ A = ∅. Let π be any permutation of �q�n such that:
+– π(x) = 0,
+– π(A) = {e1, . . . , en},
+– π(f(A)) ⊆ {1 + e1, . . . , 1 + en}.
+See Figure 4 for an illustration. Let h = π ◦ f ◦ π−1 and let us prove that G(h) = Kn. For
+all i ∈ [n], the configurations ei and 0 differ only in component i, while h(ei) ∈ {1, 2}n and
+h(0) = 0 differ in every component. Hence, G(h) has an arc from i to every component.
+Thus, G(h) = Kn and since h ∼ f we have Kn ∈ G(f).
+00
+20
+02
+x
+A
+11
+00
+20
+02
+x
+A
+11
+00
+20
+02
+x
+A
+11
+22
+00
+20
+02
+x
+A
+11
+00
+20
+02
+x
+A
+11
+22
+Figure 2: Labelling by π of complete-patterns in the first case of Lemma 1 for n = 2.
+00
+10
+01
+x
+A
+11
+20
+02
+00
+10
+01
+x
+A
+11
+20
+Figure 3: Labelling by π of complete-patterns in the second case of Lemma 1 for n = 2.
+Lemma 2. Let f ∈ F(n, q) with q ≥ 3 and n ≥ 2. If f ̸= cst, id then f has a complete-pattern.
+5
+
+00
+10
+01
+x
+A
+21
+12
+00
+10
+01
+x
+A
+21
+Figure 4: Labelling by π of complete-patterns in the third case of Lemma 1 for n = 2.
+Proof. Suppose that there exists a configuration a with at least n + 1 pre-images by f. Then
+there is a set A of size n contained in f −1(a) \ {a}. Since f ̸= cst, f has an image y ̸= a. Let
+x ∈ f −1(y). Then x and f(x) = y are not in in the union of A and f(A) = {a}, and since a /∈ A,
+f has a complete-pattern rooted in (x, A).
+Suppose now that every configuration has at most n pre-images by f. Since f ̸= id, there
+is x ∈ �q�n such that f(x) ̸= x. Let B be the images of f distinct from x and f(x). Then
+|f −1(B)| + |f −1(x)| + |f −1(f(x))| = qn and we deduce that |f −1(B)| ≥ qn − 2n which is at least
+n+1 since q ≥ 3. So there is a subset A ⊆ f −1(B)\{f(x)} of size n, and since x, f(x) /∈ A∪f(A)
+we deduce that f has a complete pattern rooted in (x, A).
+2.2
+Case n ≥ 5
+We proceed by showing that, given f ∈ F(n, q) with n ≥ 5, q ≥ 2 and f ̸= cst, id, we have
+Kn ∈ G(f) if at least one of the following three conditions holds:
+• f has at least 2n fixed points,
+• f has at least n limit cycles of length ≥ 3,
+• f has an independent set of size ≥ 2n.
+We then prove that, if f ̸= cst, id, then at least one of the three conditions must hold. Together
+this proves Theorem 1 for n ≥ 5. We start by proving that the first two conditions, involving
+fixed points and limit cycles, are sufficient for Kn ∈ G(f).
+Lemma 3. Let f ∈ F(n, q) with f ̸= id. If f has at least 2n fixed points, then Kn ∈ G(f).
+Proof. Since f ̸= id we have f(c) ̸= c for some configuration c, and since f has at least 2n fixed
+points, it has 2n − 1 fixed points distinct from f(c), say a0, a1, . . . , an, b3, . . . , bn (these are also
+distinct from c since c is not a fixed point). Let π be any permutation of �q�n such that:
+• π(a0) = 0,
+• π(ai) = ei for 1 ≤ i ≤ n,
+• π(bi) = e1 + e2 + ei for 3 ≤ i ≤ n,
+• π(c) = e1 + e2,
+• π(f(c)) = e1 + e2 + 1.
+See Figure 5 for an illustration. Let h = π ◦ f ◦ π−1 and let us prove that G(h) = Kn. For
+i ∈ [n], the configurations 0 and ei only differ in component i, and since h(0) = 0 and h(ei) = ei
+we deduce that G(h) has a loop on i. It remains to prove that G(h) has an arc from i to j for
+distinct i, j ∈ [n]. The configurations e2 and e1 +e2 only differ in component 1, while h(e2) = e2
+and h(e1 +e2) = e1 +e2 +1 differ in every component j ̸= 1. So G(h) has an arc from 1 to every
+component j ̸= 1. Similarly, the configurations e1 and e1 + e2 only differ in component 2, while
+6
+
+h(e1) = e1 and h(e1 +e2) = e1 +e2 +1 differ in every component j ̸= 2. So G(h) has an arc from
+2 to every component j ̸= 2. Finally, for 3 ≤ i ≤ n, the configurations e1 + e2 and e1 + e2 + ei
+only differ in component i, while h(e1 + e2 + ei) = e1 + e2 + ei and h(e1 + e2) = e1 + e2 + 1
+differ in every component j ̸= i. So G(h) has an arc from i to every j ̸= i. Thus, G(h) = Kn
+and since h ∼ f we have Kn ∈ G(f).
+00000
+a0
+10000
+a1
+01000
+a2
+00100
+a3
+00010
+a4
+00001
+a5
+11000
+c
+11100
+b3
+11010
+b4
+11001
+b5
+00111
+f(c)
+Figure 5: Labelling by π in Lemma 3 for n = 5.
+Lemma 4. Let f ∈ F(n, q). If f has at least n limit cycles of length ≥ 3, then Kn ∈ G(f).
+Proof. Suppose that f has n limit cycles of length ≥ 3; this implies n ≥ 4. Then there are n
+configurations a1, . . . , an that belong to distinct limit cycles of f of length ≥ 3. For i ∈ [n], let
+bi = f(ai) and ci = f(bi). Then a1, . . . , an, b1, . . . , bn, c1, . . . , cn are all distinct. Let π be any
+permutation of �q�n such that
+• π(ai) = ei−1 + ei,
+• π(bi) = ei−1,
+• π(ci) = ei−1 + 1,
+where e0 means en. See Figure 6 for an illustration. Note that the map π is indeed injective on
+the 3n selected configurations: the weights of π(ai), π(bi), and π(ci) are 2, 1, and n − 1 ≥ 4 − 1
+respectively. Let h = π◦f ◦π−1 and let us prove that G(h) = Kn. For i ∈ [n], the configurations
+ei−1 and ei−1 + ei only differ in component i, while h(ei−1) = ei−1 + 1 and h(ei−1 + ei) = ei−1
+differ in every component. So G(h) has an arc from i to every j ∈ [n]. Thus, G(h) = Kn and
+since h ∼ f we have Kn ∈ G(f).
+10001
+a1
+00001
+b1
+11110
+c1
+11000
+a2
+10000
+b2
+01111
+c2
+01100
+a3
+01000
+b3
+10111
+c3
+00110
+a4
+00100
+b4
+11011
+c4
+00011
+a5
+00010
+b5
+11101
+c5
+Figure 6: Labelling by π in Lemma 4 for n = 5.
+We now prove that the third condition given at the beginning of the subsection, involving
+independent sets, is sufficient for Kn ∈ G(f). The precise statement is as follows.
+Lemma 5. Let f ∈ F(n, q) with n ≥ 5 and f ̸= cst. If f has an independent set of size at least
+2n, then Kn ∈ G(f).
+7
+
+The proof is more involved than for fixed points and limit cycles, and so we split the proof.
+We first prove that, if n ≥ 5 and f ̸= cst, then Kn ∈ G(f) whenever at least one of the following
+condition holds:
+• f has an independent set A with |A| ≥ n + k and |f(A)| = 2k for some 1 ≤ k ≤ n.
+• f has an independent set A with |A| > n and |f(A)| = 1.
+We then prove that if f has an independent set of size at least 2n, then at least one of the two
+conditions must hold, and this proves Lemma 5.
+Lemma 6. Let f ∈ F(n, q) with n ≥ 5. If f has an independent set A such that |A| ≥ n + k
+and |f(A)| = 2k for some 1 ≤ k ≤ n, then Kn ∈ G(f).
+Proof. Suppose that f has an independent set A of f such that |A| ��� n + k and |f(A)| = 2k for
+some 1 ≤ k ≤ n, and suppose that |A| is minimal for those properties.
+(1) |A| = n + k.
+Indeed, if |A| > n+k, then |A| > 2k and thus there are distinct a, a′ ∈ A with f(a) = f(a′).
+Hence, A′ = A \ {a} is an independent set of f such that |A′| = |A| − 1 ≥ n + k and
+|f(A′)| = 2k, and this contradicts the minimality of |A|. This proves (1).
+Let us write f(A) = {a1, . . . , a2k}, and let Ap = f −1(ap)∩A for p ∈ [2k]. Hence, {A1, . . . , A2k}
+is a partition of A. We now come to the technical part of the proof.
+(2) There are disjoint subsets X1, . . . , X2k of {0, 1}n of size |A1|, . . . , |A2k| respectively, such
+that, for all i ∈ [n], there are p ∈ [k] and x ∈ X2p−1 with x + ei ∈ X2p.
+Let I1, . . . , I2k be a partition of [n] (with some members possibly empty) such that, for
+1 ≤ ℓ ≤ 2k, the size of Iℓ is |Aℓ| − 1 if ℓ is odd, and |Aℓ| otherwise; such a partition exists
+since |A1| + · · · + |A2k| = n + k by (1).
+For p ∈ [k], select a configuration x2p−1 ∈ {0, 1}n, a component j2p ∈ I2p, and let
+X2p−1
+=
+{x2p−1} ∪ {x2p−1 + ej2p + ei | i ∈ I2p−1},
+X2p
+=
+{x2p−1 + ei | i ∈ I2p}.
+Clearly, |X2p−1| = |I2p−1| + 1 = |A2p−1| and |X2p| = |I2p| = |A2p|.
+Let i ∈ [n] and let us prove that there are p ∈ [k] and x ∈ X2p−1 with x + ei ∈ X2p. We
+have i ∈ Iℓ for some 1 ≤ ℓ ≤ 2k. If ℓ = 2p − 1 then, setting x = x2p−1 + ej2p + ei, we
+have x ∈ X2p−1 and x + ei = x2p−1 + ej2p ∈ X2p. If ℓ = 2p then, setting x = x2p−1, we
+have x ∈ X2p−1 and x + ei ∈ X2p. Thus, fixing the components j2p arbitrarily, we only
+have to prove that we can choose the configurations x2p−1 so that the sets X1, . . . , X2k are
+pairwise disjoint.
+This is obvious if k = 1, since X1 and X2 are disjoint for any choice of x1. If k = 2,
+then, since n ≥ 5, the sets X1, X2, X3, X4 are disjoint by taking x1 = 0 and x3 = 1:
+configurations in X1 have weight 0 or 2; configurations in X2 are of weight 1; configurations
+in X3 are of weight n or n − 2; and configurations in X4 are of weight n − 1.
+Suppose now that k ≥ 3 and choose x2p−1 = ej2p−2 for all p ∈ [k], where j0 means j2k.
+Then each X2p−1 only contains configurations of weight 1 or 3 and each X2p only contains
+configurations of weight 2. Hence, given 1 ≤ p < q ≤ k, we have to prove that X2p−1 ∩
+X2q−1 = ∅ and X2p ∩ X2q = ∅.
+8
+
+Suppose that x ∈ X2p−1 ∩ X2q−1. If the weight of x is 1 then we deduce that x = ej2p−2 =
+ej2q−2 which is false since p ̸= q. If the weight of x is 3 then xi = 1 for some i ∈ I2p−1
+while yi = 0 for all y ∈ X2q−1, a contradiction. Thus, X2p−1 ∩ X2q−1 = ∅.
+Suppose now that x ∈ X2p ∩ X2q. Then x = ej2p−2 + ei2p = ej2q−2 + ei2q for some i2p ∈ I2p
+and i2q ∈ I2q. Thus, i2p = j2q−2 and i2q = j2p−2. Hence, j2q−2 ∈ I2p, and since p < q this
+implies q = p + 1. Also, j2p−2 ∈ I2q and since p < q this implies p = 1 and q = k, but
+then q ̸= p + 1 since k ≥ 3. Thus, X2p ∩ X2q = ∅. Hence, the sets X1, . . . , X2k are indeed
+pairwise disjoint. This proves (2).
+Let X1, . . . , X2k be as in (2). Let X = X1 ∪ · · · ∪ X2k, which is of size n + k.
+(3) There is Y ⊆ {0, 1}n with |Y | ≥ n such that Y , Y + 1 and X are pairwise disjoint.
+Let Z be any subset of {0, 1}n of size 2n−1 such that Z and Z + 1 are disjoint. Let Y be
+the set of y ∈ Z such that y, y + 1 /∈ X. Clearly, Y, Y + 1, X are pairwise disjoint, so it
+is sufficient to prove that |Y | ≥ n. Setting X0 = X ∩ Z and X1 = X ∩ (Z + 1), we have
+Y = Z \(X0 ∪ (X1 + 1)) and thus |Y | ≥ |Z| − |X0| − |X1|. Since Z and Z + 1 are disjoint,
+X0 and X1 are disjoint subsets of X, so
+|Y | ≥ |Z| − |X| = 2n−1 − (n + k) ≥ 2n−1 − 2n ≥ n.
+where the last inequality holds since n ≥ 5.
+Let Y = {y1, . . . , yk} be a subset of {0, 1}n of size k such that Y , Y + 1 and X are pairwise
+disjoint, which exists by (3). Hence, there is a permutation π of {0, 1}n such that, for all p ∈ [k]:
+• π(a2p−1) = yp,
+• π(a2p) = yp + 1,
+• π(A2p−1) = X2p−1,
+• π(A2p) = X2p.
+See Figure 7 for an illustration. Let h = π ◦ f ◦ π−1 and let us prove that G(h) = Kn. Since
+the sets X1, . . . , X2k are as in (2), for every i ∈ [n] there is p ∈ [k] and x ∈ X2p−1 with
+x + ei ∈ X2p. Since π−1(x) ∈ A2p−1 and π−1(x + ei) ∈ A2p, we have h(x) = π(a2p−1) = yp and
+h(x + ei) = π(a2p) = yp + 1. Thus, h(x) and h(x + ei) differ in every component, and so G(h)
+has an arc from i to every j ∈ [n]. Thus, G(h) = Kn and since h ∼ f we have Kn ∈ G(f).
+Lemma 7. Let f ∈ F(n, q) with n ≥ 5 and f ̸= cst. If f has an independent set A such that
+|A| > n and |f(A)| = 1, then Kn ∈ G(f).
+Proof. Let A be an independent set of f with |A| > n and |f(A)| = 1. Then f(A) = {a} for
+some configuration a ∈ �q�n. Since f ̸= cst, there is b ∈ �q�n with f(b) ̸= a, and thus b /∈ A. We
+consider three cases.
+• Case 1: f(a) ̸= a. Since |A| > n, there are n configurations a1, . . . , an in A distinct from
+f(a). Then a1, . . . , an, a, f(a) are all distinct, so there is a permutation π of �q�n such
+that:
+– π(a) = 0,
+– π(f(a)) = 1,
+– π(ai) = ei for 1 ≤ i ≤ n.
+9
+
+00001
+11001
+01001
+01000
+01100
+01010
+00100
+00101
+A
+A1
+A2
+A3
+A4
+A5
+A6
+X1
+X2
+X3
+X4
+X5
+X6
+00000
+a1
+11111
+a2
+10000
+a3
+01111
+a4
+00010
+a5
+11101
+a6
+Figure 7: Labelling by π in Lemma 6 for n = 5 and k = 3. The independent set A is of size |A| =
+n+k = 8 and its image f(A) = {a1, . . . , a6} is of size 2k = 6. Since the consecutive sizes of the set
+A1, . . . , A6 are (2, 1, 1, 2, 1, 1), we have to choose a partition I of [n] into 6 parts I1, . . . , I6 whose
+consecutive sizes are (1, 1, 0, 2, 0, 1). We take (I1, I2, I3, I4, I5, I6) = ({1}, {2}, ∅, {3, 4}, ∅, {5}).
+We have to choose one member j2, j4, j6 in X2, X4, X6 respectively. We choose j2 = 2, j4 = 3
+and j6 = 5. For these choices, we obtain the set X1, . . . , X6 described in the figure, which are
+used to label the configurations in A1, . . . , A6. Finally, we have to choose a set Y of k = 3
+configurations, such that Y, Y + 1 and X = X1 ∪ · · · ∪ X6 are pairwise disjoint. We choose
+Y = {00000, 10000, 00010}, which is used to label a1, a3, a5, and Y + 1 is then used to label
+a2, a4, a6 accordingly.
+Note that: 11001 ∈ X1 and 01001 ∈ X2 only differ in component 1;
+01000 ∈ X1 and 01001 ∈ X2 only differ in component 2; 01000 ∈ X3 and 01100 ∈ X4 only
+differ in component 3; 01000 ∈ X3 and 01010 ∈ X4 only differ in component 4; 00100 ∈ X5 and
+00101 ∈ X6 only differ in component 5.
+See Figure 8 for an illustration. Let h = π ◦ f ◦ π−1 and let us prove that G(h) = Kn. For
+i ∈ [n], h(0) = 1 and h(ei) = 0 differ in every component, therefore G(h) has an arc from
+i to every j ∈ [n]. Thus, G(h) = Kn and since h ∼ f we have Kn ∈ G(f).
+• Case 2: f(a) = a and f(b) = b. Let a1, . . . , an ∈ A, all distinct. Then a1, . . . , an, a, b are
+all distinct since b = f(b) ̸= a. So there is a permutation π of �q�n such that:
+– π(a) = 1,
+– π(b) = 0,
+– π(ai) = ei for 1 ≤ i ≤ n.
+See Figure 9 for an illustration. Let h = π ◦ f ◦ π−1 and let us prove that G(h) = Kn. For
+i ∈ [n], h(0) = 0 and h(ei) = 1 differ in every component, hence G(h) has an arc from i
+to every j ∈ [n]. Thus, G(h) = Kn and since h ∼ f we have Kn ∈ G(f).
+• Case 3: f(a) = a and f(b) ̸= b. Since |A| > n, there is A′ ⊆ A \ {f(b)} of size n. Then
+A′ ∪ {b} is an independent set of size n + 1 and |f(A′ ∪ {b})| = |{a, f(b)}| = 2, thus
+Kn ∈ G(f) by Lemma 6.
+00000
+a
+11111
+f(a)
+10000
+a1
+01000
+a2
+00100
+a3
+00010
+a4
+00001
+a5
+Figure 8: Labelling by π in the first case of Lemma 7 for n = 5.
+10
+
+11111
+a
+10000
+a1
+01000
+a2
+00100
+a3
+00010
+a4
+00001
+a5
+00000
+b
+Figure 9: Labelling by π in the second case of Lemma 7 for n = 5.
+Proof of Lemma 5. Suppose that f has an independent set A of size |A| ≥ 2n. Then we can
+choose A so that |A| = 2n, and then |f(A)| ≤ 2n. Consequently, if |f(A)| is even, we have
+|f(A)| = 2k for some k ∈ [n] and thus Kn ∈ G(f) by Lemma 6. So assume that |f(A)| is odd.
+If |f(A)| = 1 then Kn ∈ G(f) by Lemma 7. So assume that |f(A)| = 2k + 1 for some 1 ≤ k < n.
+Let us write f(A) = {a1, . . . , a2k+1}, and let Ap = f −1(ap) ∩ A for 1 ≤ p ≤ 2k + 1. Suppose,
+without loss, that |A1| ≤ |A2| ≤ · · · ≤ |A2k+1|. Then A′ = A \ A1 is an independent set with
+|f(A′)| = 2k. If |A′| ≥ n + k then Kn ∈ G(f) by Lemma 6. So suppose that |A′| < n + k. Since
+|A′| = 2n − |A1| we have |A1| > n − k. If |A2| ≥ k then A′′ = A1 ∪ A2 is an independent with
+|A′′| ≥ n + 1 and |f(A′′)| = 2 thus Kn ∈ G(f) by Lemma 6. So assume that |A2| < k. Then
+n − k < |A1| ≤ |A2| < k, so 2k > n. We deduce that
+2n = |A| = |A1| + · · · + |A2k+1| ≥ (2k + 1)|A1| > n|A1|.
+So |A1| = 1 but then |A′| = 2n−1 ≥ n+k, a contradiction. Thus, Kn ∈ G(f) in every case.
+By Lemmas 3, 4 and 5, to prove Theorem 1 for n ≥ 5 it is sufficient to prove the following.
+Lemma 8. Let f ∈ F(n, q) with f ̸= cst, id and n ≥ 5. Then at least one of the following holds:
+• f has at least 2n fixed points,
+• f has at least n limit cycles of length ≥ 3,
+• f has an independent set of size at least 2n.
+Proof. Let F be the set of fixed points of f, and let L be a minimal subset of �q�n intersecting
+every limit cycle of f of length ≥ 3. Let Γ′ be obtained from Γ(f) by deleting the vertices in
+F ∪ L; then Γ′ has only cycles of length two, thus it is bipartite. Suppose that the first two
+conditions are false, that is, |F| < 2n and |L| < n. Then Γ′ has at least N = qn −3n+2 vertices.
+Since Γ′ is bipartite, it has an independent set A of size at least N/2. Since Γ′ is an induced
+subgraph of Γ(f), A is an independent set of f, and since n ≥ 5 we have
+|A| ≥ ⌈N/2⌉ ≥ ⌈(2n − 3n + 2)/2⌉ ≥ 2n
+thus the third condition holds.
+3
+Minimum in-degree
+In this section, we will prove Theorem 2, that we restate. Given f ∈ F(n, q), we denote by δ−(f)
+the minimum integer d such that some digraph in G(f) has minimum in-degree at most d. The
+quantity δ−(n, q) defined in the introduction is then the maximum of δ−(f) for f ∈ F(n, q), and
+Theorem 2 says that, for any n, q ≥ 2 with n ≥ 3 or q ≥ 3, we have
+2 ≤ δ−(n, 2) ≤ 5
+and
+2 ≤ δ−(n, q) ≤ (2 + o(1))q.
+11
+
+Instead of considering the in-degree, it is more convenient to consider the strict in-degree,
+defined as follows. Given a digraph G on [n], the strict in-degree of i ∈ [n] in G is the number of
+in-neighbors of i in G distinct from i. The minimum strict in-degree of G is then the minimum
+strict in-degree of a vertex in G. Given f ∈ F(n, q), we denote by δ−
+s (f) the smallest integer d
+such that some digraph in G(f) has minimum strict in-degree at most d; we have δ−(f) − 1 ≤
+δ−
+s (f) ≤ δ−(f).
+Our first goal is to give, for any 0 ≤ k < n, a necessary and sufficient condition for δ−
+s (f) ≤
+n − k − 1 which is invariant by isomorphism. In this way, we have a description of δ−
+s (f) which
+only relies on the isomorphic class of f. This description is based on the following definition.
+Definition 1 (k-nice partition). Let f ∈ F(n, q) and k ∈ [n]. A k-nice partition of f is a
+partition A = {A1, . . . , Aq} of �q�n, with some parts possibly empty, such that |Ap| and |Aℓ ∩
+f −1(Ap)| are multiples of qk for all p, ℓ ∈ [q].
+A partition is balanced when its parts have all the same size. The characterization is the
+following.
+Lemma 9. For every f ∈ F(n, q) and 0 ≤ k < n, we have δ−
+s (f) ≤ n − k − 1 if and only if f
+has a balanced k-nice partition.
+Actually, the proof of Theorem 2 only uses one direction, that δ−
+s (f) ≤ n − k − 1 if f has
+a balanced k-nice partition. But the main property needed for the other direction will be used
+later, and so we give it separately (in Lemma 10 below), and then put things together to get
+the characterization. For that we use the following definitions. Given x, y ∈ �q�n, we denote
+by ∆(x, y) the set of i ∈ [n] with xi ̸= yi. Given I ⊆ [n], a set X ⊆ �q�n is I-closed if, for any
+x, y ∈ �q�n with ∆(x, y) ⊆ I, we have x ∈ X if and only if y ∈ X. It is easy to see that if X is
+I-closed then |X| is a multiple of q|I|, and that if X and Y are both I-closed then so is X ∩ Y .
+Lemma 10. Let f ∈ F(n, q) and suppose that G(f) has a vertex i with strict in-degree at most
+n − k − 1. For 1 ≤ p ≤ q, let Ap be the set of x ∈ �q�n with xi = p − 1. Then A = {A1, . . . , Aq}
+is a balanced k-nice partition of f.
+Proof. For k = 0 the result is obvious, so assume that k ≥ 1. Since A is obviously balanced, we
+just have to prove that A is k-nice. Since i is of strict in-degree at most n − k − 1, there is a set
+I ⊆ [n] of size k such that G(f) has no arc from I to i and i /∈ I. Since i /∈ I, each Ap is I-closed.
+Let x be any configuration in f −1(Ap), that is, fi(x) = p − 1. Since there are no arc from I to
+i, for every y ∈ �q�n with ∆(x, y) ⊆ I we have fi(y) = fi(x) = p − 1 and thus y ∈ f −1(Ap). As
+such, f −1(Ap) is I-closed. Since Aℓ and f −1(Ap) are I-closed for any p, ℓ ∈ [k], their intersection
+Aℓ ∩ f −1(Ap) is also I-closed, and so |Aℓ ∩ f −1(Ap)| is a multiple of qk, as desired.
+Proof of Lemma 9. For k = 0 the result is obvious, so assume that k ≥ 1.
+First, let f ∈ F(n, q) and suppose that the minimum strict in-degree of G(h) is at most
+n − k − 1 for some h ∼ f. By Lemma 10, h has a balanced k-nice partition and this trivially
+implies that f has a balanced k-nice partition. This proves the first direction.
+For the other direction, let f ∈ F(n, q) and suppose that f has a balanced k-nice partition
+A = {A1, . . . , Aq}. For all p, ℓ ∈ [q], let A−
+p = f −1(Ap) and ap,ℓ = |Ap ∩ A−
+ℓ |/qk, which is an
+integer since |Ap ∩ A−
+ℓ | is a multiple of qk. Note that, for all p ∈ [q], we have
+�
+ℓ∈[q]
+ap,ℓ = q−k �
+ℓ∈[q]
+|Ap ∩ A−
+ℓ | = q−k|Ap| = qn−k−1.
+12
+
+Hence, there is a partition Xp = {Xp,1, . . . , Xp,q} of �q�n−k−1 (with some members possibly
+empty) such that |Xp,ℓ| = ap,ℓ for all ℓ ∈ [q]. For all p, ℓ ∈ [q], let
+Yp,ℓ = {x ∈ �q�n | xn = p − 1, (x1, . . . , xn−k−1) ∈ Xp,ℓ}.
+Since each configuration in Xp,ℓ is extended into exactly qk configurations in Yp,ℓ, we have
+|Yp,ℓ| = ap,ℓqk = |Ap ∩ A−
+ℓ |. Furthermore, it is clear that Yp,ℓ ∩ Yp′,ℓ′ = ∅ for all p, ℓ, p′, ℓ′ ∈ [q]
+with (p, ℓ) ̸= (p′, ℓ′), so {Yp,ℓ | p, ℓ ∈ [q]} is a partition of �q�n. Hence, there is a permutation π
+of �q�n such that π(Ap ∩ A−
+ℓ ) = Yp,ℓ for all p, ℓ ∈ [q].
+Let h = π ◦ f ◦ π−1 and let us prove that G(h) has no arc from I = [n − k, n − 1] to n. So
+consider any x, y ∈ �q�n with ∆(x, y) ⊆ I, and let us prove that hn(x) = hn(y). Let p, ℓ ∈ [q]
+such that x ∈ Yp,ℓ. Since Yp,ℓ is I-closed, we also have y ∈ Yp,ℓ. Hence, setting x′ = π−1(x)
+and y′ = π−1(y), we have x′, y′ ∈ Ap ∩ A−
+ℓ , and thus f(x′), f(y′) ∈ Aℓ. So there are r, s ∈ [q]
+such that f(x′) ∈ Aℓ ∩ A−
+r and f(y′) ∈ Aℓ ∩ A−
+s , and we deduce that h(x) = π(f(x′)) ∈ Yℓ,r and
+h(y) = π(f(x′)) ∈ Yℓ,s. Consequently, hn(x) = hn(y) = ℓ − 1 as desired. Hence, G(h) has no arc
+from I to n, and thus the strict in-degree of n in G(h) is at most n − k − 1.
+3.1
+Upper bounds
+By the preceding characterization, to prove that δ−
+s (f) is small we have to prove that f has a
+balanced k-nice partition for some large k. We will prove that using two tools. The first is the
+following easy lemma. Given f ∈ F(n, q), we say that X ⊆ �q�n is a weakly independent set of
+f if f(x) /∈ X \ {x} for all x ∈ X (so X is an independent set that can contain loops).
+Lemma 11. Let f ∈ F(n, q) and Y ⊆ �q�n. Then f has a weakly independent set X ⊆ Y with
+|X| ≥ |Y |/3.
+Proof. For X ⊆ �q�n, we denote by Γ[X] the subgraph of Γ(f) induced by X.
+Let Z be a
+minimal subset of Y intersecting every cycle of Γ[Y ] of length at least 3. Let Y ′ = Y \ Z. Since
+|Z| ≤ |Y |/3 we have |Y ′| = |Y | − |Z| ≥ 2|Y |/3. Let Γ′ be obtained from Γ[Y ′] by removing
+loops. Then Γ′ is bipartite, as every cycle is of length 2. So Γ′ has an independent set X ⊆ Y ′
+with |X| ≥ |Y ′|/2 ≥ |Y |/3. Since X is an independent set of Γ′, there are no distinct x, y ∈ X
+such that f(x) = y and we deduce that X is a weakly independent set of f.
+The second tool comes from additive number theory and is based on the following definition.
+Definition 2 (s(m, d)). For any integers m, d ≥ 1, let s(m, d) be the smallest integer s such
+that, for any functions a1, . . . , ad : [s] → N, there is a subset X ⊆ [s] of size m such that
+�
+x∈X ai(x) is a multiple of m for every i ∈ [d].
+That s(m, d) exists, and is at most (m − 1)md − 1, is an easy consequence of the pigeonhole
+principle. Indeed, let us say that x, y ∈ [s] are equivalent if, for all i ∈ [d], we have ai(x) ≡ ai(y)
+mod m. There are at most md equivalence classes and so if s ≥ (m − 1)md + 1 then, by the
+pigeonhole principle, some class is of size at least m, and any subset X of size m in this class is
+as in the definition. But much better bounds exist. Erd¨os, Ginzburg and Ziv famously proved
+that s(m, 1) = 2m − 1 [3]. The exact value of s(m, 2) has been established by Reiher, resolving
+a longstanding conjecture of Kemnitz.
+Theorem 4 (Reiher [6]). For every m ≥ 1,
+s(m, 2) = 4m − 3.
+Hence, s(m, 1) and s(m, 2) are linear in m. Using deep arguments, Alon and Dubiner proved
+that s(m, d) is actually linear in m for every fixed d.
+13
+
+Theorem 5 (Alon-Dubiner [1]). There exists an absolute constant c such that, for all m, d ≥ 1,
+s(m, d) ≤ (cd log2 d)dm.
+We now prove that any f ∈ F(n, q) has a balanced k-nice partition for some large k, expressed
+with the quantity s(m, d). We begin with the case q = 2, where we optimize the use of s(m, d) in
+order to use the exact value given by Reiher for d = 2. The general case, where such optimization
+is not needed, follows the same approach and is presented afterwards.
+Lemma 12. Every f ∈ F(n, 2) has a balanced k-nice partition for any k such that
+3s(2k, 2) ≤ 2n−1.
+Proof. Let f ∈ F(n, 2). For X ⊆ {0, 1}n, we use X− as a shorthand for f −1(X). Let k be as in
+the statement. Observe that, by setting A1 = {0, 1}n and A2 = ∅, we obtain a k-nice partition
+{A1, A2} since |A1| = |A1 ∩ A−
+1 | = 2n and |A2| = |A1 ∩ A−
+2 | = |A2 ∩ A−
+1 | = |A2 ∩ A−
+2 | = 0. So
+there is a k-nice partition {A1, A2} with |A1| ≥ 2n−1 where A1 is minimal for this property. We
+will prove that |A1| = 2n−1 so that the partition is balanced. Suppose, for a contradiction, that
+|A1| > 2n−1.
+By Lemma 11, there is a subset Y ⊆ A1 of size at least |A1|/3 which is a weakly independent
+set. For x ∈ Y , let
+α1(x) = |A1 ∩ f −1(x)|,
+α2(x) = |A2 ∩ f −1(x)|,
+β1(x) =
+�
+1
+if f(x) ∈ A1 \ {x},
+0
+otherwise,
+Since |A1| ≥ 2n−1 we have |Y | ≥ |A1|/3 ≥ 2n−1/3 ≥ s(2k, 2). Considering the 2 functions
+a1, a2 defined on Y by a1(x) = α1(x) + β1(x) and a2(x) = α2(x) − β1(x), we deduce from the
+definition of s that there is a subset X ⊆ Y of size 2k such that �
+x∈X a1(x) and �
+x∈X a2(x) are
+multiples of 2k. Setting αi(X) = �
+x∈X αi(x) for i = 1, 2 and β1(X) = �
+x∈X β1(x) we obtain
+that α1(X) + β1(X) and α2(X) − β1(X) are multiples of 2k. Furthermore:
+αi(X) = |Ai ∩ X−| for i = 1, 2,
+(1)
+β1(X) = |X ∩ A−
+1 | − |X ∩ X−|.
+(2)
+(1) is obvious, since
+|Ai ∩ X−| = |Ai ∩ (∪x∈Xf −1(x))| =
+�
+x∈X
+|Ai ∩ f −1(x)| = αi(X).
+For (2), note that β1(X) is the number of x ∈ X with f(x) ∈ A1 \ {x}. Since X is weakly
+independent, f(x) ∈ A1 \ {x} is equivalent to f(x) ∈ A1 \ X, and thus β1(X) is the number of
+x ∈ X with f(x) ∈ A1 \ X, that is, β1(X) = |X ∩ (A1 \ X)−| = |X ∩ A−
+1 | − |X ∩ X−|.
+Consider the partition
+{B1, B2} = {A1 \ X, A2 ∪ X}.
+Recall that {A1, A2} is k-nice, hence |A1| is a multiple of 2k, and since we also have |A1| > 2n−1,
+it follows that |B1| = |A1| − 2k ≥ 2n−1.
+As a consequence, in order to obtain the desired
+contradiction, it is sufficient to prove that {B1, B2} is k-nice. First, since |X| = 2k, it is clear
+14
+
+that |B1| and |B2| are multiples of 2k. So it remains to prove that |Bi ∩ B−
+j | is a multiple of 2k
+for every 1 ≤ i, j ≤ 2. First, using (1) and (2), we have:
+|B1 ∩ B−
+1 | = |(A1 \ X) ∩ (A−
+1 \ X−)|
+= |A1 ∩ A−
+1 | − |A1 ∩ X−| − |X ∩ A−
+1 | + |X ∩ X−|
+= |A1 ∩ A−
+1 | − α1(X) − β1(X)
+|B2 ∩ B−
+1 | = |(A2 ∪ X) ∩ (A−
+1 \ X−)|
+= |A2 ∩ A−
+1 | − |A2 ∩ X−| + |X ∩ A−
+1 | − |X ∩ X−|
+= |A2 ∩ A−
+1 | − α2(X) + β1(X).
+Since |A1 ∩ A−
+1 |, |A2 ∩ A−
+1 |, α1(X) + β1(X) and α2(X) − β1(X) are multiples of 2k, we deduce
+that |B1 ∩ B−
+1 | and |B2 ∩ B−
+1 | are multiples of 2k. Then we have
+|B1 ∩ B−
+2 | = |B1| − |B1 ∩ B−
+1 |
+|B2 ∩ B−
+2 | = |B2| − |B2 ∩ B−
+1 |.
+Since |B1|, |B2|, |B1 ∩ B−
+1 | and |B2 ∩ B−
+1 | are multiples of 2k, we deduce that |B1 ∩ B−
+2 | and
+|B2 ∩ B−
+2 | are multiples of 2k. Thus, {B1, B2} is k-nice, a contradiction.
+Combining this lemma with Lemma 9 and Reiher’s theorem, which says that s(m, 2) =
+4m − 3, we obtain the first upper bound in Theorem 2.
+Lemma 13. For all n ≥ 1 we have δ−(n, 2) ≤ 5, and δ−(5, 2) ≤ 4.
+Proof. Let f ∈ F(n, 2). Using Theorem 4 we have
+3s(2n−5, 2) = 12 · 2n−5 − 9 ≤ 16 · 2n−5 = 2n−1.
+Hence, by Lemma 12, f has a balanced (n−5)-nice partition. Thus, δ−
+s (f) ≤ n−(n−5)−1 = 4
+by Lemma 9. This proves that δ−(n, 2) ≤ 5. Then, since 3s(2, 2) = 15 ≤ 24, we deduce from
+Lemma 12 that every f ∈ F(5, 2) has a 1-nice partition and thus δ−
+s (f) ≤ 3 by Lemma 9. This
+proves that δ−(5, 2) ≤ 4.
+We now extend the previous arguments to larger alphabets.
+Lemma 14. Every f ∈ F(n, q) has a balanced k-nice partition for any k such that
+3s(qk, 2q) ≤ qn−1.
+Proof. Let f ∈ F(n, q). For X ⊆ �q�n, we use X− as a shorthand for f −1(X). Let k be as in
+the statement. Given a partition A = {A1, . . . , Aq} of �q�n, we say that A is 1-dominating if
+|A1| ≥ qn−1 and |Ai| ≤ qn−1 for all 2 ≤ i ≤ q. Observe that, by setting A1 = �q�n and Ai = ∅
+for 2 ≤ i ≤ q, we obtain a 1-dominating k-nice partition: for every 2 ≤ i ≤ q and 1 ≤ j ≤ q, we
+have |A1| = |A1 ∩ A−
+1 | = qn and |Ai| = |A1 ∩ A−
+i | = |Ai ∩ A−
+j | = 0. So we can suppose that A is
+a 1-dominating k-nice partition with |A1| minimal.
+We will prove that |A1| = qn−1, which (since A is 1-dominating) forces A to be balanced.
+Suppose, for a contradiction, that |A1| > qn−1. Since A is 1-dominating there is 2 ≤ i ≤ n such
+that |Ai| < qn−1. Suppose, without loss, that |A2| < qn−1.
+By Lemma 11, there is a subset Y ⊆ A1 of size at least |A1|/3 which is a weakly independent
+set. For x ∈ Y and 1 ≤ i ≤ q let
+αi(x) = |Ai ∩ f −1(x)|,
+15
+
+βi(x) =
+�
+1
+if f(x) ∈ Ai \ {x},
+0
+otherwise,
+β′
+i(x) =
+�
+βi(x) + 1
+if i = 2 and f(x) = x,
+βi(x)
+otherwise.
+Since |A1| ≥ qn−1 we have |Y | ≥ |A1|/3 ≥ qn−1/3 ≥ s(qk, 2q). Considering the 2q functions
+αi, β′
+i, we deduce from the definition of s that there is a subset X ⊆ Y of size qk such that αi(X) =
+�
+x∈X αi(x) and β′
+i(X) = �
+x∈X β′
+i(x) are multiples of qk for all 1 ≤ i ≤ q. Furthermore:
+αi(X) = |Ai ∩ X−| for all 1 ≤ i ≤ q,
+(1)
+βi(X) = |X ∩ A−
+i | for all 2 ≤ i ≤ q,
+(2)
+β′
+i(X) = |X ∩ A−
+i | for all 3 ≤ i ≤ q,
+(3)
+β′
+1(X) = |X ∩ A−
+1 | − |X ∩ X−|,
+(4)
+β′
+2(X) = |X ∩ A−
+2 | + |X ∩ X−|.
+(5)
+(1) is obvious, since
+|Ai ∩ X−| = |Ai ∩ (∪x∈Xf −1(x))| =
+�
+x∈X
+|Ai ∩ f −1(x)| = αi(X).
+For (2), note that βi(X) is the number of x ∈ X with f(x) ∈ Ai \ {x}, which is equivalent to
+f(x) ∈ Ai since 2 ≤ i ≤ q and x ∈ A1. So βi(X) is the number of x ∈ X with f(x) ∈ Ai, that is,
+βi(X) = |X ∩ A−
+i |. (3) results from (2) since β′
+i = βi for 3 ≤ i ≤ q. For (4), note that β′
+1 = β1
+and that β1(X) is the number of x ∈ X with f(x) ∈ A1 \ {x}. Since X is weakly independent,
+f(x) ∈ A1 \ {x} is equivalent to f(x) ∈ A1 \ X, and thus β′
+1(X) is the number of x ∈ X with
+f(x) ∈ A1 \ X, that is, β′
+1(X) = |X ∩ (A1 \ X)−| = |X ∩ A−
+1 | − |X ∩ X−|. To prove (5), let ℓ be
+the number of x ∈ X with f(x) = x. It is clear that β′
+2(X) = β2(X) + ℓ, and using the second
+point we obtain β′
+2(X) = |X ∩ A−
+2 | + ℓ. But since X is weakly independent, for x ∈ X we have
+f(x) = x if and only if f(x) ∈ X, and thus ℓ = |X ∩ X−| so that β′
+2(X) = |X ∩ A−
+2 | + |X ∩ X−|.
+Consider the partition
+B = {B1, B2, B3, . . . , Bq} = {A1 \ X, A2 ∪ X, A3, . . . , Aq}.
+Recall that A is k-nice, hence |A1|, |A2| are multiples of qk, and since we also have |A1| > qn−1
+and |A2| < qn−1, it follows that |B1| = |A1| − qk ≥ qn−1 and |B2| = |A2| + qk ≤ qn−1, therefore
+B is 1-dominating. As a consequence, in order to obtain the desired contradiction, it is sufficient
+to prove that B is k-nice. First, since |X| = qk, it is clear that |Bi| is a multiple of qk for every
+1 ≤ i ≤ q. So it remains to prove that |Bi ∩ B−
+j | is a multiple of qk for every 1 ≤ i, j ≤ q. This
+is obvious for 3 ≤ i, j ≤ q since |Bi ∩ B−
+j | = |Ai ∩ A−
+j |. We show that the other intersection sizes
+are multiples of qk by expressing these sizes as a sum of (positive or negative) numbers that are
+already known to be multiples of qk. For that we use many times the properties (1),(3),(4),(5).
+We begin with the case 1 ≤ i, j ≤ 2:
+|B1 ∩ B−
+1 | = |(A1 \ X) ∩ (A−
+1 \ X−)|
+= |A1 ∩ A−
+1 | − |A1 ∩ X−| − |X ∩ A−
+1 | + |X ∩ X−|
+= |A1 ∩ A−
+1 | − α1(X) − β′
+1(X)
+|B1 ∩ B−
+2 | = |(A1 \ X) ∩ (A−
+2 ∪ X−)|
+16
+
+= |A1 ∩ A−
+2 | + |A1 ∩ X−| − |X ∩ A−
+2 | − |X ∩ X−|
+= |A1 ∩ A−
+2 | + α1(X) − β′
+2(X)
+|B2 ∩ B−
+1 | = |(A2 ∪ X) ∩ (A−
+1 \ X−)|
+= |A2 ∩ A−
+1 | − |A2 ∩ X−| + |X ∩ A−
+1 | − |X ∩ X−|
+= |A2 ∩ A−
+1 | − α2(X) + β′
+1(X)
+|B2 ∩ B−
+2 | = |(A2 ∪ X) ∩ (A−
+2 ∪ X−)|
+= |A2 ∩ A−
+2 | + |A2 ∩ X−| + |X ∩ A−
+2 | + |X ∩ X−|
+= |A2 ∩ A−
+2 | + α2(X) + β′
+2(X).
+For 1 ≤ i ≤ 2 and 3 ≤ j ≤ q we have:
+|B1 ∩ B−
+j | = |(A1 \ X) ∩ A−
+j |
+= |A1 ∩ A−
+j | − |X ∩ A−
+j |
+= |A1 ∩ A−
+j | − β′
+j(X)
+|B2 ∩ B−
+j | = |(A2 ∪ X) ∩ A−
+j |
+= |A2 ∩ A−
+j | + |X ∩ A−
+j |
+= |A2 ∩ A−
+j | + β′
+j(X).
+Finally, for 3 ≤ i ≤ q and 1 ≤ j ≤ 2, we have:
+|Bi ∩ B−
+1 | = |Ai ∩ (A−
+1 \ X−)|
+= |Ai ∩ A−
+1 | − |Ai ∩ X−|
+= |Ai ∩ A−
+1 | − αi(X)
+|Bi ∩ B−
+2 | = |Ai ∩ (A−
+2 ∪ X−)|
+= |Ai ∩ A−
+2 | + |Ai ∩ X−|
+= |Ai ∩ A−
+2 | + αi(X).
+So |Bi ∩ B−
+j | is indeed a multiple of qk for all 1 ≤ i, j ≤ q, thus B is k-nice, a contradiction.
+Combining this lemma with Lemma 9 and Alon-Dubiner’s theorem, which says that s(m, d) ≤
+(cd log2 d)dm for some constant c, we obtain the second upper bound in Theorem 2.
+Lemma 15. For all n ≥ 1 and q ≥ 2, we have δ−(n, q) ≤ (2 + o(1))q.
+Proof. Let f ∈ F(n, q), let α(q) = 2c log2(2q) where c is as in Theorem 5, and let k = ⌊ℓ⌋ with
+ℓ = n − 2q logq(qα(q)) − 1 − logq 3.
+Using Theorem 5 for the first inequality, we have
+3s(qk, 2q) ≤ 3(qα(q))2qqk ≤ 3(qα(q))2qqℓ = qn−1.
+Hence, by Lemma 14, f has a balanced k-nice partition, and thus δ−
+s (f) ≤ n−k−1 by Lemma 9.
+So δ−(f) ≤ n − k, and since k ≥ ℓ − 1 we have
+δ−(f) ≤ 2q logq(qα(q)) + 2 + logq 3 =
+�
+2 + 2 logq(α(q)) + (2 + logq 3)q−1�
+q.
+Since 2 logq(α(q)) + (2 + logq 3)q−1 tends to 0 as q tends to infinity, this proves the lemma.
+17
+
+3.2
+Lower bound
+It remains to prove the lower bound in Theorem 2, which is based on a simple construction.
+Lemma 16. For all n, q ≥ 2 with n ≥ 3 or q ≥ 3, we have δ−(n, q) ≥ 2.
+Proof. Let f ∈ F(n, q) with n, q ≥ 2 and n ≥ 3 or q ≥ 3. Suppose that f has a limit cycle
+of length ℓ = qn−1 + 1, whose configurations are x1, . . . , xℓ in order, and suppose that x1 is
+the image of the all configurations which are not in the limit cycle; so f has exactly ℓ images.
+See Figure 10 for an illustration. We will prove that G(f) has minimum in-degree at least two,
+and since f is described up to isomorphism this proves that δ−(f) ≥ 2 and the lemma follows.
+Suppose, for a contradiction, that G(f) has a vertex i with in-degree at most 1.
+If i is of in-degree 0, then fi is a constant function, which always returns some a ∈ �q�.
+Hence, xi = a for all the images x of f, and thus f has at most qn−1 < ℓ images, a contradiction.
+So i is of in-degree 1. Let j be its in-neighbor (we may have i = j). There is then a non-
+constant function g : �q� → �q� such that fi(x) = g(xj) for all x ∈ �q�n. Let Y = {x1, . . . , xℓ−1}
+and note that Y = �q�n \ f −1(x1) by construction. For a ∈ �q� we set Xi,a = {x ∈ �q�n |
+xi = a} and Xj,a = {x ∈ �q�n | xj = a}. If Xj,a is not included in Y , that is if there is a
+configuration x with xj = a and f(x) = x1, then g(a) = fi(x) = x1
+i . We deduce that if Xj,a is
+not included in Y for every a ∈ �q�, then g is a constant function (which always return x1
+i ), a
+contradiction. So Xj,a ⊆ Y for some a, and since Y is of size ℓ − 1 = qn−1 = |Xj,a| we deduce
+that Y = Xj,a. Hence, setting b = g(a), we have f(Y ) ⊆ Xi,b, and since f(Y ) = {x2, . . . , xℓ}
+is of size ℓ − 1 = qn−1 = |Xi,b| we deduce that f(Y ) = Xi,b. Consequently, Xj,a and Xi,b are
+distinct. Furthermore, they are not disjoint since Xj,a ∩ Xi,b = Y ∩ f(Y ) = {x2, . . . , xℓ−1} is
+of size ℓ − 2 = qn−1 − 1 > 0. We deduce that i ̸= j and thus |Xj,a ∩ Xi,b| = qn−2. We obtain
+qn−2 = qn−1 − 1, which is a contradiction since n ≥ 3 or q ≥ 3.
+x1
+x2
+x3
+x4
+x5
+•
+•
+•
+Figure 10: The function of Lemma 16 for n = 3 and q = 2.
+4
+Dense digraphs
+In this section, we will prove Theorem 3, that for n, q ≥ 2 there is f ∈ F(n, q) such that every
+digraph in G(f) has at least ⌊n2/4⌋ arcs. We need two lemmas. In a digraph, the sources are
+the vertices of in-degree 0, and the non-sources are the vertices of in-degree at least 1.
+Lemma 17. Let f ∈ F(n, q) and let X be the images of f. Let k ∈ [n] and suppose that, for
+all non-empty strict subsets Y of X, |f −1(Y )| is not a multiple of qk. Then every non-source of
+G(f) has strict in-degree at least n − k.
+18
+
+Proof. Suppose, for a contradiction, that G(f) has a non-source i with strict in-degree at most
+n − k − 1. For 1 ≤ p ≤ q, let Ap be the set of x ∈ �q�n with xi = p − 1. By Lemma 10,
+A = {A1, . . . , Aq} is a balanced k-nice partition of f. Let 1 ≤ p ≤ q such that X ∩ Ap ̸= ∅ and
+let Y = X ∩ Ap. If X ⊆ Ap, then it means that fi is a constant function (which always returns
+p), which is a contradiction because i has in-degree at least 1. We deduce that Y is a non-
+empty strict subset of X. Furthermore, we have |f −1(Y )| = |f −1(Ap)| = �q
+ℓ=1 |Aℓ ∩ f −1(Ap)|.
+Since A is k-nice, each term of the sum is a multiple of qk. So |f −1(Y )| is a multiple of qk, a
+contradiction.
+Lemma 18. For all n, q ≥ 2 and k ∈ [n], there is f ∈ F(n, q) such that, for every G ∈ G(f), G
+has at least k non-sources and all the non-sources of G have strict in-degree at least n − k.
+Proof. Let X be a subset of �q�n of size qk and let x∗ ∈ X. Let f ∈ F(n, q) such that for all
+x ∈ �q�n, f(x) = x if x ∈ X and f(x) = x∗ otherwise. See Figure 11 for an illustration. The
+images of f are thus exactly X. We will prove that G(f) has at least k non-sources, each of
+strict in-degree at least n − k. Since f is described up to isomorphism, this proves the lemma.
+We first prove that every non-source of G(f) has strict in-degree at least n−k. Let Y be any
+non-empty strict subset of X, and let Y − = f −1(Y ). We have 0 < |Y | < |X| = qk thus |Y | is not
+a multiple of qk. We will show that |Y −| is not a multiple of qk. Note that |f −1(x∗)| = qn−qk+1
+and |f −1(x)| = 1 for all x ∈ X \ {x∗}. Thus, if x∗ /∈ Y then |Y −| = |Y | is not a multiple of qk.
+If x∗ ∈ Y then |Y −| = |f −1(x∗)| + |Y | − 1 = qn − qk + |Y |, and since |Y | is not a multiple of qk
+we deduce that |Y −| is not a multiple of qk. Consequently, by Lemma 17, every non-source of
+G(f) has strict in-degree at least n − k.
+Let ℓ be the number of non-sources of G(f). It remains to prove that ℓ ≥ k. Let I be the
+set of sources of G(f). For each i ∈ I, fi is a constant function which always returns some
+member ai ∈ �q�. Let Z be the set of x ∈ �q�n with xi = ai for all i ∈ I. We have X ⊆ Z, so
+qk = |X| ≤ |Z| = qn−|I| = qℓ. Thus, ℓ ≥ k as desired.
+x∗
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+Figure 11: The function of Lemma 18 for n = 4 and q = k = 2.
+The proof of Theorem 3 is now immediate: by Lemma 18 applied with k = ⌊n/2⌋, there
+is f ∈ F(n, q) such that every digraph in G(f) has at least ⌊n/2⌋ vertices of in-degree at least
+n − ⌊n/2⌋, and thus at least ⌊n/2⌋(n − ⌊n/2⌋) = ⌊n2/4⌋ arcs.
+5
+Concluding remarks
+• Let us say that a digraph G is (n, q)-universal if G is isomorphic to some digraph in G(f)
+for every f ∈ F(n, q) with f ̸= id, cst. Theorem 1 says that Kn is (n, q)-universal if n ≥ 5
+or q ≥ 3. Actually, for n, q ≥ 2, any (n, q)-universal digraph is isomorphic to Kn. Indeed,
+suppose G is (n, q)-universal. First, if f ∈ F(n, q) is a bijection, then every digraph in
+G(f) has no source. Thus, G has no source. Then, by Lemma 18 applied with k = 1,
+there is f ∈ F(n, q) such that any digraph in G(f) has at least one non-source, and each
+19
+
+non-source has strict in-degree n − 1. So G has this property, and since it has no source,
+we deduce that each vertex has strict in-degree n − 1. Finally, if f ∈ F(n, q) has at least
+qn−1 + 1 fixed points, then we easily check that any digraph in G(f) has n loops. So G
+has n loops and we deduce that it is isomorphic to Kn.
+• We proved that there is no f ∈ F(n, q) such that G(f) only contains Kn when n is large
+compared to q, since we proved that δ−(n, q) ≤ cqq for some constant cq that only depends
+on q, and thus δ−(n, q) < n for n > cqq. In particular, for n ≥ 5, there is no f ∈ F(n, 2)
+such that G(f) = {Kn} since we proved that δ−(5, 2) ≤ 4 and δ−(n, 2) ≤ 5. However, for
+every q ≥ 3, there is f ∈ F(2, q) such that G(f) = {K2} since we proved that δ−(2, q) = 2.
+Does this phenomenon hold for every n, that is, for every fixed n and q large enough with
+respect to n, is there f ∈ F(n, q) such that G(f) = {Kn}? An equivalent formulation is:
+does δ−(n, q) necessarily increase with q? Also, we proved that 2 ≤ δ−(n, 2) ≤ 5, for every
+n ≥ 3, and we may ask if δ−(n, 2) is a constant, and if so which one of the four possible
+values it takes.
+• Let d(n, q) be the maximum real d ∈ [0, 1] such that there is f ∈ F(n, q) such that every
+digraph in G(f) has dn2 arcs, that is, has density d. We prove that d(n, q) ≥ 1/4 − 1/n2
+where 1/n2 tends to 0 as n tends to infinity, i.e. is o(1).
+Hence, this lower bound is
+independent of the alphabet size q, and we may ask, as above, whether d(n, q) is necessarily
+increasing with q.
+Acknowledgments
+This work was supported by the Young Researcher project ANR-18-
+CE40-0002-01 “FANs”.
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+[7] Ren´e Thomas and Marcelle Kaufman. Multistationarity, the basis of cell differentiation and
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+20
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+Best Arm Identification in Stochastic Bandits: Beyond β−optimality
+Arpan Mukherjee
+Ali Tajer *
+Abstract
+This paper focuses on best arm identification (BAI) in stochastic multi-armed bandits (MABs) in the fixed-
+confidence, parametric setting. In such pure exploration problems, the accuracy of the sampling strategy critically
+hinges on the sequential allocation of the sampling resources among the arms. The existing approaches to BAI ad-
+dress the following question: what is an optimal sampling strategy when we spend a β fraction of the samples on
+the best arm? These approaches treat β as a tunable parameter and offer efficient algorithms that ensure optimality
+up to selecting β, hence β−optimality. However, the BAI decisions and performance can be highly sensitive to the
+choice of β. This paper provides a BAI algorithm that is agnostic to β, dispensing with the need for tuning β, and
+specifies an optimal allocation strategy, including the optimal value of β. Furthermore, the existing relevant literature
+focuses on the family of exponential distributions. This paper considers a more general setting of any arbitrary family
+of distributions parameterized by their mean values (under mild regularity conditions).
+1
+Introduction
+We consider the problem of best arm identification (BAI) in stochastic multi-armed bandits in the fixed-confidence
+setting. The bandit instances are assumed to be generated by a general class of distributions parameterized by their
+mean values. Under some mild assumptions on the probability models, this model is general and subsumes the
+common classes of distributions investigated in bandit models (e.g., Gaussian, exponential, and Bernoulli bandits). In
+BAI, the objective is to identify the best arm (i.e., the arm with the largest mean value) within a pre-specified level
+of confidence with the fewest number of samples. In this paper, we focus on the open problem of BAI in the fixed-
+confidence parametric setting. Next, we review the relevant literature on BAI to provide context for specifying our
+BAI setting and discussing our contributions.
+Fixed- Confidence versus Budget.
+BAI was first studied as a pure exploration bandit problem in [1]. Subsequently,
+it has been investigated in two broad settings: the fixed-confidence setting and the fixed-budget setting. The goal in the
+fixed-confidence setting is to identify the best arm within a specified guarantee on the decision confidence while using
+as few samples as possible to arrive at a decision. Representative studies in the fixed-confidence setting include [2–7].
+On the other hand, in the fixed-budget setting, the sampling budget is pre-specified. The goal is to minimize the
+probability of error in the terminal decision. Representative studies in this regime include [1,8,9]. Our focus is on the
+fixed-confidence setting, the literature on which is discussed next.
+*The authors are with the Electrical, Computer, and System Engineering Department, Rensselaer Polytechnic Institute.
+1
+arXiv:2301.03785v1  [stat.ML]  10 Jan 2023
+
+Bayesian versus Non-Bayesian.
+BAI in the fixed-confidence setting can be categorized into Bayesian and non-
+Bayesian models. Bayesian settings assume a prior distribution on the space of parameters and make arm selection
+decisions based on the posterior distribution computed from the prior and the observed rewards. In contrast, the non-
+Bayesian settings do not use posterior sampling for arm selection. Top-two sampling was first introduced in [10] for
+the Bayesian setting. The principle of top-two sampling involves dynamically, over time, identifying a leader and a
+challenger as the top arm candidates. Subsequently, the sampling strategy randomizes between these two arms. The
+top-two Thompson sampling (TTTS) algorithm, proposed and analyzed in [10, 11], involves sampling the posterior
+for defining the leader and the challenger. Despite the simplicity of TTTS, it faces the computational challenge of
+repeatedly sampling from the posterior in defining a challenger. To mitigate this, [11] proposed a computationally
+efficient alternative called the top-two transportation cost (T3C) algorithm. The empirical performance of T3C was
+further improved using a penalized transportation cost, promoting exploration, in [12].
+In the non-Bayesian setting, [4] has proposed the track-and-stop (TaS) algorithm for BAI, with optimal perfor-
+mance in the asymptote of diminishing probability of error. More investigations on TaS-based algorithms include [13]
+and [14]. The TaS sampling strategies, in general, hinge on tracking the optimal allocation of sampling resources
+over time. Maintaining such allocation in a bandit setting with K arms necessitates solving K equations at each time
+using the bisection method. Hence, these approaches are generally computationally expensive. It was shown in [14]
+that the tracking procedure could only be performed intermittently at exponentially spaced intervals. This reduces the
+computational complexity for TaS. Nevertheless, the approach of [14] only applies to linear bandits with Gaussian
+noise. To address this computational challenge, the top-two approach has also been used to design algorithms for the
+non-Bayesian setting. Representative top-two non-Bayesian algorithms include top-two sequential probability ratio
+test (TT-SPRT) [7,12,15], top-two expected improvement (TTEI) [16], and the empirical best leader with an improved
+transportation cost (EB-TCI) [12]. While efficient, these algorithms achieve optimality performance only up to a tun-
+able parameter. Specifically, these algorithms, at their core, identify an optimal allocation of the sampling resources
+among the arms. The nature of the optimality guarantees offered by the top-two algorithms in the non-Bayesian setting
+is as follows: if a β fraction of the sampling resources are reserved for the top arm, then these algorithms can determine
+how to optimality allocate the remaining (1 − β) fraction among the rest of the arms. Hence, these algorithms are
+said to be only β−optimal, where β ∈ (0, 1) is treated as a tunable parameter. In some settings, the choice of β can
+critically affect the performance.
+Parametric versus Non-parametric.
+Algorithms for BAI can also be categorized based on whether the bandit
+instance follows parametric or non-parametric families of distributions. In the case of the non-parametric family, the
+upper confidence bound (UCB) based approaches have been investigated (e.g., [2, 5]) and shown to be optimal up to
+constant factors for the family of sub-Gaussian bandits. More recently, the study in [13] has considered the class of
+distributions satisfying a specific functional boundedness property. This study proposes a tracking-based sampling
+strategy along with a likelihood ratio-based stopping rule, which was shown to be asymptotically optimal for this
+specified class of distributions. For parametric bandits, most studies have focused on the single parameter exponential
+family (e.g., [4,7,10,12,17]). Some investigations have considered the case of Gaussian bandits with known variances
+and unknown means, e.g., [11, 16]. In both of these settings, the top-two sampling strategy is β−optimal, while the
+TaS algorithm has been shown to be asymptotically optimal, despite being computationally expensive.
+2
+
+Contributions.
+This paper is focused on fixed-confidence, non-Bayesian, and parametric settings. Hence, the di-
+rectly relevant literature to this scope includes [4,7,11,12,16]. We propose an algorithm, referred to as the transporta-
+tion cost balancing (TCB) algorithm, that addresses two open problems. First, we dispense with the need for treating
+β as a tuning parameter. Specifically, the TCB algorithm is agnostic to β and can specify its unique optimal value as a
+by-product. In order to showcase the importance of determining β and the gap between a β−optimal solution and an
+optimal solution, in Figure 1 we show how the sample complexity of the β−optimal solutions vary with respect to β,
+demonstrating (i) the sensitivity of the sample complexity to β, and (ii) a substantial gap (e.g., an order of magnitude)
+between the optimal and the β−optimal guarantees. β = 0.5 has been prescribed as a good choice for top-two algo-
+rithms [10]. Nevertheless, in many instances that can be far from the optimal choice. For instance, Figure 1 confirms
+demonstrates sub-optimality of choosing β = 0.5 in three existing approaches that ensure β−optimality. In all these
+cases, the optimal value is β ≈ 0.27. As the second contribution, we also generalize the probability models to any
+arbitrary class of parametric models (that satisfy certain regularity conditions). These models subsume the exponen-
+tial family, which is the only parametric class for which algorithms and performance guarantees are available in the
+literature.
+0.2
+0.3
+0.4
+0.5
+0.6
+5000
+6000
+7000
+8000
+9000
+10000
+11000
+12000
+TT-SPRT
+T3C
+TS-TCI
+Figure 1: Sample complexity versus β.
+Methodology: Transport Cost Balancing (TCB).
+For fixed-confidence BAI, for any bandit instance ν, the univer-
+sal lower bound on the average sample complexity is inversely proportional to a problem complexity measure Γ(ν)
+(specified later in (27)). Achieving this lower bound is predicated on sequentially determining an optimal allocation
+of the sampling resources among the arms. While the TaS algorithm in [4] computes these optimal allocations, it is
+computationally expensive. To mitigate this computational challenge, the top-two sampling rules assign a sampling
+proportion β ∈ (0, 1) to the best arm, where β is a tunable parameter and, then determine the allocation of the re-
+maining (1 − β) fraction over the rest. The analysis of the top-two methods shows that this facilitates convergence
+to the β−optimal allocation. In these methods, the sample complexity is inversely proportional to the transportation
+3
+
+cost Γβ(ν), where Γ(ν) ≥ Γβ(ν). Equality holds at the optimal value β⋆, which depends on the bandit instance
+and is unknown a priori. In our approach, essentially, we also determine the optimal value of β, which was shown
+by [10] to have a unique solution. Our algorithm guides its decisions by balancing transportation costs over time.
+These decisions lead to efficiently estimating the optimal sampling proportion. The fundamental advantage of our arm
+selection rules is that they can track the sampling proportions without having to compute the sampling proportions
+at each round. Rather, the sampling proportion is estimated by sampling from the set of under-sampled arms in each
+round.
+2
+Stochastic BAI Model and Assumption
+Stochastic Model. Denote the class of probability measures defined on any sample space Ω ⊆ R by Q(Ω). Let
+P(Ω) ⊂ Q(Ω) denote the class of probability measures that are parameterized by their mean values, i.e.,
+P(Ω) ≜ {P ∈ Q(Ω) : m(P) ∈ Θ} ,
+(1)
+where m(P) ≜ EP[X], and EP denotes the expectation under measure P. Furthermore, define M ≜ P⊗K(Ω) as the
+Cartesian product of K sets of measures in P(Ω). We denote the likelihood function associated with measure P by
+πP. We make the following assumptions on this stochastic model.
+1. Θ ⊆ R is a compact parameter space.
+2. All distributions in P(Ω) have the same support Ω.
+3. The likelihood functions πP are continuous and differentiable in m(P) for every P ∈ P(Ω).
+4. There exists σ2 > 0 such that for any P ∈ P(Ω),
+∂2 log πP(x | θ)
+∂θ2
+≤ −σ2 ,
+∀x ∈ Ω, ∀θ ∈ Θ .
+(2)
+The assumption in (2) specifies a bound on the second order derivative of the log-likelihood function over the
+parameter space Θ. For instance, for the exponential family of distributions, (2) is equivalent to the variance
+being bounded away from zero.
+5. There exists c > 0 such that for any P ∈ P(Ω), for any x, y ∈ Ω and for all θ ∈ Θ,
+����
+∂ log πP(x | θ)
+∂θ
+− ∂ log πP(y | θ)
+∂θ
+���� ≤ c|x − y| .
+(3)
+The condition in (3) is a Lipschitz-related assumption on the derivative of the log-likelihood function with
+respect to the realizations.
+6. For any θ, θ′ ∈ Θ, we assume that
+EP
+�����log πP(X | θ)
+πP(X | θ′)
+����
+2�
+< +∞ .
+(4)
+The condition in (4) implies that the log-likelihood ratio corresponding to parameters θ and θ′ has a finite second
+moment. The condition in (4) holds for a wide variety of distributions, such as the exponential family and the
+log-normal distribution.
+4
+
+7. Any P ∈ P(Ω) satisfies the L1 transportation inequality. Specifically, for any P ∈ P(Ω) and any reference
+measure Q, we assume that there exists a universal constant c′ ∈ R+ such that
+W1(P, Q) ≤
+�
+2c′DKL(P∥Q) ,
+(5)
+where W1 denotes the Wasserstein distance metric between the measures P and Q, and DKL represents the
+Kullback-Leibler (KL) divergence.
+Assumptions 1-3 ensure the existence of the maximum likelihood estimates (MLEs) for the parameters of interest (i.e.,
+the mean values) and that the KL divergence measures between any pair of distributions in M are finite. Assumptions
+3-6 are mild regularity conditions on the log-likelihood functions and are satisfied by a wide variety of distributions.
+For instance, the exponential family of distributions satisfies these assumptions, which includes important bandit
+models such as Gaussian and Bernoulli bandits. Examples not from the exponential family include the log-normal
+distribution known up to one of the mean values and mixture distributions. Finally, let (M, DTV) denote the metric
+space of the distributions in the set M endowed with the total variation distance metric DTV. The environment M
+subsumes several classes of common parametric distribution families, such as the scalar parameter exponential family,
+which has been widely investigated in the BAI literature [4,7,10,12].
+Bandit Model.
+Consider a K-armed stochastic bandit. The rewards of arm i ∈ [K] ≜ {1, · · · , K} are generated
+from Pi ∈ P(Ω). We denote the likelihood function associated with Pi by πi, and define µ(i) ≜ m(Pi). Accordingly,
+we define the bandit instance ν ≜ [P1, . . . , PK].
+Sequential Decisions.
+At each round t ∈ N, the learner chooses an action At ∈ [K], and receives a reward Xt ∼
+PAt. We denote the sequence of actions, the corresponding rewards, and the filtration generated by the sequence of
+actions and rewards by the ordered sets
+At ≜ {As : s ∈ [t]} ,
+Xt ≜ {Xs : s ∈ [t]} ,
+and
+Ft ≜ {A1, X1, · · · , At, Xt} .
+(6)
+Denote the set of rewards obtained by selecting an arm i ∈ [K] up till time t by
+X i
+t ≜ {Xs : s ∈ [t], As = i} .
+(7)
+The objective of the learner is to identify the best arm, which is assumed to be unique, and is defined as the arm with
+the largest mean, i.e.,
+a⋆ ≜ arg max
+i∈[K]
+µ(i) .
+(8)
+For the algorithm design, we use information projection measures defined as follows. For any measure P ∈ P(Ω) and
+x ∈ R, we define
+dU(P, x) ≜
+inf
+Q∈P(Ω): m(Q) ≤ x DKL(P∥Q) ,
+(9)
+and
+dL(P, x) ≜
+inf
+Q∈P(Ω): m(Q) ≥ x DKL(P∥Q) .
+(10)
+5
+
+Specifically, the information measure dU(P, x) with respect to any distribution P ∈ P(Ω) and x ∈ R is the minimum
+KL divergence between P and any distribution with mean at most x. Similarly, dL(P, x) measures the KL divergence
+between P and any distribution with mean at least x. In the fixed confidence setting, the goal is to identify the best
+arm with a pre-specified level of confidence while minimizing the number of samples in making the decision. Let τ
+denote an F-adapted stopping time, i.e., {τ = t} ∈ Ft for every t ∈ N. Corresponding to the stochastic stopping time
+τ, let ˆAτ denote the terminal decision of the learner. The δ-PAC objective of the learner is formalized next.
+Definition 1 (δ−PAC). A BAI algorithm is δ−PAC if the algorithm has a stopping time τ adapted to {Ft : t ∈ N},
+and at the stopping time with the terminal decision ˆAτ ∈ [K] it ensures
+Pν{ ˆAτ = a⋆} > 1 − δ ,
+(11)
+where Pν denotes the probability measure induced by the interaction of the BAI algorithm with the bandit instance ν.
+3
+Transportation Cost Balancing Algorithm
+In this section, we specify (i) a stopping rule that decides when to stop collecting samples and form a confident decision
+about the best arm, (ii) an arm selection rule that guides the order of sampling arms over time, and (iii) an estimation
+routine that aims to learn the unknown model parameters of interest (e.g., the mean values). The central statistic that
+guides all the three decisions in the TCB algorithm is the generalized log-likelihood ratio (GLLR). To formalize the
+GLLRs, we define K hypotheses {Hi : i ∈ [K]} such that
+Hi : arm i is the best arm.
+(12)
+For each arm i ∈ [K] at time t we next define
+Λt(i) ≜ sup
+Hi
+Pν(Xt | Hi is true) ,
+(13)
+where supHi denotes taking a supremum with the constraint that all the realizations are generated by a bandit instance
+in which arm i is the best arm. It can be readily verified that
+Λt(i) =
+sup
+µ: µ(i)>max
+j̸=i µ(j)
+�
+a∈[K]
+�
+s∈[t]:As=a
+πa(Xs | µ(a)) ,
+(14)
+where we have defined
+µ ≜ [µ(1), . . . , µ(K)] .
+(15)
+Accordingly, for any pair of arms (i, j) ∈ [K]2, at time t, we define the GLLR between arms i and j as
+Λt(i, j) ≜ log supHi Pν(Xt | Hi is true)
+supHj Pν(Xt | Hj is true)
+(16)
+= log Λt(i)
+Λt(j) .
+(17)
+Next, we characterize a closed-form expressions for the GLLRs in (17). First, we provide a few relevant definitions
+required for stating the closed-form expression.
+6
+
+1. Denote the log-likelihood ratio for arm i ∈ [K] corresponding to parameters θ, θ′ ∈ Θ by
+LLRi(θ, θ′, x) ≜ log πi(x | θ)
+πi(x | θ′) ,
+∀x ∈ Ω .
+(18)
+2. At any time t ∈ N and for any arm i ∈ [K], and based on the samples available from this arm, we denote the
+maximum likelihood estimate (MLE) of µ(i) by
+µt(i) ≜ arg max
+µ∈R
+�
+s∈[t]
+log πi(Xs | µ) · 1{As=i} ,
+(19)
+where 1 denotes the indicator function.
+3. Define the set of superior arms relative to arm i as
+Rt(i) ≜ {j ∈ [K] : µt(i) < µt(j)} .
+(20)
+4. Finally, define ˜µt(i) as the solution to
+∂
+∂µ
+�
+� �
+j∈Rt(i)
+�
+s∈[t]:As=i
+log πj(Xs | µ)
+�
+� = 0 ,
+(21)
+which denotes the MLE for the arms j ∈ Rt(i) under the hypothesis that arm i ∈ [K] is the best arm. More
+specifically, ˜µt(i) is the estimate of the mean of the arms j ∈ [K] which have a larger MLE µt(j) compared to
+µt(i), under the hypothesis Hi. As an example, for the exponential family of distributions, ˜µi(t) is simply the
+weighted combination of the MLEs of the arms j ∈ Rt(i) [4, Section 3.2].
+Next, we state a closed-form expression for Λt(i, j) in (17).
+Lemma 1. For any pair of arms (i, j) ∈ [K]2, the GLLRs Λt(i, j) can be equivalently expressed as follows.
+1. If µt(i) > µt(j):
+Λt(i, j) =
+�
+a∈Rt(i)
+�
+s∈[t]:As=a
+LLRa(˜µt(i), ˜µt(j), Xs)
++
+�
+a∈Rt(j)\Rt(i)
+�
+s∈[t]:As=a
+LLRa(µt(a), ˜µt(j), Xs) ,
+2. If µt(i) ≤ µt(j):
+Λt(i, j) =
+�
+a∈Rt(j)
+�
+s∈[t]:As=a
+LLRa(˜µt(i), ˜µt(j), Xs)
++
+�
+a∈Rt(i)\Rt(j)
+�
+s∈[t]:As=a
+LLRa(˜µt(i), µt(a), Xs) .
+Proof. See Appendix A.
+■
+GLLR-based Stopping Rule.
+We specify a GLLR-thresholding stopping criterion, which compares the GLLR
+statistic against a time-varying threshold. When the GLLR statistic exceeds the threshold, the algorithm stops col-
+lecting more samples and forms a decision about the best arm. Let us denote the maximum likelihood (ML) decision
+at time t about the top arm by atop
+t , i.e.,
+atop
+t
+≜ arg max
+i∈[K]
+µt(i) .
+(22)
+7
+
+Our stopping criterion is based on sufficiently distinguishing between the ML decision atop
+t , and the most likely
+contender compared to the best arm, which we refer to as the challenger. This challenger arm at time t is the arm
+closest to the top arm in a GLLR sense, and it is specified by
+ach
+t
+≜
+arg min
+i∈[K]\{atop
+t }
+Λt(atop
+t , i) .
+(23)
+In other words, ach
+t
+denotes the arm that is the top contender to the best arm atop
+t , where the comparison is made
+using the likelihood ratio between the arms. The stopping rule compares the GLLR between atop
+t
+and ach
+t and stops
+collecting samples when the GLLR exceeds a threshold. The threshold depends on the level of confidence δ required
+on the final decision and it is denoted by βt(δ). The stopping rule is stated next.
+τ ≜ inf
+�
+t ∈ N : Λt(atop
+t , ach
+t ) > βt(δ)
+�
+.
+(24)
+The threshold βt(δ) is specified in Theorem 1 such that it ensure the δ-PAC guarantee on the decision. Next, we
+delineate the arm selection rules. For this purpose, we first formalize the problem complexity measure, which quan-
+tifies the hardness of identification in a BAI instance. Specifically, the problem complexity captures the minimum
+distance between the given bandit instance and any other bandit instance with a different best arm. Clearly, the smaller
+the problem complexity, the larger the number of samples required for identification. We also state an equivalent
+representation of the problem complexity, which motivates our arm selection strategies.
+Problem complexity.
+Consider a bandit instance ν = [P1, · · · , PK] with the top arm a⋆. Given a⋆, we define an
+alternative set of bandit instances ¯ν = [¯P1, · · · , ¯PK] in which the top arm is not a⋆. Specifically,
+alt(a⋆) ≜ {¯ν ∈ M : m(¯Pa⋆) ≤ max
+i̸=a⋆ m(¯Pi)} .
+(25)
+Subsequently, given a⋆ and alt(a⋆), under the weight vector w = [w1, · · · , wK] ∈ ∆K, where ∆K represents the
+K-dimensional probability simplex, we define the problem complexity associated with ν as the smallest weighted KL
+divergence from ν to the set alt(a⋆). Specifically,
+Γ(ν, w) ≜
+inf
+¯ν∈alt(a⋆)
+�
+i∈[K]
+wi DKL(Pi∥¯Pi) .
+(26)
+Given any weight vector w ∈ ∆K, Γ(ν, w) captures the hardness of distinguishing ν from the closest alternate
+bandit instance, where the divergence between the arms is weighted by w. Finally, we define the problem complexity
+associated with the bandit instance ν as
+Γ(ν) ≜
+sup
+w∈∆K Γ(ν, w) .
+(27)
+It can be readily verified that Γ(ν) captures the maximum hardness in distinguishing ν from the closest bandit instance.
+Accordingly, we define the maximizer weight vector as
+w(ν) ≜ arg sup
+w∈∆K
+Γ(ν, w) .
+(28)
+The weight vector w(ν) characterizes the optimal allocation in which to sample arms, such that the sample complex-
+ity for BAI for the bandit instance ν is minimized. Next, we provide an equivalent representation of the problem
+complexity measure, which facilitates analyzing its key properties.
+8
+
+Lemma 2. The problem complexity Γ(ν, w) defined in (26) can be equivalently expressed as
+Γ(ν, w) = min
+i̸=a⋆
+inf
+x∈[µ(i),µ(a⋆)]
+�
+wa⋆dU(Pa⋆, x) + widL(Pi, x)
+�
+.
+(29)
+Proof. The proof follows similar line of arguments as [4, Lemma 3]. For completeness, we provide the proof in
+Appendix C.1.
+■
+The expression for Γ(ν, w) in (29) involves an inner minimization, which is a weighted combination of divergence
+measures from ν to an alternate bandit instance. Note that the inner minimization only depends on the divergence
+measures for the best arm a⋆ and any other arm i ̸= a⋆. Furthermore, the outer minimization acts over all the other
+arms i ̸= a⋆, establishing that Γ(ν, w) is the weighted divergence measure between the bandit instance ν and the
+closest alternate bandit instance.
+Transportation cost balancing (TCB).
+Designing the arm selection rule consists of two key components. The
+first component ensures that none of the arms remain under-explored. Specifically, the objective in this phase is to
+ensure that we have a reasonable estimate of each arm’s mean value, such that our estimates converge to the true
+mean values if the arm selection rule is allowed to collect samples without stopping. While estimating the mean
+values is not the goal in BAI, our arm selection rule performs estimation as an intermediate step to form a confident
+decision about the best arm. The second component of the arm selection rule is to track the optimal proportion w(ν)
+of arm selections defined in (28), which ensures that we minimize the average sample complexity. For this, TaS [4]
+proposes to compute the optimal sampling proportions at the current mean estimates and track the estimated sampling
+proportions. However, this is computationally expensive and requires solving K equations using the bisection method
+in each round. To circumvent this, we propose a simple sampling mechanism focusing on sampling from the set of
+under-sampled arms at each instant. In doing so, the sampling strategy is able to converge to the optimal sampling
+proportions w(ν) asymptotically. Next, we describe the sampling rule that combines these two components.
+Under-explored Arms. At any time t, the sampling rule defines a set of under-explored arms as
+Ut ≜
+�
+i ∈ [K] : Tt(i) ≤
+����
+t/K
+��
+,
+(30)
+where we have defined Tt(i) ≜ �t
+s=1 1{As = i} as the counter for the number of times arm i ∈ [K] is chosen up to
+time t. In case the set of under-explored arms is non-empty, indicating that some of the arms are under-explored, the
+arm selection strategy selects the arm that is sampled the least. Otherwise, the arm selection rule leverages the MLEs
+of the arm means to compute an empirical estimate of Γ(ν, w) defined in Lemma 2. Based on this, the goal is to select
+the next arm in a way that maximizes the estimate. We show in Lemma 10 (Appendix E) that this is equivalent to
+sampling from the set of under-sampled arms.
+Transport cost-based Estimation of Allocation.
+We provide a few measures that are used to delineate the arm
+selection rule. Define Pt,i as the distribution of arm i ∈ [K] when the mean takes a value equal to the MLE µt(i).
+Furthermore, for any arm i ∈ [K], define the interval It,i ≜ [µt(i), µt(atop
+t )], which specifies the interval for minimiza-
+tion in the empirical problem complexity defined in 31 next. Based on these, we define the minimum transportation
+cost [11], as the minimum weighted combination of divergence measures dU and dL defined in (9) and (10) of the top
+arm atop
+t
+and any other arm i ̸= atop
+t
+as follows.
+Γt(w) ≜
+min
+i∈[K]\{atop
+t } min
+x∈It,i
+�
+watop
+t dU(Pt,atop
+t , x) + widL(Pt,i, x)
+�
+.
+(31)
+9
+
+Γt(w) in (31) specifies the minimum cost of transporting the current estimated bandit instance νt ≜ [Pt,1, · · · , Pt,K]
+to an alternate bandit instance for which the best arm is not atop
+t . At any time t, let us denote the allocation obtained if
+the arm i ∈ [K] is selected in the next round by wt(i) ≜ [wt,1(i), · · · , wt,K(i)], where we have defined
+wt,j(i) ≜
+1
+t + 1
+�
+Tt(j) + 1{j = i}
+�
+,
+∀j ∈ [K] .
+(32)
+Based on the equivalent form of the problem complexity in Lemma 2, the goal of the sampling strategy is to maximize
+the lowest information measure Γt(w). This ensures that we sample the under-sampled arms in each round and move
+closer toward the optimal sampling proportion. To this end, let us define the arm with the lowest information measure
+as
+amin
+t
+≜
+arg min
+i∈[K]\{atop
+t }
+min
+x∈It,i
+�Tt(atop
+t )
+t
+dU(Pt,atop
+t , x) + Tt(i)
+t
+dL(Pt,i, x)
+�
+.
+(33)
+In (33), amin
+t
+denotes the arm that minimizes the estimate of the problem complexity at time t. We note that to
+increase the lowest information measure, we should select either the current best arm atop
+t
+or the arm with the lowest
+information measure amin
+t
+. Based on the above definitions, the arm selection for TCB is carried out as follows.
+At+1 ≜
+�
+�
+�
+�
+�
+�
+�
+arg min
+i∈Ut
+Tt(i),
+if Ut ̸= ∅
+amin
+t
+,
+if Γt(wt(amin
+t
+)) > Γt(wt(atop
+t )) and Ut = ∅
+atop
+t ,
+if Γt(wt(amin
+t
+)) < Γt(wt(atop
+t )) and Ut = ∅
+.
+(34)
+The complete procedure is presented in Algorithm 1.
+Algorithm 1: Transportation cost balancing (TCB)
+1 Initialize: t = 0, Ut = [K], µt(i) = 0 ∀ i ∈ [K], Tt(i) = 0 ∀ i ∈ [K], Λ(atop
+t , ach
+t ) = ∞, βt(δ) = 0
+2 while Λ(atop
+t , ach
+t ) ≤ βt(δ) do
+3
+t ← t + 1
+4
+Select an arm at specified by (34) and obtain reward Xt
+5
+Update µt,at and Tt(at) using (19)
+6
+atop
+t
+← arg max
+i∈[K]
+µt(i)
+7
+Compute amin
+t
+using (33)
+8
+For every i ∈ [K], compute wt(i) using (32)
+9
+Compute Γt(wt(amin
+t
+)) and Γt(wt(atop
+t )) using (31)
+10
+Compute Λt(atop
+t , i) for every i ∈ [K] \ {a1
+t}
+11
+ach
+t ←
+arg min
+i∈[K]\{atop
+t
+}
+Λt(atop
+t , i)
+12
+Update βt(δ) using (24)
+13 Output: Top arm atop
+t
+10
+
+Improved Transportation Cost Balancing (ITCB).
+A recent study by [12] shows that an additional exploration
+penalty based on the number of times that each arm is chosen, improves the empirical performance of top-two al-
+gorithms, albeit achieving the same asymptotic optimality guarantee. Specifically, [12] proposes an additive penalty
+log(Tt(i)) to the GLLRs for each arm i ∈ [K] \ {atop
+t } to promote further exploration of under-explored arms. Mo-
+tivated by this observation, we also devise a modified sampling rule that achieves the same optimality guarantee as
+TCB, with improved empirical performance. To formalize the modified sampling rule, we begin by defining the lowest
+penalized information measure as:
+Φt(w) ≜
+min
+i∈[K]\{atop
+t }
+�
+min
+x∈It,i
+�
+watop
+t dU(Pt,atop
+t , x) + widL(Pt,i, x)
+�
++ log((t + 1)wi)
+t + 1
+�
+,
+(35)
+for any w ∈ ∆K. Furthermore, let us define the arm having the lowest penalized information measure as
+bmin
+t
+≜
+arg min
+i∈[K]\{atop
+t }
+�
+min
+x∈It,i
+�
+Tt(atop
+t )
+t
+dU(Pt,atop
+t , x) + Tt(i)
+t
+dL(Pt,i, x)
+�
++ log(Tt(i))
+t
+�
+,
+(36)
+Based on this definition, the modified sampling rule in ITCB is specified next.
+At+1 ≜
+�
+�
+�
+�
+�
+�
+�
+arg min
+i∈Ut
+Tt(i),
+if Ut ̸= ∅
+bmin
+t
+,
+if Φt(wt(bmin
+t
+)) > Φt(wt(atop
+t )) and Ut = ∅
+atop
+t ,
+if Φt(wt(bmin
+t
+)) < Φt(wt(atop
+t )) and Ut = ∅
+.
+(37)
+Note that main difference in the selection rules (34) and (37) lies in the cost function Φt which has an additional
+penalty compared to Γt, that promotes additional exploration. The ITCB sampling rule follows the same principle as
+TCB, with the difference of a penalized cost function.
+4
+Performance Guarantees
+This section provides the main results pertaining to the performance of TCB and ITCB algorithms. There are two key
+results that we are interested in proving. First, we want to show that the TCB and ITCB algorithms satisfy the δ−PAC
+guarantee on the decision confidence. Next, we show that the average sample complexities of these algorithms match
+the known information-theoretic lower bound asymptotically. To this end, we will prove a few properties of TCB and
+ITCB, which will collectively establish asymptotic optimality in terms of the average sample complexity. We start by
+stating the guarantee on the probability of error and characterizing βt(δ) such that the algorithms are δ−PAC.
+Theorem 1 (δ−PAC). The stopping rule in (24) with the choice of the threshold
+βt(δ) ≜ log t1+α(K − 1)
+δ
++ log
+�
+1 + 1
+α
+�
+,
+(38)
+along with any arm selection strategy and the decision rule ˆAτ ≜ atop
+τ
+is δ−PAC for any α > 0.
+Proof. See Appendix B.
+■
+Next, we state the results related to the asymptotic optimality of TCB and ITCB in terms of the average sample com-
+plexity. We begin by establishing a few properties of the problem complexity that are useful for characterizing the
+11
+
+sample complexity. Specifically, the form of the problem complexity Γ(ν) in Lemma 2 is instrumental in charac-
+terizing the key properties of Γ(ν) and establishing an upper bound on the average sample complexity of TCB and
+ITCB. The following lemma characterizes these properties, which include the continuity of the problem complexity in
+terms of the bandit instance ν ∈ M in the metric space (M, DTV) and the characterization of an optimal allocation
+of samples that maximizes the problem complexity.
+Lemma 3 (Properties of Γ(ν)). The problem complexity Γ(ν) has the following properties:
+1. Functions dU(·, ·) and dL(·, ·) are strictly convex in their second arguments.
+2. Problem complexity Γ : M �→ R and the optimal allocation w : M �→ ∆K are continuous functions on the
+metric space (M, DTV). Furthermore, an optimal sampling proportion is given by the unique allocation that
+satisfies:
+Γi(ν, w) = Γj(ν, w) ,
+∀i, j ̸= a⋆ ,
+(39)
+where, for any i ∈ [K] \ {a⋆}, we have defined
+Γi(ν, w) ≜
+inf
+¯P∈M:m(¯Pi)≥m(¯Pa⋆)
+�
+wa⋆DKL(Pa⋆∥¯Pa⋆) + wiDKL(Pi∥¯Pi)
+�
+.
+(40)
+Proof. See Appendix C.2.
+■
+The continuity of Γ has been previously established by [13] in the metric space (M, W1). However, the continuity
+was established under the assumption that for any P ∈ P(Ω), the boundedness assumption EP[f(|X|)] < B holds
+for any convex and differentiable function f. The key difference in the proof of Lemma 3 with that of [13, Lemma
+4] is that we do not impose the boundedness assumption. Rather, we leverage the properties of the Jensen-Shanon
+divergence (details in Appendix C.2) to establish the continuity property of the problem complexity. Next, we state
+two key properties of the TCB and ITCB arm selection strategies used for characterizing an upper bound on the average
+sample complexity that coincides with the universal lower bound. First, we show that the explicit exploration phase
+ensures that each arm i ∈ [K] is sampled sufficiently often, such that the MLEs of the mean values converge to the
+true means. Let us define the time instant N ϵ
+ν as
+N ϵ
+ν ≜ inf
+�
+t ∈ N : |µs(i) − µ(i)| < ϵ, ∀i ∈ [K], ∀s ≥ t
+�
+.
+(41)
+The stochastic time N ϵ
+ν marks the convergence of the MLEs to the respective ground truths for every arm i ∈ [K]. In
+the following result, we will show that the arm selection strategies ensure that N ϵ
+ν has a finite average value.
+Theorem 2 (Convergence in mean). Under the TCB and ITCB sampling strategies we have Eν[N ϵ
+ν] < +∞.
+Proof. See Appendix D.
+■
+Leveraging the continuity property of the problem complexity proved in Lemma 3, next, we establish the convergence
+in the sampling proportions due to the TCB and ITCB sampling rules to that of the optimal proportions. This is a
+key contribution of the paper, which helps establish the asymptotic optimality of the algorithms, compared to the
+β−optimality achieved by top-two algorithms.
+12
+
+Theorem 3 (Convergence in sampling proportions). If the TCB arm selection rule in (34) is allowed to continue
+sampling without stopping, for any ϵ > 0, there exists a stochastic time instant N ϵ
+w such that
+����
+Tt(i)
+t
+− wi(ν)
+���� < ϵ ,
+∀i ∈ [K], ∀t ≥ N ϵ
+w .
+(42)
+Furthermore, Eν[N ϵ
+w] < +∞.
+Proof. See Appendix E.
+■
+Theorem 4 (Convergence in sampling proportions). If the ITCB arm selection rule in (37) is allowed to continue
+sampling without stopping, for any ϵ > 0, there exists a stochastic time instant N ϵ
+w such that
+����
+Tt(i)
+t
+− wi(ν)
+���� < ϵ ,
+∀i ∈ [K], ∀t ≥ N ϵ
+w .
+(43)
+Furthermore, Eν[N ϵ
+w] < +∞.
+Proof. See Appendix E.
+■
+Finally, leveraging Theorems 2 and 3 (or 4), we characterize an upper-bound on the average sample complexity of the
+TCB and ITCB algorithms.
+Theorem 5. The TCB and ITCB algorithms, comprising the arm selection rules in (34) and (37) and the stopping rule
+in (24), satisfy the following upper-bound on the average sample complexity.
+lim sup
+δ→0
+Eν[τ]
+log(1/δ) ≤ 1 + α
+Γ(ν) ,
+(44)
+for any α > 0.
+Proof. See Appendix F.
+■
+The upper bound in Theorem 5 matches the information-theoretic lower bound on the average sample complexity
+provided by [4] up to any α > 0. This establishes the asymptotic optimality of the TCB and ITCB algorithms.
+5
+Numerical Experiments
+Convergence in allocation.
+First, we empirically establish the convergence of the TCB and ITCB sampling rules on
+several Gaussian and Bernoulli bandit instances. These bandit instances are summarized in Table 1. More specifically,
+we have chosen (ν1, ν3) from [4] and (ν2, ν4, ν6) from [10]. We have also included ν5 as a setting with identical
+arm means. We plot the maximum deviation from the optimal allocation maxi∈[K] |Tt(i)/t − wi(ν)| over time in
+Figure 2 for TCB and Figure 3 for ITCB. Both these figures confirm the convergence in allocation for the TCB and
+ITCB selection rules to the optimal allocation.
+13
+
+Distribution
+ν
+w(ν)
+Bernoulli
+ν1 = [0.3, 0.21, 0.2, 0.19, 0.18]
+[0.3359, 0.2515, 0.1766, 0.1324, 0.1036]
+Bernoulli
+ν2 = [0.5, 0.4, 0.3, 0.2, 0.1]
+[0.4514, 0.4483, 0.0628, 0.0248, 0.0127]
+Bernoulli
+ν3 = [0.5, 0.45, 0.43, 0.4]
+[0.4188, 0.3882, 0.1360, 0.0569]
+Gaussian
+ν4 = [0.5, 0.25, 0, −0.25, −0.5]
+[0.4513, 0.4441, 0.0639, 0.0263, 0.0144]
+Gaussian
+ν5 = [1.2, 1, 1, 1, 1]
+[0.3337, 0.1666, 0.1666, 0.1666, 0.1666]
+Gaussian
+ν6 = [−1.9, −0.6, −0.5, −0.4, −0.3,
+−0.1, −0., 0.1, 0.4, 1.8]
+[0.2814, 0.1986, 0.1097, 0.0939, 0.0815,
+0.0634, 0.0566, 0.0509, 0.0461, 0.0177]
+Table 1
+100
+200
+300
+400
+500
+600
+700
+0
+0.05
+0.1
+0.15
+0.2
+Figure 2: Convergence in allocation for TCB
+100
+200
+300
+400
+500
+600
+700
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+Figure 3: Convergence in allocation for ITCB
+Comparison to the state of the art.
+We compare the TCB and ITCB algorithms against the existing strategies for
+BAI. Specifically, we perform our experiments based on two common reward distributions, Bernoulli bandits and
+Gaussian bandits. All the experiments are averaged over 2000 independent Monte Carlo trials. For the experiments,
+we compare the TCB and ITCB arm selection rules with four state-of-the-art top-two algorithms for BAI: T3C [11],
+TT-SPRT [7], TS-TCI [12] and EB-TCI [12]. TT-SPRT and EB-TCI use the empirical best arm as the leader. TT-
+SPRT uses the arm with the closest GLLR statistic to the empirical best arm as the challenger, and EB-TCI includes
+an additional exploration penalty to the GLLR statistic to determine the challenger. Distinct from TT-SPRT and EB-
+TCI, T3C and TS-TCI use Thompson sampling from the posterior distribution to identify the leader. T3C selects the
+challenger to be the arm having the closest GLLR statistic to the leader, while TS-TCI uses a penalized GLLR statistic
+to promote exploration.
+Bernoulli bandit.
+For this experiment, we use a Bernoulli bandit with mean values [0.8, 0.5, 0.3, 0.29, 0.06], and set
+δ ≜ 10−8. Figure 4 shows the performance of the TCB and ITCB arm selection rules compared to the state-of-the-art.
+We observe that the TCB and ITCB arm selection strategies in (34) and (37) are agnostic to the tuning parameter β.
+Furthermore, the ITCB outperforms all other strategies for the bandit instance for most values of β. Specifically, the
+14
+
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+200
+400
+600
+800
+1000
+1200
+1400
+TCB
+EB-TCI
+T3C
+ITCB
+TS-TCI
+TT-SPRT
+Figure 4: Bernoulli, δ = 10−8
+performance of ITCB matches that of TS-TCI and EB-TCI at β ≈ 0.4.
+Gaussian bandit.
+For the next experiment, we take a Gaussian bandit instance with mean values given by [1.2, 1, 1, 1, 1]
+and variance σ = 1. Figure 5 compares the TCB and ITCB algorithms against state-of-the-art BAI algorithms for var-
+ious values of the tuning parameter β. We set δ ≜ 10−8 for this experiment. We observe that the ITCB algorithm
+outperforms other algorithms in most regimes of β, and the performances of TS-TCI and EB-TCI match that of ITCB
+in the range 0.3 ≤ β ≤ 0.4.
+6
+Conclusions
+We have investigated the problem of fixed-confidence best-arm identification (BAI) in stochastic multi-arm bandits
+(MABs). The bandit instance is assumed to be drawn from the family of distributions parameterized by their mean
+value. We have designed the transport cost balancing (TCB) algorithm, in which the key decision is finding the optimal
+allocation of the sampling resources among different arms. This extends the existing strategies that aim to allocate
+a β fraction of the resources to the best arm and achieve β−optimality with β remaining a tuning parameter. In the
+proposed TCB algorithm, the optimal value of β is also estimated, rendering the algorithm independent of β. In
+TCB, instead of reserving a β fraction of samples for the best arm, we have devised a mechanism for tracking the
+optimal sampling proportion by sampling the under-sampled arms at each instant. As a result, we have established
+that the proposed TCB algorithm is asymptotically optimal. We have also extended the TCB algorithm by including
+an additional exploration penalty based on the number of times each arm is chosen. This algorithm, referred to as
+improved TCB (ITCB), is also shown to achieve asymptotic optimality and improved empirical performance compared
+to TCB.
+15
+
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+104
+TCB
+EB-TCI
+T3C
+ITCB
+TS-TCI
+TT-SPRT
+Figure 5: Gaussian, δ = 10−8
+A
+Proof of Lemma 1
+Under each hypothesis Hi, for i ∈ [K], we solve the following optimization problem:
+Pi
+t(µ) ≜
+�
+�
+�
+�
+�
+�
+�
+max
+µ∈RK
+K
+�
+i=1
+�
+s∈[t]:As=i
+log πi(Xs | µi)
+s.t.
+µi ≥ max
+j̸=i µj
+.
+(45)
+Let us define the augmented Lagrangian corresponding to the optimization problem in (45) as
+Li
+t(µ, u) ≜
+�
+i∈[K]
+�
+s∈[t]:As=i
+log πi(Xs | µi) +
+�
+j̸=i
+uj [µj − µi] .
+(46)
+Furthermore, denote the first derivative of the likelihood function by π′
+i(x | µi) ≜ ∂πi(x | µi)
+∂µi
+. Using the Karush Kuhn
+Tucker (KKT) first order necessary conditions, any solution to (45) for all i ∈ [K]
+∂Li
+t(µ, u)
+∂µi
+=
+�
+s∈[t]:As=i
+π′
+i(Xs | µi)
+πi(X | µi) −
+�
+j̸=i
+uj = 0 ,
+(47)
+and for all j ̸= i
+∂Li
+t(µ, u)
+∂µj
+=
+�
+s∈[t]:As=j
+π′
+j(Xs | µj)
+πj(Xs | µj) + uj = 0 ,
+(48)
+uj (µj − µi) = 0 ,
+(49)
+uj ≥ 0 .
+(50)
+16
+
+Let us denote an optimal solution to (45) by ρt(i) ≜ [ρt,1(i), · · · , ρt,K(i)]. It can be readily verified that a solution to
+(47)-(50) is given by ρt(i) such that for every j ∈ [K],
+ρt,j(i) ≜
+�
+µt(j) ,
+if j ∈ ¯Rt(i)
+˜µt(i) ,
+otherwise
+,
+(51)
+where ¯Rt(i) ≜ [K] \ Rt(i), and we have defined ˜µt(i) as a solution to:
+∂
+∂µ
+�
+� �
+j∈Rt(i)
+�
+s∈[t]:As=j
+log πj(Xs | µ)
+�
+� = 0 .
+(52)
+Lemma 1 can be readily obtained by plugging the maximizers of the mean values ρt(i) under any hypothesis i ∈ [K]
+into the GLLR defined in (17).
+B
+Proof of Theorem 1
+The probability of an incorrect decision can be bounded as follows.
+Pν (τ < +∞, ˆaτ ̸= a⋆) = Pν
+�
+∃t ∈ N : µt(atop
+t ) > µt(a⋆), Λt(atop
+t , ach
+t ) > βt(δ)
+�
+(53)
+≤ Pν
+�
+∃t ∈ N, i ̸= a⋆ : µt(i) ≥ max
+j̸=i µt(j), Λt(i, a⋆) > βt(δ)
+�
+(54)
+≤
+�
+t∈N
+�
+i∈[K]\{a⋆}
+Pν
+�
+Λt(i, a⋆) > βt(δ)
+�
+(55)
+=
+�
+t∈N
+�
+i∈[K]\{a⋆}
+Pν
+�
+�
+�
+a∈Rt(a⋆)
+�
+X∈X a
+t
+log πa(X | µt(a))
+πa(X | ˜µt(a⋆)) ≥ βt(δ)
+�
+�
+(56)
+≤
+�
+t∈N
+�
+i∈[K]\{a⋆}
+Pν
+�
+�
+�
+�
+�
+�
+�
+�
+�
+a∈[K]
+�
+X∈X a
+t
+log πa(X | µt(a))
+πa(X | µ(a))
+�
+��
+�
+≜Λ0
+t (µt∥µ)
+≥ βt(δ)
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(57)
+where (55) follows from union bound and (57) follows from the fact that ˜µt(a⋆) is the maximizer of the likelihood
+function under the hypothesis Ha⋆, and µ(a⋆) > maxi̸=a⋆ µ(i) by definition. Let us denote the measure induced by
+the interaction of the BAI algorithm with the bandit instance parameterized by the current ML estimate µt by Pt. We
+have
+Pν (τ < +∞, ˆaτ ̸= a⋆)
+(57)
+≤
+�
+t∈N
+�
+i∈[K]\{a⋆}
+Eν
+�
+1{Λ0
+t(µt∥µ) ≥ βt(δ)}
+�
+(58)
+=
+�
+t∈N
+�
+i∈[K]\{a⋆}
+�
+(Λ0
+t (µt∥µ)≥βt(δ))
+dPν(Xt)
+(59)
+=
+�
+t∈N
+�
+i∈[K]\{a⋆}
+�
+(Λ0
+t (µt∥µ)≥βt(δ))
+dPν(Xt)
+dPt(Xt) dPt(Xt)
+(60)
+=
+�
+t∈N
+�
+i∈[K]\{a⋆}
+�
+(Λ0
+t (µt∥µ)≥βt(δ))
+exp
+�
+−Λ0
+t(µt∥µ)
+�
+dPt(Xt)
+(61)
+17
+
+≤
+�
+t∈N
+�
+i∈[K]\{a⋆}
+e−βt(δ)Pt
+�
+Λ0
+t(µt∥µ) ≥ βt(δ)
+�
+(62)
+≤ (K − 1)
+�
+t∈N
+e−βt(δ) ,
+(63)
+where (61) is obtained from the fact that Λ0
+t(µt∥µ) = log dPt(Xt)
+dPν(Xt), and (63) follows from the fact that Pt(Λ0
+t(µt∥µ)) ≤ 1.
+Finally, by choosing
+βt(δ) ≜ log C(K − 1)t1+α
+δ
+,
+(64)
+and setting C ≜ 1 + α−1, we obtain
+P (τ < +∞, ˆaτ ̸= a⋆) ≤ δ .
+(65)
+C
+Problem Complexity
+C.1
+Proof of Lemma 2
+Recall that corresponding to a bandit instance ν ∈ M with the best arm a⋆, the set of alternate bandit instances is
+defined as
+alt(a⋆) ≜
+�
+¯ν ∈ M : m(¯Pa⋆) ≤ max
+i̸=a⋆ m(¯Pi)
+�
+.
+(66)
+It can be readily verified that the set of alternate bandit instances can be equivalently stated as
+alt(a⋆) =
+�
+i̸=a⋆
+�
+¯ν ∈ M : m(¯Pi) ≥ m(¯Pa⋆)
+�
+.
+(67)
+Using (67), the problem complexity can be simplified as follows:
+Γ(ν) = sup
+w∈∆K
+inf
+¯ν∈alt(a⋆)
+�
+i∈[K]
+wiDKL(Pi∥¯Pi)
+(68)
+= sup
+w∈∆K min
+i̸=a⋆
+inf
+¯ν∈M:m(¯Pi)≥m(¯Pa⋆)
+�
+wa⋆DKL(Pa⋆∥¯Pa⋆) + wiDKL(Pi∥¯Pi)
+�
+.
+(69)
+Let us define
+Γi(ν, w) ≜
+inf
+¯ν∈M:m(¯Pi)≥m(¯Pa⋆)
+�
+wa⋆DKL(Pa⋆∥¯Pa⋆) + wiDKL(Pi∥¯Pi)
+�
+.
+(70)
+It can be readily verified that
+Γ(ν, w) ≜ min
+i̸=a⋆ Γi(ν, w) .
+(71)
+Note that Γi(ν, w) can be further simplified as:
+Γi(ν, w) = inf
+x∈R
+�
+wa⋆
+inf
+¯P∈P(Ω):m(¯P)≤x DKL(Pa⋆∥¯P) + wi
+inf
+¯P∈P(Ω):m(¯P)≥x DKL(Pi∥¯P)
+�
+(72)
+= inf
+x∈R
+�
+wa⋆dU(Pa⋆, x) + widL(Pi, x)
+�
+.
+(73)
+18
+
+Note that dU is non-increasing in x and dL is non-decreasing in x. Thus, when x < µ(i), we have dU(Pa⋆, x) >
+dU(Pa⋆, µ(i)), and dL(Pi, x) > dL(Pi, µ(i)). This implies that the optimizer x⋆
+i of (73) should satisfy x⋆
+i ≥ µ(i).
+Using a similar argument, we can show that x⋆
+i ≤ µ(a⋆). Hence, (73) can be rewritten as:
+Γi(ν, w) =
+inf
+x∈[µ(i),µ(a⋆)]
+�
+wa⋆dU(Pa⋆, x) + widL(Pi, x)
+�
+.
+(74)
+Finally, the problem complexity can be equivalently expressed as:
+Γ(ν) = sup
+w∈∆K min
+i̸=a⋆
+inf
+x∈[µ(i),µ(a⋆)]
+�
+wa⋆dU(Pa⋆, x) + widL(Pi, x)
+�
+.
+(75)
+C.2
+Proof of Lemma 3
+1. First, we will prove that dU and dL are strictly convex in x. For any x ∈ R and y ∈ R, and for any λ ∈ [0, 1], let
+us define
+z ≜ λx + (1 − λ)y .
+(76)
+Furthermore, define
+ηx ≜
+arg inf
+η∈P(Ω):m(η)≤x
+DKL(Pa⋆∥η) ,
+ηy ≜
+arg inf
+η∈P(Ω):m(η)≤y
+DKL(Pa⋆∥η) ,
+and
+ηz ≜
+arg inf
+η∈P(Ω):m(η)≤z
+DKL(Pa⋆∥η) .
+(77)
+Furthermore, define κz ≜ ληx + (1 − λ)ηy. Note that
+m(κz) = λm(ηx) + (1 − λ)m(ηy)
+(78)
+≤ λx + (1 − λ)y
+(79)
+(76)
+= z .
+(80)
+Now,
+dU(Pa⋆, z) = DKL(Pa⋆∥ηz)
+(81)
+≤ DKL(Pa⋆∥κz)
+(82)
+< λDKL(Pa⋆∥ηx) + (1 − λ)DKL(Pa⋆∥ηy)
+(83)
+= λdU(Pa⋆, x) + (1 − λ)dL(Pa⋆, y) ,
+(84)
+where (82) is a result of (80), and (83) is a result of the strict convexity of KL divergence in both arguments.
+Thus, dU is strictly convex in x. Using a similar argument, we can prove that dL is also strictly convex in x.
+Thus, gi : M × R �→ R, defined as
+gi(ν, x) ≜ wa⋆dU(Pa⋆, x) + widL(Pi, x) ,
+(85)
+is strictly convex in its second argument. Thus, gi has a unique minimum in [µ(i), µ(a⋆)].
+19
+
+2. For establishing the continuity of Γ : M �→ R and w : M �→ ∆K, we will leverage the following two lemmas,
+which provide the sufficient conditions for continuity.
+Lemma 4 (Berge’s maximum theorem [18]). Suppose g is a continuous function on S × Θ and D : Θ �→ S is
+a compact-valued continuous correspondence on Θ. Let
+g⋆(θ) ≜ max
+x∈D(θ) g(x, θ)
+and
+D⋆(θ) ≜ arg max
+x∈D(θ)
+g(x, θ) .
+(86)
+Then, g⋆ is a continuous function on Θ, and D⋆ is a compact-valued upper hemicontinuous correspondence on
+Θ.
+Lemma 5 ( [19]). Let us denote the generalized Jensen-Shannon (JS) divergence between two measures P1 and
+P2 with weight α ∈ (0, 1) by
+JSα(P1∥P2) ≜ αDKL(P1∥¯Pα) + (1 − α)DKL(P2∥¯Pα) ,
+(87)
+where we have defined
+¯Pα ≜ αP1 + (1 − α)P2 .
+(88)
+Then, JSα is upper-bounded as
+JSα(P1∥P2) ≤ 1 .
+(89)
+Now, we show that Γ(ν) and w(ν) is continuous in ν. For this, let us define the correspondence D : M �→ ∆K
+such that for any ν ∈ M, D(ν) ≜ ∆K. Clearly, for any ν ∈ M, D(ν) is a compact set, and hence D is a
+compact-valued constant correspondence. Finally, we need to show that for each i ∈ [K] \ {a⋆}, Γi(ν, w) is
+continuous in ν and w, where we have defined Γi(ν, w) in (74). First, note that Γi is lower semicontinuous in ν
+due to the lower semicontinuity of KL divergence in both arguments [20]. Next, we will leverage [21, Theorem
+5.43], which provides a sufficient condition for the global continuity of convex functions.
+Lemma 6 ( [21]). For a convex function f : X �→ R on an open convex subset of a topological vector space,
+the following statements are equivalent.
+(a) f is bounded above on a neighborhood of some point in X.
+(b) f is upper semicontinuous on X.
+Note that Γi is convex in its first argument, since KL divergence is a convex function. Let us denote the interior
+of the set of distributions M by int(M). For any η ∈ int(M), there exists a neighborhood Nr(η) ⊂ int(M),
+where we have defined
+Nr(η) ≜
+�
+λ ∈ int(M) : DTV(η∥λ) < r
+�
+.
+(90)
+Furthermore, for any ν ∈ Nr(η) and w ∈ ∆K, let us define the distribution
+κa⋆,i ≜ wa⋆Pa⋆ + wiPi
+wa⋆ + wi
+.
+(91)
+20
+
+Expanding Γi, we obtain
+Γi(ν, w) =
+inf
+¯ν∈M:m(¯Pa⋆)≤m(¯Pi)
+�
+wa⋆DKL(Pa⋆∥¯Pa⋆) + wiDKL(Pi∥¯Pi)
+�
+(92)
+≤ wa⋆DKL(Pa⋆∥κa⋆,i) + wiDKL(Pi∥κa⋆,i)
+(93)
+= (wa⋆ + wi)
+�
+wa⋆
+wa⋆ + wi
+DKL(Pa⋆∥κa⋆,i) +
+wi
+wa⋆ + wi
+DKL(Pi∥κa⋆,i)
+�
+(94)
+= (wa⋆ + wi)JS
+wa⋆
+wa⋆ +wi (Pa⋆∥Pi)
+(95)
+≤ 1 ,
+(96)
+where (96) is a result of Lemma 5. Thus, leveraging Lemma 6, we obtain that Γi(ν, w) is upper semicontinuous
+in ν, which proves that Γi(ν, w) is continuous in its first argument in M. Furthermore, Γi(ν, w) is linear in
+the second argument, and hence, it is continuous. This shows that Γi and the correspondence D satisfies the
+conditions in Lemma 4, and we obtain that Γi(ν, w) is continuous in ν and w is upper hemicontinuous in ν.
+Finally, following the same line of arguments as [10, Proposition 7], we can show that for a given ν ∈ M, w is
+the unique allocation satisfying
+Γi(ν, w) = Γj(ν, w) ,
+for all
+i, j ̸= a⋆ .
+(97)
+This shows that w is continuous in ν.
+D
+Proof of Theorem 2
+Recall the definition of the stochastic time N ϵ
+ν, defined as
+N ϵ
+ν ≜ inf
+�
+t ∈ N : |µs(i) − µ(i)| < ϵ, ∀i ∈ [K], ∀s ≥ t
+�
+.
+(98)
+In order to show that Eν[N ϵ
+ν] < +∞, we leverage a concentration inequality for ML estimators, which hold under the
+assumptions provided in Section 2.
+Lemma 7 ( [22]). Consider a sequence of i.i.d. random variables {Xi : i ∈ [t]} with mean µ and distributed
+according to P. Let us denote the MLE of µ up to time t by µt. Under the Assumptions 1 − 7 in Section 2, for any
+ε ∈ R+ we have
+P (|µt − µ| > ε) ≤ 2 exp
+�
+−
+tε2
+2cc′σ2
+�
+.
+(99)
+Next, we show that as an artifact of our explicit exploration phase, we ensure the convergence of the ML estimates of
+each arm to their corresponding true values. Note that for any t ∈ N, the explicit exploration phase ensures that for
+every i ∈ [K], Tt(i) ≥
+�
+t/K − 1, which can be shown following the same steps as [7, Theorem 4]. Thus, for any
+t ∈ N, we have:
+Pν(N ϵ
+ν > t) =
+∞
+�
+s=t+1
+Pν(N ϵ
+ν = s)
+(100)
+21
+
+=
+∞
+�
+s=t
+Pν
+�
+∃i ∈ [K] : |µs(i) − µ(i)| > ϵ, and ∀u > s, ∀i ∈ [K] : |µu(i) − µ(i)| ≤ ϵ
+�
+(101)
+≤
+∞
+�
+s=t
+Pν
+�
+∃i ∈ [K] : |µs(i) − µ(i)| > ϵ
+�
+(102)
+≤
+�
+i∈[K]
+∞
+�
+s=t
+Pν
+�
+|µs(i) − µ(i)| > ϵ
+�
+(103)
+=
+�
+i∈[K]
+∞
+�
+s=t
+Pν
+�
+|µs(i) − µ(i)| > ϵ, Ts(i) >
+� s
+K − 1
+�
++
+�
+i∈[K]
+∞
+�
+s=t
+Pν
+�
+|µs(i) − µ(i)| > ϵ, Ts(i) ≤
+� s
+K − 1
+�
+�
+��
+�
+=0
+(104)
+=
+�
+i∈[K]
+∞
+�
+s=t
+∞
+�
+ℓ=√
+s/K−1
+Pν
+�
+|µs(i) − µ(i)| > ϵ, Ts(i) = ℓ
+�
+(105)
+≤ 2
+�
+i∈[K]
+∞
+�
+s=t
+∞
+�
+ℓ=√
+s/K−1
+exp
+�
+−
+ℓϵ2
+2cc′σ2
+�
+(106)
+≤ 2
+�
+i∈[K]
+∞
+�
+s=t
+� ∞
+√
+s/K−2
+exp
+�
+− xϵ2
+2cc′σ2
+�
+dx
+(107)
+= 2
+�
+i∈[K]
+∞
+�
+s=t
+exp
+�
+−
+ϵ2
+2cc′σ2 (
+�
+s/K − 2)
+�
+(108)
+≤ 2K exp
+� ϵ2
+cc′σ2
+� � ∞
+t−1
+exp
+�
+−
+ϵ2
+2cc′σ2
+� u
+K
+�
+du
+(109)
+= 8Kcc′σ2
+ϵ2
+exp
+� ϵ2
+cc′σ2
+�
+· exp
+�
+− ϵ2
+cc′σ2
+�
+t − 1
+K
+�
+�
+��
+�
+≜Bt
+,
+(110)
+where (106) is obtained from Lemma 7. Finally, we have
+Eν[N ϵ
+ν] =
+�
+t∈N
+Pν(N ϵ
+ν > t)
+(111)
+≤
+�
+t∈N
+Bt
+(112)
+< 16(Kcc′σ2)2
+ϵ4
+exp
+� ϵ2
+cc′σ2
+�
+(113)
+< +∞ .
+(114)
+E
+Proof of Theorems 3 and 4
+Let us define the set of over-sampled arms as:
+Oϵ
+t ≜
+�
+i ∈ [K] : Tt(i)
+t
+> wi(ν) + ϵ
+�
+.
+(115)
+22
+
+Furthermore, we define the set of under-sampled arms as:
+Pϵ
+t ≜
+�
+i ∈ [K] : Tt(i)
+t
+< wi(ν) + ϵ
+2
+�
+.
+(116)
+The convergence in allocation for the proposed algorithm is shown in two key steps. First, we prove that if any
+sampling strategy always samples from the set of under-sampled arms, then the sampling strategy converges to the
+optimal allocation w(ν). This step is common in the proof for both Theorem 3 and Theorem 4. In the next step, we
+show that the proposed sampling strategies always sample from the set of under-sampled arms. We show the first step
+through Lemma 8 and Lemma 9, which we provide next. Essentially, Lemma 8 shows that if the sampling strategy
+always samples from the set of under-sampled arms, then, after some time, the set of over-sampled arms becomes
+empty. Lemma 9 then shows that when the set of over-sampled arms is empty, eventually the allocation for each
+arm converges to the optimal allocation. The key distinction in the proofs of Theorem 3 and Theorem 4 arises in
+the next step. In Lemma 10, we show that the TCB arm selection rule stated in (34) always samples from the set of
+under-sampled arms. In Lemma 11, we show that the ITCB arm selection rule provided in (37) always samples from
+the set of under-sampled arms. Before stating Lemma 8, let us define the sampling proportion γt ≜ [γt,1, · · · , γt,K]
+computed at the current MLE µt as
+γt ≜ arg sup
+w∈∆K
+inf
+¯ν∈alt(a⋆)
+�
+i∈[K]
+wiDKL(Pt,i∥¯Pi) .
+(117)
+Lemma 8. There exists a stochastic time N ϵ ∈ N such that for all t > N ϵ, Oϵ
+t = ∅, and E[N ϵ] < +∞, if the
+sampling strategy satisfies
+1
+t+1Tt(at+1) < γt,at+1 + ζ for any ζ ∈ [0, ϵ
+4] and for any t > M, where M is a stochastic
+time satisfying Eν[M] < +∞.
+Proof. Let us define the time instant M ϵ
+1 such that for all t > M ϵ
+1 and for al i ∈ [K], |γt,i −wi(ν)| < ϵ/8. Leveraging
+the continuity of Γ in Lemma 3 and the convergence of the MLE in Theorem 2, we obtain that E[M ϵ
+1] < +∞.
+Furthermore, define M ϵ
+2 ≜ ⌈(8/ϵ) − 1⌉, and M ϵ ≜ max{M, M ϵ
+1, M ϵ
+2}. We have the following two cases:
+1. Oϵ
+M ϵ = ∅: In this case, we will use induction on t to show that for all t > M ϵ, Oϵ
+t = ∅. First, by our assumption,
+for t = M ϵ, Oϵ
+M ϵ = ∅. Next, assume the inductive hypothesis that for some t > M ϵ, Oϵ
+t = ∅. Then,
+Tt+1(at+1)
+t + 1
+= Tt(at+1) + 1
+t + 1
+(118)
+< γt,at+1 +
+1
+t + 1 + ζ
+(119)
+≤ wat+1(ν) + ϵ
+8 +
+1
+t + 1 + ζ
+(120)
+≤ wat+1(ν) + ϵ
+2 ,
+(121)
+where (119) holds since the sampling strategy satisfies
+1
+t+1Tt(at+1) < γt,at+1 + ζ, and (121) is obtained using
+the definition of Mϵ. Hence, Oϵ
+t+1 = ∅ and it concludes the proof.
+2. |Oϵ
+M ϵ| ≥ 1: In this case, following the same steps as (119)-(121), we can show that for all t > M ϵ, any i ∈ Pϵ
+t
+is not included in Oϵ
+t. Furthermore, for any t > M ϵ and for any j ∈ Oϵ
+t, let us define Lϵ
+j as the time that j
+leaves Oϵ
+t, i.e., for all t ∈ {M ϵ, · · · , Lϵ
+j − 1}, j ∈ Oϵ
+t. Next, defining Lϵ ≜ maxj∈[K] Lϵ
+j, for all t > Lϵ, we
+obtain that |Oϵ
+t| = 0. Finally, defining N ϵ ≜ max{M ϵ, Lϵ}, we obtain that for all t > N ϵ, Oϵ
+t = ∅.
+23
+
+■
+Lemma 9. For all t > N
+ϵ
+K , the allocation for every arm i ∈ [K] satisfies
+����
+Tt(i)
+t
+− wi(ν)
+���� ≤ ϵ .
+(122)
+Proof. We will prove (122) by contradiction. Assume that there exists j ∈ [K] such that 1
+t Tt(i) < wj(ν) − ϵ. For all
+t > N
+ϵ
+K , leveraging Lemma 8, we have
+�
+i∈[K]
+Tt(i)
+t
+=
+�
+i̸=j
+Tt(i)
+t
++ Tt(j)
+t
+(123)
+≤
+�
+i̸=j
+�
+wi(ν) + ϵ
+K
+�
++ wj(ν) − ϵ
+(124)
+= 1 − ϵ
+K ,
+(125)
+which is a contradiction. Thus, (122) holds for all t > N
+ϵ
+K .
+■
+Next, we show that our proposed sampling strategy always samples from the set of under-sampled arms, which is
+specified in Lemma 10. Let us define the minimum sub-optimality gap
+∆min ≜
+min
+i∈[K]\{a⋆} µ(a⋆) − µ(i) .
+(126)
+Lemma 10. For all t > N ∆min/4
+ν
+, the TCB sampling rule provided in (34) satisfies
+Tt(at+1)
+t + 1
+≤ γt,at+1 .
+(127)
+Proof. Note that by Theorem 2, for all t > N ∆min/4
+ν
+, the proposed sampling rule satisfies that atop
+t
+= a⋆. Hence,
+according to our sampling strategy, for any t > N ∆min/4
+ν
+, there could be the two following cases.
+1. at+1 = a⋆: Γt(w) is a minimum of linear functions, and hence, it is a concave function [23]. Furthermore, w
+belongs to a compact space ∆K, and hence Γt(w) has a unique extremum. By our definition, the extremum is γt.
+Let us assume that
+1
+t+1Tt(a⋆) > γt,a⋆. Note that by the definition of amin
+t
+, this implies that 1
+t Tt(amin
+t
+) < γt,amin
+t
+.
+To show this, let us assume that 1
+t Tt(amin
+t
+) ≥ γt,amin
+t
+. We have
+min
+x∈It,amin
+t
+�Tt(a⋆)
+t
+dU(Pt,a⋆, x) + Tt(amin
+t
+)
+t
+dL(Pt,amin
+t
+, x)
+�
+≥
+min
+x∈It,amin
+t
+�
+γt,a⋆dU(Pt,a⋆, x) + Tt(amin
+t
+)
+t
+dL(Pt,amin
+t
+, x)
+�
+(128)
+≥
+min
+x∈It,amin
+t
+�
+γt,a⋆dU(Pt,a⋆, x) + γt,amin
+t
+dL(Pt,amin
+t
+, x)
+�
+(129)
+= Γt(γt) ,
+(130)
+where (128) and (129) hold due to the fact that Γt is an increasing function in each of its arguments, keeping the
+other argument fixed [10, Lemma 2]. It can be readily verified that (130) is a contradiction, since we obtain that
+Γt(pt) ≥ Γt(γt), where we have defined pt ≜ 1
+t [Tt(1), · · · , Tt(K)], and γt is the unique maximizer. Thus,
+we have 1
+t Tt(amin
+t
+) < γt,amin
+t
+, if
+1
+t+1Tt(a⋆) > γt,a⋆.
+24
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+latexit>wamin
+t
+Figure 6: Positions of z and z′ assuming
+1
+t+1Tt(a⋆) > γt,a⋆
+Next, we will prove that
+1
+t+1Tt(a⋆) ≤ γt,a⋆ by contradiction. We proceed with our assumption that
+1
+t+1Tt(a⋆) >
+γt,a⋆, and define the points z and z′ such that for any λ1, λ2 ∈ (0, 1),
+z ≜ λ1γt + (1 − λ1)wt(a⋆) ,
+(131)
+and wt(amin
+t
+) ≜ λ2z′ + (1 − λ2)wt(a⋆) .
+(132)
+For a geometric representation of the relative position of these points, we refer to Figure 6. Owing to the
+concavity of Γt, we have
+Γt(z) ≥ λ1Γt(γt) + (1 − λ1)Γt(wt(a⋆))
+(133)
+≥ Γt(wt(a⋆)) .
+(134)
+Note that as a result of the TCB sampling rule,we have
+Γt(wt(a⋆)) > Γt(wt(amin
+t
+)) .
+(135)
+Accordingly, let us define
+ϵt ≜ Γt(wt(a⋆)) − Γt(wt(amin
+t
+)) .
+(136)
+Furthermore, for any w ∈ ∆K, let ∇Γt(w) denote the sub-gradient of the function Γt at w. Owing to the
+concavity of Γt, we have
+Γt(z′) ≥ Γt(z) − ⟨∇Γt(z), z′ − z⟩
+(137)
+≥ Γt(z) − ∥∇Γt(z)∥ ∥z′ − z∥
+(138)
+(134)
+≥
+Γt(wt(a⋆)) − ∥∇Γt(z)∥ ∥z′ − z∥
+(139)
+(136)
+=
+Γt(wt(amin
+t
+)) + ϵt − ∥∇Γt(z)∥ ∥z′ − z∥ ,
+(140)
+where (138) is obtained using the Cauchy-Schwarz inequality. Furthermore, leveraging the concavity of Γt, we
+have
+Γt(wt(amin
+t
+)) ≥ λ2Γt(z′) + (1 − λ2)Γt(wt(a⋆))
+(141)
+25
+
+(140)
+≥
+λ2
+�
+Γt(wt(amin
+t
+)) + ϵt − ∥∇Γt(z)∥ ∥z′ − z∥
+�
++ (1 − λ2)Γt(wt(a⋆)) ,
+(142)
+which implies that
+Γt(wt(amin
+t
+)) ≥ Γt(wt(a⋆)) −
+λ2
+1 − λ2
+(∥∇Γt(z)∥ ∥z′ − z∥ − ϵt) .
+(143)
+Let us set λ2 ≜ O(ϵ2
+t). Hence, (143) can be rewritten as
+Γt(wt(amin
+t
+)) ≥ Γt(wt(a⋆)) + O(ϵ2
+t)
+(144)
+(136)
+=
+Γt(wt(amin
+t
+)) + ϵt + O(ϵ2
+t) ,
+(145)
+which is a contradiction. This shows that when at+1 = a⋆, we have
+1
+t+1Tt(at+1) ≤ γt,at+1.
+2. at+1 = amin
+t
+: Let us assume that
+1
+t+1Tt(amin
+t
+) > γt,amin
+t
+. Furthermore, by our sampling strategy, we have
+Γt(wt(a⋆)) < Γt(wt(amin
+t
+)) .
+(146)
+Following similar arguments as the case when at+1 = a⋆, leveraging the concavity of Γt, we can arrive at a
+contradiction. Thus, in this case, when at+1 = amin
+t
+, we have
+1
+t+1Tt(at+1) ≤ γt,at+1.
+■
+Lemma 11. For any ζ ∈ R+, there exists a stochastic time M ζ
+w such that Eν[M ζ
+w] < +∞, and for all t > M ζ
+w, the
+ITCB sampling rule provided in (37) satisfies
+Tt(at+1)
+t + 1
+≤ γt,at+1 + ζ .
+(147)
+Proof. Let us recall that
+bmin
+t
+≜
+arg min
+i∈[K]\{atop
+t }
+�
+min
+x∈It,i
+�Tt(atop
+t )
+t
+dU(Pt,atop
+t , x) + Tt(i)
+t
+dL(Pt,i, x)
+�
++ log(Tt(i))
+t
+�
+.
+(148)
+Note that for all t > N ∆min/4
+ν
+, we have atop
+t
+= a⋆. Furthermore, for all t > ⌈
+1
+(Kϵ2)⌉ and for all i ∈ [K], we almost
+surely have
+log Tt(i)
+t
+≤ log(
+�
+t/K − 1)
+t
+(149)
+≤ log(
+�
+t/K) + 1
+t
+(150)
+≤
+�
+t/K
+t
+(151)
+≤ ϵ ,
+(152)
+where (149) is a result of the fact that Tt(i) ≥
+�
+t/K − 1 for all i ∈ [K] [7, Theorem 4], (151) holds due to
+the fact that log(1 + x) ≤ x, and (152) is obtained from the fact that t > ⌈1/(Kϵ2)⌉.
+Let us define M ϵ
+3 ≜
+max{N ∆min/4
+ν
+, 1/(Kϵ2)⌉}. For a sufficiently small ϵ ∈ R+, for any t > M ϵ
+3, we almost surely have
+bmin
+t
+=
+arg min
+i∈[K]\{a⋆}
+�
+min
+x∈It,i
+�Tt(a⋆)
+t
+dU(Pt,a⋆, x) + Tt(i)
+i
+dL(Pt,i, x)
+�
++ ϵ
+�
+(153)
+= amin
+t
+.
+(154)
+We have the following two cases.
+26
+
+1. at+1 = a⋆ : Let us assume that
+1
+t+1Tt(a⋆) > γt,a⋆ + ζ. Due to the ITCB sampling strategy in (37), for all
+t > M ϵ
+3, we have
+Γt(wt(a⋆)) + log(Tt(amin
+t
+))
+t + 1
+≥ Γt(wt(amin
+t
+)) + log(Tt(amin
+t
+) + 1)
+t + 1
+,
+(155)
+or, equivalently
+Γt(wt(a⋆)) − Γt(wt(amin
+t
+)) ≥
+1
+t + 1 log
+�
+1 +
+1
+Tt(amin
+t
+)
+�
+.
+(156)
+This implies that
+Γt(wt(a⋆)) − Γt(wt(amin
+t
+)) > 0 .
+(157)
+Following the same argument as Lemma 10, (157) implies that
+1
+t+1Tt(a⋆) ≤ γt,a⋆ + ζ for any ζ ≥ 0.
+2. at+1 = amin
+t
+: Let us assume that Tt(amin
+t
+)/(t+1) > γt,amin
+t
++ζ. We will show that this is a contradiction, if the
+condition in the ITCB sampling strategy in (37) holds. Specifically, according to the ITCB sampling strategy,
+we have
+Γt(wt(amin
+t
+)) − Γt(wt(a⋆)) ≥ −
+1
+t + 1 · log
+�
+1 +
+1
+Tt(amin
+t
+)
+�
+�
+��
+�
+≜ϵt
+.
+(158)
+We may have the following two cases.
+• Γt(wt(amin
+t
+)) − Γt(wt(a⋆)) > 0: In this case, following the same line of arguments as in Lemma 10 we
+obtain that Tt(amin
+t
+)/(t + 1) ≤ γt,amin
+t
++ ζ for any ζ ≥ 0.
+• Γt(wt(amin
+t
+)) ∈ [Γt(wt(a⋆))−ϵt/(t+1), Γt(wt(a⋆))]: Let us define the vector ei ≜ [e1(i), · · · , eK(i)]⊤,
+where, for any j ∈ [K] we have defined
+ej(i) ≜
+�
+�
+�
+�
+�
+�
+�
+1,
+if j = a⋆
+−1,
+if j = i
+0,
+otherwise
+.
+(159)
+Leveraging the concavity of Γt, we have
+Γt(wt(a⋆)) ≥ Γt(wt(amin
+t
+)) −
+�
+∇Γt(wt(a⋆)), wt(amin
+t
+) − wt(a⋆)
+�
+�
+��
+�
+<0
+,
+(160)
+which implies that
+Γt(wt(a⋆)) − Γt(wt(amin
+t
+)) ≥
+���
+∇Γt(wt(a⋆)), wt(a⋆) − wt(amin
+t
+)
+���
+(161)
+=
+1
+t + 1 ·
+���
+�
+∇Γt(wt(a⋆)), eamin
+t
+���� .
+(162)
+Combining (158) and (162), we obtain
+ϵt ≥
+���
+�
+∇Γt(wt(a⋆)), eamin
+t
+���� .
+(163)
+27
+
+Next, for any i ∈ [K] \ {a⋆}, let us define the set
+Mζ(i) ≜
+�
+w ∈ ∆K : |γt,i − wi| > ζ , |⟨∇Γt(w), ei⟩| > 0
+�
+.
+(164)
+It can be readily verified that wt(a⋆) ∈ Mζ(amin
+t
+). Thus, from (163), we obtain that
+ϵt ≥
+inf
+w∈Mζ(amin
+t
+)
+���
+�
+∇Γt(w), eamin
+t
+���� ,
+(165)
+where we have used the convention that inf{∅} = +∞. (165) can be further lower-bounded as
+ϵt ≥
+min
+i∈[K]\{a⋆}
+inf
+w∈Mζ(i) |⟨∇Γt(w), ei⟩| .
+(166)
+Furthermore, define the instant M ϵ
+4 ≜ max{⌈8K/ϵ2, 3K/(3 − 2
+√
+2)⌉}. For all t > M ϵ
+4, we have
+ϵt ≤
+1
+Tt(amin
+t
+)
+(167)
+≤
+1
+�
+t/K − 1
+(168)
+≤
+�
+2K
+t
+(169)
+≤ ϵ
+2 ,
+(170)
+where (167) is obtained from the fact that log(1 + x) ≤ x, (168) follows from the property of forced
+exploration that Tt(i) ≥
+�
+t/K − 1 for all i ∈ [K] [7, Theorem 4], and (169) and (170) follow from the
+fact that t > M ϵ
+4. Furthermore, setting
+ϵ = ϵ(ζ) ≜
+min
+i∈[K]\{a⋆}
+inf
+w∈Mζ(i) |⟨∇Γt(w), ei⟩| ,
+(171)
+it can be readily verified that (163) is a contradiction. This imples that for all t > M ϵ(ζ)
+4
+, assuming
+that
+1
+t+1Tt(amin
+t
+) > γt,amin
+t
+, the ITCB sampling condition (158) does not hold, and hence at+1 ̸= amin
+t
+.
+Finally, defining M ζ
+w ≜ max{M ϵ
+3, M ϵ(ζ)
+4
+}, it satisfies
+1
+t+1Tt(amin
+t
+) ≤ γt,amin
+t
++ ζ for any ζ ∈ R+.
+■
+F
+Proof of Theorem 5
+The upper bound on the average sample complexity is obtained by leveraging the convergence of the empirical
+problem complexity Γt( 1
+t Tt) as a result of our sampling strategy, to the true value Γ(ν), where we have defined
+Tt ≜ [Tt(1), · · · , Tt(K)]. This is stated in Lemma 12. Next, Lemma 13 states the relationship between the GLLR
+Λt(atop
+t , ach
+t ) and the empirical problem complexity Γt( 1
+t Tt). Finally, leveraging Lemma 12 and Lemma 13, we ob-
+tain an upper bound on the average sample complexity as a result of the TCB and ITCB sampling rules (34) and (37),
+and the stopping rule (24).
+Lemma 12. Under TCB and ITCB, for any ϵ ∈ R+, there exists Nϵ such that for all t ≥ Nϵ, we have
+����Γ(ν) − Γt
+�1
+t Tt
+����� ≤ ϵ ,
+(172)
+and Eν[Nϵ] < +∞.
+28
+
+Proof. For any ϵ′ > 0, let us define the time N ϵ′
+1 ≜ max{N ϵ′
+ν , N ϵ′
+w}. For all t > N ϵ′
+1 , we have:
+1. µt(i) ∈ [µ(i) − ϵ′, µ(i) + ϵ′] for every arm i ∈ [K].
+2.
+1
+t Tt(i) ∈ [wi(ν) − ϵ′, wi(ν) + ϵ′] for every arm i ∈ [K].
+3. Let νt ≜ [Pt,1, · · · , Pt,K] denote the bandit instance characterized by the mean values m(νt) = [µt,1, · · · , µt,K].
+As a result of the continuity of Γ(ν, w) in its first argument established in Lemma 3, for all t > N ϵ′
+1 , there exists
+ϵ′′ such that we have |Γ(ν, w) − Γ(νt, w)| < ϵ′′.
+Thus, for all t > N ϵ′
+1 , we have
+Γt(T/t) = min
+i̸=a⋆ min
+x∈It,i
+�Tt(a⋆)
+t
+dU(Pt,a⋆, x) + Tt(i)
+t
+dL(Pt,i, x)
+�
+(173)
+≤ min
+i̸=a⋆ min
+x∈It,i
+�
+(wa⋆(ν) + ϵ′) dU(Pt,a⋆, x) + Tt(i)
+t
+dL(Pt,i, x)
+�
+(174)
+≤ min
+i̸=a⋆ min
+x∈It,i {(wa⋆(ν) + ϵ′dU(Pt,a⋆, x)) + (wi(ν) + ϵ′) dL(Pt,i, x)}
+(175)
+= Γ(νt, w(ν)) + O(ϵ′)
+(176)
+≤ Γ(ν) + ϵ′′ + O(ϵ′)
+�
+��
+�
+≜ ϵ
+,
+(177)
+where (173) follows from the fact that atop
+t
+= a⋆ for all t > N ϵ′
+1 , (174) and (175) follow from the fact that Γi(ν, w)
+is an increasing function in each coordinate wi, keeping the other coordinates fixed [10, Lemma 2], and (177) follows
+from the fact that t > N ϵ′
+1 . Following similar steps as (173)-(177), we can show that
+Γt
+�1
+t Tt
+�
+≥ Γ(ν) − ϵ .
+(178)
+The proof is completed by setting Nϵ ≜ N ϵ′
+1 .
+■
+Lemma 13. Under the TCB and ITCB sampling rules, there exists a stochastic time N ϵ
+2 such that for all t > N ϵ
+2, the
+GLLR statistic satisfies
+1
+t Λt(atop
+t , ach
+t ) ≥ Γt
+�1
+t Tt
+�
+− ϵ ,
+(179)
+and Eν[N ϵ
+2] < +∞.
+Proof. We use the notion of r-quick convergence, which we define below.
+Definition 2 (r-quick convergence [24]). Consider the sequence of i.i.d. zero-mean random variables {Zt : t ∈ N}.
+Let ¯Zt ≜ 1
+t
+�t
+s=1 Zt denote the empirical mean. Furthermore, for any ϵ ∈ R+ define
+Tϵ ≜ sup
+�
+t ∈ N : | ¯Zt| > ϵ
+�
+.
+(180)
+Then, {Zt : t ∈ N} converges r-quickly for r > 0, if E[T r
+ϵ ] < +∞.
+We leverage r-quick convergence for r = 1 to establish the convergence of the normalized GLLR statistic to the
+corresponding KL divergence measure. For this, we first state the necessary and sufficient condition for r-quick
+convergence to hold.
+29
+
+Lemma 14 (Corollary 4, [24]). The i.i.d. sequence {Zt : t ∈ N} converges r-quickly if and only if E[|Zt|r+1] < +∞.
+In our setting, for any arm i ∈ Rt(ach
+t ) and any s ∈ [t] such that As = i, let us set
+Zs ≜ log πi(Xs | µt(i))
+πi(Xs | ˜µt(ach
+t )) − DKL(Pt,i∥˜Pt,ach
+t ) ,
+(181)
+where ˜Pt,ach
+t denotes the distribution of the arm ach
+t when the mean value is equal to ˜µt(ach
+t ). Furthermore, leveraging
+Assumption 6, we have
+Eν[|Zs|2] < +∞ .
+(182)
+Hence, using 1-quick convergence, we obtain that Eν[N ϵ
+KL] < +∞, where we have defined
+N ϵ
+KL ≜ sup
+�
+�
+�t ∈ N :
+������
+1
+Tt(i)
+�
+s∈[t]:As=i
+log πi(Xs | µt(i))
+πi(Xs | ˜µt(ach
+t )) − DKL
+�
+Pt,i∥˜Pt,ach
+t
+�
+������
+> ϵ
+�
+�
+� .
+(183)
+Thus, for any t > N ϵ
+KL, we have
+1
+t Λt(atop
+t , ach
+t ) ≥ 1
+t
+�
+i∈Rt(ach
+t )
+Tt(i)
+�
+DKL(Pt,i∥˜Pt,ach
+t ) − ϵ
+�
+(184)
+= 1
+t
+�
+�
+�
+i∈Rt(ach
+t )
+Tt(i)DKL(Pt,i∥˜Pt,ach
+t ) − Tt(i)ϵ
+�
+�
+(185)
+≥
+�
+i∈Rt(ach
+t )
+Tt(i)
+t
+DKL(Pt,i∥˜Pt,ach
+t ) − ϵ .
+(186)
+Furthermore, for any ϵ ∈ (0, ∆min/4) define the time instant N ϵ
+2 ≜ max{N ϵ
+KL, N ϵ
+ν}. Thus, for any t > N ϵ
+2 and for
+any i ∈ Rt(ach
+t ), we have
+1
+t Λt(atop
+t , ach
+t ) ≥ Tt(a⋆)
+t
+DKL(Pt,a⋆∥˜Pt,ach
+t ) + Tt(i)
+t
+DKL(Pt,i∥˜Pt,ach
+t ) − ϵ
+(187)
+≥
+min
+x∈[µt(ach
+t ),µt(a⋆)]
+�Tt(a⋆)
+t
+dU(Pt,a⋆, x) + Tt(i)
+t
+dL(Pt,i, x)
+�
+− ϵ
+(188)
+= min
+i̸=a⋆
+min
+x∈[µt(i),µt(a⋆)]
+�Tt(a⋆)
+t
+dU(Pt,a⋆, x) + Tt(i)
+t
+dL(Pt,i, x)
+�
+− ϵ
+(189)
+= Γt
+�1
+t T
+�
+− ϵ ,
+(190)
+where (187) holds since a⋆ ∈ Rt(ach
+t ) for any t > N ϵ
+2 and (188) holds since µt(ach
+t ) < µt(i) for any i ∈ Rt(ach
+t ).
+Finally, it can readily verified that N ϵ
+2 < N ϵ
+KL + N ϵ
+ν, and hence, Eν[N ϵ
+2] < +∞.
+■
+Next, in order to prove Theorem 5, let us define the time instant N ϵ
+3 ≜ max{Nϵ, N ϵ
+2}. Expanding the time instant just
+before stopping, we have
+τ − 1 = (τ − 1)1{τ−1≤N ϵ
+3} + (τ − 1)1{τ−1>N ϵ
+3}
+(191)
+≤ N ϵ
+3 + (τ − 1)1{τ−1>N ϵ
+3} .
+(192)
+30
+
+Leveraging Lemma 12 and Lemma 13, along with the fact that at τ − 1, Λτ−1(atop
+τ−1, ach
+τ−1) ≤ βτ−1(δ), if τ − 1 > N ϵ
+3
+we have
+Γ(ν) − 2ϵ ≤ Γτ−1
+�
+1
+τ − 1Tτ−1
+�
+− ϵ ≤ Λτ−1(atop
+τ−1, ach
+τ−1)
+τ − 1
+≤ βτ−1(δ)
+τ − 1
+.
+(193)
+Subsequently, following the same steps as [7, Theorem 8], we obtain
+lim
+δ→0
+E[τ]
+log(1/δ) ≤
+1 + α
+Γ(ν) − 2ϵ .
+(194)
+Finally, taking infimum with respect to ϵ in (194), we obtain the desired result.
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+32
+

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+arXiv:2301.02202v1  [math.AC]  5 Jan 2023
+ON THE EQUALITY OF TEST IDEALS
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+Abstract. We provide a natural criterion which implies equality of the finitistic test ideal
+and test ideal in local rings of prime characteristic. Most notably, we show that the criterion
+is met by every local weakly F-regular ring whose anti-canonical algebra is Noetherian on
+the punctured spectrum.
+1. Introduction
+Let R be a Noetherian ring of prime characteristic p > 0 and let R◦ be the set of elements
+which avoid all minimal primes of R. Let I ⊆ R be an ideal of R and denote by I[pe] the
+expansion of I along the eth iterate of the Frobenius endomorphism. The tight closure of
+I is the ideal I∗ consisting of elements x ∈ R such that there exists an element c ∈ R◦
+with the property that cxpe ∈ I[pe] for all e ≫ 0. Unlike integral closure of ideals, the tight
+closure of an ideal does not commute with localization, [BM10]. Brenner’s and Monsky’s
+counterexample to the localization problem leaves open the intriguing problem if the property
+of tight closure being a trivial operation on ideals commutes with localization.
+Continue to let R be a Noetherian ring of prime characteristic p > 0. The ring R is
+called weakly F-regular if every ideal is tight closed, that is I = I∗ for every ideal I.1 A
+ring is called F-regular if every localization of R is weakly F-regular. Let F e
+∗R denote the
+restriction of scalars of R along the eth iterate Frobenius endomorphism F e : R → R. We
+say that R is strongly F-regular if for each nonzero element c ∈ R there exists e ∈ N such
+that the R-linear map R → F e
+∗R defined by 1 �→ F e
+∗c is pure. Every strongly F-regular ring
+is weakly F-regular and the property of being strongly F-regular passes to localization. It
+is conjectured that all three notions of F-regularity agree.
+Conjecture 1.1 (The weak implies strong conjecture). If R is an excellent weakly F-regular
+ring of prime characteristic p > 0 then R is strongly F-regular.
+Williams proved Conjecture 1.1 for the class of 3-dimensional rings, [Wil95]. Every ex-
+cellent 4-dimensional F-regular ring of prime characteristic p > 5 is strongly F-regular by
+pairing [AP22, Corollary 4.4] with [BMP+20, Corollary K]. The purpose of this article is to
+extend the results of [AP22] to rings of arbitrary dimension. In particular, if the results of
+the prime characteristic minimal model program in dimension 3 established in [BMP+20]
+are valid in all dimensions, then the classes of excellent F-regular and excellent strongly
+F-regular rings are equivalent.
+Polstra was supported in part by NSF Grant DMS #2101890 and by a grant from the Simons Foundation,
+Grant Number 814268, MSRI.
+1An important property of tight closure theory is that every regular ring is weakly F-regular (and therefore
+also F-regular).
+1
+
+2
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+A prime characteristic ring R is weakly F-regular if and only if Rm is weakly F-regular
+for every maximal ideal m ∈ Spec(R). Moreover, an excellent local ring is weakly F-regular
+if and only if its completion is weakly F-regular. We therefore restrict our attention to the
+class of weakly F-regular local rings which admit a canonical module. Every local weakly
+F-regular ring is a normal domain and therefore has divisor class group cl(R).
+Theorem A. Let (R, m, k) be an excellent normal domain of prime characteristic p > 0, of
+Krull dimension d, and I ⊆ R an anti-canonical ideal.2 Suppose that there exists an m ∈ N
+such that I(m) is principal when localized at a each height 2 prime3 and for each 1 ≤ j ≤ d−2
+there exists an ideal aj of height d − j + 1 such that
+ape
+j Hj
+m
+�
+R
+I(mpe)
+�
+= 0
+for every e ∈ N. If R is weakly F-regular then R is strongly F-regular.
+Remark 1.2. The Matlis dual of the local cohomology module Hj
+m(R/I(mpe)) is the com-
+pletion of Extd−j
+R (R/I(mpe), J), a module which is not supported in codimension d − j if
+j ≤ d − 2. Hence Hj
+m(R/I(mpe)) is annihilated by an ideal of height d − j + 1. The criterion
+of Theorem A is therefore reasonable as it is natural to anticipate that the annihilators of
+Hj
+m(R/I(mpe)) are of linear comparisons as e → ∞.
+Remark 1.3. Let (R, m, k) be as in Theorem A and assume that R is Cohen-Macaulay.
+Suppose that ER(k) is an injective hull of the residue field. Let 0∗
+ER(k) and 0∗fg
+ER(k) denote
+the tight closure and finitisitic tight closure respectively of the 0-submodule of ER(k), see
+Section 3 for definitions. Then 0∗fg
+ER(k) = 0∗
+ER(k) under the hypotheses of Theorem A, see The-
+orem 4.10. Therefore the test ideal and big test ideal of R agree by [HH90, Proposition 8.23]
+and [AE03, Theorem 3.2], c.f. [LS01, Theorem 7.1 and Theorem 7.2]. By definition, the test
+ideal of R is the unit ideal if and only if R is weakly F-regular and the big test ideal of R is
+the unit ideal if and only if R is strongly F-regular. Therefore Theorem A is a consequence
+of Theorem 4.10.
+Conjecture 1.1 is valid for rings R which are standard graded over a field, [LS99].
+It
+would be interesting to know if such rings satisfy the hypotheses of Theorem A. Without the
+standard graded assumption, most established cases of Conjecture 1.1 require an assumption
+on R that is akin to being Gorenstein. Hochster and Huneke proved Conjecture 1.1 for the
+class of Gorenstein rings, [HH94]. Building upon Williams’ proof of Conjecture 1.1 for the
+class of 3-dimensional rings, [Wil95], MacCrimmon proved the weak implies strong conjecture
+for rings which are Q-Gorenstein on the punctured spectrum, [Mac96]. Singh announced that
+Conjecture 1.1 is valid for rings whose anti-canonical algebra4 is Noetherian. Singh’s result
+was never published, but has since been recaptured by others, [CEMS18].
+2An ideal I ⊆ R is an anti-canonical ideal if it represents the inverse of the canonical divisor in the class
+group of R. Equivalently, there exists a canonical ideal J ⊆ R, with components disjoint from that of I, so
+that I ∩ J is a principal ideal.
+3Every normal F-pure ring admits an m ≥ 1 with this property, see [Har98, Section 4].
+4Suppose that R is a normal domain and I ⊆ R is an anti-canonical ideal. The anti-canonical algebra of R
+is the symbolic Rees algebra R ⊕ I ⊕ I(2) ⊕ · · · , an algebra unique up to R-algebra isomorphism.
+
+ON THE EQUALITY OF TEST IDEALS
+3
+Singularities of prime characteristic rings are related to KLT singularities of the complex
+minimal model program through the process of reduction to prime characteristic, [HW02,
+Tak04]. Theorems of the complex minimal model program establish that if R is essentially
+of finite type over C with at worst KLT singularities, then the symbolic Rees algebras
+associated to ideals of pure height 1 are Noetherian. It is therefore natural to expect the
+same in strongly F-regular rings and that the hypotheses of Singh’s Theorem are vacuous.
+Conjecture 1.4. If R is an excellent strongly F-regular ring of prime characteristic p > 0
+and I ⊆ R an ideal of pure height 1. Then the symbolic Rees algebra of I is Noetherian.
+Progress around Conjecture 1.4 is quite limited. An elementary and (mostly) algebraic
+proof of Conjecture 1.4 for the class of 2-dimensional F-regular rings can be derived from
+[Pol22, Corollary 3.2]. Recent progress progress of the minimal model program establishes
+Conjecture 1.4 for the class of 3-dimensional F-regular rings, see [BMP+20, Corollary K]
+and [AP22, Proof of Corollary 4.5] for necessary details.
+In light of Conjecture 1.4, it would be desirable to remove the assumption that the anti-
+canonical algebra of R is Noetherian in Singh’s Theorem and replace it with the milder
+hypothesis that the anti-canonical algebra is assumed to be Noetherian at non-closed points
+of Spec(R). Such a step puts forth a much needed inductive program to establish Conjec-
+ture 1.4, or at the very least establish that the class of F-regular and strongly F-regular
+rings agree. This is what we accomplish and is the main contribution of this article.
+Theorem B. Let (R, m, k) be an excellent weakly F-regular ring of prime characteristic
+p > 0, of Krull dimension d, and I ⊆ R an anti-canonical ideal. Suppose that the anti-
+canonical algebra of R is Noetherian on the punctured spectrum. There exists m ∈ N so that
+I(m) is principal when localized at each height 2 prime and for each 1 ≤ j ≤ d−2 there exists
+an ideal aj of height d − j + 1 such that
+ape
+j Hj
+m
+�
+R
+I(mpe)
+�
+= 0
+for every e ∈ N. In particular, the ring R is strongly F-regular by Theorem A.
+2. Annihilators of Local Cohomology
+This section is devoted to proving Theorem B. Let (R, m, k) be an excellent local normal
+domain of Krull dimension d ≥ 3 and I ⊆ R an ideal of pure height 1. Let W = R \
+�
+P ∈min(I) P and for each n ∈ N let I(n) = InRW ∩ R denote the nth symbolic power of
+the ideal I. To study the annihilators of Hi
+m(R/I(n)) we will approximate the ideals I(n) by
+ideals of the form (y1, . . . , yh)n where h is “small,” y1, . . . , yh ∈ I, and J denotes the integral
+closure of an ideal J ⊆ R.
+Let J ⊆ R be an ideal and n ∈ N. There are short exact sequences
+0 → Jn−1
+Jn
+→ R
+Jn →
+R
+Jn−1 → 0,
+and so there are exact sequences of local cohomology modules
+Hi
+m
+�Jn−1
+Jn
+�
+→ Hi
+m
+� R
+Jn
+�
+→ Hi
+m
+� R
+Jn−1
+�
+.
+
+4
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+Our aim is to establish uniform annihilators of the local cohomology modules Hi
+m(Jn−1/Jn)
+that are independent of n. For the sake of convenience, we adopt the following notation:
+• R[Jt] = �
+n≥0 Jntn is the Rees algebra of J;
+• R[Jt, t−1] = �
+n∈N Jntn is the extended Rees algebra of J, i.e. R[Jt, t−1] agrees with
+the Rees algebra R[Jt] in positive degree and contains copies of R in negative degree;
+• R is the integral closure of the Rees algebra R[Jt] in R[t]; R is N-graded and the
+nth graded piece of R is Jn;
+• R[t−1] is the integral closure of the extended Rees algebra R[Jt, t−1] in R[t, t−1]. If
+n ≥ 0 then the nth graded piece of R[t−1] is Jn. The algebra R[t−1] contains copies
+of R in negative degrees.
+If x ∈ R then xHi
+m(Jn−1/Jn) = 0 for all n ∈ N if and only if
+xHi
+m(R[t−1]/(t−1R[t−1]) = 0.
+The Faltings Annihilator Theorem, later generalized by Brodmann, provides a criterion to
+establish such annihilation properties.
+Theorem 2.1 ([BS13, Theorem 9.5.1]). Let S be a Noetherian ring which is the homomor-
+phic image of a regular ring, M a finitely generated S-module, and let a, b ⊆ R be ideals.
+Then
+min{i ∈ N |̸ ∃C : aCHi
+b(M) = 0} = min{depth(MP) + height((b + P)/P) | P ̸∈ V (a)}.
+Our first step towards proving Theorem B is the following lemma.
+Lemma 2.2. Let (R, m, k) be an excellent local Cohen-Macaulay normal domain of Krull
+dimension d and J ⊆ R an ideal generated by at most h elements. Suppose that the associated
+graded algebra �
+n≥0 Jn/Jn+1 ⊗R Rx is Cohen-Macaulay. Then there exists a constant C so
+that
+xCHi
+m(Jn/Jn+1) = 0
+for every 0 ≤ i ≤ d − h − 1 and n ∈ N.
+Proof. Let S = R[t−1] and G = S/t−1S. The lemma is equivalent to the assertion that there
+exists a constant C so that xCHi
+mS(G) = 0 for every 1 ≤ i ≤ d − h − 1. By Theorem 2.1, it
+suffices to show that if P ∈ Spec(S) \ V (xS) then
+depth(GP) + height
+�mS + P
+P
+�
+≥ d − h.
+If P ̸∈ V (xS) then GP is Cohen-Macaulay. Therefore
+depth(GP) = dim(GP) = heightS(P) − 1.
+Then, because S is catenary,
+depth(GP) + height
+�mS + P
+P
+�
+= heightS(P) − 1 + dim(S/P) − dim(S/mS + P)
+= heightS(P) − 1 + d + 1 − heightS(P) − dim(S/mS + P)
+= d − dim(S/mS + P).
+
+ON THE EQUALITY OF TEST IDEALS
+5
+Recall that S is the integral closure of R[t−1] in R[t, t−1]. It follows that S/mS is a finite
+extension of the fiber cone of J, an R/m-algebra of Krull dimension at most h. Therefore
+dim(S/mS + P) ≤ dim(S/mS) = h
+and so depth(GP) + height
+�
+mS+P
+P
+�
+≥ d − h as needed.
+□
+Remark 2.3. If we are only interested in annihilation properties of H1
+m(Jn/Jn+1), then
+many of the assumptions of Lemma 2.2 are not necessary. One only needs to assume that
+R is an excellent normal domain and J is generated by at most d − 2 elements to conclude
+that there exists a constant C so that mC annihilates H1
+m(Jn/Jn+1) for every n ∈ N. Indeed,
+height(mS + P/P) ≥ 1 for all P ∈ Spec(S) \ V (mS). Thus, to show
+depth(GP) + height(mS + P/P) ≥ 2
+for every P ∈ Spec(S) \ V (mS), it suffices to show that height(mS + P/P) ≥ 2 whenever
+depth(GP) = 0. If depth(GP) = 0 then P ∈ Spec(S) is an associated prime of t−1S. The
+ring S is normal and t−1 is a nonzerodivisor. Therefore every associated prime of t−1S is
+minimal and so dim(GP) = 0. One can now proceed as in the proof of Lemma 2.2 to show
+that height(mS + P/P) ≥ 2.
+Corollary 2.4. Let (R, m, k) be an excellent local Cohen-Macaulay normal domain of Krull
+dimension d and J ⊆ R an ideal generated by at most h elements. Suppose that the ring
+� Jn/Jn+1 ⊗R Rx is Cohen-Macaulay. Then there exists a constant C so that
+xCnHi
+m(R/Jn) = 0
+for every 0 ≤ i ≤ d − h − 1 and n ∈ N.
+Proof. For every i ≥ 0 and for every n ∈ N there are exact sequences of local cohomology
+modules
+Hi
+m(Jn/Jn+1) → Hi
+m(R/Jn+1) → Hi
+m(R/Jn).
+By Lemma 2.2, if i ≤ d − h − 1, then there exists a constant C so that xC annihilates the
+left most module of the above exact sequences for all n ≥ 0. By induction, xCn annihilates
+Hi
+m(R/Jn) for every n ∈ N.
+□
+Remark 2.5. As in the proof of Lemma 2.2, if we are only interested in annihilation prop-
+erties of H1
+m(R/Jn), then we only need to assume that R is an excellent local normal domain
+and J is generated by at most d − 2 elements.
+Lemma 2.6. Let (R, m, k) be an excellent Noetherian local normal domain with infinite
+residue field, I ⊆ R an ideal, P1, . . . , Pt ∈ Spec(R) a finite collection of non-comparable
+prime ideals, and W = R \
+�t
+i=1 Pi. Suppose that ℓRPi(IRPi) ≤ h for every 1 ≤ i ≤ t. Then
+there exist elements y1, . . . , yh ∈ I and x ∈ W with the following properties:
+(1) (y1, . . . , yh)RW ⊆ IRW is a reduction of IRW;
+(2) xnIn ⊆ (y1, . . ., yh)n for all n ∈ N.
+Proof. Recall the following: Suppose (S, n, ℓ) is a local ring and J ⊆ I are ideals. Then J is
+reduction of I if and only if S[Jt] ⊗S ℓ → S[It] ⊗S ℓ is finite, see [SH06, Proposition 8.2.4].
+
+6
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+In particular, if J′ ⊆ I is an ideal such that J′ ≡ J + nI and J is a reduction of I then J′ is
+a reduction of I.
+To prove the lemma start by choosing elements y1,i, . . . , yh,i ∈ I so that (y1,i, . . . , yh,i)RPi
+forms a reduction of the ideal IRPi. Choose elements rj ∈ Pj \ �
+i̸=j Pi and set yi = � rjyj,i.
+Then (y1, . . . , yh)RPi forms a reduction of IRPi for each 1 ≤ i ≤ t by the above discussion.
+Therefore (y1, . . ., yh)Rw forms a reduction of IRW by [SH06, Propositions 8.1.1.].
+Let J = (y1, . . . , yh). Then JRW = IRW and so there exists an element x ∈ W such
+that xI ⊆ J, in particular xI ⊆ J. Raising the containment to the nth power we find that
+xnIn ⊆ Jn for every n ∈ N. We claim that xnIn ⊆ Jn. Let r ∈ In, then there exists a t ∈ N
+and an equation
+rt + a1rt−1 + · · · + at−1r + at = 0
+such that aj ∈ Inj for each 1 ≤ j ≤ t. Multiplying by xnt we find that there is an equation
+(xnr)t + xna1(xnr)t−1 + · · · xn(t−1)at−1 + xntat = 0.
+The elements xnjaj belong to Jnj and therefore xnr ∈ Jn = Jn.
+□
+Theorem 2.7. Let (R, m, k) be an excellent local Cohen-Macaulay normal domain of Krull
+dimension d ≥ 3 and I ⊆ R an ideal of pure height 1 with the following properties:
+• InRP = I(n)RP for every P ∈ Spec(R) \ {m} and every n ∈ N;
+• If P ∈ Spec(R) \ {m} and G = R[t−1]/t−1R[t−1] is the associated graded ring of I
+then GP is Cohen-Macaulay.
+Then there exists a system of parameters x1, x2, . . . , xd such that for every 3 ≤ t ≤ d;
+(xn
+1, . . ., xn
+t )Hj
+m(R/I(n)) = 0
+for every 0 ≤ j ≤ d − (t − 1) and n ∈ N.
+Proof. The ideal I is locally principal at its height 1 components because R is normal.
+Therefore In ⊆ I(n). Our assumptions inform us that I(n)/In is 0-dimensional for every
+integer n. Therefore for every integer i ≥ 1
+Hi
+m(R/In) ∼= Hi
+m(R/I(n)).
+Start by choosing x1 ∈ I. Then clearly xn
+1 ∈ In and therefore xn
+1 annihilates Hi
+m(R/In) for
+all integers i and n. If W1 is the complement of the union of the minimal primes of x1R then
+IRW1 is a principal ideal. By Lemma 2.6 there exists an element y ∈ I and x ∈ W1 so that
+xnIn ⊆ ynR for every n ∈ N. There are short exact sequences
+0 → In
+ynR →
+R
+ynR → R
+In → 0
+and so Hj
+m(R/In) ∼= Hj+1
+m
+(In/ynR) if j ≤ d−3 and there is an injective map Hd−2
+m
+(R/In) →
+Hd−1
+m
+(In/ynR). Therefore xn annihilates Hj
+m(R/In) for every j ≤ d − 2 and we take x2 = x.
+If W2 is the complement of the union of the minimal primes of (x1, x2) then IRW2 has
+analytic spread at most 1, see [CHS10, Proof of Theorem 1.5]. The ring RW2 is normal, every
+principal ideal in a normal ring is integrally closed, and therefore IRW2 is a principal ideal.
+We therefore proceed as before to find an element x3 so that xn
+3 annihilates Hj
+m(R/In) for
+every j ≤ d − 2 as needed.
+
+ON THE EQUALITY OF TEST IDEALS
+7
+Inductively, suppose that we have found parameter elements x1, . . ., xi, with i ≥ 3, so that
+if 3 ≤ t ≤ i then
+(xn
+1, . . . , xn
+t )Hj
+m(R/In) = 0
+for every 0 ≤ j ≤ d − (t − 1). It is important that t ≥ 3 in the inductive step of the proof. If
+t = 2 then it is not the case that (xn
+1, xn
+2) annihilates Hj
+m(R/In) for every 0 ≤ j ≤ d−(2−1) =
+d − 1. Indeed, the annihilator of the top local cohomology module Hd−1
+m
+(R/In) is the height
+1 ideal I(n) and (xn
+1, xn
+2) ̸⊆ I(n). If i = d then we are done. Suppose that i ≤ d − 1. Our aim
+is to find a parameter element xi+1 so that
+xn
+i+1Hj
+m(R/In) = 0
+for every j ≤ d − i.
+Let W be the complement of the union of the minimal primes of the parameter ideal
+(x1, . . . , xi). Then InRW = I(n)RW for all integers n and so the localization of IRW at
+a maximal ideal of RW is an ideal of analytic spread at most i − 1, see [CHS10, Proof of
+Theorem 1.5]. By Lemma 2.6 there exists elements y1, . . . , yi−1 ∈ I and x′
+i+1 ∈ W so that
+(1) (y1, . . . , yi−1)RW ⊆ IRW is a reduction of IRW;
+(2) (x′
+i+1)nIn ⊆ (y1, . . . , yi−1)n for all n ∈ N.
+Let J = (y1, . . ., yi−1) and consider the short exact sequences
+0 → In
+Jn → R
+Jn → R
+In → 0.
+The element (x′
+i+1)n annihilates the left-most module in the above short exact sequence and
+there are exact sequences of local cohomology modules
+Hj
+m
+� R
+Jn
+�
+→ Hj
+m
+� R
+In
+�
+→ Hj+1
+m
+� In
+Jn
+�
+.
+The element (x′
+i+1)n annihilates the right-most module. By our hypothesis that the asso-
+ciated graded ring of I is Cohen-Macaulay on the punctured spectrum of R, Corollary 2.4
+implies that there exists a constant C so that (x′
+i+1)Cn annihilates Hj
+m(R/Jn) for every
+j ≤ d − (i − 1) − 1 = d − i. Therefore (x′
+i+1)C(n+1) annihilates Hj
+m(R/In) for every j ≤ d − i.
+Therefore xi+1 = (x′
+i+1)2C has the desired annihilation properties.
+□
+Theorem B is a consequence of the following theorem.
+Corollary 2.8. Let (R, m, k) be an excellent local Cohen-Macaulay normal domain of prime
+characteristic p > 0 and Krull dimension d ≥ 3. Suppose that R is a splinter on the punc-
+tured spectrum of R and that the anti-canonical algebra of R is Noetherian on the punctured
+spectrum. Then there exists an ideal I ⊆ R of pure height 1 and parameters x1, . . . , xd with
+the following properties:
+(1) I ∼= ω(−m)
+R
+for some m ≥ 1;
+(2) For each 1 ≤ j ≤ d − 2, the ideal aj := (x1, . . ., xd−j+1) is such that
+a[pe]
+j
+Hj
+m(R/I(pe)) = 0
+for each e ∈ N.
+
+8
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+Proof. Start by choosing an ideal I ⊆ R of pure height 1 so that I ∼= ω(−1)
+R
+is an anti-canonical
+ideal.
+We are assuming that the anti-canonical algebra is Noetherian on the punctured
+spectrum. Therefore if P ∈ Spec(R) \ {m} then there exists an integer m such that the
+Symbolic Rees algebra of I(m)RP is standard graded. The set of prime ideals ∪ Ass(R/I(m)n)
+is a finite set by [Bro79], see also [HS15]. By prime avoidance there exists an s ∈ R\P which
+is contained in each non-minimal member of ∪ Ass(R/I(m)n). Then I(m)nRs = I(mn)Rs for
+every n ∈ N. The space Spec(R\{m} is quasi-compact. Therefore there exists finitely many
+open sets D(s1), . . . , D(St) covering Spec(R \ {m} and integers mi, 1 ≤ i ≤ s, so that for all
+n ∈ N I(mi)nRsi = I(min)Rsi. If m is a common multiple of m1, . . . , ms and then the Symbolic
+Rees algebra of I(m) is standard graded on the punctured spectrum, i.e. I(m)nRP = I(mn)RP
+for all n ∈ N and P ∈ Spec(R) \ {m}. We replace I by I(m). By Theorem 2.7, it suffices to
+show that R is Cohen-Macaulay and that if G = GrI(R) is the associated graded ring of I
+then GP is a Cohen-Macaulay algebra for all P ∈ Spec(R) \ {m}.
+For each non-maximal prime P the localized ring RP is strongly F-regular by [CEMS18,
+Corollary 5.9], see also [Wat94, Theorem 0.1]. Therefore the localized (symbolic) Rees alge-
+bras R[It] ⊗ RP are Cohen-Macaulay for all P ∈ Spec(R) \ {m}, see [CEMS18, Lemma 6.1].
+We may now conclude that GP is a Cohen-Macaulay algebra for all P ∈ Spec(R) \ {m} by
+[Hun82, Proposition 1.1].
+□
+3. Tight closure, local cohomology, and local cohomology bounds
+3.1. Tight closure. Let R be a ring of prime characteristic p > 0 and let R◦ be the
+complement of the union of the minimal primes of R. The eth Frobenius functor, or the
+eth Peskine-Szpiro functor, is the functor F e : Mod(R) → Mod(R) obtained by extending
+scalars along the eth iterate of the Frobenius endomorphism. If N ⊆ M are R-modules and
+m ∈ M, then m is in the tight closure of N relative to M if there exists a c ∈ R◦ such that
+for all e ≫ 0 the element m is in the kernel of the following composition of maps:
+M → M/N → F e(M/N) ·c−→ F e(M/N).
+In particular, if we consider an inclusion of R-modules of the form I ⊆ R then F e(R/I) ∼=
+R/I[pe] where I[pe] = (rpe | r ∈ I), and an element r ∈ R is in the tight closure of I relative
+to R if there exists a c ∈ R◦ such that crpe ∈ I[pe] for all e ≫ 0. The tight closure of the
+module N relative to the module M is denoted N∗
+M. In the case that M = R and N = I is
+an ideal then we denote the tight closure of I relative to R as I∗. We say that N is tightly
+closed in M if N = N∗
+M. If an ideal is tightly closed in R then we simply say that the ideal
+is tightly closed. The finitistic tight closure of N ⊆ M is denoted N∗fg
+M
+and is the union of
+(N ∩ M′)∗
+M′ where M′ runs over all finitely generated submodules of M.
+The notions of weak F-regularity and strong F-regularity can be compared by studying
+the finitistic tight closure and tight closure of the zero submodule of the injective hull of a
+local ring by [HH90, Proposition 8.23] and [Smi93, Proposition 7.1.2]. Suppose that (R, m, k)
+is complete local and ER(k) is the injective hull of the residue field. The finitistic test ideal
+of R is τfg(R) =
+�
+I⊆R AnnR(I∗/I) and agrees with AnnR(0∗fg
+ER(k)). The (big) test ideal of R
+is τ(R) = �
+N⊆M∈Mod(R) AnnR(N∗
+M/N) and agrees with AnnR(0∗
+ER(k)). The ring R is weakly
+F-regular if and only if τfg(R) = R and R is strongly F-regular if and only if τ(R) = R.
+
+ON THE EQUALITY OF TEST IDEALS
+9
+Thus to prove the conjectured equivalence of weak and strong F-regularity it is enough to
+show 0∗
+ER(k) = 0∗fg
+ER(k) under hypotheses satisfied by rings which are weakly F-regular.
+To explore the tight closure of the zero submodule of ER(k) we exploit the structure of
+ER(k) as a direct limit of 0-dimensional Gorenstein quotients of R described in [Hoc77].
+Suppose (R, m, k) is a complete local Cohen-Macaulay domain of Krull dimension d and
+J1 ⊊ R a canonical ideal. Let 0 ̸= x1 ∈ J1, x2, . . . , xd ∈ R a parameter sequence, and
+for each t ∈ N let It = (xt−1
+1
+J1, xt
+2, . . . , xt
+d). The sequences of ideals {It} form a decreasing
+sequence of irreducible m-primary ideals cofinal with {mt}. Moreover, the direct limit system
+lim
+−→ R/It
+·x1···xd
+−−−−→ R/It+1 is isomorphic to ER(k). The following lemma uses this description of
+the injective hull of the residue field to describe any potential difference between the modules
+0∗
+ER(k) and 0∗fg
+ER(k). We refer the reader to the discussion at the beginning of [Abe02, Section 2]
+for details.
+Lemma 3.1. Let (R, m, k) be a complete Cohen-Macaulay local ring of prime characteristic
+p > 0 and of Krull dimension d. Let J1 ⊊ R be a choice of canonical ideal and x1, . . . , xd a
+system of parameters such that x1 ∈ J. Make the following identification of ER(k):
+ER(k) ∼= lim
+−→
+�
+R
+(xt−1
+1
+J1, xt
+2, · · · xt
+d)
+·x1···xd
+−−−−→
+R
+(xt
+1J1, xt+1
+2
+, · · · xt+1
+d
+)
+�
+If η = [r + (xt−1
+1
+J1, xt
+2, · · ·xt
+d)] ∈ ER(k) then
+(1) η ∈ 0∗fg
+ER(k) if and only if there exists a c ∈ R◦ and s ∈ N such that for all e ∈ N
+c(r(x1x2 · · · xd)s)pe ∈ (xs+t−1
+1
+J1, xs+t
+2
+, . . . , xs+t
+d
+)[pe];
+(2) η ∈ 0∗
+ER(k) if and only if there exists a c ∈ R◦ such that for all e ∈ N there exists an
+s = s(e) such that
+c(r(x1x2 · · · xd)s)pe ∈ (xs+t−1
+1
+J1, xs+t
+2
+, . . . , xs+t
+d
+)[pe].
+3.2. Local Cohomology Bounds. We will relate the modules 0∗fg
+ER(k) and 0∗
+ER(k) in Lemma 3.1
+through the language of local cohomology bounds. To this end, suppose that M is a module
+over a ring R and x = x1, . . ., xd a sequence of elements. For each j ∈ N let xj = xj
+1, . . ., xj
+d
+and for each pair of integers j1 ≤ j2 let ˜α•
+M;xi;j1;j2 be the map of Kosul cocomplexes
+0
+� M
+·xj1
+i
+�
+=
+�
+M
+�
+·xj2−j1
+i
+�
+0
+0
+� M
+·xj2
+i
+� M
+� 0
+Let ˜α•
+M;x;j1;j2 be the following product:
+˜α•
+M;x;j1;j2 := ˜α•
+R;x1;j1;j2 ⊗ ˜α•
+R;x2;j1;j2 ⊗ · · · ⊗ ˜α•
+R;xd;j1;j2 ⊗ M.
+Then ˜α•
+M;x;j1;j2 is a map of Koszul cocomplexes
+K•(xj1; M)
+˜α•
+M;x;j1;j2
+−−−−−→ K•(xj2; M).
+
+10
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+Let αi
+M;x;j1;j2 denote the induced map of Koszul cohomologies
+Hi(xj1; M)
+αi
+M;x;j1;j2
+−−−−−→ Hi(xj2; M).
+Then
+lim
+−→
+j1≤j2
+�
+Hi(xj1; M)
+αi
+M;x;j1;j2
+−−−−−→ Hi(xj2; M)
+�
+∼= Hi
+(x)R(M)
+by [BH93, Theorem 3.5.6].
+Denote by αi
+M;x;j;∞ the map
+Hi(xj; M)
+αi
+M;x;j;∞
+−−−−−→ Hi
+(x)A(M).
+Observe that η ∈ Ker(αi
+M;x;j;∞) if and only if there exists some k ≥ 0 such that η ∈
+Ker(αi
+M;x;j;j+k). If η ∈ Ker(αi
+M;x;j;∞) we let
+ǫi
+xj(η) = min{k | η ∈ Ker(αi
+M;x;j;j+k)}.
+Definition 3.2. Let R be a ring, x = x1, . . ., xd a sequence of elements in R, and M an
+R-module. The ith local cohomology bound of M with respect to the sequence of elements
+x is
+lcbi(x; M) = sup{ǫi
+xj(η) | η ∈ Ker(αi
+M;x;j;∞) for some j} ∈ N ∪ {∞}.
+Observe that if M is an R-module and x is a sequence of elements, then lcbi(x; M) = N <
+∞ simply means that if η ∈ Hi(xj; M) represents the 0-element in the direct limit
+lim
+−→
+j1≤j2
+�
+Hi(xj1; M)
+αi
+M;x;j1;j2
+−−−−−→ Hi(xj2; M)
+�
+∼= Hi
+(x)R(M)
+then αi
+M;x;j;j+N(η) is the 0-element of the Koszul cohomology group Hi(xj+N; M). There-
+fore finite local cohomology bounds correspond to a uniform bound of annihilation of zero
+elements in a choice of direct limit system defining a local cohomology module. It would be
+interesting to understand when local cohomology bounds are finite.
+3.3. Basic properties of local cohomology bounds. Our study of local cohomology
+bounds begins with two useful observations.
+Lemma 3.3. Let R be a commutative Noetherian ring, M an R-module, and x = x1, . . ., xd
+a sequence of elements, then lcbi(xj; M) ≤ lcbi(x; M). Furthermore, lcbi(x; M) ≤ jm for
+some integers j, m if and only if lcbi(xj; M) ≤ m where xj is the sequence of elements
+xj
+1, . . . , xj
+d.
+Proof. One only has to observe that αi
+M;xj;k,k+m = αi
+M;x;jk,jk+jm.
+□
+If x1, . . . , xd is a sequence of elements in a ring R and if x1M = 0 for some R-module M
+then the short exact sequence of Koszul cocomplexes
+0 → K•(x2, . . . , xd; M)(−1) → K•(x1, x2, . . . , xd; M) → K•(x2, . . . , xd; M) → 0
+
+ON THE EQUALITY OF TEST IDEALS
+11
+is split and therefore Hi(x1, x2, . . . , xd; M) ∼= Hi(x2, . . ., xd; M) ⊕ Hi−1(x2, . . . , xd; M). The
+content of the following lemma is a description of the behavior of the maps αi
+M;x1,x2,...,xd;j,j+k
+with respect to these isomorphisms of Koszul cohomologies.
+Lemma 3.4. Let R be a commutative Noetherian ring, M an R-module, and x1, x2, . . ., xd
+a sequence of elements such that x1M = 0. If i, j, k ∈ N then
+Hi(xj
+1, xj
+2, . . . , xj
+d; M) ∼= Hi(xj
+2, . . . , xj
+d; M) ⊕ Hi−1(xj
+2, . . . , xj
+d; M)
+and the map αM;x1,x2,...,xd;j,j+k is the direct sum of αi
+M;x2,...,xd;j,j+k and the 0-map.
+Proof. Let (F •, ∂•) be the Koszul cocomplex K•(xj
+2, . . . , xj
+d; R) and let (G•, δ•) be the Koszul
+cocomplex K•(xj
+1; R). Let
+(L•, γ•) = K•(xj
+1, xj
+2, . . . , xj
+d; R) ∼= K•(xj
+1; R) ⊗ K•(xj
+2, . . . , xj
+d; R).
+Then Li ∼= (G0 ⊗ F i) ⊕ (G1 ⊗ F i−1) ∼= F i ⊕ F i−1. We abuse notation and let ·xj
+1 denote the
+multiplication map on F i. The map γi can be thought of as
+γi =
+�
+∂i
+0
+±xj
+1
+∂i−1
+�
+: F i ⊕ F i−1 → F i+1 ⊕ F i.
+In particular, if we apply −⊗RM the map ·±xj
+1⊗M is the 0-map and therefore the ith map of
+the Koszul cocomplex Ki(xj
+1, xj
+2, . . ., xj
+d; M) is the direct sum of maps (∂i ⊗M)⊕(∂i−1 ⊗M).
+In particular
+Hi(xj
+1, xj
+2, . . . , xj
+d; M) ∼= Hi(xj
+2, . . ., xj
+d; M) ⊕ Hi−1(xj
+2, . . . , xj
+d; M).
+To see that αM;x1,x2,...,xd;j,j+k is the direct sum of αi
+M;x2,...,xd;j,j+k and the 0-map is similar to
+above argument but uses the fact that
+˜α•
+M;x1,x2,...,xd;j;j+k = ˜α•
+R;x2,...,xd;j;j+k ⊗ ˜α•
+R;x1;j;j+k ⊗ M
+and ˜α1
+R;x1;j;j+k ⊗ M = 0.
+□
+A particularly useful corollary of Lemma 3.4 is the following:
+Corollary 3.5. Let R be a commutative Noetherian ring and M an R-module. Suppose
+x1, . . . , xd is a sequence of elements, 1 ≤ i ≤ d, and (x1, . . . , xd−i)M = 0. If j, k ∈ N then
+αℓ
+M;x1,...,xd;j,j+k : Hℓ(xj
+1, . . . , xj
+d; M) → Hℓ(xj+k
+1
+, . . ., xj+k
+d
+; M)
+is the 0-map for all ℓ ≥ i + 1. In particular, lcbℓ(x1, . . . , xd; M) = 1 for all ℓ ≥ i + 1.
+Proof. By multiple applications of Lemma 3.4 it is enough to observe that
+Hℓ(xj
+d−i+1, . . . , xj
+d; M) = 0.
+This is clearly the case since xj
+d−i+1, . . . , xj
+d is a list of i elements and we are examining an
+ℓ ≥ i + 1 Koszul cohomology of M with respect to this sequence.
+□
+
+12
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+Suppose 0 → M1 → M2 → M3 → 0 is a short exact sequence of R-modules. The next two
+properties of local cohomology bounds we record allow us to compare the local cohomology
+bounds of the modules appearing in the short exact sequence. Proposition 3.6 allows us to
+effectively compare the local cohomology bounds of two of the terms in the sequence provided
+a subset of the elements in the sequence of elements defining Koszul cohomology annihilates
+the third. Proposition 3.7 compares the the local cohomology bounds of two of the terms in
+the short exact whenever the sequence of elements defining Koszul cohomology is a regular
+sequence on the third module.
+Proposition 3.6. Let (R, m, k) be a local ring and
+0 → M1 → M2 → M3 → 0
+a short exact sequence of finitely generated R-modules. Let x = x1, . . . , xd be a sequence of
+elements of R.
+(1) If (x1, . . ., xd−j)M1 = 0 then for all ℓ ≥ j + 1
+lcbℓ(x; M2) ≤ lcbℓ(x; M3) + 1.
+(2) If (x1, . . ., xd−j)M2 = 0 then for all ℓ ≥ j + 1
+lcbℓ(x; M3) ≤ lcbℓ+1(x; M1) + 1.
+(3) If (x1, . . ., xd−j)M3 = 0 then for all ℓ ≥ j + 1
+lcbℓ(x; M1) ≤ lcbℓ(x; M2) + 1.
+Proof. For each integer j ∈ N let xj denote the sequence of elements xj
+1, xj
+2, . . ., xj
+d. For (1)
+we consider the following commutative diagram, whose middle row is exact:
+Hℓ(xj; M2)
+Hℓ(xj; M3)
+Hℓ(xj+k; M1)
+Hℓ(xj+k; M2)
+Hℓ(xj+k; M3)
+Hℓ(xj+k+1; M1)
+Hℓ(xj+k+1; M2)
+αℓ
+M2;x;j;j+k
+αℓ
+M3;x;j;j+k
+αℓ
+M1;x;j+k;j+k+1
+αℓ
+M2;x;j+k;j+k+1
+By Corollary 3.5 the map αℓ
+M1;x;j+k;j+k+1 is the 0-map for all ℓ ≥ j + 1. A straightforward
+diagram chase of the above diagram, which follows an element η ∈ Ker(αℓ
+M2;x;j;j+k′) for some
+k′, shows that η ∈ Ker(αℓ
+M2;x;j;j+k+1) whenever k ≥ lcbℓ(x; M3). In particular, lcbℓ(x; M2) ≤
+lcbℓ(x; M3) + 1.
+Statements (2) and (3) follow in a similar manner and the details are left to the reader.
+□
+Proposition 3.7. Let R be a commutative Noetherian ring, 0 → M1 → M2 → M3 → 0 a
+short exact sequence of R-modules, and x = x1, . . . , xd a sequence of elements in R.
+(1) If x is a regular sequence on M1 then lcbi(x; M2) = lcbi(x; M3) for all i ≤ d − 1.
+(2) If x is a regular sequence on M2 then lcbi(x; M3) = lcbi+1(x; M1) for all i ≤ d − 1.
+(3) If x is a regular sequence on M3 then lcbi(x; M1) = lcbi(x; M2) for all i ≤ d.
+
+ON THE EQUALITY OF TEST IDEALS
+13
+Proof. The proofs of the three statements are very similar to one another and we only provide
+the details of (1).
+Proof of (1): For i < d we have Hi(xj; M1) = 0 and therefore if i ≤ d − 2 there are
+commutative diagrams
+Hi(xj; M2)
+Hi(xj; M3)
+Hi(xj+k; M2)
+Hi(xj+k; M3)
+∼
+=
+αi
+M2;x;j;j+k
+αi
+M3;x;j;j+k
+∼
+=
+whose horizontal arrows are isomorphisms. It readily follows that lcbi(x; M2) = lcbi(x; M3)
+whenever i ≤ d−2. Because x is a regular sequence on M1 we have that the maps αd
+M1,x,j,j+k
+are injective. Conside the following commutative diagrams whose rows are exact:
+0
+Hd−1(xj; M2)
+Hd−1(xj; M3)
+Hd(xj; M1)
+0
+Hd−1(xj+k; M2)
+Hd−1(xj+k; M3)
+Hd(xj+k; M1)
+πj
+αd−1
+M2;x;j;j+k
+δj
+αd−1
+M3;x;j;j+k
+αd
+M1;x;j;j+k
+πj+k
+δj+k
+If η ∈ Ker(αd−1
+M2;x;j,j+k) then πj(η) ∈ Ker(αd−1
+M3;x;j,j+k). The maps πj+k are injective. Therefore
+αd−1
+M2;x;j,j+k(η) = 0 whenever k ≥ lcbd−1(x; M3) and hence lcbd−1(x; M2) ≤ lcbd−1(x; M3).
+To show that lcbd−1(x; M2) ≥ lcbd−1(x; M3) consider an element η ∈ Ker(αd−1
+M3;x;j;j+k). Then
+δj(η) ∈ Ker(αd
+M1;x;j;j+k). But the maps αd
+M1;x;j;j+k are injective and therefore δj(η) = 0. In
+particular, η = πj(η′) for some η′ ∈ Hd−1(xj; M2). The maps πj+k are all injective. Therefore
+η′ ∈ Ker(αd−1
+M1;x;j;j+k) and it follows that αd−1
+M2;x;j;j+k(η) = 0 whenever k ≥ lcbd−1(x; M2).
+Therefore lcbd−1(x; M2) ≥ lcbd−1(x; M3) and hence lcbd−1(x; M2) = lcbd−1(x; M3).
+This
+completes the proof of (1).
+□
+4. Equality of test ideals
+Theorem A is a consequence of Theorem 4.2 and Theorem 4.9. Theorem 4.2 is an explicit
+relationship between local cohomology bounds and equality of test ideals.
+Theorem 4.9,
+when paired with Proposition 4.8, provides the needed local cohomology bounds described
+in Theorem 4.9 whenever we are able to linearly compare the annihilators of Hi
+m(R/I(n)) as
+n → ∞ and I ∼= ω(−m)
+R
+is a multiple of an anti-canonical ideal.
+4.1. Local Cohomology bounds and equality of test ideals. The content of the fol-
+lowing lemma can be pieced together by work of the first author in [Abe02]. We refer the
+reader to [PT18, Lemma 6.7] for a direct presentation of the lemma.5
+Lemma 4.1. Suppose that (R, m, k) is a Cohen-Macaulay local normal domain of dimension
+d, and J ⊆ R an ideal of pure height 1. Let x1, . . . , xd ∈ R be a system of parameters for R,
+assume that x1 ∈ J, and fix e ∈ N.
+5In [PT18, Lemma 6.7] there is an assumption that R is complete. But observe that since R → �R is faithfully
+flat the claims of the lemma can be checked after completion.
+
+14
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+(1) If x2J ⊆ a2R for some a2 ∈ J, then for any non-negative integers N2, . . . , Nd with
+N2 ≥ 2, we have that
+((J(pe), xN2pe
+2
+, xN3pe
+3
+, . . . , xNdpe
+d
+) : x(N2−1)pe
+2
+)
+=
+((J[pe], xN2pe
+2
+, xN3pe
+3
+, . . ., xNdpe
+d
+) : x(N2−1)pe
+2
+)
+=
+((J[pe], x2pe
+2 , xN3pe
+3
+, . . . , xNdpe
+d
+) : xpe
+2 ).
+(2) Suppose xm
+3 J(m) ⊆ a3R ⊆ J(m) for some n ∈ N, then for any non-negative integers
+N2, . . . , Nd with N3 ≥ 2, we have that
+((J(pe), xN2pe
+2
+, xN3pe
+3
+. . . , xNdpe
+d
+) : x(N3−1)pe
+3
+)
+⊆
+((J(pe), xN2pe
+2
+, x2pe
+3 , . . . , xNdpe
+d
+) : xm
+1 xpe
+3 ).
+Theorem 4.2. Let (R, m, k) be a local normal Cohen-Macaulay domain of Krull dimension
+d and of prime characteristic p > 0. Assume that R has a test element. Let J1 ⊆ R be a
+choice of canonical ideal. Suppose x1, . . ., xd is a system of parameters of R, x1 ∈ J1, and
+suppose that the following conditions are met:
+• There exists element a2 ∈ J1 and a3 ∈ J(m)
+1
+such that x2J1 ⊆ a2R and xm
+3 J(m)
+1
+⊆ a3R;6
+• For each e ∈ N there exists an integer ℓ such that
+lcbd−1(xℓ
+2, xℓ
+3, x4, . . . , xd; R/J(mpe+1)
+1
+) ≤ pe + 1.
+Then 0∗fg
+ER(k) = 0∗
+ER(k).
+Proof. Identify ER(k) as
+ER(k) ∼= lim
+−→
+�
+R
+(xt−1
+1
+J1, xt
+2, . . . , xt
+d)
+·x1x2···xd
+−−−−−→
+R
+(xt
+1J1, xt+1
+2
+, . . . , xt+1
+d
+)
+�
+.
+Suppose that η = [r + (xt−1
+1
+J1, xt
+2, . . . , xt
+d)] ∈ 0∗
+ER(k). Equivalently, there exists a c ∈ R◦ such
+that for all e ∈ N
+0 = cηpe = [crpe + (xt−1
+1
+J1, xt
+2, . . . , xt
+d)] ∈ F e(ER(k)) ∼= lim
+−→
+�
+R
+(xt−1
+1
+J1, xt
+2, . . ., xt
+d)[pe]
+�
+.
+Let J = xt−1
+1
+J1 and consider the local cohomology module
+Hd−1
+m
+� R
+J[pe]
+�
+= lim
+−→
+�
+R
+J[pe] + (xt
+2, . . . , xt
+d)
+·x2···xd
+−−−−→
+R
+J[pe] + (xt+1
+2
+, . . . , xt+1
+d
+)
+�
+.
+Claim 4.3. Let
+αpe = [rpe + (xtpe
+2 , . . . , xtpe
+d )] ∈ Hd−1
+m
+� R
+J[pe]
+�
+,
+then cαpe = [crpe + (xt
+2, . . . , xt
+d)[pe]] is the 0-element of Hd−1
+m
+(R/J[pe]).
+6This property is automatic if RP is Q-Gorenstein for each height 2 prime ideal P ∈ Spec(R). Recall that a
+local normal Cohen-Macaulay domain R with canonical ideal J ⊆ R is Q-Gorenstein if there exists a m ≥ 1
+such that J(m) is a principal ideal.
+
+ON THE EQUALITY OF TEST IDEALS
+15
+Proof of Claim. The element [crpe + (xt−1
+1
+J1, xt
+2, . . . , xt
+d)] is the 0-element of
+lim
+−→
+�
+R
+(xt−1
+1
+J1, xt
+2, . . ., xt
+d)[pe]
+�
+.
+Therefore there exists an s ∈ N such that
+crpe(x1x2 · · · xd)spe ∈ (xt+s−1
+1
+J1, xt+s
+2
+, . . ., xt+s
+d
+)[pe] = (x(t+s−1)pe
+1
+J[pe]
+1
+, x(t+s)pe
+2
+, . . . , x(t+s)pe
+d
+).
+So there exists an element j1 ∈ J[pe]
+1
+such that
+crpe(x1x2 · · · xd)spe − x(t+s−1)pe
+1
+j1 ∈ (x(t+2)pe
+2
+, . . . , x(t+s)pe
+d
+).
+The sequence x1, x2, . . . , xd is a regular sequence and so
+crpe(x2 · · · xd)spe − x(t−1)pe
+1
+j1 ∈ (x(t+s)pe
+2
+, . . . , x(t+s)pe
+d
+).
+Hence
+crpe(x2 · · · xd)spe ∈ (x(t−1)pe
+1
+J[pe]
+1
+, x(t+s)pe
+2
+, . . . , x(t+s)pe
+d
+) = (J[pe], x(t+s)pe
+2
+, . . . , x(t+s)pe
+d
+),
+which proves the claim.
+□
+Choose e0 ∈ N≥1 so that pe ≥ mpe−e0 + 1 for all e ≫ 0. If e ≫ 0 then
+J[pe] ⊆ J(pe) ⊆ J(mpe−e0+1).
+Fix e ≫ 0 and consider the local cohomology module
+Hd−1
+m
+�
+R
+J(mpe−e0+1)
+�
+∼= lim
+−→
+�
+R
+(J(mpe−e0+1), xt
+2, . . ., xt
+d)
+�
+.
+Let ˜αpe denote the image of αpe in Hd−1
+m
+(R/J(mpe−e0+1)) induced by the projection R/J[pe] →
+R/J(mpe−e0+1). By Claim 4.3
+0 = c˜αpe = [crpe +(xtpe
+2 , . . ., xtpe
+d )] ∈ Hd−1
+m
+�
+R
+J(mpe−e0+1)
+�
+∼= lim
+−→
+�
+R
+J(mpe−e0+1) + (xt
+2, . . . , xt
+d)
+�
+.
+There are short exact sequences
+0 →
+R
+J(mpe−e0+1)
+1
+·x(t−1)(mpe−e0 +1)
+1
+−−−−−−−−−−→
+R
+J(mpe−e0+1) →
+R
+x(t−1)(mpe−e0 +1)
+1
+R
+→ 0.
+Let ℓ be a choice of integer, which depends on e − e0, as in the statement of the theo-
+rem. The sequence xℓ
+2, xℓ
+3, x4, . . . , xd is a regular sequence on R/x(t−1)(mpe−e0 +1)
+1
+R. By (3) of
+Proposition 3.7 we have that
+lcbd−1(xℓ
+2, xℓ
+3, x4, . . . , xd; R/J(mpe−e0+1)
+1
+) = lcbd−1(xℓ
+2, xℓ
+3, x4, . . . , xd; R/J(mpe−e0+1)),
+and so by assumption
+lcbd−1(xℓ
+2, xℓ
+3, x4, . . . , xd; R/J(mpe−e0+1)) ≤ pe−e0 + 1 ≤ pe.
+Recall that
+0 = [crpe + (xtpe
+2 , . . . , xtpe
+d )] = [crpe(xt
+2xt
+3)(ℓ−1)pe + (xtℓpe
+2
+, xtℓpe
+3
+, xtpe
+4
+. . . , xtpe
+d )]
+
+16
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+as an element of Hd−1
+m
+(R/J(mpe−e0+1)). By Lemma 3.3,
+lcbd−1(xtℓ
+2 , xtℓ
+3 , xt
+4, . . . , xt
+d; R/J(mpe−e0+1)) ≤ lcbd−1(xℓ
+2, xℓ
+3, x4, . . . , xd; R/J(mpe−e0+1)) ≤ pe,
+so
+(4.1)
+c(rxt
+4 · · · xt
+d)pe(xt
+2xt
+3)ℓpe ∈ (J(mpe−e0+1), xt(ℓ+1)pe
+2
+, xt(ℓ+1)pe
+3
+, x2tpe
+4
+, . . . , x2tpe
+d
+).
+Notice that xtpe0
+1
+∈ J(pe0) and so
+xtpe0
+1
+J(mpe−e0+1) ⊆ xtpe0
+1
+J(pe−e0) ⊆ J(pe).
+We therefore multiply the containment (4.1) by xtpe0
+1
+and obtain that
+cxtpe0
+1
+(rxt
+4 · · ·xt
+d)pe(xt
+2xt
+3)ℓpe ∈ (J(pe), xt(ℓ+1)pe
+2
+, xt(ℓ+1)pe
+3
+, x2tpe
+4
+, . . . , x2tpe
+d
+).
+Therefore
+cxtpe0
+1
+(rxt
+4 · · · xt
+d)pe(xt
+2)ℓpe ∈ (J(pe), (xt
+2)(ℓ+1)pe, (xt
+3)(ℓ+1)pe, x2tpe
+4
+, . . . , x2tpe
+d
+) : xtℓpe
+3
+.
+We utilize the assumption that xm
+3 J(m)
+1
+⊆ a2R ⊆ J(m)
+1
+to conclude that;
+xm
+3 J(m) = xm
+3 (xt−1
+1
+J1)(m) = x(t−1)m
+1
+xm
+3 J(m)
+1
+⊆ x(t−1)m
+1
+a2R ⊆ x(t−1)m
+1
+J(m)
+1
+⊆ J(m).
+Therefore xtm
+3 J(m) ⊆ x(t−1)m
+1
+a2R ⊆ J(m) and we apply (2) of Lemma 4.1 with respect to xt
+3
+and N3 = ℓ + 1 to conclude that
+cxtpe0
+1
+(rxt
+4 · · · xt
+d)pe(xt
+2)ℓpe ∈ (J(pe), (xt
+2)(ℓ+1)pe, x2tpe
+3
+, x2tpe
+4
+, . . . , x2tpe
+d
+) : xtm
+1 xtpe
+3 .
+Equivalently,
+cxt(m+pe0)
+1
+(rxt
+3xt
+4 · · · xt
+d)pe(xt
+2)ℓpe ∈ (J(pe), (xt
+2)(ℓ+1)pe, x2tpe
+3
+, x2tpe
+4
+, . . . , x2tpe
+d
+).
+Similarly, we are able to apply (1) of Lemma 4.1 with respect to the element xt
+2 and obtain
+that
+cxt(m+pe0 )
+1
+(rxt
+2xt
+3xt
+4 · · · xt
+d)pe ∈ (J[pe], x2tpe
+2
+, x2tpe
+3
+, x2tpe
+4
+, . . ., x2tpe
+d
+) = (J, x2t
+2 , x2t
+3 , . . . , x2t
+d )[pe].
+The element cxt(m+pe0)
+1
+does not depend on e and therefore
+rxt
+2xt
+3xt
+4 · · · xt
+d ∈ (J, x2t
+2 , x2t
+3 , . . . , x2t
+d )∗.
+In particular,
+η = [r + (xt−1
+1
+J1, xt
+2, . . . , xt
+d)] = [rxt
+2xt
+3xt
+4 + (J, x2t
+2 , . . . , x2t
+d )] ∈ 0∗fg
+ER(k)
+as claimed.
+□
+
+ON THE EQUALITY OF TEST IDEALS
+17
+4.2. S2-ification, higher Ext-modules, and local cohomology. We begin with two
+lemmas that experts may already be aware of.
+Lemma 4.4. Let (S, m, k) be a Cohen-Macaulay local domain and M a finitely generated
+S-module such that ht(AnnS(M)) = h. Then Exth
+S(M, S) is an (S2)-module over its support.
+Proof. Let (F•, ∂•) be the minimal free resolution of M, let (−)∗ denote HomS(−, S), and
+consider the dual complex (F ∗
+• , ∂∗
+•). Because ht(AnnS(M)) = h we have that the following
+complex is exact:
+0 → F ∗
+0
+∂∗
+1
+−→ F ∗
+1 → . . . → F ∗
+h−1
+∂∗
+h
+−→ F ∗
+h → Coker(∂∗
+h) → 0.
+In particular, depth(Coker(∂∗
+h)) = d − h. Moreover, there is a short exact sequence
+0 → Exth
+S(M, S) → Coker(∂∗
+h) → Im(∂∗
+h+1) → 0.
+The module Im(∂∗
+h+1) is torsion-free and therefore has depth at least 1. If d − h ≥ 2 then
+Exth
+S(M, S) has depth at least 2. If d−h = 1 then the depth of Exth
+S(M, S) is 1. If d−h = 0
+then M is 0-dimensional. Therefore if ht(AnnS(M)) = h then Exth
+S(M, S) is an (S2)-module
+over its support.
+□
+Continue to consider the ring S, the module M, and the resolution (F•, ∂•) as above.
+Suppose further S is a regular local ring and hence every finitely generated S-module has
+a finite free resolution.
+Consider the minimal free resolution (G•, δ•) of Exth
+S(M, S).
+If
+depth(M) = d − h is maximal, then Exth
+S(M, S) = Coker(∂∗
+h) and therefore (G•, δ•) is the
+complex
+0 → F ∗
+0
+∂∗
+1
+−→ F ∗
+1 → . . . → F ∗
+h−1
+∂∗
+h
+−→ F ∗
+h → 0.
+In particular, if depth(M) = d − h then Exth
+S(Exth
+S(M, S), S) ∼= M. Suppose depth(M) <
+d − h and let (F ∗
+• , ∂∗
+•)tr be the complex obtained by truncating (F ∗
+• , ∂∗
+•) at the hth spot.
+That is (F ∗
+• , ∂∗
+•)tr is the minimal free resolution of Coker(∂∗
+h). Then the natural inclusion
+Exth
+S(M, S) ⊆ Coker(∂∗
+h) lifts to a map of complexes (G•, δ•) → (F ∗
+• , ∂∗
+•)tr and therefore there
+is an induced natural map M → Exth
+S(Exth
+S(M, S), S). The map M → Exth
+S(Exth
+S(M, S), S)
+is an isomorphism whenever M is a (maximal) Cohen-Macaulay module over its support.
+Lemma 4.5. Let (R, m, k) be a complete local normal domain of dimension at least 3 and
+J ⊆ R a pure height 1 ideal.
+Suppose (S, n, k) is a regular local ring mapping onto R,
+R ∼= S/P, and ht(P) = h. Then for every integer i the kernel of the natural map R/Ji →
+Exth+1
+S
+(Exth+1
+S
+(R/Ji, S), S) is J(i)/Ji. In particular, for every integer i there is a natural
+inclusion R/J(i) ⊆ Exth+1
+S
+(Exth+1
+S
+(R/Ji, S), S). Moreover, the natural inclusion R/J(i) ⊆
+Exth+1
+S
+(Exth+1
+S
+(R/Ji, S), S) is an isomorphism whenever localized at prime ideal p ∈ V (J)
+such that (R/J(i))p is Cohen-Macaulay.
+Proof. It only remains to show that the kernel of R/Ji → Exth+1
+S
+(Exth+1
+S
+(R/Ji, S), S) is
+J(i)/Ji. But this follows from the observation that the map
+R/Ji → Exth+1
+S
+(Exth+1
+S
+(R/Ji, S), S)
+is an isomorphism when localized at any minimal component of J by the discussion proceed-
+ing the statement of the lemma.
+□
+
+18
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+We record a corollary of Lemma 4.5 for future reference.
+Corollary 4.6. Let (R, m, k) be a complete local Cohen-Macaulay domain, which is Q-
+Gorenstein in codimension 2, and J1 ⊊ R a choice of canonical ideal. Let m ∈ N be an
+integer such that J(m)
+1
+is principal in codimension 2.
+Suppose (S, n, k) is a regular local
+ring mapping onto R, R ∼= S/P, and ht(P) = h.
+Then for every integer i the natural
+inclusion R/J(mi+1)
+1
+→ Exth+1
+S
+(Exth+1
+S
+(R/Jmi+1
+1
+, S), S) is an isomorphism whenever localized
+at a prime ideal of R of height 2 or less.
+Proof. Immediate by Lemma 4.5 since J(mi+1)
+1
+Rp ∼= J1Rp is a canonical ideal whenever p is
+a prime of R of height 2 or less.
+□
+The next proposition and theorem provide the linear bound of top local cohomology
+bounds of the family of R-modules
+�
+R/J(mpe+1)
+1
+�
+described in Theorem 4.2 whenever there
+exists an ideal I ⊆ R of pure height 1 and parameters x1, . . . , xd with the following properties:
+(1) I ∼= ω(−h)
+R
+for some h ≥ 1 and I is principal in codimension 2;
+(2) For each 1 ≤ j ≤ d − 2, the ideal aj := (x1, . . ., xd−j+1) is such that
+a[pe]
+j
+Hj
+m(R/I(pe)) = 0
+for each e ∈ N.
+We first provide a lemma. In the following lemma we let (−)∨ denote the Matlis dual functor.
+Lemma 4.7. Let (R, m, k) be a local normal Cohen-Macaulay domain of Krull dimension
+d and Q-Gorenstein in codimension 2. Assume that R has a test element. Let J1 ⊆ R be a
+choice of canonical ideal and m ∈ N such that J(m)
+1
+is principal in codimension 2. Suppose S
+is a regular local ring of Krull dimension d + h mapping onto R, R ∼= S/P, and ht(P) = h.
+Suppose that I1 ⊆ R is an ideal of pure height 1 with components disjoint from those of J1,
+I1 ∩ J1 = x1R is principal. Then
+Hj−1
+m
+� R
+I(mi)
+1
+�
+∼=
+�
+Extd+h−j
+S
+�
+Exth+1
+S
+�
+R
+J(mi+1)
+1
+, S
+�
+, S
+��∨
+for all j ≤ d − 2.
+Proof. There are isomorphisms
+Exth+1
+S
+�
+R
+J(mi+1)
+1
+, S
+�
+∼= ωR/J(mi+1)
+1
+∼= Ext1
+R
+�
+R
+J(mi+1)
+1
+, J1
+�
+.
+(4.2)
+Consider the short exact sequences
+0 → J(mi+1)
+1
+→ R →
+R
+J(mi+1)
+1
+→ 0.
+Then there is a resulting short exact sequence
+0 → J1 → HomR(J(mi+1)
+1
+, J1) → Ext1
+R
+�
+R
+J(mi+1)
+1
+, J1
+�
+→ 0.
+(4.3)
+
+ON THE EQUALITY OF TEST IDEALS
+19
+But I1 ∩ J1 is principal, I1 and J1 have disjoint components, therefore J1 ∼= I(mi)
+1
+∩ J(mi+1)
+1
+and so
+HomR(J(mi+1)
+1
+, J1) ∼= HomR(J(mi+1)
+1
+, I(mi)
+1
+∩ J(mi+1)
+1
+) ∼= I(mi)
+1
+.
+(4.4)
+The ideal J1 is a maximal Cohen-Macaulay R-module. By (4.2), (4.3), and (4.4), if j ≤ d−2
+then
+Extd+h−j
+S
+�
+Exth+1
+S
+�
+R
+J(mi+1)
+1
+, S
+�
+, S
+�
+∼= Extd+h−j
+S
+�
+I(mi)
+1
+, S
+�
+.
+Consider the short exact sequence
+0 → I(mi)
+1
+→ R →
+R
+I(mi)
+1
+→ 0.
+Then
+Extd+h−j
+S
+�
+I(mi)
+1
+, S
+� ∼= Extd+h−(j−1)
+S
+� R
+I(mi)
+1
+, S
+�
+.
+An application of Matlis duality now completes the proof as
+�
+Extd+h−(j−1)
+S
+� R
+I(mi)
+1
+, S
+��∨
+∼= Hj−1
+m
+� R
+I(mi)
+1
+�
+.
+□
+Proposition 4.8. Let (R, m, k) be a local normal Cohen-Macaulay domain of Krull dimen-
+sion d and Q-Gorenstein in codimension 2. Let p > 0 be a natural number. Let J1 ⊆ R be a
+choice of canonical ideal and m ∈ N such that J(m)
+1
+is principal in codimension 2. Suppose S
+is a regular local ring mapping onto R, R ∼= S/P, and ht(P) = h. Suppose that I1 ⊆ R is an
+ideal of pure height 1 with components disjoint from those of J1, I1 ∩ J1 = x1R is principal,
+and parameters x1, x2 . . . , xd with the property that for each 1 ≤ j ≤ d − 2, the parameter
+ideal (x2, . . . , xd−j+1) is such that
+(xpe
+2 , . . . , xpe
+d−j+1)Hj
+m(R/I(mpe)
+1
+) = 0
+for each e ∈ N. Then for all e ∈ N
+lcbd−1(xd−1
+2
+, xd−1
+3
+, . . ., xd−1
+d
+; Exth+1
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S)) ≤ pe.
+Proof. Let (F•, ∂•) be the minimal free S-resolution of Exth+1
+S
+(R/J(mpe+1)
+1
+, S). Denote by
+(−)∗ the functor HomS(−, S) and consider the dualized complex (F ∗
+• , ∂∗
+•). For every j ≥ 1
+there are short exact sequences
+0 → Exth+j
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S) → Coker(∂∗
+h+j) → Im(∂∗
+h+j+1) → 0
+and
+0 → Im(∂∗
+h+j+1) → F ∗
+h+j+1 → Coker(∂∗
+h+j+1) → 0.
+Let Je denote the preimage of J(mpe+1)
+1
+in S, an ideal of height h + 1.
+The S-module
+Coker(∂∗
+h+1) has projective dimension h + 1 and the ideal Je annihilates the submodule
+Exth+1
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S). By prime avoidance, and abuse of notation, we may lift
+
+20
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+x = x2, . . ., xd to elements of S and assume that x is a regular sequence on Coker(∂∗
+h+1) and
+the free S-modules F ∗
+i .
+The module Exth+1
+S
+(R/J(mpe+1)
+1
+, S) is an (S2)-module over its support, see Lemma 4.4. In
+particular,
+Exth+d
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S) = Exth+d−1
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S) = 0
+and
+Coker(∂∗
+h+d−2) ∼= Exth+d−2
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S).
+Consider the short exact sequence
+0 → Im(∂∗
+h+d−2) → F ∗
+h+d−2 → Exth+d−2
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S) → 0.
+By our assumptions and by Lemma 4.7,
+(xpe
+2 , xpe
+3 , . . . , xpe
+d ) Exth+d−2
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S) = 0
+for every e ∈ N.
+By (2) of Proposition 3.7 and (3) of Proposition 3.6 we have that
+lcb2(x; Im(∂∗
+h+d−2)) ≤ pe. Next, we consider the short exact sequence
+0 → Exth+d−3
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S) → Coker(∂∗
+h+d−3) → Im(∂∗
+h+d−2) → 0.
+We established lcb2(x; Im(∂∗
+h+d−2)) ≤ pe. By assumption and Lemma 4.7
+(xpe
+2 , . . . , xpe
+d−1) Exth+d−3
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S) = 0
+for every e ∈ N. By (1) of Proposition 3.6 we have
+lcb2(x; Coker(∂∗
+h+d−3)) ≤ pe + pe = 2pe.
+Next consider the short exact sequence
+0 → Im(∂∗
+h+d−3) → F ∗
+h+d−3 → Coker(∂∗
+h+d−3) → 0.
+By (2) of Proposition 3.7 and knowing that lcb2(x; Coker(∂∗
+h+d−3)) ≤ 2pe we see that
+lcb3(x; Im(∂∗
+h+d−3)) ≤ 2pe.
+Inductively, we find that
+lcbj(x; Im(∂∗
+h+d−j)) ≤ (j − 1)pe
+and
+lcbj(x; Coker(∂∗
+h+d−j−1)) ≤ jpe
+for each 2 ≤ j ≤ d − 1. In particular,
+lcbd−1(x; Exth+1
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S)) ≤ (d − 1)pe.
+By Lemma 3.3 the parameter sequence xd−1 = xd−1
+2
+, . . . , xd−1
+d
+on R/J1 satisfies
+lcbd−1(xd−1; Exth+1
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S))) ≤ pe
+for each e ∈ N.
+□
+
+ON THE EQUALITY OF TEST IDEALS
+21
+Theorem 4.9. Let (R, m, k) be a local normal Cohen-Macaulay domain of Krull dimension
+d ≥ 4 and Q-Gorenstein in codimension 2. Let p > 0 be a natural number. Let J1 ⊆ R be a
+choice of canonical ideal and m ∈ N such that J(m)
+1
+is principal in codimension 2. Suppose S
+is a regular local ring mapping onto R, R ∼= S/P, and ht(P) = h. Suppose that I1 ⊆ R is an
+ideal of pure height 1 with components disjoint from those of J1, I1 ∩ J1 = x1R is principal,
+and parameters x1, x2 . . . , xd with the following properties:
+(1) J1Rx2 and J(m)
+1
+Rx3 are principal in their respective localizations;
+(2) For every 1 ≤ j ≤ d − 2, the parameter ideal (x1, . . . , xd−j+1) is such that
+(x2, . . . , xd−j+1)peHj
+m(R/I(mpe)
+1
+) = 0
+for each e ∈ N.
+Then the following hold:
+(1) For each e ∈ N there exists ℓ ∈ N such that
+lcbd−1(xℓ(d−1)
+2
+, xℓ(d−1)
+3
+, xd−1
+4
+, . . . , xd−1
+d
+; R/J(mpe+1)
+1
+) ≤ pe + 1;
+(2) For each e ∈ N there exists ℓ ∈ N such that
+lcbd−1(xℓ(d−1)
+2
+, xℓ(d−1)
+3
+, xd−1
+4
+, . . . , xd−1
+d
+; R/Jmpe+1
+1
+) ≤ pe + 2.
+Proof. For each e ∈ N let Ce be the cokernel of
+R/Jmpe+1 → Exth+1
+S
+(Exth+1
+S
+(R/Jmpe+1
+1
+, S), S) ∼= Exth+1
+S
+(Exth+1
+S
+(R/J(mpe+1)
+1
+, S), S)
+and consider the short exact sequences
+0 → R/J(mpe+1)
+1
+→ Exth+1
+S
+(Exth+1
+S
+(R/Jmpe+1
+1
+, S), S) → Ce → 0,
+see Lemma 4.5 for details.
+By Lemma 4.5 the module Ce is 0 when either x2 or x3 is inverted. Hence for each e ∈ N
+there exists an integer ℓ such that (xℓ
+2, xℓ
+3)Ce = 0. Because d ≥ 4 we have that d −1 ≥ 3 and
+(3) of Proposition 3.6 implies
+lcbd−1(xℓ(d−1)
+2
+,xℓ(d−1)
+3
+, xd−1
+4
+, . . . , xd−1
+d
+; R/J(mpe+1)
+1
+) ≤
+lcbd−1(xℓ(d−1)
+2
+, xℓ(d−1)
+3
+, xd−1
+4
+, . . ., xd−1
+d
+; Exth+1
+S
+(Exth+1
+S
+(R/Jmpe+1
+1
+, S, S)) + 1.
+Statement (1) follows by Proposition 4.8.
+To prove (2) let Ke = J(mpe+1)
+1
+/Jmpe+1
+1
+and consider the short exact sequences
+0 → Ke → R/Jmpe+1
+1
+→ R/J(mpe+1)
+1
+→ 0.
+The module Ke is 0 when either x2 or x3 are inverted. Hence for each e ∈ N there exists an
+integer ℓ such that (xℓ
+2, xℓ
+3)Ke = 0. By (1) of Proposition 3.6 we have that
+lcbd−1(xℓ
+2, xℓ
+3, x4, . . ., xd; R/Jmpe+1
+1
+) ≤ lcbd−1(xℓ
+2, xℓ
+3, x4, . . . , xd; R/J(mpe+1)
+1
+) + 1 ≤ pe + 2.
+□
+Theorem A is a consequence of the following theorem.
+
+22
+IAN ABERBACH, CRAIG HUNEKE, AND THOMAS POLSTRA
+Theorem 4.10. Let (R, m, k) be an excellent local Cohen-Macaulay normal domain of prime
+characteristic p > 0, of Krull dimension d ≥ 4, I1 ⊆ R an anti-canonical ideal, and ER(k)
+an injective hull of the residue field. Suppose further that there exists an m ∈ N so that I(m)
+1
+is principal in codimension 2 and for each 1 ≤ j ≤ d − 2 there exists an ideal aj of height
+d − j + 1 such that
+ape
+j Hj
+m
+�
+R
+I(mpe)
+1
+�
+= 0
+for every e ∈ N. Then 0∗fg
+ER(k) = 0∗
+ER(k).
+Proof. Our strategy is to employ Theorem 4.9 and then Theorem 4.2 to conclude 0∗fg
+ER(k) =
+0∗
+ER(k). But first, we change the ideals aj, if necessary, so that there are inclusions ad−2 ⊆
+ad−3 ⊆ · · · ⊆ a1 and so that there exists parameter elements xm
+1 , x2, x3 ∈ aj for all 1 ≤ j ≤
+d − 2 with the property that x1R = I1 ∩ J1 for some canonical ideal J1 and the ideals I1Rx2
+and I(m)
+1
+Rx3 are principal in their respective localizations.
+The ideal aj ∩ aj−1 ∩ · · · ∩ a1 has height at least d − j + 1. We can replace the ideal aj
+with aj ∩ aj−1 ∩ · · · ∩ a1 and may assume that
+ad−2 ⊆ ad−3 ⊆ · · · ⊆ a1.
+Start by choosing x1 ∈ I1 a generic generator so that x1R = I1 ∩ J1 and the ideals I1 and
+J1 have disjoint components. Clearly xmpe
+1
+annihilates Hj
+m(R/I(mpe)
+1
+) for every e ∈ N. The
+ideal I1 is principal in codimension 1, the ideal I(m)
+1
+is principal in codimension 2. Therefore
+there exists part of a system of parameters x2, x3 of R/x1R so that I1Rx2 and I(m)
+1
+Rx3 are
+principal in their respective localizations. Moreover, we can replace x2 and x3 by suitable
+powers and can assume that there exists elements a, b ∈ I1 so that x2I1 ⊆ aR ⊆ I1 and
+x3I(m)
+1
+⊆ bR ⊆ I(m)
+1
+. Therefore xmpe
+2
+I(mpe)
+1
+⊆ ampeR ⊆ I(mpe)
+1
+and xpe
+3 I(mpe)
+1
+⊆ bpeR ⊆ I(mpe)
+1
+.
+Consider the short exact sequences
+0 → I(mpe)
+1
+ampeR →
+R
+ampeR →
+R
+I(mpe)
+1
+→ 0
+and
+0 → I(mpe)
+1
+bmpeR →
+R
+bmpeR →
+R
+I(mpe)
+1
+→ 0.
+The elements xmpe
+2
+and xpe
+3 annihilate I(mpe)
+1
+/ampeR and I(mpe)
+1
+/bmpeR respectively. Exam-
+ining the resulting long exact sequences of local cohomology informs us that xmpe
+2
+and xpe
+3
+annihilate Hj
+m(R/I(mpe)
+1
+) for every 1 ≤ j ≤ d − 2. Replace the element x2 by xm
+2 . Then
+(xpe
+2 , xpe
+3 ) annihilates Hj
+m(R/I(mpe)
+1
+) for every 1 ≤ j ≤ d − 2.
+For each e ∈ N the ideal
+((aj + (xm
+1 , x2, x3))4pe is generated by elements which live in either ape
+j or (xm
+1 , x2, x3)[pe] and
+therefore annihilate Hj
+m(R/I(mpe)
+1
+). We replace aj by the ideal (aj + (xm
+1 , x2, x3))4.
+The ideal aj has height at least d − j + 1 and xm
+1 , x2, x3 ∈ aj. We can extend xm
+1 , x2, x3
+to a parameter sequence xm
+1 , x2, x3, . . . , xd−j+1 in aj ⊆ aj−1 ⊆ · · · ⊆ a1. By Theorem 4.9, for
+
+ON THE EQUALITY OF TEST IDEALS
+23
+each e ∈ N there exists an ℓ so that
+lcbd−1(xℓ(d−1)
+2
+, xℓ(d−1)
+3
+, xd−1
+d
+, . . . , xd−1
+d
+; R/J(mi+1)
+1
+) ≤ pe + 1.
+Therefore 0∗fg
+ER(k) = 0∗
+ER(k) by Theorem 4.2.
+□
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+Department of Mathematics, University of Virginia, Charlottesville, VA 22903 USA
+Email address: [email protected]
+Department of Mathematics, University of Alabama, Tuscaloosa, AL 35401 USA
+Email address: [email protected]
+

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+arXiv:2301.00097v1  [math.DG]  31 Dec 2022
+REALIZATIONS OF INNER AUTOMORPHISMS
+OF ORDER FOUR AND FIXED POINTS SUBGROUPS
+BY THEM ON THE CONNECTED COMPACT
+EXCEPTIONAL LIE GROUP 퐸8, PART III
+By
+Toshikazu Miyashita
+Abstract. The compact simply connected Riemannian 4-symmetric spaces were clas-
+sified by J.A. Jiménez according to type of the Lie algebras. As homogeneous mani-
+folds, these spaces are of the form 퐺/퐻, where 퐺 is a connected compact simple Lie
+group with an automorphism ˜훾 of order four on 퐺 and 퐻 is a fixed points subgroup
+퐺훾 of 퐺. According to the classification by J.A. Jiménez, there exist seven compact
+simply connected Riemannian 4-symmetric spaces 퐺/퐻 in the case where 퐺 is of
+type 퐸8. In the present article, we give the explicit form of automorphisms ˜휔4, ˜휅4 and
+˜휀4 of order four on 퐸8 induced by the 퐶-linear transformations 휔4, 휅4 and 휀4 of the
+248-dimensional퐶-vector space 픢8퐶, respectively. Further, we determine the structure
+of these fixed points subgroups (퐸8) 휔4, (퐸8)휅4 and (퐸8) 휀4 of 퐸8. These amount to
+the global realizations of three spaces among seven Riemannian 4-symmetric spaces
+퐺/퐻 above corresponding to the Lie algebras 픥 = 픰픲(2) ⊕ 푖푹 ⊕ 픢6,푖푹 ⊕ 픰픬(14) and
+픥 = 픰픲(2) ⊕ 푖푹 ⊕ 픰픬(12), where 픥 = Lie(퐻). With this article, the all realizations
+of inner automorphisms of order four and fixed points subgroups by them have been
+completed in 퐸8.
+1. Introduction
+Let 퐺 be a Lie group and 퐻 a compact subgroup of 퐺. A homogeneous space 퐺/퐻
+with 퐺-invariant Riemannian metric 푔 is called a Riemannian 4-symmetric space if
+there exists an automorphism ˜훾 of order four on 퐺 such that (퐺훾)0 ⊂ 퐻 ⊂ 퐺훾, where
+퐺훾 and (퐺훾)0 are the fixed points subgroup of 퐺 by ˜훾 and its identity component,
+respectively.
+Now, for the exceptional compact Lie group of type 퐸8, as in Table below, there exist
+seven cases of the compact simply connected Riemannian 4-symmetric spaces which
+were classified by J.A. Jiménez as mentioned in abstract ([2]). Accordingly, our interest
+is to realize the groupfication for the classification as Lie algebra.
+Our results of groupfication corresponding to the Lie algebra 픥 in Table are given as
+follows.
+2010 Mathematics Subject Classification.
+53C30, 53C35, 17B40.
+Key words. 4-symmetric spaces, exceptional Lie groups.
+1
+
+2
+TOSHIKAZU MIYASHITA
+Case
+픥
+˜훾
+퐻 = 퐺훾
+1
+픰픬(6) ⊕ 픰픬(10)
+˜휎′
+4
+(푆푝푖푛(6) × 푆푝푖푛(10))/풁4
+2
+푖푹 ⊕ 픰픲(8)
+˜푤4
+(푈(1) × 푆푈(8))/풁24
+3
+푖푹 ⊕ 픢7
+˜휐4
+(푈(1) × 퐸7)/풁2
+4
+픰픲(2) ⊕ 픰픲(8)
+˜휇4
+(푆푈(2) × 푆푈(8))/풁4
+5
+픰픲(2) ⊕ 푖푹 ⊕ 픢6
+˜휔4
+(푆푈(2) × 푈(1) × 퐸6)/(풁2 × 풁3)
+6
+푖푹 ⊕ 픰픬(14)
+˜휅4
+(푈(1) × 푆푝푖푛(14))/풁4
+7
+픰픲(2) ⊕ 푖푹 ⊕ 픰픬(12)
+˜휀4
+(푆푈(2) × 푈(1) × 푆푝푖푛(12))/(풁2 × 풁2)
+In [3] and [4], the author has already realized the groupfication for Case 1 and Cases
+2, 3, 4 in Table, respectively. In the present article, we state the realizations of the group
+퐻 for Cases 5,6 and Case 7.
+Finally, the author would like to say that the feature of this article is to give elementary
+proofs of the isomorphism of groups by using the homomorphism theorem, and the all
+global realizations have been completed in 퐸8.
+This article is a continuation of [4], hence we start from Section 7. The readers refer
+to [3] for preliminary results and also to [3],[4],[5] or [7] for notations. Note that we
+change the numbering of Case 5 and Case 6 in [3] to the numbering of Case 3 and Case
+4 in the present article, respectively.
+7. Case 5. The automorphism ˜휔4 of order four and the group (퐸8)휔4
+In this section (also in Section 8 and Section 9), we use the 248-dimensional퐶-vector
+space 픢8퐶 used in Case 1 ([3]) and the simply connected compact exceptional Lie group
+of type 퐸8 constructed by T. Imai and I. Yokota ([1]).
+We define a 퐶-linear transformation 휔4 of 픢8퐶 by
+휔4(훷, 푃, 푄, 푟, 푠, 푡) = (휄훷휄−1, −휄푃, −휄푄, 푟, 푠, 푡),
+where 휄 on the right hand side is the 퐶-linear transformation of 픓퐶 defined in [7,
+Definition of Subsection 4.10 (p.131)] (the space 픓퐶, called the Freudenthal 퐶-vector
+space, is defined in [3, Preliminaries (p.94)]). Note that 휔4 is the composition transfor-
+mation of 휄, 휐 ∈ 퐸7 ⊂ 퐸8, where 휐 is the 퐶-linear transformation of 픢8퐶 defined in [7,
+Definition of Subsection 5.7(p.174)]. Moreover since 휄, 휐 are expressed as elements of
+퐸8 by
+휄 = exp
+�2휋푖
+4 ad(훷(0, 0, 0, 3), 0, 0, 0, 0, 0)
+�
+,
+휐 = exp
+�2휋푖
+4 ad(훷(0, 0, 0, 6), 0, 0, 0, 0, 0)
+�
+,
+
+REALIZATIONS OF INNER AUTOMORPHISMS OF ORDER FOUR ON 퐸8 PART III
+3
+respectively and together with [훷(0, 0, 0, 3),훷(0, 0, 0, 6)] = 0, we have
+휔4 = exp
+�2휋푖
+4 ad(훷(0, 0, 0, 9), 0, 0, 0, 0, 0)
+�
+.
+Hence it follows from above that 휔4 ∈ 퐸8 and (휔4)4 = 1, (휔4)2 = 휐, so that 휔4 induces
+the inner automorphism ˜휔4 of order four on 퐸8: ˜휔4(훼) = 휔4훼휔4
+−1, 훼 ∈ 퐸8.
+Now, we will study the subgroup (퐸8)휔4 of 퐸8:
+(퐸8)휔4 =
+�
+훼 ∈ 퐸8
+�� 휔4훼 = 훼휔4
+�
+.
+The aim of this section is to determine the structure of the group (퐸8)휔4. Before
+that, we prove the lemma and propositions needed later.
+Lemma 7.1. The Lie algebra (픢8)휔4 of the group (퐸8)휔4 is given by
+(픢8)휔4 =
+�
+ad(푅) ∈ Der(픢8)
+�� 휔4ad(푅) = ad(푅)휔4, 푅 ∈ 픢8
+�
+� {푅 ∈ 픢8 | 휔4푅 = 푅}
+= {푅 = (훷, 0, 0, 푟, 푠, −휏푠) | 훷 ∈ (픢7)휄 � 픲(1) ⊕ 픢6, 푟 ∈ 푖푹, 푠 ∈ 퐶} ,
+where (픢7)휄 is the Lie algebra of the group (퐸7)휄 := {훼 ∈ 퐸7 | 휄훼 = 훼휄}.
+In particular, we have dim((픢8)휔4) = (1 + 78) + 1 + 2 = 82.
+Proof. By doing straightforward computation, we can obtain the required results. The
+isomorphism (픢7)휄 � 픲(1) ⊕ 픢6 as Lie algebras follows from [7, Theorem 4.10.2].
+□
+Proposition 7.2. The group (퐸8)휔4 contains the group (퐸7)휄 which is isomorphic to
+the group (푈(1) × 퐸6)/풁3, 풁3 = {(1, 1), (휔, 휙(휔2)), (휔2, 휙(휔))}: (퐸8)휔4 ⊃ (퐸7)휄 �
+(푈(1) × 퐸6)/풁3, where 휔 = (−1/2) + (
+√
+3/2)푖.
+Proof. Immediately, we have (픢7)휄 ⊂ (픢8)휔4 from Lemma 7.1. Moreover since both
+of the groups (퐸8)휔4 and (퐸7)휄 are connected, we have (퐸7)휄 ⊂ (퐸8)휔4. However, we
+will prove it directly. Let 훼 ∈ (퐸7)휄. Note that −1 ∈ 푧(퐸7) (the center of 퐸7), it follows
+that
+휔4훼(훷, 푃, 푄, 푟, 푠, 푡) = 휔4(훼훷훼−1, 훼푃, 훼푄, 푟, 푠, 푡)
+= (휄훼훷훼−1휄−1, −휄훼푃, −휄훼푄, 푟, 푠, 푡)
+= (훼(휄훷휄−1)훼−1, 훼(−휄푃), 훼(−휄푄), 푟, 푠, 푡)
+= 훼휔4(훷, 푃, 푄, 푟, 푠, 푡), (훷, 푃, 푄, 푟, 푠, 푡) ∈ 픢퐶
+8 ,
+that is, 휔4훼 = 훼휔4. Hence we have 훼 ∈ (퐸8)휔4, so that the first half is proved.
+As for the proof of the second half, we define a mapping 휑휄 : 푈(1) × 퐸6 → (퐸7)휄 by
+휑휄(휃, 훽) = 휙(휃)훽,
+wherethemapping 휙 : 푈(1) = {휃 ∈ 퐶 | (휏휃)휃 = 1} → 퐸7 is defined by 휙(휃)(푋,푌, 휉, 휂)
+= (휃−1푋, 휃푌, 휃3휉, 휃−3휂), (푋,푌, 휉, 휂) ∈ 픓퐶.
+Then the mapping 휑휄 induces the required isomorphism (see [7, Theorem 4.10.2] in
+detail).
+□
+
+4
+TOSHIKAZU MIYASHITA
+Let the special unitary group 푆푈(2) = {퐴 ∈ 푀(2, 퐶) | (휏 푡퐴)퐴 = 퐸, det퐴 = 1} and
+we define an embedding 휙휐 : 푆푈(2) → 퐸8 by
+휙휐(
+�푎
+−휏푏
+푏
+휏푎
+�
+) =
+����������
+�
+1
+0
+0
+0
+0
+0
+0
+휏푎1
+푏1
+0
+0
+0
+0
+−휏푏1
+푎1
+0
+0
+0
+0
+0
+0
+(휏푎)푎 − (휏푏)푏
+(휏푎)(휏푏)
+푎푏
+0
+0
+0
+−2(휏푎)푏
+(휏푎)2
+−푏2
+0
+0
+0
+2푎(휏푏)
+−(휏푏)2
+푎2
+����������
+�
+.
+Proposition 7.3. [7, Theorem 5.7.4] The group (퐸8)휔4 contains a subgroup
+휙휐(푆푈(2)) = {휙휐(퐴) ∈ 퐸8 | 퐴 ∈ 푆푈(2)}
+which is isomorphic to the group 푆푈(2) = {퐴 ∈ 푀(2, 퐶) | (휏 푡퐴)퐴 = 퐸, det퐴 = 1}.
+Proof. For 퐴 =
+�푎
+−휏푏
+푏
+휏푎
+�
+:= exp
+�−푖휈
+−휏휚
+휚
+푖휈
+�
+∈ 푆푈(2), where
+�−푖휈
+−휏휚
+휚
+푖휈
+�
+∈ 픰픲(2),
+we have 휙휐(퐴) = exp(ad(0, 0, 0, 푖휈, 휚, −휏휚)) ∈ (퐸8)휔4 (Lemma 7.1).
+□
+Note that there are some errata in definition of 휑3(퐴) of [7, Theorem 5.7.4].
+Here we need the following result in the proof of Theorem 7.5 below.
+Theorem 7.4. ([7, Theorem 5.7.6]) The group (퐸8)휐 is isomorphic to the group
+(푆푈(2) × 퐸7)/풁2, 풁2 = {(퐸, 1), (−퐸, −1)}: (퐸8)휐 � (푆푈(2) × 퐸7)/풁2.
+Proof. We define a mapping 휑 : 푆푈(2) × 퐸7 → (퐸8)휐 by
+휑(퐴, 훿) = 휙휐(퐴)훿.
+Then the mapping 휑 induces the required isomorphism.
+□
+Now, we determine the structure of the group (퐸8)휔4.
+Theorem 7.5. The group (퐸8)휔4 is isomorphic to the group (푆푈(2) ×푈(1) × 퐸6)/(풁2
+× 풁3), 풁2 = {(퐸, 1, 1), (−퐸, −1, 1)}, 풁3 = {(퐸, 1, 1), (퐸, 휔, 휙(휔2)), (퐸, 휔2, 휙(휔))}:
+(퐸8)휔4 � (푆푈(2) × 푈(1) × 퐸6)/(풁2 × 풁3).
+Proof. We define a mapping 휑휔4 : 푆푈(2) × 푈(1) × 퐸6 → (퐸8)휔4 by
+휑휔4 (퐴, 휃, 훽) = 휙휐(퐴)휑휄(휃, 훽)(= 휑(퐴, 휑휄(휃, 훽))),
+where 휑 is defined in the proof of Theorem 7.4.
+First, it is clear that 휑휔4 is well-defined from Propositions 7.2, 7.3 and since the
+mapping 휑휔4 is the restriction of the mapping 휑 : 푆푈(2) × 퐸7 → (퐸8)휐, 휑휔4 is a
+homomorphism.
+Next, we will provethat 휑휔4 is surjective. Let 훼 ∈ (퐸8)휔4. From (휔4)2 = 휐, we easily
+see (퐸8)휔4 ⊂ (퐸8)휐. Hence there exist 퐴 ∈ 푆푈(2) and 훿 ∈ 퐸7 such that 훼 = 휑(퐴, 훿)
+(Theorem 7.4). Moreover, from the condition 휔4훼휔4
+−1 = 훼, that is, 휔4휑(퐴, 훿)휔4
+−1 =
+휑(퐴, 훿), we have 휑(퐴, 휄훿휄−1) = 휑(퐴, 훿). Indeed, note that the formula 휔4훿휔4
+−1 = 휄훿휄−1
+
+REALIZATIONS OF INNER AUTOMORPHISMS OF ORDER FOUR ON 퐸8 PART III
+5
+follows from 훿 ∈ 퐸7, so that it follows from 휔4휙휐(퐴)휔4
+−1 = 휙휐(퐴) (Proposition 7.3)
+that
+휔4휑(퐴, 훿)휔4
+−1 = 휔4(휙휐(퐴)훿)휔4
+−1
+= (휔4휙휐(퐴)휔4
+−1)(휔4훿휔4
+−1)
+= 휙휐(퐴)(휄훿휄−1)
+= 휑(퐴, 휄훿휄−1),
+that is, 휑(퐴, 휄훿휄−1) = 휑(퐴, 훿).
+Thus, since Ker휑 = {(퐸, 1), (−퐸, −1)} (Theorem 7.4), we have the following
+� 퐴 = 퐴
+휄훿휄−1 = 훿
+or
+� 퐴 = −퐴
+휄훿휄−1 = −훿.
+In the latter case, this case is impossible because of 퐴 ≠ 푂, where 푂 is the zero matrix.
+In the former case, 훿 ∈ (퐸7)휄 follows from the second condition, so that there exist
+휃 ∈ 푈(1) and 훽 ∈ 퐸6 such that 훿 = 휑휄(휃, 훽) (Proposition 7.2). Hence there exist
+퐴 ∈ 푆푈(2), 휃 ∈ 푈(1) and 훽 ∈ 퐸6 such that 훼 = 휑(퐴, 휑휄(휃, 훽)) = 휑휔4 (퐴, 휃, 훽). The
+proof of surjective is completed.
+Finally, we will determine Ker 휑휔4. From the definition of kernel, we have
+Ker 휑휔4 = {(퐴, 휃, 훽) ∈ 푆푈(2) × 푈(1) × 퐸6 | 휑휔4(퐴, 휃, 훽) = 1}
+= {(퐴, 휃, 훽) ∈ 푆푈(2) × 푈(1) × 퐸6 | 휑(퐴, 휑휄(휃, 훽)) = 1}.
+Here, the mapping 휑휔4 is the restriction of the mapping 휑 and together with Ker 휑 =
+{(퐸, 1), (−퐸, −1)} (Theorem 7.4), we will find the elements (퐴, 휃, 훽) ∈ 푆푈(2)×푈(1)×
+퐸6 satisfying the following
+� 퐴 = 퐸
+휑휄(휃, 훽) = 1
+or
+� 퐴 = −퐸
+휑휄(휃, 훽) = −1.
+In the former case, from Ker 휑휄 = {(1, 1), (휔, 휙(휔2)), (휔2, 휙(휔))} (Proposition 7.2),
+we have the following
+
+
+퐴 = 퐸
+휃 = 1
+훽 = 1,
+
+
+퐴 = 퐸
+휃 = 휔
+훽 = 휙(휔2)
+or
+
+
+퐴 = 퐸
+휃 = 휔2
+훽 = 휙(휔).
+In the latter case, the second condition can be rewritten as 휙(−휃)훽 = 1 from −1 =
+휙(−1). Hence, as in the former case, we have the following
+
+
+퐴 = −퐸
+휃 = −1
+훽 = 1,
+
+
+퐴 = −퐸
+휃 = −휔
+훽 = 휙(휔2)
+or
+
+
+퐴 = −퐸
+휃 = −휔2
+훽 = 휙(휔).
+
+6
+TOSHIKAZU MIYASHITA
+Hence we can obtain
+Ker 휑휔4 =
+� (퐸, 1, 1), (퐸, 휔, 휙(휔2)), (퐸, 휔2, 휙(휔)),
+(−퐸, −1, 1), (−퐸, −휔, 휙(휔2)), (−퐸, −휔2, 휙(휔))
+�
+= {(퐸, 1, 1), (−퐸, −1, 1)} × {(퐸, 1, 1), (퐸, 휔, 휙(휔2)), (퐸, 휔2, 휙(휔))}
+� 풁2 × 풁3.
+Therefore we have the required isomorphism
+(퐸8)휔4 � (푆푈(2) × 푈(1) × 퐸6)/(풁2 × 풁3)
+□
+8. Case 6. The automorphism ˜휅4 of order four and the group (퐸8)휅4
+In this section, our aim is to prove main theorem: (퐸8)휅4 � (푈(1) × 푆푝푖푛(14))/풁4.
+In [5, Theorem 5.10], the author has proved that the group (퐸8퐶)휅3 is isomorphic to
+the group (퐶∗ × 푆푝푖푛(14,퐶))/풁4 using the automorphism ˜휅3 of order three on 퐸8퐶:
+(퐸8퐶)휅3 � (퐶∗ × 푆푝푖푛(14, 퐶))/풁4, where the group (퐸8퐶)휅3 is defined as (퐸8퐶)0
+in [5, Theorem 5.10] and ˜휅3, ˜휅4 are defined later. As in the proof of [5, Theorem
+5.10], using the automorphism ˜휅4 of order four on 퐸8퐶, we first will prove that the
+group (퐸8퐶)휅4 is isomorphic to the group (퐶∗ × 푆푝푖푛(14, 퐶))/풁4 in Subsection 8.1:
+(퐸8퐶)휅4 � (퐶∗×푆푝푖푛(14, 퐶))/풁4, and using the result above, the structure of the group
+(퐸8)휅4 will be determined in Subsection 8.2. We often will use the results obtained
+and definitions in [5, Section 5.3 ], then note that we change several signs used in [5,
+Section 5.3]. For example, 휅3 in [5, Section 5.3 (p.37)] is changed to 휈3 in this section.
+At the end of the preface of this section, since the content of this section is related to
+the subalgebra 픤0 of the simple graded Lie algebras 픤 of second kind
+픤 = 픤−2 ⊕ 픤−1 ⊕ 픤0 ⊕ 픤1 ⊕ 픤2, [픤푘, 픤푙] ⊂ 픤푘+푙,
+we will review about that. In a simple graded Lie algebra 픤 of second kind, we know that
+there exists 푍 ∈ 픤0, called a characteristic element, such that each 픤푘 is the 푘-eigenspace
+of ad 푍 : 픤 → 픤, so that
+픤0 = {푋 ∈ 픤 | (ad 푍)푋 = 0} .
+Here, set ˜훾3 := exp
+�2휋푖
+3 ad 푍
+�
+as a inner automorphism of order three on 픤, then we
+have
+픤0 = {푋 ∈ 픤 | ˜훾3푋 = 푋} =: (픤)훾3.
+Indeed, let 푋 ∈ 픤0. Then it follows from (ad 푍)푋 = 0 that
+˜훾3푋 =
+�
+exp
+�2휋푖
+3 ad 푍
+��
+푋 =
+� ∞
+�
+푛=0
+1
+푛!
+�2휋푖
+3 ad 푍
+����
+푋 = 푋.
+
+REALIZATIONS OF INNER AUTOMORPHISMS OF ORDER FOUR ON 퐸8 PART III
+7
+Hence we have 푋 ∈ (픤)훾3. Conversely, let 푋 ∈ (픤)훾3. Then we easily see 푋 ∈ 픤푘 for
+some 푘 ∈ {−2, −1, 0, 1, 2}, so that (ad 푍)푋 = 푘푋 holds. Hence it follows from
+푋 = ˜훾3푋 =
+�
+exp
+�2휋푖
+3 ad 푍
+��
+푋 =
+� ∞
+�
+푛=0
+1
+푛!
+�2휋푖
+3 ad 푍
+�푛�
+푋
+=
+� ∞
+�
+푛=0
+1
+푛!
+�2휋푖
+3 푘
+�푛�
+푋 =
+�
+exp
+�2휋푖
+3 푘
+��
+푋
+that exp
+�2휋푖
+3 푘
+�
+= 1. Thus we have 푘 = 3푛, 푛 ∈ 풁, however since |푘| ≤ 2, we see
+푘 = 0, that is, 푋 ∈ 픤0. With above, 픤0 = (픤)훾3 is proved. This result will be useful later.
+8.1. The group (퐸8퐶)휅4
+We define 퐶-linear transformations 휅3 and 휅4 of 픢8퐶 by
+휅3(훷, 푃, 푄, 푟, 푠, 푡) = (휈3훷휈3
+−1, 휔2휈3푃, 휔휈3푄, 푟, 휔푠, 휔2푡),
+휅4(훷, 푃, 푄, 푟, 푠, 푡) = (휈4훷휈4
+−1, −푖휈4푃, 푖휈4푄, 푟, −푠, −푡),
+where 휔 = (−1/2) + (
+√
+3/2)푖 and 휈3, 휈4 ∈ 퐸7 on the right hand side are defined by
+휈3(푋,푌, 휉, 휂) = (��
+�
+휔2휉1
+푥3
+푥2
+푥3
+휔휉2
+휔푥1
+푥2
+휔푥1
+휔휉3
+��
+�
+, ��
+�
+휔휂1
+푦3
+푦2
+푦3
+휔2휂2
+휔2푦1
+푦2
+휔2푦1
+휔2휂3
+��
+�
+, 휔2휉, 휔휂),
+휈4(푋,푌, 휉, 휂) = (��
+�
+−푖휉1
+푥3
+푥2
+푥3
+푖휉2
+푖푥1
+푥2
+푖 푥1
+푖휉3
+��
+�
+, ��
+�
+푖휂1
+푦3
+푦2
+푦3
+−푖휂2
+−푖푦1
+푦2
+−푖 푦1
+−푖휂3
+��
+�
+, −푖휉,푖휂), (푋,푌, 휉, 휂) ∈ 픓퐶,
+respectively. Then 휅3, 휅4 can be expressed by
+휅3 = exp
+�2휋푖
+3 ad 휅
+�
+, 휅4 = exp
+�2휋푖
+4 ad 휅
+�
+,
+respectively, where 휅 := (훷(−2퐸1 ∨ 퐸1, 0, 0, −1), 0, 0, −1, 0, 0) ∈ 픢8퐶, 퐸1 ∨ 퐸1 :=
+(1/3)(2퐸1 − 퐸2 − 퐸3)∼ ∈ 픢6퐶. Besides, 휈3, 휈4 can be also expressed by
+휈3 = exp
+�2휋푖
+3 훷(−2퐸1 ∨ 퐸1, 0, 0, −1)
+�
+, 휈4 = exp
+�2휋푖
+4 훷(−2퐸1 ∨ 퐸10, 0, −1)
+�
+,
+respectively. Hence it follows from above that 휅3, 휅4 ∈ 퐸8 ⊂ 퐸8퐶 and (휅3)3 = (휅4)4 =
+1, so that 휅3 induces the inner automorphism ˜휅3 of order three on 퐸8:
+˜휅3(훼) =
+휅3훼휅3
+−1, 훼 ∈ 퐸8 and 휅4 induces the inner automorphism ˜휅4 of order four on 퐸8:
+˜휅4(훼) = 휅4훼휅4
+−1, 훼 ∈ 퐸8, so are on 퐸8퐶.
+Now, we will study the subgroup (퐸8퐶)휅4 of 퐸8퐶:
+(퐸8퐶)휅4 =
+�
+훼 ∈ 퐸8퐶 �� 휅4훼 = 훼휅4
+�
+.
+We prove the following lemma needed in the proof of theorem below.
+
+8
+TOSHIKAZU MIYASHITA
+Lemma 8.1. The Lie algebra (픢8퐶)휅4 of the group (퐸8퐶)휅4 coincides with the Lie algebra
+(픢8퐶)휅3 of the group (퐸8퐶)휅3:(픢8퐶)휅4 = (픢8퐶)휅3.
+Proof. By an argument similar to the proof of 픤0 = (픤)훾3 in the beginning of this
+section, 픤0 = (픤)훾4 := {푋 ∈ 픤 | ˜훾4푋 = 푋} is proved, where ˜훾4 := exp
+�2휋푖
+4 ad 푍
+�
+.
+Hence, by replacing 픤 and 훾3, 훾4 with 픢8퐶 and 휅3, 휅4, respectively, we obtain
+(픢8퐶)휅4 = 픤0 = (픢8퐶)휅3,
+where the Lie algebra 픤0 above is the same one obtained in [7, Theorem 5.7].
+□
+Now, we determine the structure of the group (퐸8퐶)휅4.
+Theorem 8.2. The group (퐸8퐶)휅4 is isomorphic to the group (퐶∗ × 푆푝푖푛(14, 퐶))/풁4,
+풁4 = {(1, 1), (−1, 휙(−1)), (푖, 휙(−푖)), (−푖, 휙(푖))}: (퐸8퐶)휅4 � (퐶∗ × 푆푝푖푛(14, 퐶))/풁4.
+Proof. First, we will prove that the group (퐸8퐶)휅4 coincides with the group (퐸8퐶)휅3.
+Since 퐸8퐶 is the simply connected complex Lie group type 퐸8, both of the groups
+(퐸8퐶)휅4 and (퐸8퐶)휅3 are connected (([3, Preliminaries Lemma 2.2]) in [6]), moreover
+together with (픢8퐶)휅4 = (픢8퐶)휅3 (Lemma 8.1), we have (퐸8퐶)휅4 = (퐸8퐶)휅3.
+Here, we define a mapping 휑 : 퐶∗ × 푆푝푖푛(14, 퐶) → (퐸8퐶)휅4 by the same mapping
+defined in the proof of [5, Theorem 5.10] as follows:
+휑(푎, 훽) = 휙(푎)훽,
+where 휙 is defined in [5, Subsection 5.3 (p.45)] and 푆푝푖푛(14, 퐶) is constructed in [5,
+Proposition 5.8.7].
+Therefore, from (퐸8퐶)휅4 = (퐸8퐶)휅3, we have the required isomorphism
+(퐸8퐶)휅4 � (퐶∗ × 푆푝푖푛(14, 퐶))/풁4.
+□
+8.2. The group (퐸8)휅4
+In this subsection, as for the construction of 푆푝푖푛(14, 퐶), we will give a briefly
+explain based on [5, Section 5.3], and note that several signs in [5, Section 5.3] are
+changed as mentioned in the beginning of this section. After that, we will construct the
+group 푆푝푖푛(14) in 퐸8 and determine the structure of the group (퐸8)휅4.
+Besides, as for the embedding sequence of the exceptional compact Lie groups:
+푆푝푖푛(8) ⊂ 퐹4 ⊂ 퐸6 ⊂ 퐸7 ⊂ 퐸8, see [7, Theorems 2.7.1, 3.7.1, 4.7.2, 5.7.3 ] in detail.
+We define 14-dimensional 퐶-vector subspaces (픢8퐶)2, (픢8퐶)−2 of 픢8퐶 by
+(픢8퐶)2 =
+�
+푅 ∈ 픢8퐶 | (ad휅)푅 = 2푅
+�
+=
+
+
+푅 = (훷, 0, 푄, 0, 0, 푡)
+��������
+훷 = 훷(0, 0, 휚1퐸1, 0), 휚1 ∈ 퐶,
+푄 = (휉2퐸2 + 휉3퐸3 + 퐹1(푥1), 휂1퐸1, 0, 휂),
+휉푘, 휂1, 휂 ∈ 퐶, 푥1 ∈ ℭ퐶,
+푡 ∈ 퐶
+
+
+,
+
+REALIZATIONS OF INNER AUTOMORPHISMS OF ORDER FOUR ON 퐸8 PART III
+9
+(픢8퐶)−2 =
+�
+푅 ∈ 픢8퐶 �� (ad휅)푅 = −2푅
+�
+=
+
+
+푅 = (훷, 푃, 0, 0, 푠, 0)
+��������
+훷 = 훷(0, 휐1퐸1, 0, 0), 휐1 ∈ 퐶,
+푃 = (휉1퐸1, 휂2퐸2 + 휂3퐸3 + 퐹1(푦1), 휉, 0),
+휉1, 휂푘, 휉 ∈ 퐶, 푦1 ∈ ℭ퐶,
+푠 ∈ 퐶
+
+
+,
+respectively, where 휅 = (훷(−2퐸1∨퐸1, 0, 0, −1), 0, 0, −1, 0, 0) ∈ 픢8퐶 is used in previous
+subsection .
+We define a 퐶-linear mapping 휁 : 픢8퐶 → 픢8퐶 by
+휁 (훷, 푃, 푄, 푟, 푠, 푡) = (휁1훷휁1−1, 푖휁1푄, 푖휁1푃, −푟, 푡, 푠),
+where the 퐶-linear transformation 휁1 of 픓퐶 on the right hand side is defined by
+휁1(푋,푌, 휉, 휂) = (��
+�
+푖휂
+푥3
+푥2
+푥3
+푖휂3
+−푖푦1
+푥2
+−푖 푦1
+푖휂2
+��
+�
+, ��
+�
+푖휉
+푦3
+푦2
+푦3
+푖휉3
+−푖푥1
+푦2
+−푖 푥1
+푖휉2
+��
+�
+, 푖휂1, 푖휉1),
+In particular, the restriction of the mapping 휁 to (픢8퐶)−2 induces a mapping (픢8퐶)−2 →
+(픢8퐶)2, so the explicit form of its restriction mapping is given by
+휁 (훷(0, 휐1퐸1, 0, 0), (휉1퐸1, 휂2퐸2 + 휂3퐸3 + 퐹1(푦1), 휉, 0), 0, 0, 푠, 0)
+= (훷(0, 0, 휐1퐸1, 0), 0, (−휂3퐸2 − 휂2퐸3 + 퐹1(푦1), −휉퐸1, 0, −휉1), 0, 0, 푠).
+Note that the restriction of the mapping 휁 to (픢8퐶)2 is also denoted by the same sign.
+Moreover, we define a 퐶-linear mapping 훿 : (픢8퐶)2 → (픢8퐶)2
+훿(훷(0, 0, 휚1퐸1, 0), 0, 푄, 0, 0, 푡) = (훷(0, 0, −푡퐸1, 0), 0, 푄, 0, 0, −휚1).
+We denote the composition mapping 훿휁 : (픢8퐶)−2 → (픢8퐶)2 of 휁 and 훿 by 휁훿, then the
+explicit form of the mapping 휁훿 is given by
+휁훿(훷(0, 휐1퐸1, 0, 0), (휉1퐸1, 휂2퐸2 + 휂3퐸3 + 퐹1(푦1), 휉, 0), 0, 0, 푠, 0)
+= (훷(0, 0, −푠퐸1, 0), 0, (−휂3퐸2 − 휂2퐸3 + 퐹1(푦1), −휉퐸1, 0, −휉1), 0, 0, −휐1),
+in addition, the explicit form of the inverse mapping 휁훿−1 : (픢8퐶)2 → (픢8퐶)−2 is given
+by
+휁훿−1(훷(0, 0, 휚1퐸1, 0), 0, (휉2퐸2 + 휉3퐸3 + 퐹1(푥1), 휂1퐸1, 0, 휂), 0, 0, 푡)
+= (훷(0, −푡퐸1, 0, 0), (−휂퐸1, −휉3퐸2 − 휉2퐸3 + 퐹1(푥1), −휂1, 0), 0, 0, −휚1, 0).
+Now, we define a subgroup (퐺14)퐶 of 퐸8퐶 by
+(퐺14)퐶 :=
+�
+훽 ∈ 퐸8퐶 �� (ad휅)훽 = 훽(ad휅), 휁훿훽푅 = 훽휁훿푅, 푅 ∈ (픢8퐶)−2
+�
+.
+Then, from [5, Proposition 5.8.7], the group (퐺14)퐶 is isomorphic to the group
+푆푝푖푛(14, 퐶) as the universal covering group of 푆푂(14, 퐶):
+(퐺14)퐶 � 푆푝푖푛(14, 퐶).
+
+10
+TOSHIKAZU MIYASHITA
+In particular, note that (퐺14)퐶, (픢8퐶)−2 above are denoted by 퐺14, (푉퐶)14 in [5, Propo-
+sition 5.8.7], respectively.
+We prove the following lemma needed below and later.
+Lemma 8.3. The 퐶-linear transformation 휏휆휔 satisfies the formula (ad휅)휏휆휔 = −휏휆휔
+(ad휅) and commutes with the 퐶-linear transformation 휅4:(휏휆휔)휅4 = 휅4(휏휆휔), where
+휏휆휔 is a composition transformation of 휏 and 휆휔 defined in [3, Preliminaries (p.96)].
+Proof. First, we denote 훷(−2퐸1 ∨ 퐸1, 0, 0, −1) by 훷휅 only in this lemma: 훷휅 :=
+훷(−2퐸1 ∨ 퐸1, 0, 0, −1). Then, note that 휏휆훷휅 = −훷휅휏휆, it follows that
+(ad휅)휏휆휔(훷, 푃, 푄, 푟, 푠, 푡)
+= ad휅(휏휆훷휆−1휏, 휏휆푄, −휏휆푃, −휏푟, −휏푡, −휏푠)
+= ([훷휅, 휏휆훷휆−1휏],훷휅(휏휆푄) − 휏휆푄, −훷휅(휏휆푃) − 휏휆푃, 0, 2휏푡, −2휏푠)
+= −(휏휆[훷휅,훷]휆−1휏, 휏휆(훷휅푄 + 푄), −휏휆(훷휅푃 − 푃), 0, −2(휏푡), 2(휏푠))
+= −휏휆휔(ad휅)(훷, 푃, 푄, 푟, 푠, 푡), (훷, 푃, 푄, 푟, 푠, 푡) ∈ 픢퐶
+8 ,
+that is, (ad휅)휏휆휔 = −휏휆휔(ad휅). The first half is proved.
+The second half is easily proved by doing straightforward computation under the
+definitions of 휏휆휔 and 휅4.
+□
+Here, in order to prove the proposition below , we use the following lemma.
+Lemma 8.4. (1) For 푅 ∈ (픢8퐶)−2, the formula (휏휆휔)휁훿푅 = 휁훿−1(휏휆휔)푅 holds.
+(2) For 훽 ∈ 푆푝푖푛(14, 퐶), 훽 satisfies the formula 휁훿−1훽푅′ = 훽휁훿−1푅′, 푅′ ∈ (픢8퐶)2.
+Proof. (1) Under the definition of 휏휆휔 and the mappings 휁훿, 휁훿−1 mentioned above,
+we do straightforward computation of both sides:
+(휏휆휔)휁훿푅
+= (휏휆휔)휁훿(훷(0, 휐1퐸1, 0, 0), (휉1퐸1, 휂2퐸2 + 휂3퐸3 + 퐹1(푦1), 휉, 0), 0, 0, 푠, 0)
+= 휏휆휔(훷(0, 0, −푠퐸1, 0), 0, (−휂3퐸2 − 휂2퐸3 + 퐹1(푦1), −휉퐸1, 0, −휉1), 0, 0, −휐1)
+= (휏휆훷(0, 0, −푠퐸1, 0)휆−1휏, 휏휆(−휂3퐸2 − 휂2퐸3 + 퐹1(푦1), −휉퐸1, 0, −휉1), 0, 0, 휏휐1, 0)
+= (훷(0, 휏푠퐸1, 0, 0), (−휏휉퐸1, 휏휂3퐸2 + 휏휂2퐸3 − 퐹1(휏푦1), −휏휉1, 0), 0, 0, 휏휐1, 0),
+휁훿−1(휏휆휔)푅
+= 휁훿−1(휏휆훷(0, 휐1퐸1, 0, 0)휆−1휏, 0, −휏휆(휉1퐸1, 휂2퐸2 + 휂3퐸3 + 퐹1(푦1), 휉, 0), 0, 0, −휏푠)
+= 휁훿−1(훷(0, 0, −휏휐1퐸1, 0), 0, (−휏휂2퐸2 − 휏휂3퐸3 − 퐹1(휏푦1), 휏휉1퐸1, 0, 휏휉), 0, 0, −휏푠)
+= (훷(0, 휏푠퐸1, 0, 0), (−휏휉퐸1, 휏휂3퐸2 + 휏휂2퐸3 − 퐹1(휏푦1), −휏휉1, 0), 0, 0, 휏휐1, 0).
+With above, the required formula is proved.
+(2) Let 푆푝푖푛(14, 퐶) as the group (퐺14)퐶. In the formula 휁훿훽푅 = 훽휁훿푅, 푅 ∈ (픢8퐶)−2,
+the required formula is proved by setting 휁훿푅 = 푅′, 푅′ ∈ (픢8퐶)2.
+□
+
+REALIZATIONS OF INNER AUTOMORPHISMS OF ORDER FOUR ON 퐸8 PART III
+11
+Proposition 8.5. The 퐶-linear transformation 휏휆휔 induces the involutive inner auto-
+morphism of the group 푆푝푖푛(14,퐶): ˜휏휆휔(훽) = (휏휆휔)훽(휆휔휏), 훽 ∈ 푆푝푖푛(14, 퐶).
+Proof. Let 푆푝푖푛(14, 퐶) as the group (퐺14)퐶. We define a mapping 푔 : (퐺14)퐶 →
+(퐺14)퐶 by
+푔(훽) = (휏휆휔)훽(휆휔휏).
+We will prove 푔(훽) ∈ (퐺14)퐶. First, it follows from the first half of Lemma 8.3 that
+(ad휅)푔(훽) = (ad휅)(휏휆휔)훽(휆휔휏)
+= −(휏휆휔)(ad휅)훽(휆휔휏)
+= −(휏휆휔)훽(ad휅)(휆휔휏)
+= −(휏휆휔)훽(−휆휔휏(ad휅))
+= (휏휆휔)훽(휆휔휏)(ad휅)
+= 푔(훽)(ad휅),
+that is, (ad휅)푔(훽) = 푔(훽)(ad휅).
+Next, as for 휁훿푔(훽)푅 = 푔(훽)휁훿푅, 푅 ∈ (픢8퐶)−2, it follows from Lemma 8.4 (1), (2)
+that
+휁훿푔(훽)푅 = 휁훿(휏휆휔훽휆휔휏)푅
+((휏휆휔훽휆휔휏)푅 ∈ (픢8퐶)−2)
+= (휏휆휔)휁훿−1(휏휆휔)(휏휆휔훽휆휔휏푅)
+= 휏휆휔휁훿−1훽휆휔휏푅
+(휆휔휏푅 ∈ (픢8)2)
+= 휏휆휔훽휁훿−1휆휔휏푅
+= 휏휆휔훽휏휆휔휁훿푅
+= (휏휆휔훽휆휔휏)휁훿푅
+= 푔(훽)휁훿푅,
+that is, 휁훿푔(훽)푅 = 푔(훽)휁훿푅, 푅 ∈ (픢8퐶)−2.
+Hence we have 푔(훽) ∈ (퐺14)퐶. The proof of this proposition is completed.
+□
+From Proposition 8.5, we can define a subgroup (푆푝푖푛(14, 퐶))휏휆휔 of 푆푝푖푛(14, 퐶):
+(푆푝푖푛(14, 퐶))휏휆휔 = {훽 ∈ 푆푝푖푛(14, 퐶) | (휏휆휔)훽 = 훽(휏휆휔)} .
+We prove the following lemma needed in the proof of theorem below .
+Lemma 8.6. The Lie algebra (픰픭픦픫(14, 퐶))휏휆휔 of the group (푆푝푖푛(14, 퐶))휏휆휔 is given
+by
+(픰픭픦픫(14, 퐶))휏휆휔 =
+�
+푅 ∈ (픤14)퐶 �� (휏휆휔)푅 = 푅
+�
+
+12
+TOSHIKAZU MIYASHITA
+=
+
+
+푅 = (훷, 푃, −휏휆푃, 푟, 0, 0)
+�����������������
+훷 = 훷(퐷 + ˜퐴(푑1)
++푖(휏1퐸1 + 휏2퐸2 + 휏3퐸3 + 퐹1(푡1))∼, 퐴, −휏퐴, 휈),
+퐷 ∈ 픰픬(8), 푑1, 푡1 ∈ ℭ,
+휏푘 ∈ 푹, 휏1 + 휏2 + 휏3 = 0,
+퐴 = 훼2퐸2 + 훼3퐸3 + 퐹1(푎1) ∈ 픍퐶, 휈 ∈ 푖푹,
+푃 = (휌2퐸2 + 휌3퐸3 + 퐹1(푝1), 휌1퐸1, 0, 휌) ∈ 픓퐶,
+푟 ∈ 푖푹,
+푖휏1 + (2/3)휈 + 2푟 = 0
+
+
+.
+where the Lie algebra (픤14)퐶 is defined as 픤14 in [5, Lemma 5.8.3].
+In particular, we have dim((픰픭픦픫(14, 퐶)휏휆휔) = (28+16+2+20+1)+24+1−1 = 91.
+Proof. Let 푅 = (훷(휙, 퐴, 퐵, 휈), 푃, 푄, 푟, 0, 0) ∈ (픤14)퐶, where
+휙 = 퐷 + ˜퐴1(푑1) + (휏1퐸1 + 휏2퐸2 + 휏3퐸3 + 퐹1(푡1))∼, 퐷 ∈ 픰픬(8, 퐶),
+푑1, 푡1 ∈ ℭ퐶, 휏푘 ∈ 퐶, 휏1 + 휏2 + 휏3 = 0,
+퐴 = 훼2퐸2 + 훼3퐸3 + 퐹1(푎1) ∈ 픍퐶, 훼푘 ∈ 퐶, 푎1 ∈ ℭ퐶,
+퐵 = 훽2퐸2 + 훽3퐸3 + 퐹1(푏1) ∈ 픍퐶, 훽푘 ∈ 퐶, 푏1 ∈ ℭ퐶,
+휈 ∈ 퐶,
+푃 = (휌2퐸2 + 휌3퐸3 + 퐹1(푝1), 휌1퐸1, 0, 휌) ∈ 픓퐶, 휌푘, 휌 ∈ 퐶, 푝1 ∈ ℭ퐶,
+푄 = (휁1퐸1, 휁2퐸2 + 휁3퐸3 + 퐹1(푧1), 휁1퐸1, 휁, 0) ∈ 픓퐶, 휁푘, 휁 ∈ 퐶, 푧1 ∈ ℭ퐶,
+푟 ∈ 퐶,
+휏1 + (2/3)휈 + 2푟 = 0.
+Then it follows from
+(휏휆휔)푅 = (휏휆휔)(훷(휙, 퐴, 퐵, 휈), 푃, 푄, 푟, 0, 0, )
+= (휏휆훷(휙, 퐴, 퐵, 휈)휆−1휏, 휏휆푄, −휏휆푃, −휏푟, 0, 0)
+that
+휏휆훷(휙, 퐴, 퐵, 휈)휆−1휏 = 훷(휙, 퐴, 퐵, 휈), 휏휆푄 = 푃, −휏휆푃 = 푄, −휏푟 = 푟.
+Hence, using the formula 휏휆훷(휙, 퐴, 퐵, 휈)휆−1휏 = 훷(−휏 푡휙휏, −휏��, −휏퐴, −휏휈), the
+required result is obtained.
+□
+From Lemma 8.6, we have
+(픰픭픦픫(14, 퐶))휏휆휔 ⊂ 픢8 =
+�
+푅 = (훷, 푃, −휏휆푃, 푟, 푠, −휏푠)
+����
+훷 ∈ 픢7, 푃 ∈ 픓퐶,
+푟 ∈ 푖푹, 푠 ∈ 퐶
+�
+.
+Now, we construct the spinor group 푆푝푖푛(14) in 퐸8.
+Theorem 8.7. The group (푆푝푖푛(14, 퐶))휏휆휔 is isomorphic to the group 푆푝푖푛(14) :
+(푆푝푖푛(14, 퐶))휏휆휔 � 푆푝푖푛(14).
+
+REALIZATIONS OF INNER AUTOMORPHISMS OF ORDER FOUR ON 퐸8 PART III
+13
+Proof. First, since the group 푆푝푖푛(14, 퐶) is the simply connected Lie group, the group
+(푆푝푖푛(14, 퐶))휏휆휔 is connected (([3, Preliminaries Lemma 2.2]) in [6]). Hence, since
+both of the groups (푆푝푖푛(14, 퐶))휏휆휔 and 퐸8 are connected and (픰픭픦픫(14, 퐶))휏휆휔 ⊂ 픢8,
+we confirm (푆푝푖푛(14, 퐶))휏휆휔 ⊂ 퐸8.
+We define a 14-dimensional 푹-vector space 푉14 by
+푉14 =
+�
+푅 ∈ (픢8퐶)−2
+�� (휏휆휔)휁훿푅 = −푅
+�
+=
+
+
+푅 = (훷(0, 휐퐸1, 0, 0), (휉퐸1, 휂퐸2 − 휏휂퐸3 + 퐹1(푦), 휏휉, 0), 0, 0, −휏휐, 0)
+��������
+휐 ∈ 퐶,
+휉 ∈ 퐶,
+휂 ∈ 퐶,
+푦 ∈ ℭ
+
+
+with the norm
+(푅, 푅)휁 = 1
+30퐵8(휁훿푅, 푅) = 4(휏휐)휐 + (휏휂)휂 + 푦푦 + (휏휉)휉,
+where 퐵8 is the Killing form of 픢8퐶 (as for the Killing form 퐵8, see [7, Theorem 5.3.2]
+in detail). Obviously, the group (푆푝푖푛(14, 퐶))휏휆휔 acts on 푉14.
+Here, let the orthogonal group
+푂(14) = 푂(푉14) =
+�
+훼 ∈ Iso푹(푉14)
+�� (훼푅, 훼푅)휁 = (푅, 푅)휁
+�
+.
+We consider the restriction 훼
+��
+푉14 of 훼 ∈ (푆푝푖푛(14, 퐶))휏휆휔 to 푉14, then we see 훼
+��
+푉14∈
+푂(14) = 푂(푉14). Indeed, the group (푆푝푖푛(14, 퐶))휏휆휔 acts on 푉14, so that we have
+훼
+��
+푉14∈ Iso푹(푉14). Moreover, for 푅 ∈ 푉14, it follows that
+(훼
+��
+푉14 푅, 훼
+��
+푉14 푅)휁 = 1
+30퐵8(휁훿훼
+��
+푉14 푅, 훼
+��
+푉14 푅)
+= 1
+30퐵8(휁훿훼푅, 훼푅)
+= 1
+30퐵8(훼휁훿푅, 훼푅)
+= 1
+30퐵8(휁훿푅, 푅)
+= (푅, 푅)휁.
+Hence we can define a homomorphism 휋 : (푆푝푖푛(14, 퐶))휏휆휔 → 푂(14) = 푂(푉14) by
+휋(훼) = 훼
+��
+푉14 .
+Moreover, since the mapping 휋 is continuous and the group (푆푝푖푛(14, 퐶))휏휆휔 is con-
+nected, 휋 induces a homomorphism
+휋 : (푆푝푖푛(14, 퐶))휏휆휔 → 푆푂(14) = 푆푂(푉14).
+We will determine Ker 휋. First, from the definition of kernel, we have
+Ker 휋 =
+�
+훽 ∈ (푆푝푖푛(14, 퐶))휏휆휔 �� 휋(훽) = 1
+�
+=
+�
+훽 ∈ (푆푝푖푛(14, 퐶))휏휆휔 �� 훽
+��
+푉14= 1
+�
+(⊂ 퐸8).
+
+14
+TOSHIKAZU MIYASHITA
+Then let 훽 ∈ Ker 휋. Sinceit follows from (훷(0, −퐸1, 0, 0), 0, 0, 0, 1, 0), 푖(훷(0, 퐸1, 0, 0),
+0, 0, 0, 1, 0) ∈ 푉14 that
+훽(훷(0, −퐸1, 0, 0), 0, 0, 0, 1, 0) = (훷(0, −퐸1, 0, 0), 0, 0, 0, 1, 0),
+훽푖(훷(0, 퐸1, 0, 0), 0, 0, 0, 1, 0) = 푖(훷(0, 퐸1, 0, 0), 0, 0, 0, 1, 0),
+note that 훽 ∈ Iso퐶 (픢8퐶), we have 훽(0, 0, 0, 0, 1, 0) = (0, 0, 0, 0, 1, 0).
+Here, if
+훽(0, 0, 0, 0, 1, 0) = (0, 0, 0, 0, 1, 0), we have 훽(0, 0, 0, 0, 0, 1) = (0, 0, 0, 0, 0, 1). In-
+deed, it follows from (휏휆휔)훽 = 훽(휏휆휔) that
+(0, 0, 0, 0, 0, 1) = 휏휆휔(0, 0, 0, 0, −1, 0) = −휏휆휔(0, 0, 0, 0, 1, 0)
+= −(휏휆휔)훽(0, 0, 0, 0, 1, 0) = −훽(휏휆휔)(0, 0, 0, 0, 1, 0)
+= −훽(0, 0, 0, 0, 0, −1)
+= 훽(0, 0, 0, 0, 0, 1),
+that is, 훽(0, 0, 0, 0, 0, 1) = (0, 0, 0, 0, 0, 1). Hence we have 훽 ∈ 퐸7 ⊂ (퐸8)(0,0,0,0,0,1)
+([7, Theorem 5.7.3]).
+Moreover, since it follows from (0, (퐸1, 0, 1, 0), 0, 0, 0, 0), 푖(0, (−퐸1, 0, 1, 0), 0, 0, 0, 0)
+∈ 푉14 that
+훽(0, (퐸1, 0, 1, 0), 0, 0, 0, 0) = (0, (퐸1, 0, 1, 0), 0, 0, 0, 0)
+훽푖(0, (−퐸1, 0, 1, 0), 0, 0, 0, 0) = 푖(0, (−퐸1, 0, 1, 0), 0, 0, 0, 0),
+again note that 훽 ∈ Iso퐶 (픢8퐶), we have 훽(0, (0, 0, 1, 0), 0, 0, 0, 0) = (0, (0, 0, 1, 0), 0, 0,
+0, 0) and 훽(0, (퐸1, 0, 0, 0), 0, 0, 0, 0) = (0, (퐸1, 0, 0, 0), 0, 0, 0, 0), that is, 훽(0, 0, 1, 0) =
+(0, 0, 1, 0) and 훽퐸1 = 퐸1 in 픓퐶. Hence we have 훽 ∈ 퐸6 ⊂ (퐸7)(0,0,1,0) ([7, Theorem
+4.7.2]) and 훽 satisfies 훽퐸1 = 퐸1.
+In addition, since (0, (0, 퐸2 − 퐸3, 0, 0), 0, 0, 0, 0), (0, (0, 푖(퐸2 + 퐸3), 0, 0), 0, 0, 0, 0) ∈
+푉14, 훽 satisfies 훽(퐸2 − 퐸3) = 퐸2 − 퐸3 and 훽(퐸2 + 퐸3) = 퐸2 + 퐸3 in 픍퐶 by an
+argument similar to above, together with 훽퐸1 = 퐸1 above, we have 훽퐸푖 = 퐸푖, 푖 =
+1, 2, 3, so we have 훽 ∈ 퐹4 ⊂ (퐸6)퐸 ([7, Theorem 3.7.1]), and moreover we see
+훽 ∈ 푆푝푖푛(8) ⊂ (퐹4)퐸1,퐸2,퐸3 ([7, Theorem 2.7.1]).
+Here, we denote 훽 ∈ 푆푝푖푛(8)
+by 훽 = (훽1, 훽2, 훽3) ∈ 푆푂(8) × 푆푂(8) × 푆푂(8), then since 훽1 satisfies the condition
+훽1푦 = 푦 for all 푦 ∈ ℭ, we have 훽1 = 1. Hence, from the Principle of triality on 푆푂(8),
+we have the following
+훽 = (1, 1, 1) = 1
+or
+훽 = (1, −1, −1) =: 휎.
+Hence we have Ker 휋 ⊂ {1, 휎} and vice versa, so that Ker 휋 = {1, 휎} � 풁2.
+Finally, since the group 푆푂(14) is connected and Ker 휋 is discrete, together with
+dim((픰픭픦픫(14, 퐶)휏휆휔) = 91 = dim(픰픬(14)) (Lemma 8.6), 휋 is surjective. Thus we
+have the isomorphism (푆푝푖푛(14, 퐶))휏휆휔/풁2 � 푆푂(14).
+Therefore the group (푆푝푖푛(14, 퐶))휏휆휔 is isomorphic to the group 푆푝푖푛(14) as the
+universal covering group of 푆푂(14):
+(푆푝푖푛(14, 퐶))휏휆휔 � 푆푝푖푛(14).
+
+REALIZATIONS OF INNER AUTOMORPHISMS OF ORDER FOUR ON 퐸8 PART III
+15
+□
+Let the mapping 휙 : 퐶∗ → 퐸8퐶 defined in [5, Subsection 5.3 (p.45)].
+Then we prove the following lemma needed in the proof of theorem below.
+Lemma 8.8. For 푎 ∈ 퐶∗, the action of 휙(푎) on 픢8퐶 is given by
+휙(푎)(훷, 푃, 푄, 푟, 푠, 푡) = (휓(푎)훷휓(푎)−1, 푎휓(푎)푃, 푎−1휓(푎)푄, 푟, 푎2푠, 푎−2푡),
+where the action to 픓퐶 of 휓(푎) ∈ 퐸7퐶 on right hand side is defined by
+휓(푎)(푋,푌, 휉, 휂) = (��
+�
+푎휉1
+푥3
+푥2
+푥3
+푎−1휉2
+푎−1푥1
+푥2
+푎−1푥1
+푎−1휉3
+��
+�
+, ��
+�
+푎−1휂1
+푦3
+푦2
+푦3
+푎휂2
+푎푦1
+푦2
+푎푦1
+푎휂3
+��
+�
+, 푎휉, 푎−1휂).
+Moreover, we have the following formula
+(휏휆휔)휙(푎)(휆휔휏) = 휙((휏푎)−1).
+Proof. As for the first half, its proof is proved in [5, Lemma 5.9]. As for the second
+half, we first have (휏휆)휓(푎)(휆−1휏) = 휓((휏푎)−1). Indeed, it follows that
+(휏휆)휓(푎)(휆−1휏)(푋,푌, 휉, 휂)
+= (휏휆)휓(푎)(−휏푌, 휏푋, −휏휂, 휏휉)
+= (휏휆)(��
+�
+−푎휏휂1
+−휏푦3
+−휏푦2
+−휏푦3
+−푎−1휏휂2
+−푎−1휏푦1
+−휏푦2
+−푎−1휏푦1
+−푎−1휏휂3
+��
+�
+, ��
+�
+푎−1휏휉1
+휏푥3
+휏푥2
+휏푥3
+푎휏휉2
+푎휏푥1
+휏푥2
+푎휏푥1
+푎휏휉3
+��
+�
+, −푎휏휂, 푎−1휏휉)
+= (��
+�
+(휏푎−1)휉1
+푥3
+푥2
+푥3
+(휏푎)휉2
+(휏푎)푥1
+푥2
+(휏푎)푥1
+(휏푎)휉3
+��
+�
+, ��
+�
+(휏푎)휂1
+푦3
+푦2
+푦3
+(휏푎−1)휂2
+(휏푎−1)푦1
+푦2
+(휏푎−1)푦1
+(휏푎−1)휂3
+��
+�
+, (휏푎−1)휉, (휏푎)휂)
+= 휓((휏푎)−1).
+Hence, using 휏휆휔(훷, 푃, 푄, 푟, 푠, 푡) = (휏휆훷휆−1휏, 휏휆푄, −휏휆푃, −휏푟, −휏푡, −휏푠) and the
+formula (휏휆)휓(푎)(휆−1휏) = 휓((휏푎)−1) shown above, we can obtain the formula
+(휏휆휔)휙(푎)(휆휔휏) = 휙((휏푎)−1) by doing straightforward computation.
+□
+Now, we determine the structure of the group (퐸8)휅4.
+Theorem 8.9. The group (퐸8)휅4 is isomorphic to the group (푈(1) ×푆푝푖푛(14))/풁4, 풁4
+= {(1, 1), (−1, 휙(−1)), (푖, 휙(−푖)), (−푖, 휙(푖))}: (퐸8)휅4 � (푈(1) × 푆푝푖푛(14))/풁4.
+Proof. Let 푈(1) := {푎 ∈ 퐶 | (휏푎)푎 = 1}(⊂ 퐶∗) and 푆푝푖푛(14)(⊂ 푆푝푖푛(14, 퐶)) as the
+group (푆푝푖푛(14, 퐶))휏휆휔 (Proposition 8.7). Then we can define a mapping 휑휅4 : 푈(1) ×
+푆푝푖푛(14) → (퐸8)휅4 by the restriction of the mapping 휑 : 퐶∗×푆푝푖푛(14, 퐶) → (퐸8퐶)휅4:
+휑휅4(푎, 훽) = 휑(푎, 훽) = 휙(푎)훽 (Theorem 8.2).
+First, we will prove that 휑휅4 is well-defined. It is clear 휙(푎) ∈ (퐸8퐶)휅4, so that
+we have 휙(푎) ∈ (퐸8)휅4. Indeed, since 푎 satisfies the condition (휏푎)푎 = 1, we have
+(휏휆휔)휙(푎)(휆휔휏) = 휙(푎) by Lemma 8.8, that is, 휙(푎) ∈ ((퐸8퐶)휅4)휏휆휔, moreover it
+
+16
+TOSHIKAZU MIYASHITA
+follows from (휏휆휔)휅4 = 휅4(휏휆휔) (Lemma 8.3) and (퐸8퐶)휏휆휔 = 퐸8 ([3, Preliminaries
+(p.96)]) that
+휙(푎) ∈ ((퐸8퐶)휅4)휏휆휔 = ((퐸8퐶)휏휆휔)휅4 = (퐸8)휅4.
+By an argument similar to above, it follows from Lemma 8.3 that
+훽 ∈ 푆푝푖푛(14) = (푆푝푖푛(14, 퐶))휏휆휔 ⊂ ((퐸8퐶)휅4)휏휆휔 = ((퐸8퐶)휏휆휔)휅4 = (퐸8)휅4,
+that is, 훽 ∈ (퐸8)휅4. Hence 휑휅4 is well-defined. Subsequently, we will prove that 휑휅4 is
+a homomorphism, however since 휑휅4 is the restriction of the mapping 휑, it is clear.
+Next, we will prove that 휑휅4 is surjective. Let 훼 ∈ (퐸8)휅4 ⊂ (퐸8퐶)휅4, there exist
+푎 ∈ 퐶∗ and 훽 ∈ 푆푝푖푛(14, 퐶) such that 훼 = 휑(푎, 훽) (Theorem 8.2). Moreover, from
+the condition (휏휆휔)훼(휆휔휏) = 훼, that is, (휏휆휔)휑(푎, 훽)(휆휔휏) = 휑(푎, 훽), we have
+휑((휏푎)−1, (휏휆휔)훽(휆휔휏)) = 휑(푎, 훽). Indeed, it follows from Lemma 8.8 that
+(휏휆휔)휑(푎, 훽)(휆휔휏) = (휏휆휔)휙(푎)훽(휆휔휏)
+= (휏휆���)휙(푎)(휆휔휏)(휏휆휔)훽(휆휔휏)
+= 휙((휏푎)−1)(휏휆휔)훽(휆휔휏)
+= 휑((휏푎)−1, (휏휆휔)훽(휆휔휏))
+that 휑((휏푎)−1, (휏휆휔)훽(휆휔휏)) = 휑(푎, 훽).
+Hence, since Ker휑 = {(1, 1), (−1, 휙(−1)), (푖, 휙(−푖)), (−푖, 휙(푖))}, we have the fol-
+lowing
+(i)
+� (휏푎)−1 = 푎
+(휏휆휔)훽(휆휔휏) = 훽,
+(ii)
+� (휏푎)−1 = −푎
+(휏휆휔)훽(휆휔휏) = 휙(−1)훽,
+(iii)
+� (휏푎)−1 = 푖푎
+(휏휆휔)훽(휆휔휏) = 휙(−푖)훽,
+(iv)
+� (휏푎)−1 = −푖푎
+(휏휆휔)훽(휆휔휏) = 휙(푖)훽.
+Case (i). From (휏푎)−1 = 푎, we have 푎 ∈ 푈(1) = {푎 ∈ 퐶∗ | (휏푎)푎 = 1}. From
+(휏휆휔)훽(휆휔휏) = 훽, we have 훽 ∈ 푆푝푖푛(14) (Theorem 8.7). Hence there exist 푎 ∈ 푈(1)
+and 훽 ∈ 푆푝푖푛(14) such that 훼 = 휑(푎, 훽) = 휑휅4(푎, 훽).
+Case (ii). From (휏푎)−1 = −푎, we have (휏푎)푎 = −1. However, this case is impossible
+because of (휏푎)푎 > 0.
+Case (iii). From (휏푎)−1 = 푖푎, we have (휏푎)푎 = 푖. As in Case (ii), this case is also
+impossible.
+Case (iv). From (휏푎)−1 = −푖푎, we have (휏푎)푎 = −푖. As in Case (ii), this case is also
+impossible.
+With above, the proof of surjective is completed.
+Finally, we will determine Ker 휑휅4. Since 휑휅4 is the restriction of the mapping 휑, we
+have Ker 휑휅4 = Ker 휑, that is, Ker 휑휅4 = {(1, 1), (−1, 휙(−1)), (푖, 휙(−푖)), (−푖, 휙(푖))} �
+풁4.
+Therefore we have the required isomorphism
+(퐸8)휅4 � (푈(1) × 푆푝푖푛(14))/풁4.
+
+REALIZATIONS OF INNER AUTOMORPHISMS OF ORDER FOUR ON 퐸8 PART III
+17
+□
+9. Case 7. The automorphism ˜휀4 of order four and the group (퐸8) 휀4
+We define a 퐶-linear transformations 휀4 of 픢8퐶 by
+휀4(훷, 푃, 푄, 푟, 푠, 푡) = (휈4훷휈4
+−1, −휈4푃, −휈4푄, 푟, 푠, 푡),
+where 휈4 ∈ 퐸7 on the right hand side is the same one as that defined in previous section.
+Note that 휀4 is the composition mapping of 휈4, 휐 ∈ 퐸7 ⊂ 퐸8, moreover since 휈4, 휐 are
+expressed as elements of 퐸8 by
+휈4 = exp
+�2휋푖
+4 ad(훷(−2퐸1 ∨ 퐸1, 0, 0, −1), 0, 0, 0, 0, 0)
+�
+,
+휐 = exp
+�2휋푖
+4 ad(훷(0, 0, 0, 6), 0, 0, 0, 0, 0)
+�
+,
+respectively and together with [훷(−2퐸1 ∨ 퐸1, 0, 0, −1),훷(0, 0, 0, 6)] = 0, we have
+휀4 = exp
+�2휋푖
+4 ad(훷(−2퐸1 ∨ 퐸1, 0, 0, 5), 0, 0, 0, 0, 0)
+�
+.
+Hence it follows from above that 휀4 ∈ 퐸8 and (휀4)4 = 1, so that 휀4 induces the inner
+automorphism ˜휀4 of order four on 퐸8: ˜휀4(훼) = 휀4훼휀4
+−1, 훼 ∈ 퐸8.
+Now, we will study the subgroup (퐸8)휀4 of 퐸8:
+(퐸8)휀4 =
+�
+훼 ∈ 퐸8
+�� 휀4훼 = 훼휀4
+�
+.
+The aim of this section is to determine the structure of the group (퐸8)휀4. Before that,
+we make some preparations. First, in order to prove the proposition below, we use the
+following proposition and theorem.
+Proposition 9.1. The group (퐸7)휈4 contains a group
+휙푈(1) (푈(1)) =
+�
+휙푈(1) (휃)
+��� 휃 ∈ 푈(1)
+�
+which is isomorphic to the group 푈(1) = {휃 ∈ 퐶 | (휏휃)휃 = 1}, where 휙푈(1) is the
+restriction of the mapping 휑2 defined in [7, Theorem 4.11.13]. For 휃 ∈ 푈(1), the
+mapping 휙푈(1) (휃) : 픓퐶 → 픓퐶 is given by
+휙푈(1) (휃)(푋,푌, 휉, 휂)
+= 휑2(
+�휃
+0
+0
+휏휃
+�
+)(��
+�
+휉1
+푥3
+푥2
+푥3
+휉2
+푥1
+푥2
+푥1
+휉3
+��
+�
+, ��
+�
+휂1
+푦3
+푦2
+푦3
+휂2
+푦1
+푦2
+푦1
+휂
+��
+�
+, 휉, 휂)
+= (��
+�
+휃휉1
+푥3
+푥2
+푥3
+(휏휃)휉2
+(휏휃)푥1
+푥2
+(휏휃)푥1
+(휏휃)휉3
+��
+�
+, ��
+�
+(휏휃)휂1
+푦3
+푦2
+푦3
+휃휂2
+휃푦1
+푦2
+휃푦1
+휃휂3
+��
+�
+, 휃휉, (휏휃)휂).
+
+18
+TOSHIKAZU MIYASHITA
+Proof. From the definition of 휙푈(1), it is clear 휙푈(1) ∈ 퐸7.
+Moreover, since 휈4 is
+expressed by 휙푈(1) (−푖): 휈4 = 휙푈(1) (−푖), it is also clear 휈4휙푈(1) (휃) = 휙푈(1) (휃)휈4. Hence
+we have 휙푈(1) (휃) ∈ (퐸7)휈4.
+□
+Theorem 9.2. ([7, Theorem 4.11.15]) The group (퐸7)휎 is isomorphic to the group
+(푆푈(2) ×푆푝푖푛(12))/풁2, 풁2 = {(퐸, 1), (−퐸, −휎)}: (퐸7)휎 � (푆푈(2) ×푆푝푖푛(12))/풁2.
+Proof. We define a mapping 휑 : 푆푈(2) × 푆푝푖푛(12) → (퐸7)휎 by
+휑(퐴, 훽) = 휑2(퐴)훽,
+where 휑2 is defined in [7, Theorem 4.11.13].
+Then the mapping 휑 induces the required isomorphism. As for the 푹-linear trans-
+formation 휎 of 픓퐶, see [7, Subsection 4.11(p.133)] in detail.
+□
+Proposition 9.3. The group (퐸8)휀4 contains the group (퐸7)휈4 which is isomorphic
+to the group (푈(1) × 푆푝푖푛(12))/풁2, 풁2 = {(1, 1), (−1, −휎)}: (퐸8)휀4 ⊃ (퐸7)휈4 �
+(푈(1) × 푆푝푖푛(12))/풁2.
+Proof. Let 훼 ∈ (퐸7)휈4. Note that −1 ∈ 푧(퐸7) (the center of 퐸7), we have
+휀4훼(훷, 푃, 푄, 푟, 푠, 푡) = 휀4(훼훷훼−1, 훼푃, 훼푄, 푟, 푠, 푡)
+= (휈4훼훷훼−1휈4
+−1, −휈4훼푃, −휈4훼푄, 푟, 푠, 푡)
+= (훼(휈4훷휈4
+−1)훼−1, 훼(−휈4푃), 훼(−휈4푄), 푟, 푠, 푡)
+= 훼휀4(훷, 푃, 푄, 푟, 푠, 푡), (훷, 푃, 푄, 푟, 푠, 푡) ∈ 픢8퐶,
+that is, 휀4훼 = 훼휀4. Hence we have 훼 ∈ (퐸8)휀4, so the first half is proved.
+Next, we will move the proof of the second half. Let 푈(1) = {휃 ∈ 퐶 | (휏휃)휃 = 1}
+and 푆푝푖푛(12) be constructed in [7, Theorem 4.11.11]. We define a mapping 휑휈4 :
+푈(1) × 푆푝푖푛(12) → (퐸7)휈4 by
+휑휈4 (휃, 훽) = 휙푈 (1) (휃)훽.
+Note that 휑휈4 is the restriction of the mapping 휑 : 푆푈(2) × 푆푝푖푛(12) → (퐸7)휎 defined
+in the proof of Theorem 9.2.
+First, we will prove that 휑휈4 is well-defined. From Proposition 9.1, we have 휙푈(1) (휃) ∈
+(퐸7)휈4. Since 휑2(퐴) and 훽 ∈ 푆푝푖푛(12) are commutative, we have 휈4훽 = 훽휈4 because
+of 휈4 = 휙푈(1) (−푖) = 휑2(diag(−푖, 푖)), that is, 훽 ∈ (퐸7)휈4. Hence 휑휈4 is well-defined.
+Subsequently, we will prove that 휑휈4 is a homomorphism. However, since the mapping
+휑휈4 is the restriction of the mapping 휑, it is clear.
+Next, we will prove that 휑휈4 is surjective. Let 훼 ∈ (퐸7)휈4. Then it follows from
+(휈4)2 = (휙푈 (1) (−푖))2 = 휙푈 (1) (−1) = −휎 that 훼 ∈ (퐸7)−휎 = (퐸7)휎. Hence there exist
+퐴 ∈ 푆푈(2) and 훽 ∈ 푆푝푖푛(12) such that 훼 = 휑(퐴, 훽) (Theorem 9.2). Moreover from
+the condition 휈4훼휈4
+−1 = 훼, that is, 휈4휑(퐴, 훽)휈4
+−1 = 휑(퐴, 훽), using 휈4 = 휙푈 (1) (−푖) we
+have 휑(
+� 푎
+−푏
+−푐
+푑
+�
+, 휈4훽휈4
+−1) = 휑(
+�푎
+푏
+푐
+푑
+�
+, 훽) as 퐴 :=
+�푎
+푏
+푐
+푑
+�
+.
+
+REALIZATIONS OF INNER AUTOMORPHISMS OF ORDER FOUR ON 퐸8 PART III
+19
+Thus, since Ker휑 = {(퐸, 1), (−퐸, −1)}, we have the following
+
+
+� 푎
+−푏
+−푐
+푑
+�
+=
+�푎
+푏
+푐
+푑
+�
+휈4훽휈4
+−1 = 훽
+or
+
+
+� 푎
+−푏
+−푐
+푑
+�
+=
+�−푎
+−푏
+−푐
+−푑
+�
+휈4훽휈4
+−1 = −휎훽.
+In the latter case, from 휈4훽 = 훽휈4, we have 훽 = −휎훽.
+Hence the latter case is
+impossible. Indeed, if there exists 훽 ∈ 푆푝푖푛(12) such that 훽 = −휎훽. Then apply 훽−1
+on both side of 훽 = −휎훽, we have 1 = −휎. This is contradiction.
+In the former case. From the first condition
+� 푎
+−푏
+−푐
+푑
+�
+=
+�푎
+푏
+푐
+푑
+�
+∈ 푆푈(2), we have
+퐴 =
+�푎
+0
+0
+휏푎
+�
+, (휏푎)푎 = 1, that is, 푎 ∈ 푈(1), and it is trivial that 훽 ∈ 푆푝푖푛(12). Thus
+there exist 휃 ∈ 푈(1) and 훽 ∈ 푆푝푖푛(12) such that 훼 = 휑(diag(휃, 휏휃), 훽) = 휑휈4 (휃, 훽).
+With above, the proof of surjective is completed.
+Finally, we will determine Ker 휑휈4. Since 휑휈4 is restriction of the mapping 휑, we
+have Ker 휑휈4 = Ker 휑, that is, Ker 휑휈4 = {(1, 1), (−1, −휎)} � 풁2.
+Therefore we have the required isomorphism
+(퐸7)휈4 � (푈(1) × 푆푝푖푛(12))/풁2
+□
+Lemma 9.4. (1) The Lie algebra (픢8)휐 of the group (퐸8)휐 is given by
+(픢8)휐 = {ad(푅) ∈ Der(픢8) | 휐ad(푅) = ad(푅)휐}
+� {푅 ∈ 픢8 | 휐푅 = 푅}
+= {푅 = (훷, 0, 0, 푟, 푠, −휏푠) | 훷 ∈ 픢7, 푟 ∈ 푖푹, 푠 ∈ 퐶} .
+In particular, we have dim((픢8)휐) = 133 + 1 + 2 = 136.
+(2) The Lie algebra (픢8)휀4 of the group (퐸8)휀4 is given by
+(픢8)휀4 =
+�
+ad(푅) ∈ Der(픢8)
+�� 휀4ad(푅) = ad(푅)휀4
+�
+�
+�
+푅 ∈ 픢8
+�� 휀4푅 = 푅
+�
+= {푅 = (훷, 0, 0, 푟, 푠, −휏푠) | 훷 ∈ (픢7)휈4 � 픲(1) ⊕ 픰픬(12), 푟 ∈ 푖푹, 푠 ∈ 퐶} .
+In particular, we have dim((픢8)휀4) = (1 + 66) + 1 + 2 = 70.
+Proof. By doing straightforward computation, we can prove this lemma. The Lie-
+isomorphism (픢7)휈4 � 픲(1) ⊕ 픰픬(12) follows from the group isomorphism (퐸7)휈4 �
+(푈(1) × 푆푝푖푛(12))/풁2 (Proposition 9.3) .
+□
+Proposition 9.5. The group (퐸8)휀4 contains a subgroup
+휙휐(푆푈(2)) = {휙휐(퐴) ∈ 퐸8 | 퐴 ∈ 푆푈(2)}
+which is isomorphic to the group 푆푈(2) = {퐴 ∈ 푀(2, 퐶) | (휏 푡퐴)퐴 = 퐸, det퐴 = 1},
+where 휙휐 is defined in Proposition 7.3.
+
+20
+TOSHIKAZU MIYASHITA
+Proof. For 퐴 =
+�푎
+−휏푏
+푏
+휏푎
+�
+:= exp
+�−푖휈
+−휏휚
+휚
+푖휈
+�
+∈ 푆푈(2), where
+�−푖휈
+−휏휚
+휚
+푖휈
+�
+∈ 픰픲(2),
+we have 휙휐(퐴) = exp(ad(0, 0, 0, 푖휈, 휚, −휏휚)) ∈ (퐸8)휀4 (Lemma 9.4 (2)).
+□
+Now, we determine the structure of the group (퐸8)휀4.
+Theorem 9.6. The group (퐸8)휀4 is isomorphic to the group (푆푈(2)×푈(1)×푆푝푖푛(12))/
+(풁2 × 풁2), 풁2 = {(퐸, 1, 1), (퐸, −1, −휎)}, 풁2 = {(퐸, 1, 1), (���퐸, 1, −1)}: (퐸8)휀4 �
+(푆푈(2) × 푈(1) × 푆푝푖푛(12))/(풁2 ×풁2).
+Proof. We define a mapping 휑휀4 : 푆푈(2) × 푈(1) × 푆푝푖푛(12) → (퐸8)휀4 by
+휑휀4 (퐴, 휃, 훽) = 휙휐(퐴)휑휈4 (휃, 훽)(= 휑(퐴, 휑휈4 (휃, 훽))).
+Note that this mapping is the restriction of the mapping 휑 : 푆푈(2) × 퐸7 → (퐸8)휐
+defined in the proof of Theorem 7.4.
+First, we will prove that 휑휀4 is well-defined. However, from Propositions 9.3,9.5, it is
+clear that 휑휀4 is well-defined. Subsequently, we will prove that 휑휀4 is a homomorphism.
+Since the mapping 휑휀4 is the restriction of the mapping 휑 and 휑휈4 is a homomorphism
+(Proposition 9.3), 휑휀4 is a homomorphism.
+Next, we will prove that 휑휀4 is surjective. Let 훼 ∈ (퐸8)휀4. Then, since the group 퐸8
+is the simply connected Lie group, both of the groups (퐸8)휀4 and (퐸8)휐 are connected
+(([3, Preliminaries Lemma 2.2]) in [6]), together with (픢8)휀4 ⊂ (픢8)휐 (Lemma 9.4
+(1), (2)), we have 훼 ∈ (퐸8)휀4 ⊂ (퐸8)휐. Hence there exist 퐴 ∈ 푆푈(2) and 훿 ∈ 퐸7
+such that 훼 = 휑(퐴, 훿) (Theorem 7.4). Moreover, from the condition 휀4훼휀4
+−1 = 훼,
+that is, 휀4휑(퐴, 훿)휀4
+−1 = 휑(퐴, 훿), we have 휑(퐴, 휈4훿휈4
+−1) = 휑(퐴, 훿). Indeed, from
+휀4(0, 0, 0, 푖휈, 휚, −휏휚) = (0, 0, 0, 푖휈, 휚, −휏휚) and 휙휐(퐴) = exp(ad(0, 0, 0, 푖휈, 휚, −휏휚)),
+we have 휀4휙휐(퐴)휀4
+−1 = 휙휐(퐴) by a computation similar to that in the proof of Theorem
+7.5. In addition, 휀4훿휀4
+−1 = 휈4훿휈4
+−1 follows from 훿 ∈ 퐸7. Hence it follows from
+휀4휑(퐴, 훿)휀4
+−1 = 휀4(휙휐(퐴)훿)휀4
+−1
+= (휀4휙휐(퐴)휀4
+−1)(휀4훿휀4
+−1)
+= 휙휐(퐴)(휈4훿휈4
+−1)
+= 휑(퐴, 휈4훿휈4
+−1)
+that 휑(퐴, 휈4훿휈4
+−1) = 휑(퐴, 훿).
+Thus, since Ker휑 = {(퐸, 1), (−퐸, −1)}, we have the following
+� 퐴 = 퐴
+휈4훿휈4
+−1 = 훿
+or
+� 퐴 = −퐴
+휈4훿휈4
+−1 = −훿.
+In the latter case, this case is impossible because of 퐴 ≠ 푂, where 푂 is the zero matrix.
+In the former case, 훿 ∈ (퐸7)휈4 follows from the second condition, so that there exist
+휃 ∈ 푈(1) and 훽 ∈ 푆푝푖푛(12) such that 훿 = 휑휈4 (휃, 훽) (Proposition 9.3). Hence there
+
+REALIZATIONS OF INNER AUTOMORPHISMS OF ORDER FOUR ON 퐸8 PART III
+21
+exist 퐴 ∈ 푆푈(2), 휃 ∈ 푈(1) and 훽 ∈ 푆푝푖푛(12) such that 훼 = 휑(퐴, 휃, 휑휈4 (휃, 훽)) =
+휑휀4 (퐴, 휃, 훽). The proof of surjective is completed.
+Finally, we will determine Ker 휑휀4. From the definition of kernel, we have
+Ker 휑휀4 = {(퐴, 휃, 훽) ∈ 푆푈(2) × 푈(1) × 푆푝푖푛(12) | 휑휀4 (퐴, 휃, 훽) = 1}
+= {(퐴, 휃, 훽) ∈ 푆푈(2) × 푈(1) × 푆푝푖푛(12) | 휑(퐴, 휑휈4 (휃, 훽)) = 1}.
+Here, the mapping 휑휀4 is the restriction of the mapping 휑 and together with Ker 휑 =
+{(퐸, 1), (−퐸, −1)} (Theorem 7.4), we will find the elements (퐴, 휃, 훽) ∈ 푆푈(2)×푈(1)×
+푆푝푖푛(12) satisfying the following
+�
+퐴 = 퐸
+휑휈4 (휃, 훽) = 1
+or
+�
+퐴 = −퐸
+휑휈4 (휃, 훽) = −1.
+In the former case, from Ker 휑휈4 = {(1, 1), (−1, −휎)} (Proposition 9.3), we have the
+following
+
+
+퐴 = 퐸
+휃 = 1
+훽 = 1
+or
+
+
+퐴 = 퐸
+휃 = −1
+훽 = −휎.
+In the latter case, the second condition can be rewritten as 휑휈4 (휃, −훽) = 1 from
+−1 ∈ 푧(퐸7), moreover −훽 ∈ 푆푝푖푛(12). Hence, as in the former case, we have the
+following
+
+
+퐴 = −퐸
+휃 = 1
+훽 = −1
+or
+
+
+퐴 = −퐸
+휃 = −1
+훽 = 휎.
+Hence we can obtain
+Ker 휑휀4 = {(퐸, 1, 1), (퐸, −1, −휎), (−퐸, 1, −1), (−퐸, −1, 휎)}
+= {(퐸, 1, 1), (퐸, −1, −휎)} × {(퐸, 1, 1), (−퐸, 1, −1)}
+� 풁2 × 풁2.
+Therefore we have the required isomorphism
+(퐸8)휀4 � (푆푈(2) × 푈(1) × 푆푝푖푛(12))/(풁2 × 풁2).
+□
+References
+[1] T. Imai and I. Yokota, Simply connected compact simple Lie group 퐸8(−248) of type 퐸8, J. Math.
+Kyoto Univ. 21(1981), 741-762.
+[2] J.A. Jiménez, Riemannian 4-symmetric spaces, Trans. Amer. Math. Soc. 306 (1988), 715-734.
+[3] T. Miyashita, Realizations of inner automorphisms of order 4 and fixed points subgroups by them
+on the connected compact exceptional Lie group 퐸8, Part I, Tsukuba J. Math. 41-1(2017), 91-166.
+
+22
+TOSHIKAZU MIYASHITA
+[4] T. Miyashita, Realizations of inner automorphisms of order four and fixed points subgroups by them
+on the connected compact exceptional Lie group 퐸8, Part II, Tsukuba J. Math. 43-1(2019), 1-22.
+[5] T. Miyashita and I. Yokota, 2-graded decompositions of exceptional Lie algebras 픤 and group
+realizations of 픤푒푣, 픤0, Part III, 퐺 = 퐸8, Japanese J. Math. 26-1(2000), 31-50.
+[6] Raˇsevskii, P.K., A theorem on the connectedness of a subgroup of a simply connected Lie group
+commuting with any of its automorphisms, Trans. Moscow Math. Soc, 30 (1974), 3-22.
+[7] I. Yokota, Exceptional Lie groups, arXiv:math/0902.0431vl[mathDG](2009).
+1365-3 Bessho onsen
+Ueda City
+Nagano Prefecture 386-1431
+Japan
+Email address: [email protected]
+

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+Improved analytical modeling of the non-linear power spectrum in modified gravity
+cosmologies
+Suhani Gupta,∗ Wojciech A. Hellwing, and Maciej Bilicki
+Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotnik´ow 32/46, 02-668 Warsaw, Poland
+(Dated: January 31, 2023)
+Reliable analytical modeling of the non-linear power spectrum (PS) of matter perturbations is
+among the chief pre-requisites for cosmological analyses from the largest sky surveys. This is espe-
+cially true for the models that extend the standard general-relativity paradigm by adding the fifth
+force, where numerical simulations can be prohibitively expensive. Here we present a method for
+building accurate PS models for two modified gravity (MG) variants: namely the Hu-Sawicki f(R),
+and the normal branch of the Dvali-Gabadadze-Porrati (nDGP) braneworld. We start by modifying
+the standard halo model (HM) with respect to the baseline Lambda-Cold-Dark-Matter (ΛCDM)
+scenario, by using the HM components with specific MG extensions. We find that our P(k)HM
+retains 5% accuracy only up to mildly non-linear scales (k ≲ 0.3 h/ Mpc) when compared to PS
+from numerical simulations. At the same time, our HM prescription much more accurately captures
+the ratio Υ(k) = P(k)MG/P(k)ΛCDM up to non-linear scales. We show that using HM-derived Υ(k)
+together with a viable non-linear ΛCDM P(k) prescription (such as halofit), we render a much
+better and more accurate PS predictions in MG. The new approach yields considerably improved
+performance, with modeled P(k)MG being now accurate to within 5% all the way to non-linear scales
+of k ≲ 2.5 − 3 h/ Mpc. The magnitude of deviations from GR as fostered by these MG models is
+typically O(10%) in these regimes. Therefore reaching 5% PS modeling is enough for forecasting
+constraints on modern-era cosmological observables.
+I.
+INTRODUCTION
+The standard model of cosmology, the Lambda-Cold-
+Dark-Matter (ΛCDM), has been remarkably well-tested
+observationally in the last two decades.
+Presently, it
+is our best approximation of the real Universe [1–5].
+The precise observations of the Cosmic Microwave Back-
+ground (CMB) radiation [2, 6], large-scale galaxy cluster-
+ing [4, 7–9], and the abundance of massive galaxy clusters
+[10] among others, form a long list where the standard
+cosmological model predictions are successful.
+So far, the bulk of these ΛCDM observational tests
+concerns the linear regime, the large scales, and/or early
+times. But it is the mildly non-linear and fully non-linear
+density fluctuation regimes where the vast majority of the
+modifications to ΛCDM are expected to deviate signifi-
+cantly from the standard model predictions [11–16]. It
+is also in this regime, stretching usually from hundreds
+down to a few Megaparsecs, where the present and up-
+coming cosmological surveys like desi [17], lsst [18], and
+euclid [19] aim to measure various statistics concerning
+the large-scale structure to a percent level accuracy. As
+a result, with the influx of data from these surveys, the
+level of the statistical errors can get so small that the
+measurements start to be more sensitive to systematic
+effects. If both the new level of accuracy of base-level
+predictions as well as the control of the known system-
+atics will be successfully implemented, these new large-
+scale surveys will yield new unprecedentedly accurate es-
+timates and constraints on cosmological parameters: like
+the DE equation of state, the growth rate of structure,
+∗ [gupta, hellwing, bilicki]@cft.edu.pl
+or parameters quantifying possible departures from the
+standard General Relativity (GR)-based structure forma-
+tion scenario.
+In this context, one of the most useful and widely used
+theoretical quantities is the power spectrum (PS) of den-
+sity fluctuations, P(k). This statistic generally charac-
+terizes the properties of large-scale structures across vast
+cosmological epochs and scales. Not only it can be used
+as an end-goal model prediction on its own, but it is
+also a basic quantity that is used to model and fore-
+cast a number of other useful LSS observables, includ-
+ing galaxy clustering measures, cluster abundance, weak-
+lensing shear and convergence, the amplitude of the bulk
+peculiar galaxy motions, and many others [1, 4, 18, 20–
+23].
+Since the PS forms a basis for the predictions of many
+cosmological LSS observational statistics, the accuracy,
+and scales to which we know the input PS limits our
+abilities to forecast the derived observables. Thus, ob-
+taining robust estimates of the PS beyond the linear the-
+ory regime (i.e.
+scales of k > 0.1 h/ Mpc) became of
+paramount importance to modern cosmology, and has
+been a subject of extensive effort in recent years [24–
+27]. A classical approach is to either directly use the re-
+sults of a number of N-body simulations of LSS evolution
+to predict PS, or use the simulation results for calibra-
+tion of more or less approximate models [24–26, 28, 29].
+Recently, machine-learning-based-emulators are also em-
+ployed for computing non-linear PS [e.g. 27, 30]. This ap-
+proach especially depends on the growing computational
+power.
+In recent years, the progress in modeling the PS has
+been truly significant. The resulting current state-of-the-
+art PS models for ΛCDM are already, or close to, at-
+arXiv:2301.12016v1  [astro-ph.CO]  27 Jan 2023
+
+2
+taining sub-percent accuracy in the non-linear regime, as
+required for the success of the cosmological tests offered
+by the incoming big survey data. However, this amazing
+progress has been mostly limited to the ΛCDM alone.
+When it comes to many interesting extensions and modi-
+fications of the standard model, such as the whole family
+of beyond-GR Modified Gravity (MG) scenarios, the cur-
+rent accuracy, and versatility of PS modeling is still very
+much lacking. The reasons for this are both higher theo-
+retical complications of such models, and their increased
+levels of non-linearity [16, 31–35]. For MG models, N-
+body simulations play an even more important role in
+fully assessing the effect of the fifth force, and are crucial
+for disentangling pure MG effects from the standard GR-
+based scenarios [36–38]. This is connected with the richer
+phenomenology of such models [39–48]. Given the fact
+that the MG simulations are usually many times more
+expensive than the standard ΛCDM case [14, 15], it be-
+comes computationally prohibitive to obtain simulation
+libraries of the same volume and precision for MG, as is
+possible for ΛCDM. However, such libraries are neces-
+sary to be applied to the proven state-of-the-art emulat-
+ing or fitting methods to achieve the same precision, and
+success in modeling MG effects, as we have for the case
+of ΛCDM.
+In this work, we attempt to remedy the deficit of ac-
+curate MG PS modeling. To circumnavigate the prob-
+lem of prohibitively expensive MG simulations, we ex-
+plore a different approach. Instead of trying to model
+the absolute MG PS predictions, we take ΛCDM to al-
+ways be our baseline, and build a semi-analytical model
+for the relative MG effects on the ΛCDM PS. We build
+our model on the basis of a more general Halo Model
+(HM) approach ([49], reviewed by [e.g. 24]). Next, we
+demonstrate how various degrees of modeling freedom
+can be calibrated and constrained already by a relatively
+small library of N-body simulations, to achieve an un-
+precedented level of PS modeling in the MG scenarios
+studied here.
+There are many models that can be considered beyond-
+GR structure formation scenarios. Most of the viable,
+and at the same time cosmologically interesting ones usu-
+ally involve some extra couplings to the metric in the
+Einstein-Hilbert action that manifests themselves as ad-
+ditional degrees of freedom (d.o.f.).
+The propagation
+(gradient) of this d.o.f. induces an additional gravita-
+tional force component, called as the fifth force, which
+acts on top of the Newtonian gravitational force on the
+cosmological scales. However, propagation of a signifi-
+cant fifth force both on small galaxy scales, and in the
+strong field regime is tightly constrained observation-
+ally [50–55]. Thus, only MG models that exhibit some
+kind of a fifth force screening mechanism, which, as the
+name suggests, would screen the fifth force in these ob-
+servationally tested regimes are viable MG candidates
+[41, 46, 56–58].
+The clockwork of MG models and their involved
+screening mechanisms can differ in many ways.
+From
+our point of view, however, we can significantly simplify
+the subject by focusing just on phenomenological effec-
+tive modifications to the density fluctuations PS. As our
+test-case models, we choose variants of two popular MG
+set-ups: namely f(R) [44] and nDGP gravity [59], which
+will serve as a good representative for their whole respec-
+tive families. Further in the text, we offer a more detailed
+description and definitions of these models.
+Most of the works that have considered computing the
+non-linear PS in MG models either rely on simulations
+[31, 32], post-Friedmann (PPF) formalism [33], or per-
+turbation theory focusing on quasi-linear scales [16, 34].
+In Ref.
+[60], the non-linear PS is computed using the
+hmcode [26] for a variety of extensions to the standard
+cosmological model, including f(R) and nDGP. The level
+of this prediction is however significantly limited by a
+number of approximations.
+For example, a simplified
+spherical collapse theoretical formalism is used there to
+estimate DM halo properties. From another perspective,
+mg-halofit was proposed in [61] as an extension of stan-
+dard halofit for f(R) gravity models, but [62] showed
+that the former has limited applicability and accuracy.
+The HM formalism has been used to model non-linear
+MG PS in [34, 35, 63–67], which is mainly based on
+the theoretical spherical collapse model, and is explic-
+itly solved for each MG variant. In our approach, how-
+ever, we rely on the calibration of phenomenological com-
+ponents of HM to N-body simulations.
+An additional
+strength of our approach is that it is general enough to
+be quite straightforwardly extended, not only to a wider
+part of the model parameters space but also, in principle,
+to other modified structure formation models.
+This paper is organized as follows: In Section II, we
+describe the MG models, numerical data sets, and sim-
+ulations.
+In Section III, we elaborate on the HM for-
+malism and describe the empirical halo properties: halo
+mass function (III A), halo bias (III B) and halo density
+profile (III C). In Section IV, we discuss the results ob-
+tained from extending the standard HM predictions to
+the MG models considered in this work (IV A), and from
+our new approach (IV B). In (IV C), we test our approach
+on another suite of MG simulations, and the final Section
+V includes our conclusions, discussion, and future work
+prospects. Details of the Appendices are mentioned in
+the respective sub-sections.
+II.
+MODIFIED GRAVITY MODELS,
+NUMERICAL DATA SETS AND TOOLS
+As our main data for calibration of the non-linear PS
+amplitude, we take the elephant (Extended LEnsing
+PHysics using ANalytic ray Tracing) suite of N-body sim-
+ulations [32]. These simulations provide a good test bed
+to model the impact of f(R) and nDGP physics on for-
+mation of the large-scale structure.
+In f(R), the fifth force is manifested as a result of addi-
+tional degrees of freedom from the interaction between an
+
+3
+auxiliary scalar field (or scalaron) and matter. This addi-
+tional force appears as a non-linear function of the Ricci
+scalar, R in the Einstein-Hilbert action, hence the term
+f(R). We work with the Hu-Sawicki form of f(R) gravity,
+where Chameleon screening screens the fifth force [68]. In
+this screening, the scalaron becomes very massive in the
+high curvature (and high matter density) regimes, and
+the fifth force exponentially decays above the length scale
+determined by the inverse of the mass of the scalaron.
+This length scale is termed the Compton wavelength. As
+a result of this decay, the scalar interaction diminishes
+above the Compton wavelength, and GR is recovered [69].
+In the nDGP model, gravity, unlike other standard
+forces, mediates from 4D brane to 5D Minkowski space-
+time [59, 70]. In this model, the scalar is identified as the
+brane-bending mode which describes the deformation of
+the 4D brane in the 5D bulk spacetime. The brane bend-
+ing mode has a second-order term in the equation of mo-
+tion. On small scales, this term dominates over the linear
+term. As a result, the coupling between the scalar field
+and matter is suppressed, and the solutions for metric
+perturbations approach GR. This is referred to as the
+Vainshtein screening [71].
+In elephant, along with ΛCDM, two f(R) variants
+have been employed, with their free parameter |fR0| (the
+strength of the scalar field today), taken to be 10−6 and
+10−5 (increasing order of deviation from ΛCDM) dubbed
+as F6 and F5, respectively. For nDGP gravity, we have
+two variants with the model parameter rcH0 = 5 and 1
+(which is the dimensionless crossing-over scale character-
+izing transition from 4D to 5D gravity), marked conse-
+quently as N5 and N1, respectively.
+The simulations were run from zini = 49 to zfin = 0 em-
+ploying the ecosmog code [14, 72, 73], each using 10243
+N-body particles in a cubic box of a size 1024 Mpc/h.
+The mass of a single particle is mp = 7.798×1010 M⊙/h,
+and the comoving force resolution is ε = 15 kpc/h.
+Each set of simulations has five independent realizations,
+evolved from the same set of initial conditions. The cos-
+mological parameters of the fiducial background model
+are given as Ωm= 0.281 (fractional matter density), Ωb
+= 0.046 (fractional baryonic density), ΩΛ = 0.719 (frac-
+tional cosmological constant density), Ων = 0 (relativistic
+species density), h = 0.697 (dimensionless Hubble con-
+stant), ns = 0.971 (primordial spectral index), and σ8 =
+0.842 (power spectrum normalization). These parame-
+ters apply to background cosmologies in the simulations
+of all the gravity models. For further processing, we take
+simulation snapshots saved at z = 0, 0.3, 0.5 and 1.
+As indicated above, the elephant-suite will be our
+main calibration data set. To test the accuracy of our
+PS modeling and the general quality of extrapolation, we
+also use different N-body data. For these additional tests,
+we take the MG simulations for F5 and N1, described in
+[74].
+These simulations have background cosmological
+parameters different from our parent elephant simula-
+tions, with Ωm = 0.3111, Ωb = 0.049, ΩΛ = 0.6889, Ων
+= 0, h = 0.6766, ns = 0.9665 and σ8 = 0.8245. This
+simulation set is run using mg-cola [13] in a 500 Mpc/h
+box. For each model, we build an ensemble based on five
+independent realizations.
+Linear matter power spectra, P(k)lin, used in this work
+were calculated using a modified version of the camb cos-
+mological code [75], which includes a module implement-
+ing both the f(R) and nDGP models. The simulation
+power spectra, P(k)sim, were computed using powmes
+[76]. In what follows, by P(k) we will be denoting the
+fully non-linear matter power spectrum, unless indicated
+otherwise.
+III.
+HALO MODEL FORMALISM
+As a baseline prediction and our starting point, we
+take the halo model (HM) approach. It has been pro-
+posed as an attempt to analytically model the variance of
+density fluctuations into the non-linear regime using the
+properties and clustering of halos as main input param-
+eters. HM describes the statistics of the density field up
+to the mildly non-linear regimes (i.e. k <
+∼ 0.5 h/ Mpc).
+Despite its inferior accuracy compared to heavy N-body
+simulations, the HM has been successfully used for mod-
+eling observables and constraining cosmological parame-
+ters [24, 49, 77].
+In HM, the main presumption is that all contributions
+to the cosmic density field variance come from the matter
+collapsed into halos. This allows for moderately accurate
+modeling of the non-linear two-point clustering statistics,
+although HM can be used to compute the density field
+at even higher levels of the n-point hierarchy [78].
+Following HM, the total matter power spectrum
+P(k)HM can be described as a sum of two contributions:
+P(k)HM = P(k)1h + P(k)2h
+(1)
+where P(k)1h models the contribution from the matter
+clustered inside halos (called the one-halo term) and
+P(k)2h is the contribution from clustering of separate ha-
+los (the two-halo term). In practice, the one-halo term
+dominates at small scales (i.e. k >
+∼ 1h/ Mpc) and sat-
+urates to a constant value at larger scales, where the
+two-halo term becomes the dominant component of the
+power spectrum.
+These contributions are further defined as:
+P(k)1h =
+� ∞
+0
+dM|˜u(k|M)|2
+�M
+¯ρ
+�2
+n(M)
+(2)
+and
+P(k)2h = I2
+m(k)P(k)lin
+(3)
+where,
+Im(k) = 1
+¯ρ
+� ∞
+0
+dM|˜u(k|M)|Mn(M)b(M)
+(4)
+and Im → 1 for k → 0 in order to match the linear theory
+predictions at large scales. The integrals in Eq. (2) and
+
+4
+Eq. (4) should in principle cover all possible halo mass
+ranges, but in practice, some Mmin and Mmax are intro-
+duced (these mass limits are discussed in more details in
+the next sub-sections).
+Here, ¯ρ corresponds to the mean density of the uni-
+verse, P(k)lin is the linear theory matter power spectrum,
+n(M) is the halo mass function, and b(M) is the linear
+halo bias. The term |˜u(k|M)| is the normalized Fourier
+transform of the internal density profile of a halo of mass
+M, such that ˜u(k → 0, M) → 1. The above HM build-
+ing blocks are intrinsically redshift-dependent functions,
+which, in principle, allows one to obtain HM prediction
+at any redshift for which the integrands are well-defined.
+All the components of the HM can be varied inde-
+pendently from each other, and each specific choice of
+fitting functions, formulae, or tabulated data creates a
+unique realization.
+Thus, HM is a general framework
+under which one can create many different families of
+PS models. Motivated by literature and our own studies
+for each of our cosmological models (i.e. ΛCDM, and all
+MG variants), we find an optimal combination of analytic
+formulae and fitting functions to describe the input prop-
+erties of halo mass function, halo bias, and halo concen-
+trations. Below we provide a more detailed description
+of the particular choices we make. For a quick summary
+and look-up, we refer the reader to Table I which contains
+a concise list and references of all the fitting functions for
+the halo properties used in this work, and for each model.
+A.
+Halo mass function
+The halo mass function (HMF), n(M), quantifies the
+number of halos per unit mass per unit comoving volume.
+The most commonly adopted theoretical formulation of
+the HMF is via the Extended Press–Schechter (EPS) for-
+malism [87, 88], in which HMF is given by:
+n(M) ≡ dn
+dM =
+ρ
+M 2 F(σ)
+����
+d ln σ
+d ln M
+����
+(5)
+The halo multiplicity function, F(σ) = νF(ν) denotes
+the fraction of matter collapsed into halos, in a loga-
+rithmic bin around the peak height, ν = δc(z)/σ(M, z).
+Here, δc(z) is the spherical collapse density threshold,
+and σ(M, z) is the linear variance in the density fluctua-
+tion field smoothed using a top-hat filter. This scaling re-
+lation has been modeled extensively in the literature and
+it has been shown to be approximately universal across
+redshifts for ΛCDM [79, 82, 89–91]. In our earlier work
+[80], we have shown that after simple re-scaling, the F(σ)
+in both f(R) and nDGP also exhibits a similar degree of
+universality as in the ΛCDM-case.
+Following our previous study, we will model MG HMF
+as a fractional deviation, ∆MG from the ΛCDM fiducial
+baseline. We have shown that such an approach allows
+for achieving quite a good accuracy (5 − 10%), which
+also holds for different background cosmologies.
+How-
+ever, to obtain such precision, a careful choice of the
+baseline ΛCDM HMF model is paramount.
+Thus, for our baseline ΛCDM, we tested various HMF
+models in the literature (e.g. [79, 82, 89–93]), as these
+functions can in principle be extrapolated to desired halo
+mass ranges. We found that the fitting function proposed
+in Watson et al. 2013 [79, hereafter W13] proved to be
+optimal for HM power-spectrum forecasting. Therefore,
+we used W13 for our ΛCDM HMF computations.
+For completeness, we now recall the essential steps of
+Ref. [80]. Here the target MG HMF is modeled as:
+n(σM)MG = ∆MG(σM) · n(σM)ΛCDM ,
+(6)
+where σM ≡ σ(M) is simply the linear mass variance at
+the Lagrangian top-hat halo mass scale, M.
+For f(R) gravity models, the fractional deviation fit is
+expressed as:
+∆MG ≡ ∆f(R) = 1 + a exp
+�
+−(X − b)2
+c2
+�
+,
+(7)
+X ≡ ln(σ−1). Here, (a, b, c) are parameters of the fit that
+were calibrated using simulations. They depend on the
+variant of f(R) gravity model under consideration. See
+Table I for the specific values that we use in this work.
+For nDGP gravity models:
+∆MG ≡ ∆nDGP = p + q arctan (s X + r).
+(8)
+Here, X is the re-scaled mass density variance, X ≡
+ln(�σ−1), �σ = σ/Ξ(z). Again, (p, q, r, s) are the parame-
+ters of the fit, whose values are determined by the variant
+of the nDGP gravity model.
+The resolution of our simulations allowed us to probe
+only intermediate- and large-mass halos to compute the
+HMF. In this mass regime, HMF in MG increases w.r.t.
+ΛCDM, as small-mass halos accrete matter and merge
+faster to form larger structures. However, this enhanced
+structure formation at large halo mass-end is happening
+at the expense of the abundance of smaller halos used
+up in this process [see e.g. 80, 94–96].
+Thus, we can
+expect that there should be a simultaneous decrease in
+the number of small-mass halos in the MG models when
+compared to ΛCDM.
+Equation (8) for nDGP allows the possibility of
+∆nDGP < 1 for small mass halos. However, our fit for
+∆f(R) is never below 1. To admit for low-mass halo defi-
+ciency also in the f(R), we impose an artificial decrease
+in f(R) HMF for M < 1011 M⊙/h, when compared to
+ΛCDM results.
+For low halo masses, we assume that
+∆f(R) is a linear function of ln(σ−1), and is given by:
+∆f(R) → (m ln(σ−1) + n) × ∆f(R)
+(9)
+We tested for different combinations of the (m, n) pa-
+rameters values. The combination (m, n) = (0.06, 0.99)
+turned-out to be optimal for our both f(R) variants.
+Thus we use these values in this work. A note of caution
+
+5
+TABLE I. Compilation of the fitting functions used in this work for the halo properties in HM build-up, for both ΛCDM and
+MG models.
+Halo properties
+Fitting functions
+Notes
+Halo mass function, HMF
+ΛCDM : Watson et al. 2013 [79]
+f(σ)ΛCDM = A
+��
+β
+σ
+�α + 1
+�
+e−γ/σ2
+For MG(=f(R), nDGP),
+A = 0.282, α = 2.163, β = 1.406 and γ = 1.210.
+f(σ)MG = ∆MG × f(σ)ΛCDM
+f(R): Gupta et al. 2022 [80]
+∆f(R) = 1 + a exp
+�
+− (X−b)2
+c2
+�
+Additional cut-off expression at
+For F5: a = 0.230, b = 0.100 and c = 0.360
+low-mass scales for f(R) (Eq. (9))
+For F6: a = 0.152, b = -0.583 and c = 0.375
+X ≡ ln(σ−1)
+nDGP: Gupta et al. 2022 [80]
+∆nDGP = p + q arctan (s X + r)
+Ξ(z): nDGP force enhancement
+For N1: p = 1.35, q = 0.258, r = 5.12, s = 4.05
+w.r.t. GR [81].
+For F6: p = 1.06, q = 0.0470, r = 11.8, s = 4.19
+X ≡ ln(�σ−1), �σ = σ/Ξ(z)
+Linear halo bias, b(M)
+All models: Sheth et al. 2001 [82]
+b(M) =
+1
+√aδc(z) (√a(aν2) + √ab(aν2)1−c
+This expression has been proposed
+−
+(aν2)c
+(aν2)c+b(1−c)(1−c/2) )
+for ΛCDM. We extrapolated
+a = 0.707, b = 0.5 and c = 0.6.
+the relation for MG.
+Concentration-mass relation, c(M)
+ΛCDM: Ludlow et al. 2016 [83]
+c(ν)ΛCDM = c0
+�
+ν
+ν0
+�−γ1
+�
+1 +
+�
+ν
+ν0
+�1/β�−β(γ2−γ1)
+For MG(=f(R), nDGP),
+c0 = 3.395 × (1 + z)−0.215
+c(M)MG = ∆c(M),MG × c(M)ΛCDM
+β = 0.307 × (1 + z)0.540
+γ1 = 0.628 × (1 + z)−0.047
+γ2 = 0.317 × (1 + z)−0.893
+ν0 = (4.135 − 0.564a−1 − 0.210a−2
++0.0557a−3 − 0.00348a−4) × D(z)−1
+f(R): Mitchell et al. 2019 [84]
+y(x) = 1
+2
+�
+λ
+ωs φ(x′)
+�
+1 + erf
+�
+αx′
+√
+2
+��
++ γ
+�
+(1 − tanh(ωt[x + ξt]))
+y = log10(∆c(M),f(R))
+x′ = (x − ξs)/ωs
+x = log10(M500/10p2)
+p2 = 1.5 log10
+�
+¯
+fR(z)
+1+z
+�
++ 21.64 [85]
+λ = 0.55 ± 0.18
+ξs = −0.27 ± 0.09
+For M ≤ 1012M⊙/h,
+ωs = 1.7 ± 0.4
+c(M)f(R),nDGP = c(M)ΛCDM
+α = −6.5 ± 2.4
+γ = −0.07 ± 0.04
+ωt = 1.3 ± 1.0
+ξt = 0.1 ± 0.3
+nDGP: Mitchell et al. 2021 [86]
+∆c(M),nDGP = [A − Blog10(M200M⊙h−1)](H0rc)−0.71±0.05 + 1
+A = (0.35 ± 0.01)(H0rc)−0.71±0.05
+B = (0.0302 ± 0.0008)(H0rc)−0.71±0.05
+is in place here. There is no clear physical justification
+for our particular choice of both m and n, other than that
+they are providing optimal HM power spectrum predic-
+tions. An interested reader can play around and search
+for a different choice of (m, n). However, the overall im-
+pact of the particular (m, n) choice on the resulting HM
+remains small.
+B.
+Halo bias
+The relation between the clustering amplitude of the
+underlying DM density field and halos is quantified in
+terms of the linear halo bias relation, δh(M) = b(M)δ. In
+the context of power spectra, it is convenient to consider
+the following Fourier-space estimator of the halo bias:
+ˆb(k, M) = Phm(k, M)
+P(k)
+(10)
+Here, Phm(k, M) is the halo-matter cross power spec-
+trum, and P(k) is the matter power spectrum. One can
+find an optimal value of the linear bias by taking a limit,
+or an average of this estimator at the smallest possible
+k’s. We consider such a power-spectrum-based bias es-
+timator to use results from elephant suite for testing
+and finding optimal analytic bias formula for HM.
+For this purpose, we tested various b(M) fitting func-
+tions for ΛCDM [97–99]. Sheth et al. 2001 b(M) [82,
+hereafter S01] gave the best match to the simulations.
+Thus, this will be our choice for the b(M) computations
+in this work.
+In the Appendix A, we show the performance of S01,
+both in capturing the ratio of MG b(M) versus ΛCDM
+(Fig. 5), and the absolute b(M) relation (Fig. 6) . We
+find that S01 gives reasonable predictions in both cases.
+Given that b(M) impacts only the two-halo term, which
+by construction matches the P(k)lin on large scales, the
+choice of b(M) does not impact the HM results to a great
+extent.
+
+6
+C.
+Halo density profile: concentration-mass
+relation
+The scale-free nature of structure formation in CDM
+scenarios results in self-similar density profiles for indi-
+vidual DM halos, which was first pointed out by Navarro,
+Frenk, and White in [100, hereafter NFW]. As a result,
+DM density profiles are re-scaled by a characteristic cen-
+tral density, ρs, and radial scale, rs, (or mass M and
+concentration c(M), respectively). The c(M) relation is
+defined as the ratio of the virial radius, Rv of the halo to
+rs, and determines the density profile of NFW halos.
+To obtain relatively unbiased and good-quality NFW
+fits, the simulated halos need to be well-resolved. The
+convergence of the halo density profile depends on the
+simulation’s force and mass resolution. Thus c(M) can
+be reliably estimated only for a limited halo mass range,
+usually for halo with masses corresponding to at least
+a few × 103 particles [see e.g. 101]. The resolution of
+the elephant suite allow only for probing the c(M ≥
+1013M⊙/h). Because of this, we need to resort to the
+fitting functions for c(M) here.
+We use relations proposed in [84] and [86] to com-
+pute the c(M) relation in f(R) and nDGP gravity
+models, respectively.
+In these works, direct NFW
+fitting was used to compute the halo density pro-
+files, and functional forms were derived for the ratio
+c(M)MG(=f(R),nDGP)/c(M)ΛCDM (refer to Table I for ex-
+plicit expressions). The MG c(M) can be therefore ob-
+tained as a product of this ratio times the concentration-
+mass relation for ΛCDM, for which we use the form pro-
+posed in [83]. Considering the ratio instead of absolute
+MG c(M) would eliminate the leading-order systematic
+uncertainties coming from the background cosmology.
+The authors in [84] proposed functional form for
+log(c(M)f(R)/c(M)ΛCDM).
+When expressed as a func-
+tion of M500/10p2, this ratio is independent of the back-
+ground scalar field and z. The parameter p2 defined in
+[85], encapsulates these dependencies, and in turn allows
+different variants of f(R) gravity model to be studied in
+a unified way.
+For the case of the nDGP gravity model, in ref. [86],
+the ratio c(M)nDGP/c(M)ΛCDM is fitted as a decreasing
+function of M200.
+This fitting also captures the z de-
+pendence, hence making the ratio only dependent on the
+nDGP parameter, rcH0.
+The halo mass range probed in both Mitchell et al. [84,
+86] is confined to ≥ 1012 M⊙/h. Therefore, we restrict the
+use of their fitting functions to the calibrated mass range,
+and artificially impose c(M)f(R), nDGP = c(M)ΛCDM for
+M < 1012 M⊙/h.
+IV.
+RESULTS
+In this section, we combine all the HM components to
+give an analytical prediction for matter overdensity PS.
+As a reference case to gauge our results against, we al-
+ways take the PS from elephant simulations. At large
+scales, the linear perturbation theory gives accurate and
+reliable predictions both for ΛCDM and MG PS. Hence,
+we focus here only on the scales corresponding to mildly
+and fully non-linear regimes. In practice, we will be in-
+terested in the performance of our models for k ≥ 0.1
+h/ Mpc.
+A.
+Halo model predictions for modified gravity
+We start by testing the standard set-up for HM, which
+aims to yield a theoretical prediction for the PS am-
+plitude in a given cosmology.
+For ΛCDM alone, this
+approach has at best limited accuracy, since the clas-
+sical HM fails to accurately capture PS already in the
+mildly non-linear regime, i.e.
+k
+>
+∼ 0.2 − 0.3 h/ Mpc
+[25, 26, 102, 103]. Thus, we do not expect that it will per-
+form better in MG cosmologies, which have even richer
+phenomenology. However, it is still an illustrative exer-
+cise, since we will be using this basic HM set-up to obtain
+much more accurate PS predictions for MG.
+Using the inputs of the HMF, b(M), and c(M) relation
+in their MG versions discussed in the previous section, we
+compute the resultant power spectra for a number of red-
+shifts. For this, we employ Eqs. (2)-(4), integrating from
+Mmin = 1 M⊙/h to Mmax = 1016 M⊙/h. We choose a
+sufficiently broad halo mass range so as to account for the
+maximum possible halo masses that still have an impact
+on the resulting PS.
+For the integral in Eq. (4) to approach unity at large
+scales, the bias needs to attain unity when integrated
+over all the halo masses, i.e.
+1
+¯ρ
+� ∞
+0
+b(M)n(M)dM = 1
+(11)
+In practice, this integral yields a value below unity, even
+when the integration is taken over the maximum possi-
+ble halo mass range. Changing the high mass limit for
+the integration does not impact the results to a great ex-
+tent, because on these scales, halos become exponentially
+rare which makes their contribution to the total power
+negligible. On the other hand, we expect a significant
+contribution from the low-mass regime.
+However, ow-
+ing to resolution limits, the properties of low mass halos
+cannot be properly calibrated using simulations.
+Therefore, to add the contribution of the low-mass ha-
+los to HM computations, we use the correction proposed
+in [104, 105]. This correction adds the contribution of the
+missing halos to the two-halo term, in order to recover
+P(k)lin at large scales.
+The correction term is simply
+yielded by:
+A = 1 − 1
+¯ρ
+� Mmax
+Mmin
+b(M)n(M)dM,
+(12)
+and it is used as an additive component in the two-halo
+
+7
+term:
+C = A˜u(k|Mmin)
+Mmin
+(13)
+Here, ˜u(k|Mmin) is the normalized Fourier transform of
+the density profile for the lowest resolved mass Mmin.
+Equation (3) is then modified and the resultant two-halo
+term is given by:
+P(k)2h = P(k)lin(Im + C)2
+(14)
+One could instead replace the P(k)2h term with
+P(k)lin, as the former differs from the latter only for
+k ≥ 1 h/ Mpc, where already P(k)1h takes over as the
+dominant contributor.
+However, for completeness, we
+use the full above expression for the two-halo term.
+The results of such direct HM computations for our
+MG models are illustrated in Fig. 1, where we compare
+P(k)HM (solid lines), as well as linear theory P(k)lin (dot-
+ted lines), with the elephant simulations for all our
+models at z = 0. The shaded region corresponds to the
+uncertainty in the elephant results, which is the inverse
+of the square root of the number of statistically indepen-
+dent modes contributing to each k-bin, and the horizontal
+dashed lines correspond to the 5% accuracy regime. The
+performance of HM in these MG models is similar to the
+ΛCDM results and is not much better than the actual
+linear theory. With respect to the simulation prediction,
+P(k)HM gives better than 5% accuracy for k ≤ 0.2 − 0.3
+h/ Mpc, and stays within 10% for k ≤ 0.4 − 0.5 h/ Mpc.
+An interesting exception is the F5 f(R) variant, where
+better than 2% accuracy is kept all the way to k ∼ 0.2
+h/ Mpc.
+In all the models,
+we also encounter an under-
+prediction w.r.t.
+the simulation results for k ≈ 0.5
+h/ Mpc. This is a well-known problem of the HM for-
+malism in ΛCDM [26, 103], and further propagates to
+the MG scenarios (also seen for Galileon models in [65]).
+Similar behavior is observed also for other redshifts that
+our simulations probe, but we do not show them here for
+brevity.
+These results clearly show that the standard HM can-
+not be used in MG, similarly as in ΛCDM, if one is in-
+terested in attaining the accuracy that we expect from
+future LSS surveys.
+However, noticing the above, the
+positive result here is that HM can be actually employed
+to yield predictions for MG power spectra with the same-
+level accuracy as for ΛCDM. This is a somewhat sur-
+prising result because the standard HM does not include
+any room for extra MG physics (like the fifth-force and
+screening).
+Yet it seems that self-consistent modifica-
+tions of HMF, b(M), and c(M) are enough to obtain the
+usual ΛCDM HM-level predictions also for different MG
+cosmologies. This is very encouraging, and as we show
+below this can be used as a strong advantage to build an
+even better and more accurate PS model for MG.
+FIG. 1. Comparison of the power spectrum from halo-model,
+P(k)HM (solid lines) and linear theory, P(k)lin (dotted lines)
+with elephant simulations P(k)sim, for ΛCDM and the vari-
+ants of MG models, as in the legend, for z = 0. The horizontal
+grey dashed lines correspond to 5% accuracy regime.
+B.
+An improved model for MG power spectrum
+In the previous section, we have shown that when HM
+is applied to model the PS amplitude, it offers limited
+accuracy, and is comparable to what can be achieved for
+the standard ΛCDM. In this section, we will demonstrate
+that we can build a much more accurate PS model for
+MG. This can be realized when we apply HM to estimate
+the fractional departure from the ΛCDM baseline, rather
+than trying to predict the absolute amplitude of PS alone.
+Our starting point will be the ratio of the power spec-
+tra:
+Υ(k) ≡ P(k)MG/P(k)ΛCDM .
+(15)
+By modeling this ratio, rather than the MG PS itself, we
+can benefit from a number of properties, namely: (i) the
+dependence on the background cosmological parameters
+(such as Ωm, H0, or σ8) should cancel out from the ra-
+tio to the leading order; and (ii) the scale of significant
+departure from ΛCDM (i.e. from Υ = 1) is naturally
+determined in terms of the ΛCDM baseline, rather than
+some arbitrary non-linear amplitude or scale.
+In Fig. 2, we compare the ratios Υ(k) estimated from
+elephant simulations (solid line), linear theory (dotted
+line), and HM (dashed line), for both f(R) (left pan-
+els), and nDGP (right panels), at z = 0 (top panel)
+and z = 0.5 (bottom panel). Naturally, both the sim-
+ulation and HM results for Υ(k) are expected to match
+the linear theory prediction at large scales for both
+models.
+As we approach smaller scales, the departure
+from linear predictions increases (namely, linear theory
+runs away for f(R) and stays constant for nDGP), and
+trends peculiar to each model emerge.
+This is a well-
+known result, which highlights the fact that these fam-
+
+0.15
+Halo Model .... Linear Theory
+0.10
+0.05
+P(k)/P(k)sIM-1
+0.00
+-0.05
+ΛCDM
+F5
+F6
+-0.10
+N1
+N5
+-0.15
+10-2
+10-1
+100
+k (h/Mpc)8
+FIG. 2. The ratio Υ(k) ≡ P(k)MG/P(k)ΛCDM obtained from linear theory (dotted lines), elephant simulation (solid lines)
+and the halo model (dashed lines), at z = 0 (top panels) and z = 0.5 (bottom panels). The left panels correspond to f(R)
+gravity variants: F5 and F6, and the right panels correspond to nDGP gravity variants: N1 and N5. Shaded regions are the
+propagated errors from P(k)sim.
+ily of MG models usually exhibits an increased degree
+of non-linearity of the density field, owing to both the
+fifth force and their respective screening mechanisms
+[16, 31, 38, 44, 56, 106, 107]. For f(R) gravity models,
+PS approaches ΛCDM on the large-scales, and we see a
+monotonic increase in the ratio with k (although slower
+than what the linear theory would predict). Whereas,
+for nDGP, Υ(k) enhancement is maximum at the inter-
+mediate scales, and this enhancement decreases for large
+k.
+A crucial observation from our study is that HM pre-
+diction agrees qualitatively with the simulations.
+We
+note that the agreement is far from perfect, especially
+around the peak-like features, but the HM captures the
+essential shape and scales of the PS ratios.
+As mentioned in the previous section, one perennial
+problem with the HM has been the ‘transition’ region,
+where both two- and one-halo terms have a similar mag-
+nitude, and both contribute equivalently to the predicted
+signal. In general, the HM under-predicts the strength of
+clustering in this region, with the exact amount depend-
+ing on redshift and cosmology [26]. We also highlight a
+similar problem with the HM-based MG predictions in
+Fig. 1 at k ≈ 0.5 h/ Mpc. These scales are also called the
+‘quasi-linear’ regime because the evolution of perturba-
+tions at these scales is not exactly governed by linear
+perturbation theory.
+For standard ΛCDM, the inaccuracies of the HM in
+this transition regime are addressed by devising empiri-
+cal fitting functions. One of the earliest, yet successful
+examples was halofit [25], which is motivated by the
+principles of HM, and calibrated using N-body simula-
+tions. It was later improved, in particular by [28] who
+updated its fitting functions from higher resolution sim-
+ulations and ameliorated the modeling for dark energy
+cosmologies. Methods and prescriptions to predict the
+non-linear PS in ΛCDM are numerous, but in this work
+we will use halofit as it is sufficiently accurate for our
+purposes.
+Having seen that the ratio Υ(k) between HM-derived
+PS for MG and ΛCDM correctly captures the simulation
+trends, we propose to use it to obtain the fully non-linear
+PS in MG. This is obtained by multiplying ΥHM with an
+accurate model for the ΛCDM baseline P(k). Therefore,
+we characterize the beyond-ΛCDM PS (P(k)MG) as:
+P(k)MG = Υ(k)HM × P(k)ΛCDM
+(16)
+In this prescription, P(k)ΛCDM is the non-linear ΛCDM
+PS, for which we take the halofit predictions using the
+parameters of a given background cosmology.
+The results of applying our proposed methodology are
+
+[F(R)
+nDGP
+ Simulation Y(k)sim
+Halo Model Y(k)HM
+F5
+F6
+N1
+N5
+... Linear Y(k)Linear
+z=0
+z=O
+1.25
+1.25
+1.20]
+1.20
+P(k)MG/P(k)ACDM
+1.15
+1.15
+1.10]
+[1.10
+1.05
+1.05
+1.00
+1.00
+1.40z=0.5
+z=0.5
+1.40
+II
+1.32
+1.32
+(X)人
+1.24
+1.24
+1.16
+1.16
+1.08
+1.08
+1.00
+1.00
+10-1
+100
+10-2
+100
+10-2
+10-1
+k (h/Mpc)9
+FIG. 3. Top panel: Matter power spectra obtained from our new approach (P(k)MG from Eq. (16)) for all the MG variants
+considered in this work, at redshifts as indicated in the legends. Bottom panel: Comparison of P(k)MG, derived with our new
+method, with P(k)sim. The shaded region in all the plots corresponds to the uncertainty in the P(k)sim, and the horizontal
+dotted lines shows 5% accuracy regime.
+illustrated in Fig. 3, where we plot P(k)MG obtained
+using HM for Υ(k) multiplied by the halofit ΛCDM-
+baseline. The top panels present the power spectra di-
+rectly: P(k)sim from elephant (dots), and P(k)MG de-
+rived with Eq. (16) (solid lines). In the bottom panels, we
+show departures of thus-obtained P(k)MG from P(k)sim
+treated as reference. These new results, when compared
+with the standard HM predictions from Fig. 1, clearly
+perform much better. The standard HM reaches 5% ac-
+curacy only up to k ≤ 0.2 − 0.3 h/ Mpc. Now, by us-
+ing HM only for predicting Υ(k), and combining it with
+halofit ΛCDM-baseline, we improve the scale at which
+modeling is accurate within 5% by an order of magni-
+tude, reaching up to k ≤ 0.5 − 2.5 h/ Mpc (depending
+on the model and redshift).
+We note that the perfor-
+mance of P(k)MG generally worsens for higher redshifts,
+but still remains significantly improved when compared
+to the standard HM.
+More generally, the accuracy of P(k)MG will depend on
+the user input of baseline P(k)ΛCDM. As already men-
+tioned, other approaches are being developed to further
+improve the limited accuracy of halofit, especially for
+models departing from the flat Planck-based ΛCDM. We
+tested one such alternative way of deriving the non-linear
+ΛCDM PS, that goes into our P(k)MG prediction (16):
+the so-called hmcode [26, 108]. The results, detailed in
+Appendix B, indicate that both halofit and hmcode
+give similar accuracy, however with different trends at
+different scales and redshifts.
+Given the fact that we have calibrated our MG HM
+with a limited-resolution elephant simulation suite, it
+is encouraging that this allowed for already an order-of-
+magnitude improvement of the scale at which we can ob-
+tain accurate PS predictions. Obtaining accurate MG PS
+into the fully non-linear regime at k ≥ 1 h/ Mpc with so
+straightforward modifications to HM opens up an avenue
+for even better PS predictions for the MG phenomenol-
+ogy. Even better PS accuracy could be achieved using
+possible improvements to HM, better informed about the
+clustering and properties of small halo mass regime in
+
+F5
+104
+P(k) (Mpc/h)3
+103
+O=Z
+102
+z=0.3
+z=0.5
+101
+z=1
+0.15
+0.10
+P(k)/P(k)sim-1
+0.05
+0.00
+-0.05
+-0.10
+-0.15
+10-2
+10-1
+100
+k (h/Mpc)F6
+104
+P(k) (Mpc/h)3
+103
+O=Z
+102
+z=0.3
+z=0.5
+101
+z=1
+0.15
+0.10
+P(k)/P(k)sim-1
+0.05
+0.00
+-0.05
+-0.10
+-0.15
+10-2
+10-1
+100
+k (h/Mpc)N1
+104
+P(k) (Mpc/h)3
+103
+102
+Z=0
+z=0.3
+101
+z=0.5
+0.15
+0.10
+P(k)/P(k)sim-1
+0.05
+0.00
+-0.05
+-0.10
+-0.15
+10-2
+10-1
+100
+k (h/Mpc)N5
+104
+P(k) (Mpc/h)3
+103
+102
+Z=0
+z=0.3
+101
+z=0.5
+0.15
+0.10
+P(k)/P(k)sim-1
+0.05
+0.00
+-0.05
+-0.10
+-0.15
+10-2
+10-1
+100
+k (h/Mpc)10
+FIG. 4. Comparison of our P(k)MG modeling, with simulation results from mg-cola [13, 74], for two MG variants: N1 (left)
+and F5 (right). The redshifts are as indicated in the legends. Dotted grey lines are the 5% accuracy regime. The shaded region
+is the simulation error, which is the standard deviation obtained from five realizations at each redshift.
+MG.
+C.
+Testing non-linear P(k)MG beyond ELEPHANT
+In this subsection, we extend our work beyond the
+elephant simulations to test the performance of our
+new approach. For this purpose, we consider different
+N-body simulation runs for the F5 and N1 models, per-
+formed using mg-cola [13], and described in [74]. The
+most important factors for us are that these runs have
+a different background cosmology than elephant (see
+Section II), and were run using different codes. Unlike
+standard N-body approach, these simulations employ the
+COLA method [109], that can straightforwardly trade
+accuracy at small-scales in order to gain computational
+speed without sacrificing accuracy at large scales. On one
+hand, this approach is much faster than the standard N-
+body, but the price to pay is the approximations made,
+which do not allow us to use these suite of simulations
+as the calibration data. Hence, we use these simulations
+but only as a test-bed.
+Here, the HM ingredients were calculated using the
+same methodology and setup as above, described in Sec-
+tion III and summarized in Table I. The main difference
+with respect to Sec. IV B was that different background
+cosmological parameters were used in the linear power
+spectra that go into the particular ingredients of the HM
+build-up, namely HMF, b(M) and c(M). Everything else,
+including the halo mass integration ranges for the HM
+components, were the same as before.
+Using the HM outputs and ΛCDM halofit predic-
+tions for the background cosmology of this alternative
+simulation suite, we computed P(k)MG. A comparison
+of our results with the simulation predictions is in Fig. 4,
+for both N1 (left plot) and F5 (right plot). Given the
+small box size of these simulations (L = 500 Mpc/h), we
+obtain a discrepancy > 5% with the simulation predic-
+tions on large scales, for k < 0.1 − 0.2 h/ Mpc. Now,
+contrary to the elephant results, for N1 our new PS
+model performs better than before. However, for F5 the
+performance of our approach decreases with increasing
+redshift.
+Overall, we see a similar performance of the
+new P(k)MG in both the simulations that we tested, with
+5% accuracy from mildly non-linear to non-linear scales
+(k ≤ 0.5 − 2.5 h/ Mpc).
+This test with a different simulation and cosmology re-
+assures us that our new approach is a valid technique to
+compute the non-linear PS in these MG scenarios, and
+can be successfully extended to simulations and cosmolo-
+gies beyond our original data used for calibration and
+fitting.
+V.
+DISCUSSION AND CONCLUSION
+In this work, we combined Halo Model (HM) pre-
+dictions with an accurate ΛCDM baseline for building
+an analytical framework to compute the non-linear power
+spectrum (PS) in modified gravity (MG) scenarios, where
+structure formation differs from that in ΛCDM.
+For
+calibration and testing, we used the elephant suite –
+a set of N-body simulations, which incorporates stan-
+dard ΛCDM and two MG models:
+Hu-Sawicki f(R)
+and the normal branch of the Dvali-Gabadadze-Porrati
+braneworld (nDGP). HM has been extensively studied
+for ΛCDM [49, 77], and we further extended it to these
+MG cosmologies. This formalism is advantageous as a
+quick and reliable tool to obtain predictions for statistics
+of density fields well into the regimes, where linear and
+perturbation theory fails to reproduce simulation results.
+The HM framework requires the input of three main
+halo properties: halo mass function (HMF), which quan-
+tifies the number density of halos; linear halo bias b(M),
+describing the relation between halos and the underlying
+DM density field; and the concentration-mass relation
+c(M), which describes the internal distribution of mass
+in halos. For the HM framework, we needed to compute
+these quantities over large range of halo masses, that go
+much beyond the range of our simulations. As a result,
+we relied on fitting functions for the halo properties in
+these MG scenarios (Table I).
+Using these three inputs, we obtained the HM-based
+predictions, P(k)HM for these two MG models.
+We
+
+0.15
+N1
+0.10
+P(k)/P(k)sim-1
+0.05
+0.00
+-0.05
+Z=O
+z=0.3
+-0.10
+Z=0.5
+Z=1
+-0.15
+10-1
+100
+k (h/Mpc)0.15
+F5
+0.10
+P(k)/P(k)sim-1
+0.05
+0.00
+-0.05
+O=Z
+z=0.3
+-0.10
+Z=0.5
+z=1
+-0.15
+10-1
+100
+k (h/Mpc)11
+showed that P(k)HM is within 5-15% of the simulation re-
+sults across the k-ranges, from k = 0.01 to k = 1 h/ Mpc.
+However, MG signatures from these models that quantify
+deviations from GR are typically in itself a factor of a few
+dozen per cent. Hence, we cannot use HM predictions in
+its standard form to complement the expected accuracy
+from future LSS surveys in order to detect these MG sig-
+nals.
+Additionally, similar to the case of ΛCDM, HM
+also faces the consistent problem of under-prediction of
+power in the transition regime for both f(R) and nDGP.
+These scales correspond to k ≈ 0.5 h/ Mpc.
+To get a better PS model, we further investigated, us-
+ing HM, the relative ratio Υ(k) = P(k)MG/P(k)ΛCDM,
+instead of employing the absolute PS amplitudes alone.
+From this, we obtained new analytical PS by taking a
+product of Υ(k)HM, with the non-linear prediction for
+ΛCDM, P(k)ΛCDM (Eq. (16)). For the latter, we used
+halofit [25, 28], as it has been a successful approach
+for ΛCDM to circumvent the HM under-prediction of PS
+in the intermediate scales, and is widely used to ana-
+lytically compute non-linear ΛCDM PS. One could use
+other approaches for the input non-linear P(k)ΛCDM to
+multiply our ratio ΥHM(k) with, and we tested one of
+them: hmcode [26, 108]. It gave comparable accuracy
+as halofit for our cosmology.
+Using this approach, we significantly improved the ac-
+curacy of PS modeling compared to standard HM predic-
+tion. For HM, we obtained results within 5% accuracy
+with the simulation predictions for k ≤ 0.2−0.3 h/ Mpc.
+With our new approach, we now matched simulations
+within this accuracy for k between 0.5−2.5 h/ Mpc, with
+the performance of the method depending on the MG
+model and redshift.
+The k-range probed in this work
+corresponds to the mildly non-linear and the fully non-
+linear regime: a range of scales crucial to constrain mod-
+ern era cosmological observables. The sensitivity of these
+observables to changes in the matter PS will be very im-
+portant for making powerful observational cosmological
+tests of the theory of gravity, or dark energy.
+The main advantage of our approach over using simu-
+lations is that it is computationally inexpensive. The two
+main inputs, HM and halofit (or e.g. hmcode) can be
+flexibly applied to a different background cosmologies,
+whereas, in simulations, we need to perform a new run
+for each new set of parameters. HM also gives the flexi-
+bility of employing different combinations of HMF, b(M)
+and c(M) that is best suited to probe a particular cos-
+mology, scale, halo mass range, or redshift.
+To test the limits and accuracy of our approach, we
+applied Eq. (16) to another suite of MG N-body simula-
+tions, run with mg-cola [13, 74]. Using the same fitting
+functions as described for the elephant simulations, we
+computed halo properties for the mg-cola cosmology,
+and in turn P(k)HM, Υ(k)HM and P(k)MG.
+We com-
+pared P(k)MG with the P(k)sim results, and obtained
+similar accuracy as with the original data of elephant.
+For N1, P(k)MG gives consistency with simulation results
+within 5% for k between 0.5 − 2.5 h/ Mpc. However, for
+the case of F5, overall performance of our approach de-
+creases with z.
+We need to appreciate that another promising solution
+for analytical modeling of the MG PS is via the fast and
+reliable emulation techniques [e.g 30, 110, 111].For MG
+models, emulators have been proposed in e.g. [112–116].
+This approach is sophisticated and promising, however,
+is still in its infancy, and has limitations. For instance,
+predictions from emulators are confined to the parameter
+space defined in the starting base grid of the calibrating
+simulations.
+In addition, new extensions in emulators
+(e.g.
+new degrees of freedom, or additional screening
+mechanisms in MG models) often requires one to sub-
+stantially adapt the base grid of simulations used to build
+the emulator, which can in itself be computationally ex-
+pensive.
+On the other hand, HM potentially provides a sim-
+ple, physically-motivated semi-analytical picture of the
+clustering of matter. We showed that HM, in its stan-
+dard form, can be qualitatively used to predict estimates
+for MG signatures in cosmological observables which re-
+late to matter perturbations.
+Furthermore, using HM
+for modeling the PS ratio Υ(k), and combining it with
+a high-quality baseline ΛCDM predictions yields signif-
+icantly better results. This method is advantageous as
+contrary to MG scenarios, we have much tighter con-
+straints on ΛCDM physics, and the field of modeling
+ΛCDM PS is much more sophisticated and advanced [25–
+28]. As a result, more precise ΛCDM results will provide
+MG PS with similarly improved performance. Here we
+present our results by incorporating the halofit and
+hmcode predictions for ΛCDM. These results in them-
+selves give a percent level of accuracy in both quasi-linear
+and non-linear regimes.
+In order to further improve HM modeling in the MG
+variants studied here, we need to probe deeper into the
+non-linear scales. For this, the behavior of halo density
+profiles and HMF in both f(R) and nDGP at low halo
+masses requires deeper investigation, as the full effect of
+the respective screening mechanisms comes to play in the
+non-linear regime of gravitational collapse. As mentioned
+above, the accuracy of c(M) fitting functions for both
+f(R) and nDGP has not been tested for Mhalo < 1012
+M⊙/h.
+Additionally, we also extrapolated our earlier
+HMF fits for these MG models [80] to small halo mass
+scales, which are not resolved by our N-body simulations
+(the limit being Mhalo ≲ 8×1012 M⊙/h). Both c(M) and
+HMF are important ingredients in modeling the one-halo
+term, which is the dominant non-linear contributor in the
+HM approach. Such a study will require a completely
+new set of high-resolution MG N-body simulations, and
+we plan it as a future project.
+We also note that in this work, we focus on modeling
+only the dark matter PS. At our scales of interest (k be-
+tween 0.1−2.5 h/ Mpc), PS is not significantly influenced
+by baryons, as baryonic suppression in PS is of the order
+of a few percent for k < 1 − 5 h/ Mpc [117–119]. How-
+ever, [105, 108, 117] have shown that HM provides the
+
+12
+flexibility, which allows it to add additional parameters
+that can incorporate baryonic effects from hydrodynam-
+ical simulation. Accounting for such effects in our MG
+PS modeling is a significant endeavor, and is well beyond
+the scope of this work.
+The data used in this work is publicly available from
+our website1. We provide Υ(k) for a wide range of z,
+from z = 0 to z = 2 for each MG model considered in
+this work. A description of the data set is also enclosed
+in the directory. Also, the data used to make the figures
+in this article are available on request to the authors.
+ACKNOWLEDGMENTS
+We would like to thank Hans A. Winther for kindly
+providing us with mgcamb version for specific forms of
+µ(a, k) and γ(a, k) functions implementing our f(R) and
+nDGP models. The authors also thank Krishna Naidoo
+for giving his suite of mg-cola simulations that we used
+as a testbed in our work.
+We also acknowledge con-
+structive discussions with Marika Asgari and Alberto
+Acuto.
+Matter power spectrum calculations to com-
+pute linear halo bias have been done using pylians2.
+This work is supported via the research project “VEr-
+TIGO” funded by the National Science Center, Poland,
+under agreement no.
+2018/30/E/ST9/00698.
+We
+also acknowledge the support from the Polish Na-
+tional
+Science
+Center
+within
+research
+projects
+no.
+2018/31/G/ST9/03388, 2020/39/B/ST9/03494 (WAH &
+MB), 2020/38/E/ST9/00395 (MB), and the Polish Min-
+istry of Science and Higher Education (MNiSW) through
+grant DIR/WK/2018/12.
+This project also benefited
+from numerical computations performed at the Interdis-
+ciplinary Centre for Mathematical and Computational
+Modeling (ICM), the University of Warsaw under grants
+no GA67-17 and GB79-7.
+Appendix A: Comparison of theoretical and
+simulation halo bias
+Here we compare the simulation results for the linear
+halo bias, b(M), with the theoretical predictions from
+Sheth et al. 2001 [82, hereafter S01]. The formula pro-
+posed by S01 is given by:
+b(M) =
+1
+√aδc(z)(√a(aν2) + √ab(aν2)1−c
+−
+(aν2)c
+(aν2)c + b(1 − c)(1 − c/2),
+(A1)
+with the parameters a = 0.707, b = 0.5 and c = 0.6.
+1 https://data.cft.edu.pl/UPSILON_PK/UpsilonPk.tar.gz
+2 https://pylians3.readthedocs.io/en/master/
+To apply the S01 expression to our MG variants,
+we used P(k)lin to compute σ(M, z), and then ν =
+δc(z)/σ(M, z), specific to each MG model. For that, we
+used standard ΛCDM spherical collapse based δc values.
+We stay with the ΛCDM δc baseline since we have found
+that using slightly different values suggested for either
+f(R) [120], or for the nDGP model [67] impacts the final
+HM results by less than a sub-percent.
+The results of our substitution are shown in Fig. 5,
+where we plot the bias ratios between MG and ΛCDM,
+as a function of halo mass, M200. Here, we include the
+two models most departing from ΛCDM: N1 (left col-
+umn) and F5 (right column). These variants illustrate
+the most extreme behavior in b(M) for the two MG mod-
+els we work with.
+Points illustrate simulation results
+(with error bars corresponding to the propagated stan-
+dard deviation from simulations). For comparison, ratios
+of S01 predictions for MG and ΛCDM are also shown,
+but they are extended outside of the M200 ranges probed
+by our simulations, to show the asymptotic behavior at
+small and large halo masses. Depending on the redshift
+and the model, departures in MG b(M) from ΛCDM can
+reach up to ∼ 10%. Contrary to the HMF, MG-induced
+increase in the strength of gravity lowers the bias, as a
+result of enhanced DM clustering. Similar trends have
+also been reported in [67, 118, 121]. The ratios predicted
+analytically from the S01 framework do not match the
+simulation amplitudes exactly, but they still qualitatively
+capture the trends.
+Further, in Fig. 6, we plot the b(M) in these MG mod-
+els as a function of M200. The top panels present ab-
+solute b(M) values, while the bottom ones include the
+ratio between S01 predictions and the simulation-based
+bias. Here, we can clearly see that the analytical model
+matches the simulation results within 5 − 10%. This af-
+firms our approach in extending S01 to beyond ΛCDM,
+for the f(R) and nDGP models we study.
+As for the HM build-up, we need a bias prescription for
+a much wider halo mass range than what our simulations
+cover.
+This overall consistency between the analytical
+and simulation results is sufficient for us, and thus we can
+use the S01 modeling for b(M) also in our MG variants.
+Appendix B: Comparison of HALOFIT and
+HMCODE results
+As discussed in Sec. IV B, for our baseline PS model-
+ing in MG we multiply the HM-based ratio Υ(k) with
+halofit PS derived for ΛCDM. Here we test our ap-
+proach for the case where the ΛCDM PS is obtained
+from the hmcode [26] instead.
+Similarly to halofit,
+hmcode is also built on the principles of HM, incorpo-
+rating however additional corrections in the standard HM
+build-up, owing to physical constraints. The parameters
+of the corrections are based on high-resolution simulated
+ΛCDM power spectra from the emulator introduced in
+Ref. [122]. Here we use the latest ‘hmcode-2020’ ver-
+
+13
+FIG. 5. Ratios of halo bias, b(M), between MG and ΛCDM for N1 (left plot), and F5 (right plot), across range of redshifts as
+indicated in the legends. Solid lines are the analytical results from Sheth et al. 2001 [82], and the respective dots of the same
+color are from simulations. Error bars illustrate the propagated standard deviation from simulations.
+FIG. 6. Top panels: Linear halo bias, b(M), as a function of halo mass, M200 for N1 (left column) and F5 (right column).
+The solid lines correspond to theoretical Sheth et al. 2001 [82, S01] predictions, and the respective dots of the same color
+are the simulation results obtained using Eq. (10). Error bars correspond to the standard deviation across five realizations of
+the simulation box. Bottom panels: Ratio between S01 and simulation linear halo bias predictions. Grey dotted lines are 5%
+accuracy regimes.
+sion3 [108].
+We compute P(k)MG by multiplying Υ(k)HM with
+both halofit and hmcode inputs for P(k)ΛCDM. Then,
+in Fig. 7, we compare both predictions with the ele-
+phant simulation results.
+Here we see a similar per-
+formance of both the methods, with some exceptions
+at small scales and high-z, where hmcode occasionally
+performs better. Interestingly, at z = 0, the halofit
+framework seems to lead to better results for a range of
+k-scales. We emphasize however that as our simulations
+were done for one particular set of cosmological parame-
+ters, these trends between halofit and hmcode-based
+predictions could change for other background cosmolo-
+gies. In any case, as what we provide is the ratio Υ(k)
+to be multiplied by the ΛCDM PS prediction, one can
+employ any best-fit P(k)ΛCDM for the latter to possibly
+improve the final accuracy of P(k)MG.
+3 https://github.com/alexander-mead/HMcode
+
+N1
+14
+TI
+z=0
+Sheth et al. 2001
+TI
+z=0.3
+Simulation
+12
+TO
+z=0.5
+10
+T
+z=1
+(M)
+8
+6
+4
+2
+0
+b(M)s01/b(M)sIM
+0.9
+1013
+1014
+1015
+M200 (h-1Mo)F5
+14
+TI
+z=0
+Sheth et al. 2001
+TI
+z=0.3
+Simulation
+12
+TO
+z=0.5
+10
+T
+z=1
+(M)
+8
+6
+6
+4
+2
+0
+b(M)s01/b(M)sIM
+0.9
+1013
+1014
+1015
+M200 (h-1Mo)N1
+F5
+Z=O
+z=O
+1.050
+1.050
+z=0.3
+Z=0.3
+z=0.5
+Z=0.5
+[1.025
+1.025
+z=1
+1.000
+1.000
+0.975
+0.950
+0.925
+0.925
+0.900
+0.900
+0.875
+0.875
+Sheth et al. 2001
+Sheth et al. 2001
+Simulation
+Simulation
+0.850
+0.850
+1012
+1013
+1014
+1015
+1013
+1014
+1015
+1012
+M200 (h-1Mo)14
+FIG. 7. Comparison of P(k)MG obtained from the input of P(k)ΛCDM, HMCODE (solid lines) and P(k)ΛCDM, HALOFIT (dashed
+lines), with the elephant simulation results, for a range of redshifts as indicated in the legends. The error contours correspond
+to the uncertainty in the simulation PS results, and the vertical grey dotted line is the 5% accuracy regime.
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S9FKT4oBgHgl3EQfkC7A/content/tmp_files/2301.11848v1.pdf.txt

@@ -0,0 +1,822 @@
+Combination of Multi-Fidelity Data Sources For Uncertainty Quantification: A
+Lightweight CNN Approach
+Minghan Chu * 1 Weicheng Qian * 2
+Department of Mechanical and Materials Engineering 1 Department of Computer Science 2
+Queen’s University 1 University of Saskatchewwan 2
+[email protected] 1 [email protected] 2
+Abstract
+Reynolds Averaged Navier Stokes (RANS) mod-
+elling is notorious for introducing the model-form
+uncertainty due to the Boussinesq turbulent vis-
+cosity hypothesis. Recently, the eigenspace pertur-
+bation method (EPM) has been developed to esti-
+mate the RANS model-form uncertainty. This ap-
+proach estimates model-form uncertainty through
+injecting perturbations to the predicted Reynolds
+stress tensor. However, there is a need for a reli-
+able machine learning method for estimating the
+perturbed amplitude of the Reynolds stress tensor.
+Machine learning models are often too complex
+and data intensive for this application. We pro-
+pose a lightweight convolutional neural network
+(CNN) approach to learn a correction function for
+RANS from paired-samples of RANS and DNS
+simulation results. The CNN learned RANS cor-
+rection function successfully facilitates the RANS
+uncertainty quantification (UQ), and our findings
+suggest that the lightweight CNN approach is ef-
+fective in combining RANS and DNS simulation
+results to enrich the existing perturbation method
+in estimating the RANS UQ more precisely.
+1. Introduction
+Turbulent flows are phenomena that manifest over a wide
+range of length and time scales. Such length scales vary
+from micro-meters to kilo-meters. Furthermore these mo-
+tions are coupled across length scales and exhibit strong
+interactions. A similar scenario exists for the range of time
+scales as well. Any numerical simulation of a turbulent
+flow needs to either explicitly resolve or model these ranges
+of motions. There are different approaches to such sim-
+ulations, varying by the ranges of length scales that they
+resolve and those that they model. For example direct nu-
+merical simulation (DNS) computes all the scales of motion,
+while large eddy simulation (LES) computes only the largest
+scales while modelling the smaller length scales of motion
+in the flow. Both DNS and LES methods can yield high
+fidelity results, while also incurring a considerable increase
+in computational cost. Reynolds Averaged Navier Stokes
+(RANS) based simulations use simplified models for all
+scales. Due to this, RANS-based simulations are relatively
+computationally inexpensive and are widely used in industry
+and academia for analyzing turbulent flows. However, most
+RANS models are often inaccurate when predicting com-
+plex flows with Boussinesq turbulent viscosity hypothesis
+(TVH) adopted (Craft et al., 1996). Therefore, a compro-
+mise is to stick with the low-fidelity RANS simulations at
+low computational cost with uncertainties properly quanti-
+fied.
+Uncertainty studies can be roughly classified into two cat-
+egories: 1) aleatory UQ and 2) epistemic UQ. Aleatory
+uncertainties are introduced in the imprecision of a sys-
+tem (Duraisamy et al., 2019). Studies that have focused on
+the aleatory uncertainties replace model parameters with
+random variables (Loeven & Bijl, 2008; Ahlfeld & Mon-
+tomoli, 2017), defined a domain region as a random field
+(Dow & Wang, 2015; Doostan et al., 2016), and quantify
+the irreducible aleatory uncertainties associated with the
+boundary conditions (Pecnik et al., 2011). Epistemic uncer-
+tainties are intrinsic in proposed turbulence models, hence
+known as model-form uncertainty (Duraisamy et al., 2019).
+The model-form uncertainties have often been ignored in
+uncertainty studies, while such uncertainties are at a higher-
+level uncertainty than the aleatory ones (Duraisamy et al.,
+2017). Less than a decade ago, Iaccarino et al. proposed
+a physics-based eigenspace perturbation approach (Emory
+et al., 2013; Iaccarino et al., 2017) to estimate the model-
+form uncertainty introduced in RANS-based models via
+sequential perturbations to the amplitude (turbulent kinetic
+energy), shape (eigenvalues), and orientation (eigenvectors)
+of the predicted Reynolds stress tensor. The Eigenspace Per-
+turbation Method has been applied with considerable suc-
+cess for Civil engineering applications (Gorl´e et al., 2019),
+Aerospace engineering applications (Mishra et al., 2019b;
+Mishra & Iaccarino, 2017; Mishra et al., 2019a), Robust
+and Reliability Based Designs (Cook et al., 2019; Mishra
+arXiv:2301.11848v1  [physics.flu-dyn]  27 Jan 2023
+
+Submission and Formatting Instructions for ICML 2023
+et al., 2020), besides others. However a central shortcoming
+of the Eigenspace Perturbation Method is that it always pre-
+supposes the worst case scenario and thus leads to overly
+conservative uncertainty estimates. A central contribution of
+this investigation is to ensure that the uncertainty estimates
+are sharp and well calibrated.
+Related Works
+Data driven approaches like Machine
+Learning models have found wide application in physics in
+general and turbulence modelling specifically(Duraisamy
+et al., 2019; Ihme et al., 2022; Brunton et al., 2020). Recent
+machine learning models that have specifically focused on
+the estimation of RANS model uncertainty with improved
+accuracy (Xiao et al., 2016; Wu et al., 2016; Xiao et al.,
+2017; Wang et al., 2017b;a; Wu et al., 2018; Heyse et al.,
+2021b;a; Zeng et al., 2022) are often complex and demand a
+large size of training data. Complex machine learning mod-
+els not only require additional computational resources in
+training but also become less comprehensive to researchers.
+This hinders the understanding and shrinks the room for
+improvement in the existing theories. In general, machine
+learning models in RANS UQ can be classified into two
+categories: 1) physics-based Bayesian inference and 2) data-
+driven random forest. There are relatively few studies for re-
+ducing the model-form uncertainty using the convolutional
+neural network (CNN) approach. Besides, most studies have
+focused on reducing the model-form uncertainty intrinsic
+to eigenvalues and eigenvectors, except very recently, Chu
+et al. (Chu et al., 2022a) used polynomial regressions to
+quantify the model-form uncertainty in turbulence kinetic
+energy.
+Therefore, the purpose of this paper is to advance the under-
+standing of the performance of CNN approach for reducing
+the model-form uncertainty introduced in turbulence kinetic
+energy in RANS simulations with complex flow features
+such as separation and reattachment. The novelty of this
+work is the presentation of a lightweight CNN to learn cor-
+rection function for RANS simulation from both RANS and
+DNS results. This CNN-based correction function is help-
+ful to the physics-based eigenspace perturbation framework
+(Emory et al., 2013; Iaccarino et al., 2017). Our CNN-based
+correction function method is non-intrusive, meaning that
+no modifications need to be made to the RANS-based turbu-
+lence models.
+2. Method
+2.1. Eigenspace Perturbation
+Most RANS models have adopted the Boussinesq turbulent
+viscosity hypothesis (Pope, 2001) that assumes Reynolds
+stresses are proportional to the rate of mean strain:
+�
+uiuj
+�
+= 2
+3kδij − 2vt
+�
+Sij
+�
+,
+(1)
+where k is the turbulence kinetic energy, δij is the Kronecker
+delta, νt is the turbulent viscosity, and
+�
+Sij
+�
+is the rate of
+mean strain tensor.
+The model-form uncertainty introduced in RANS modelling
+can be quantified via the eigenspace perturbation approach
+(Emory et al., 2013; Iaccarino et al., 2017), the perturbed
+Reynolds stresses are defined as
+�
+uiuj
+�∗ = 2k∗
+�1
+3δij + v∗
+inˆb∗
+nlv∗
+jl
+�
+,
+(2)
+where k∗ is the perturbed turbulence kinetic energy, ˆb∗
+kl is
+the perturbed eigenvalue matrix, v∗
+ij is perturbed eigenvector
+matrix.
+k∗ = f ∗(x, y)
+(3)
+A better estimated k∗ can improve UQ. Correction function
+of RANS is often used to correct RANS towards DNS,
+resulting a better estimated k∗.
+2.2. Correction Function for RANS
+Low-fidelity RANS simulations only focus on the mean
+flow quantities, and hence, are computationally inexpensive.
+In general RANS simulations are capable of generating
+satisfactory results for simple shear flows (Pope, 2001);
+however, RANS simulations are inaccurate in predicting
+complex flow features such as the separation bubble. On
+the other hand, DNS simulations give high-fidelity results
+by resolving all scales of fluid motion, while DNS requires
+tremendous amount of computational resources. Therefore,
+it is beneficial to correct low-fidelity RANS simulations with
+the accuracy of high-fidelity DNS. In the present study, we
+adopt a correction function to correct RANS results towards
+DNS, namely correction function.
+Owing to the importance of RANS model accuracy for en-
+gineering design, reliability, and safety, correction function
+for RANS models has seen a rapid growth of interest in the
+last few years, such as a linear form (Ahlfeld et al., 2016)
+and additive and multiplicative form (Voet et al., 2021).
+For both the RANS and DNS simulation, we can summarize
+their results as the function of the perturbed turbulence
+kinetic energy k:
+k = f(x, y)
+(4)
+
+Submission and Formatting Instructions for ICML 2023
+where x and y are coordinates in a two-dimensional compu-
+tational domain, and f is the mapping from every coordinate
+(x, y) to k, embedded in triples (x, y, k) from simulation
+results.
+Without assuming a specific form, the correction function
+for RANS is a mapping between two functions:
+Z : f RANS(x, y) → f DNS(x, y)
+with kDNS = f DNS(x, y) and kRANS = f RANS(x, y), we
+can rewrite Z as a mapping ζ between points that comprises
+f RANS and f DNS
+ζ : (x, y, kRANS) → (x, y, kDNS)
+Consider the model error for RANS and DNS in terms of
+kinetic energy, we have
+pRANS(Kg | x, y) = p(kg = kRANS | x, y)
+pDNS(Kg | x, y) = p(kg = kDNS | x, y)
+where Kg is the unknown ground truth of kinetic energy at
+(x, y).
+Kinetic energy resulted from DNS simulation results pRANS
+can be estimated with kinetic energy from RANS simulation
+pDNS and its correction function g as
+pDNS(Kg | x, y) = g(kRANS, x, y)p(kRANS | x, y)
+Because kDNS = f DNS(x, y) and kRANS = f RANS(x, y),
+at each x,
+we have that kDNS
+x
+=
+f DNS
+x
+(y) and
+kRANS
+x
+=
+f RANS
+x
+(y),
+assuming both f RANS
+x
+and
+f DNS
+x
+are continuous, that is, ∀ϵ
+>
+0, ∃δ
+>
+0,
+s.t. ∀|d| < δ,
+��fx(y + d) − fx(y)
+�� < ϵ. We can approx-
+imate g(kRANS, x, y) with ˆg(kRANS
+x,y,δ ), where kRANS
+x,y,δ
+=
+[kRANS
+x,y0
+, kRANS
+x,y1
+, · · · ]⊤ and y0, y1, · · · ∈ [y − δ, y + δ]. In
+another word, we can learn ˆg with paired (kRANS
+x,y,δ , kDNS
+x,y,δ).
+2.3. CNN-based Correction Function
+We employed a one-dimensional convolutional neural net-
+work (1D-CNN) to learn the correction function ˆg from
+paired RANS and DNS simulation estimated kinetic en-
+ergy (kRANS
+x,y,δ , kDNS
+x,y,δ). Because our approximated correc-
+tion function ˆg only depends on the neighbor of kRANS
+and coordinates (x, y) are only used to group neighbors of
+kRANS, we grouped simulation data by x and transformed
+(y, k) at x into kRANS
+x,y,δ via a rolling window parameterized
+by window size. Our 1D-CNN has four-layers and in to-
+tal 86 parameters: a single model for all zones at any x to
+correct RANS towards DNS.
+3. Experiments Setup and Data Sources
+We experimented our lightweight CNN-based approach
+to approximate the correction function for RANS on two
+datasets: the in-house RANS/DNS (Zhang, 2021; Chu et al.,
+2022a) dataset and the public RANS/DNS dataset (Voet
+et al., 2021). The in-house RANS/DNS dataset (Zhang,
+2021) was obtained by considering the flow around an
+SD7003 airfoil, with a separation bubble formed on the
+suction side of the airfoil due to the adverse pressure gra-
+dient. The public RANS/DNS dataset (Voet et al., 2021)
+was generated from the two-dimensional channel flow over
+periodically arranged hills. Similar to the flow over the air-
+foil, the flow experiences adverse pressure gradient when
+encountering the curved surface of the hill, which causes
+the formation of a separation bubble behind the hill. The in-
+house DNS dataset (Zhang, 2021) was obtained by consid-
+ering the flow around an SD7003 airfoil, with a separation
+bubble formed on the suction side of the airfoil due to the
+adverse pressure gradient. The public dataset (Voet et al.,
+2021) was generated from DNS flows over periodically ar-
+ranged hills. Similar to the flow over the airfoil, the flow
+experiences adverse pressure gradient when encountering
+the curved surface of the hill, which results in the formation
+of a separation bubble behind the hill.
+We split x-coordinate grouped pairs of (kRANS
+x,y,δ , kDNS
+x,y,δ) into
+train set and validation set by their group key x.
+For
+both the in-house DNS and the public dataset, we choose
+x at only three positions from the beginning, the mid-
+dle and the end of all paired x values. For the in-house
+dataset, x = 0.4, 0.56, 0.58; for the public dataset, the
+x = 0, 0.046, 0.116, 0.128. For each dataset, we have a
+80%–20% split as training–testing dataset.
+For both datasets, we validated our trained 1D-CNN
+by comparing the L1 loss of RANS, denoted as
+L1
+c(rans)
+=
+��CF RANS
+k
+− CF DNS
+k
+�� with the L1 loss
+of 1D-CNN corrected RANS, denoted as L1
+c(pred) =
+��CF CNN
+k
+− CF DNS
+k
+��.
+4. Results
+Our CNN-based correction function is validated at all paired
+x locations, Figures 2 and 3 are four typical x locations for
+each dataset. For both datasets, our CNN-based correction
+function results in RANS results located closer to the DNS
+results at any zone (x). The CNN predicted DNS profiles
+for k shows overall good resemblance to the DNS dataset,
+although an over-prediction exists at the beginning of the ab
+zone. From the Figure 2, the series of CNN predicted DNS
+
+Submission and Formatting Instructions for ICML 2023
+Correction Function 
+RANSx(y, k) → CorrectedRANSx(y, k) 
+RANS 
+Simulation
+RANSx(y, k)
+DNS 
+Simulation
+DNSx(y, k)
+Training/Validating 
+1D-CNN
+Corrected RANS 
+RANSx(y, k)
+Testing
+Sampled Pairs
+(y, k); x
+Figure 1. Data-flow Diagram for Experiments. Blue path is the training/validating path, and the red path is the validation path. Thickness
+of the path indicates the amount of data through the flow.
+profiles in the first row are then smoothed with the moving
+average with a window size of six consecutive estimations.
+Our CNN predicted DNS profiles resemble the ground truth
+DNS despite being trained with only a few pairs of RANS
+and DNS results. From Figure 2, the discrepancy in general
+reduces as the flow proceeds further downstream. From the
+Figure 2, the second row shows the computed L1 error of
+the baseline solution and the CNN predicted DNS. It is clear
+that the L1 error for CNN-based correction function can
+significantly reduce the L1 error in magnitude compared to
+that for the original RANS.
+5. Discussion
+We proposed a CNN approach to approximate the correction
+function that corrects RANS simulation towards DNS sim-
+ulation. We further examined our method on two datasets:
+1) one flow being considered is over a SD7003 airfoil at
+8◦ angle of attack and the Reynolds number based on the
+cord length of Rec = 60000 (Zhang, 2021). A laminar
+separation bubble evolves on the suction side of the airfoil
+whereby the flow undergoes transition to turbulence, 2) an-
+other is generated from the DNS two-dimensional channel
+flow over periodically arranged hills (Voet et al., 2021). The
+flow experiences adverse pressure gradient when encoun-
+tering the curved surface of the hill. It should be noted that
+a separation bubble occurs for both datasets. The RANS
+results deviate from the DNS data in both flow scenarios
+and our CNN-based correction function can significantly
+reduce the L1 error of RANS- from DNS-simulations.
+For both datasets, the CNN-based correction function is
+trained with paired RANS-DNS simulated turbulence ki-
+netic energy using less than 20% positions on the x-axis,
+but the trained correction function is effective for the re-
+maining 80% positions on the x-axis. Furthermore, our
+lightweight CNN model uses the y-axis only for grouping
+RANS-simulated turbulence kinetic energy within a neigh-
+bor. The results of our CNN-based correction function
+suggests that RANS results might be improved by leverag-
+ing information embedded in the positions within a close
+neighbor, which is independent of the absolute coordinates
+(x, y).
+There are relatively few studies for correcting the perturbed
+turbulence kinetic energy. Very recently, the study of Chu
+et al. (Chu et al., 2022a) assessed the effect of polynomial
+regression on the estimation of the perturbed turbulence ki-
+netic energy. Our CNN-based correction method has readily
+implications on practical applications, such as, to be cou-
+pled to the eigenspace perturbation approach of Emory et al.
+(Emory et al., 2013). The eigenspace perturbation approach
+has been implemented within the OpenFOAM framework
+to construct a marker function for the perturbed turbulence
+kinetic energy (Chu et al., 2022a). Our CNN-based correc-
+tion method can be used as a new marker function to predict
+the perturbed turbulence kinetic energy.
+5.1. Application on UQ for airfoil
+Our CNN-based correction function method can be applied
+to different flow cases to correct RANS towards DNS. In
+this section, the CNN-based correction function is applied to
+the SD7003 airfoil case to predict the perturbed turbulence
+kinetic energy.
+The CNN corrected RANS and ground truth profiles for the
+
+Submission and Formatting Instructions for ICML 2023
+0.06
+0.08
+0.10
+0.00
+0.05
+0.10
+0.15
+0.20
+CFk
+x/c = 0.17
+dns
+pred
+0.06
+0.08
+0.10
+0.00
+0.05
+0.10
+0.15
+0.20
+x/c = 0.25
+dns
+pred
+0.06
+0.08
+0.10
+0.00
+0.05
+0.10
+0.15
+0.20
+x/c = 0.32
+dns
+pred
+0.06
+0.08
+0.00
+0.05
+0.10
+0.15
+0.20
+x/c = 0.44
+dns
+pred
+0.06
+0.08
+0.10
+10−5
+10−3
+10−1
+101
+L1
+c(CFk)
+L1
+c(rans)
+L1
+c(pred)
+0.06
+0.08
+0.10
+10−5
+10−3
+10−1
+101
+L1
+c(rans)
+L1
+c(pred)
+0.06
+0.08
+0.10
+10−5
+10−3
+10−1
+101
+L1
+c(rans)
+L1
+c(pred)
+0.06
+0.08
+10−5
+10−3
+10−1
+101
+L1
+c(rans)
+L1
+c(pred)
+y
+Figure 2. Results for Selig-Donovan 7003 airfoil. First row: CNN corrected DNS (pred) compared with ground truth (dns). Second
+row: Validation of 1D-CNN by comparing L1 loss between L1
+c(rans) and L1
+c(pred).
+0
+1
+2
+0.00
+0.05
+0.10
+0.15
+0.20
+CFk
+x/c = 0.0
+dns
+pred
+0.00
+0.25
+0.50
+0.75
+0.00
+0.05
+0.10
+0.15
+0.20
+x/c = 0.046
+dns
+pred
+0.00
+0.25
+0.50
+0.75
+0.00
+0.05
+0.10
+0.15
+0.20
+x/c = 0.116
+dns
+pred
+0.0
+0.5
+1.0
+0.00
+0.05
+0.10
+0.15
+0.20
+x/c = 0.128
+dns
+pred
+0
+1
+2
+10−5
+10−3
+10−1
+101
+L1
+c(CFk)
+L1
+c(rans)
+L1
+c(pred)
+0.00
+0.25
+0.50
+0.75
+10−5
+10−3
+10−1
+101
+L1
+c(rans)
+L1
+c(pred)
+0.00
+0.25
+0.50
+0.75
+10−5
+10−3
+10−1
+101
+L1
+c(rans)
+L1
+c(pred)
+0.0
+0.5
+1.0
+10−5
+10−3
+10−1
+101
+L1
+c(rans)
+L1
+c(pred)
+y
+Figure 3. Results for data set from Voet et al. . First row: CNN corrected DNS (pred) compared with ground truth (dns). Second row:
+Validation of 1D-CNN by comparing L1 loss between L1
+c(rans) and L1
+c(pred).
+turbulence kinetic energy normalized with the freestream
+velocity squared, k∗/U 2
+∞ and k/U 2
+∞ are shown in Figs. 4
+(a) and (b), respectively. The k∗/U 2
+∞ and k/U 2
+∞ profiles are
+equally spaced for the ab and cd zone with x/c = 0.01, and
+a uniform spacing of x/c = 0.02 is used for the ef zone.
+It is clear that the k∗/U 2
+∞ and k/U 2
+∞ profiles are more
+
+Submission and Formatting Instructions for ICML 2023
+Figure 4. (a) CNN corrected RANS (CNN DNS) (solid-dotted lines) of the normalized perturbed turbulence kinetic energy and (b) ground
+truth (CNN RANS) along the suction side of the SD7003 airfoil (geometry depicted by gray line): from left to right are zone ab, zone cd
+and zone ef. There are 32 positions on the suction side of the airfoil.
+densely packed for the ab and cd zone, within which the
+flow features are complex due to the presence of separation
+and reattachment.
+From Figs. 4 (a) and (b), the CNN corrected DNS profiles
+in general exhibit a similar trend as that for the ground truth
+dataset, as both profiles show a gradual increase in the ab
+and cd zone. Then a reduction of the profile is observed
+further downstream in the ef zone.
+In Fig. 4 (a), CNN corrected RANS profiles in general in-
+crease in magnitude as the flow moves further downstream,
+which is qualitatively similar to the ground truth profiles.
+Further, it should be noted that the CNN corrected RANS
+profiles increase in a somewhat larger magnitude than that
+for the ground truth in the ab zone. The discrepancy is more
+than 50% at the beginning of the ab zone and gradually
+reduces as the flow moves further downstream, which indi-
+cates that a better accuracy of our CNN model is yielded
+further downstream. This behavior becomes more clear for
+the cd and ef zone. In the region where the end of the cd
+zone meets the beginning of the ef zone, the ground truth
+profiles are clustered due to the complex flow feature of
+the reattachment (Chu et al., 2022b), as shown in Figs 4
+(b). This clustering behavior is successfully captured by
+our CNN model, as shown in Fig. 4 (a). In the ef zone,
+our CNN model gives overall accurate predictions for the
+k∗/U 2
+∞ profiles, i.e., the CNN corrected RANS profiles and
+the ground truth profiles are almost identical.
+6. Conclusion
+To the best of our knowledge, we are among the first to
+examine the projection from RANS to DNS using the CNN
+approach. Our experiment results suggest that the CNN-
+based correction function corrects the RANS predictions for
+perturbed turbulence kinetic energy towards the in-house
+
+Submission and Formatting Instructions for ICML 2023
+DNS data. A projection that can approximate the in-house
+DNS data reasonably well from RANS might exist inde-
+pendent of x. Our methodology can be easily extended to
+analyze flows over different types of airfoils.
+Our findings are subject to following limitations: our CNN
+model was trained with only two datasets. Future work
+may include validation with more datasets using different
+flow cases, e.g., different types of airfoils. In addition, our
+CNN-based correction function will be integrated with the
+eigenspace perturbation framework to result in accurate
+perturbations and hence improved estimation of RANS UQ.
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+

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+A Fully Quantum-Mechanical Treatment for Kaolinite
+Sam Shepherd,1 Gareth A. Tribello,1 and David M. Wilkins1, a)
+Centre for Quantum Materials and Technologies, School of Mathematics and Physics, Queen’s University Belfast,
+Belfast BT7 1NN, Northern Ireland, United Kingdom
+(Dated:)
+Neural network potentials for kaolinite minerals have been fitted to data extracted from density functional
+theory calculation that were performed using the revPBE + D3 and revPBE + vdW functionals. These
+potentials have then been used to calculate static and dynamic properties of the mineral. We show that
+revPBE + vdW is better at reproducing the static properties. However, revPBE + D3 does a better job
+of reproducing the experimental IR spectrum. We also consider what happens to these properties when a
+fully-quantum treatment of the nuclei is employed. We find that nuclear quantum effects (NQEs) do not make
+a substantial difference to the static properties. However, when NQEs are included the dynamic properties
+of the material change substantially.
+I.
+INTRODUCTION
+Clay minerals represent some of the most abun-
+dant, yet experimentally frustrating minerals on the
+planet.
+They are employed in a large number of
+applications1 ranging from catalysis2–5 and aqueous con-
+taminant removal6–9, to those involving the modification
+of the clay minerals via intercalation between the clay
+layers10,11.
+Due to the diversity of these applications
+and the range of clay minerals used within them, interest
+in these processes is ongoing and expanding.
+The ex-
+perimental understanding of these processes is however,
+incomplete. Numerous methods have been used to inves-
+tigate clay minerals12–17, but unearthing the properties
+and structures of individual minerals and surface motifs
+remains a difficult task15,18,19.
+One such clay mineral, kaolinite, finds practical use
+in a large number of fields, ranging from applications
+within medicine20,21 and catalysis22 to the creation of
+nano-hybrid devices23. Kaolinite is also an illustrative
+example of the incomplete picture currently available
+experimentally24,25. The small (sub-µm) size and rough-
+ness of the grain surface makes it difficult to deter-
+mine the mineral structure using standard experimental
+methods26.
+To aid experimental understanding, theoretical ap-
+proaches offer us a method to reconcile the difficulties en-
+countered when using experimental techniques. Compu-
+tational techniques have been used to investigate the for-
+mation of kaolinite particles27, ice formation on layer sur-
+faces of kaolinite28 and halloysite nanotube formation29.
+These studies employed classical force fields30–32 that al-
+low one to study processes on long length and time scales.
+These timescales are necessary to enable thorough inves-
+tigation of the vibrational dynamics33 of the system and
+interactions with aqueous interlayer species31. Classical
+force fields are typically fitted to reproduce experimen-
+tal results or to quantum mechanical calculations and
+a)Electronic mail: [email protected]
+hence, may not work well when the system moves to
+parts of phase space that lie outside of the region where
+they are fitted. Other research has thus used electronic
+structure theory to study the mechanisms via which tran-
+sition metals and organic molecules34–36 adsorb on the
+surface and enter the interlayer regions37–42.
+In these
+more-accurate simulations, small system sizes are stud-
+ied over short length scales, which limits available insight
+into processes which occur over longer timescales.
+When studying other materials, researchers have lever-
+aged the power of machine learning43–51. In one such ap-
+plication, an ab-initio method is used to build a database
+of structures and their associated energies and atomic
+forces.
+A less computationally-expensive potential is
+then fitted from this database. One can thus run long
+length and time scale simulations with ab-initio accuracy
+without incurring the additional computational expense
+associated with the reference method. In this paper we
+therefore extend our previous work52 in which we inves-
+tigated how nuclear quantum effects (NQEs) impact the
+structural and dynamical properties of the clay mineral
+kaolinite and pyrophyllite. However, instead of employ-
+ing the classical force field, CLAYFF30,31,33 we use neural
+network potentials53,54 that were trained from ab-initio
+data.
+We use these potentials to perform path integral ab
+initio molecular dynamics55 simulations at the ab-initio
+level and thus study the impact nuclear quantum effects
+have on the system when the electrons are modelled ab-
+initio.
+As we observed in our previous simulations52, nuclear
+quantum effects do not change the static properties of the
+material. However, they have a substantial effect on the
+dynamical properties. The fact that we observe similar
+differences when the electrons are modelled classically
+and when they are modelled using ab initio techniques
+tells us that nuclear quantum effects are important and
+must be included. The discrepancies that we observed
+between the experimental spectra and the spectra that
+was obtained from the classical simulation in our previ-
+ous work is not simply a consequence of an inaccurate
+forcefield. There is important physics missing from the
+arXiv:2301.04505v1  [physics.comp-ph]  11 Jan 2023
+
+2
+model when the nuclei are treated classically.
+The remainder of this paper is laid out as follows. In
+section II we provide some background information on
+the structure of kaolinite and the previous work that has
+been done on this material using ab initio simulation.
+We then describe how the machine learning potentials
+were trained in section III A. We detail how molecular
+dynamics simulations were run and analysed in section
+III B. A comparison of the structural and dynamic prop-
+erties we obtain from our simulations using our machine
+learned potentials when nuclear quantum effects are both
+included and not included is then presented in sections
+IV A and IV B.
+II.
+BACKGROUND
+The complexity involved with the computational mod-
+elling of kaolinite is predominantly due to the layered
+structure of the mineral. As shown in figure 1, Kaolinite
+is a 1:1 type smectite clay, with each individual layer com-
+prised of an octahedral aluminol sheet covalently bonded
+to a tetrahedral silica sheet24,56. These layers are held
+together through hydrogen bonds and dispersion inter-
+actions between the partially positive aluminol surface
+and the partially negative silica surface. In the case of
+1:1 type clay minerals like kaolinite, these interactions
+are dominated by hydrogen bonding between the termi-
+nal O-H groups of the aluminol layer and the interfacial
+oxygen atoms of the silica layer. Ensuring these interac-
+tions are accounted for when studying clay materials is
+extremely important for the accurate modelling of the
+mineral8,57–59.
+The combination of hydrogen bonding
+and dispersion interactions ensures that any theoretical
+study into kaolinite using ab initio methods is a compu-
+tationally expensive undertaking.
+In spite of this expense, traditional electronic struc-
+ture methods like density functional theory (DFT) have
+been used extensively in the study of kaolinite38,39,41 and
+other clay minerals60,61. These have allowed researchers
+to gain insight into numerous properties of kaolinite,
+mainly regarding the absorption mechanisms41 and vi-
+brational properties12,62–64 of the mineral. These studies
+have been limited in both time and length scale, but have
+shown that the accuracy of the calculations depend quite
+sensitively on the level of theory used when modelling
+the system. Both dispersion corrections57–59 and a full
+quantum mechanical treatment of the nuclei52 are neces-
+sary to reproduce the relevant physics. The necessity of
+including all of the aforementioned physics has hitherto
+limited the ability of researchers to study these minerals
+in greater detail.
+III.
+COMPUTATIONAL METHODS
+A.
+Neural Network Potential Creation
+We used the kaolinite structure from neutron scatter-
+ing experiments24 as a starting point. This structure was
+replicated twice in each direction to create the 2 x 2 x 2
+supercell that is pictured in Fig. 1.
+To create an initial dataset for the training of a ma-
+chine learning potential, we performed a number of both
+canonical (NVT) and constant-stress (NST) ensemble
+molecular dynamics simulations at the density functional
+tight binding65–67 (DFTB) level of theory with the pa-
+rameters of Frenzel et al68. These simulations were per-
+formed using CP2K69. These simulations were all started
+from the structure shown in figure 1 and run a range of
+temperatures (200 K - 400 K). The final dataset that
+was used for training was then assembled using farthest
+point sampling (FPS)70 to select the most diverse con-
+figurations from all available data.
+The total energy and the forces were calculated for
+a subset of the configurations within the database us-
+ing density functional theory (DFT). Two potentials
+were trained, which we will refer to as ClayNN-D3 and
+ClayNN-vdW. The revPBE functional71,72was used when
+calculating reference energies for both potentials. The
+difference was in the way dispersion interactions are in-
+corporated. In particular, Grimme’s D3 corrections73,74
+was used for the ClayNN-D3 potential, while Dion et
+al’s dispersion interactions75 were used for the ClayNN-
+vdW potential.
+These two different dispersion recipes
+were employed because previous work by Tunega et al57
+using the revPBE functional argued that dispersion cor-
+rections must be included if the non-bonded interactions
+within clay minerals are to be reproduced accurately.
+Grimme’s D3 corrections were selected based on the abil-
+ity of revPBE + D3 to accurately model dispersion inter-
+actions within water76. Dion’s corrections were selected
+due to previous work by Crasto de Lima8 who utilized
+the revPBE + vdW functional to study the adsorption
+mechanisms of contaminants onto clay mineral surfaces.
+More recently, Kobayashi et al.77 were able to achieve
+excellent agreement for a wide array of structural, dy-
+namical and mechanical properties of kaolinite by using
+the meta-GGA functional, SCAN without the explicit
+addition of dispersion corrections.
+These findings sug-
+gest that the SCAN functional incorporates dispersion
+interactions correctly. It is still important to use differ-
+ent treatments of dispersion in this work, however, as we
+are using a lower level of theory.
+To train the machine learning potentials we used
+the well-established method based on high-dimensional
+neural networks that was introduced by Behler and
+Parinello54,78. An extensive description of the construc-
+tion, architecture and symmetry functions we used is in-
+cluded in the SI. Initial neural network potentials (NNP)
+were trained using N2P279,80 with a training dataset con-
+
+3
+sisting of 6000 frames. Expansion of these potentials was
+completed using the ‘query by committee’43 procedure,
+which evaluated the total energy and atomic forces pre-
+dicted by the driving NNP using a 10 membered com-
+mittee of similar (but importantly, non-identical) poten-
+tials. If the standard deviation of the committees average
+atomic force prediction was below a set range (20 - 50
+meV/ ˚A) the configuration was well represented within
+the training dataset and would contribute nothing upon
+its addition to said training dataset. In the event of a
+configuration producing a relatively high error (with a
+preset upper limit, to avoid the addition of non-physical
+systems), this configuration was added to the training
+data of the potentials. The energies and forces for this
+configuration were then recomputed using the reference
+method for the potential.
+The
+final
+potentials
+presented
+within
+this
+work
+(ClayNN-D3 and ClayNN-vdW) consist of a total of 7748
+kaolinite configurations sampled from initial DFTB sim-
+ulations and NNP driven NVT and NST simulations. All
+DFT calculations used throughout this work were carried
+out using CP2K interfaced with AiiDA81,82. Example in-
+put files for theses computations can be found within our
+github repository.
+ClayNN-D3 and ClayNN-vdW were validated by com-
+paring the predicted system energy and atomic forces to
+reference values that were computed using revPBE + D3
+and revPBE + vdW calculations in CP2K. Both poten-
+tials were validated using a similar strategy to that of
+Schran et al.44,83, with structural and dynamical proper-
+ties of the system used to validate the potentials as well
+as the predicted and calculated energies and forces. The
+structural properties used included the computed and
+predicted radial distribution functions and the phonon
+density of states (PDOS). The way in which both the
+energies and the forces were used for validation are pre-
+sented within the SI.
+FIG. 1: Experimental structure of kaolinite used within
+this work. Initial structure was obtained from Bish24
+and replicated in all directions to create a 2× 2 supercell
+to serve as a suitable starting point for simulations.
+B.
+Molecular Dynamic Simulations
+To equilibrate the system initial MD simulations in the
+NVT ensemble were run for 500 ps at 300 K prior to a fur-
+ther 500 ps equilibration in the NST ensemble at 300 K.
+The timestep in these simulations was set equal to 0.25 fs.
+Furthermore, in these NST simulations the diagonal ele-
+ments of the stress tensor were set equal to 1 bar and all
+off-diagonal components were set equal to zero. Temper-
+ature was controlled using a Langevin thermostat84 with
+a friction of 1 ps. To obtain sufficient statistics for the
+structural properties of the clay system, a subsequent 1
+ns of NST simulation was performed.
+To study the vibrational dynamics of the system, we
+ran 500 ps simulations at temperatures ranging from 10
+K to 300 K in the NST ensmble. Ten configurations from
+each of these runs were then used as starting points for
+microcanonical (NVE) simulations. Separate estimates
+of the vibrational density of states (VDOS) were cal-
+culated from each of these simulations by finding the
+Fourier transform of the velocity-velocity autocorrelation
+function. The final VDOS results at each temperature
+that we show in the results section are thus averages over
+10 independent estimates.
+Path integral molecular dynamics (PIMD) was used
+to account for nuclear quantum effects. All simulations
+using PIMD were run using 16 replicas at 300 K. This
+number of replicas is justified as we observed a change
+of less than 0.5 % of the total system energy when the
+number of replicas was increased from 16 to 32.
+When the system was modelled with PIMD it was al-
+lowed to equilibrate for 200 ps in the NVT ensemble. It
+was then subsequently equilibrated for a further 100 ps
+in the NST ensemble.
+Static properties for the kaoli-
+nite system were then obtained from a 100 ps produc-
+tion simulation. The global path integral Langevin equa-
+tion (PILE)85 thermostat with global velocity rescaling
+was employed in all these simulations. When consider-
+ing dynamical properties for the system, we employed
+both thermostatted ring polymer molecular dynamics
+(TRPMD)86,87 and partially adiabatic centroid molec-
+ular dynamics (PACMD)88. We provide input files for
+both of these simulations. We present the results from
+the PACMD simulations in the manuscript as they were
+superior.
+However, information on the results we ob-
+tained with TRPMD are provided in the SI for complete-
+ness.
+All molecular dynamics simulations (both classical and
+path-integral) were performed using LAMMPS89 inter-
+faced with N2P2 and driven by the i-PI90 code. Example
+input files for all simulations are available in our github
+repository.
+
+L4
+IV.
+RESULTS AND DISCUSSION
+A.
+Structural Properties
+Table.
+I shows the unit cell parameters for kaolin-
+ite that were obtained from experiments performed at
+1.5 K24. These parameters are compared with the re-
+sults from our simulations. The first part of the table
+shows the cell parameters that are obtained from geome-
+try and cell optimization calculations. Geometry optimi-
+sations were performed using the reference DFT function-
+als (revPBE + D3 and revPBE + vdW) directly and us-
+ing the fitted neural network potentials (ClayNN-D3 and
+ClayNN-vdW). The δNNP column within Table I shows
+the differences in the cell parameters that are obtained
+when the optimisation is performed using the functional
+directly and when the optimisation is performed using
+the neural network potential. The maximum difference
+between the cell parameters is 0.7% for the ClayNN-D3
+(this difference is in the β parameter) and 1.14% for
+ClayNN-vdW (this difference is in the α parameter) re-
+spectively. There is thus an overall excellent agreement
+between the the reference DFT methods and the NNPs.
+Table I show that the optimisations that were per-
+formed using revPBE + vdW have cell parameters that
+more closely match those in experiment. For revPBE +
+D3 the percentage difference from the experiment are -
+0.058 ˚A (1.12%), -0.114 ˚A (1.27%) and 0.25◦ (0.28%) for
+A, B and γ respectively. For revPBE + vdW these dif-
+ferences take lower values of 0.041 ˚A (0.79%), 0.078 ˚A
+(0.87%) and 0.09◦ (0.1%). On these parameters, how-
+ever, the agreement is excellent for both functionals.
+More marked differences in the parameters are observed
+for the other cell parameters. The C, α and β param-
+eters for revPBE + D3 differ from experiment by 0.122
+˚A (1.62%), 2.64◦ (2.79%) and 3.16◦ (2.92%) respectively.
+For revPBE + vdW these differences are 0.048 ˚A (0.65%),
+0.67◦ (0.74%) and -0.120◦ (0.11%). The differences be-
+tween the two functionals is more marked for these pa-
+rameters because these are the cell directions that are
+more strongly affected by the treatment of dispersion.
+There is a slight underbinding between the hydrogen
+bonded layers when the D3 dispersion correction is used.
+Meanwhile, the vdW correction predicts cell parameters
+that are in line with those obtained using the dispersion
+corrected functionals that were considered by Tunega et
+al.57 However, the strength of the dispersion interaction
+is still underestimated slightly with this correction as the
+layers sit further apart than they would in experiment.
+The second part of Table I shows the values of the cell
+parameters that are obtained from MD and PIMD simu-
+lations that were run using the fitted, neural network po-
+tentials ClayNN-D3 and ClayNN-vdW. These estimates
+of the cell parameters were obtained from NST simula-
+tions of a 2 × 2 supercell.
+Comparing the parameters
+in the right half of the table with those in the left tells
+us that the unit cell size increases with temperature as
+would be expected.
+We have performed simulations with and without
+NQEs using two different treatments for dispersion. The
+molecular dynamics results in table I show that changing
+the way dispersion is modelled has a much larger effect
+on the structure than including an explicit treatment of
+NQEs. We summarise this by calculating the percentage
+cell volume change between classical and path integral
+simulations. The cell volume increased by 7.26% when
+the vdW correction is used in place of the D3 in a classi-
+cal model of the nuclei. By contrast, the cell volume in-
+creased by 0.60% when NQEs are included in the model
+with the vdW correction. The percentage volume change
+when the D3 correction is used was 0.29%, slightly less
+than the vdW value. This result is surprising as individ-
+ual cell parameters when described using ClayNN-D3 are
+more keenly affected by NQEs than ClayNN-vdW. The
+largest increase due to NQEs was calculated as 0.103 ˚A,
+observed for the C cell parameter when using ClayNN-
+D3.
+This value is an order of magnitude larger than
+that observed for the C cell parameter when using vdW
+corrections, which remains the most affected parameter.
+This result is hardly surprising for either potential, as
+the C parameter is the one most acutely affected by dis-
+persion interactions.
+The discrepancy between poten-
+tials we attribute to the less accurate reproduction of
+experimental results by ClayNN-D3. The comparatively
+small effect incurred by including NQEs is perhaps sur-
+prising given how much hydrogen bonding occurs within
+the structure of kaolinite. This observation is however,
+in agreement with our previous simulations with classical
+potentials52, where the percentage cell volume increase
+due to NQEs was 0.63%.
+To investigate whether the discrepancies between the
+experimental and simulated cell parameters can be at-
+tributed to a poor description of the interaction between
+a particular pair of atom types, we calculated radial dis-
+tribution functions (RDFs). Fig .2 shows three of these
+RDFs: gO-H(r), gSi-O(r), and gAl-O(r). RDFs for all other
+atom pairs can be found within the SI.
+Figure 2 also shows average interatomic distances for
+the corresponding pairs of atoms that are obtained from
+experiments24.
+We note the good agreement between
+these experimentally obtained values and the positions
+of the first peak in gO-H(r) and gAl-O(r). By contrast,
+the first peak in the gSi-O(r) sits at a value of r that is
+2.47% larger than the experimental average bond length.
+The Si-O interaction is thus weaker than it should be
+in our model. This under-binding of Si-O bonds by the
+studied dispersion corrected functionals is consistent with
+findings from Tunega et al57.
+Lastly, figure 2 shows a comparison of the RDF that is
+obtained when the simulation includes nuclear quantum
+effects and when it does not.
+It is clear from the fig-
+ure that including NQEs does not affect the positions of
+the peaks. However, the peaks in gO-H(r) are broadened
+when nuclear quantum effects are included. One would
+expect this RDF to be most impacted by the inclusion of
+
+5
+TABLE I: Unit cell parameters for kaolinite obtained using various means. The left part of the table contains results
+obtained from geometry optimization calculations using the reference DFT functionals (revPBE + D3 and revPBE
++ vdW) and the neural network potentials (ClayNN-D3 and ClayNN-vdW). The right part of the table contains
+results from ClayNN-D3 and ClayNN-vdW driven classical (CL) and PI (NQE) MD calculations in the
+constant-stress (NST) ensemble. The δNQE column in this right part contains the differences between the results
+obtained with and without nuclear quantum effects. The δD3−vdW column contains the differences in the cell
+parameters for the two models of dispersion. This difference was calculated for simulations in which the nuclei were
+treated classically. The experimental values are given in the column labelled EXP and are taken from Bish24.
+Geometry Optimization
+Molecular Dynamics at 300K
+revPBE + D3
+revPBE + vdW
+revPBE + D3
+revPBE + vdW
+Property
+EXP
+DFT
+NNP δNNP
+DFT
+NNP δNNP CLD3 NQED3 δNQE-D3 CLvdW NQEvdW δNQE-vdW δD3-vdW
+A(˚A)
+5.153
+5.095
+5.115
+0.02
+5.194
+5.235 0.039 5.128
+5.137
+0.011
+5.243
+5.252
+0.009
+-0.115
+B(˚A)
+8.942
+8.828
+8.818 -0.01 9.020
+9.059 0.039 8.879
+8.887
+0.008
+9.073
+9.088
+0.005
+-0.194
+C(˚A)
+7.391
+7.513
+7.523
+0.01
+7.439
+7.525 0.086 7.773
+7.876
+0.103
+7.586
+7.605
+0.019
+0.187
+α(◦)
+91.93
+94.57
+95.10
+0.53
+92.61
+91.42 -1.19 103.34 106.24
+2.90
+91.15
+91.23
+0.08
+12.19
+β(◦)
+105.05 108.21 108.97 0.76 104.93 104.96 0.03 108.31 107.94
+-0.37
+104.53
+104.49
+-0.04
+3.78
+γ(◦)
+89.80
+90.05
+90.37
+0.32
+89.89
+90.03
+0.04
+90.61
+90.61
+0.00
+90.09
+90.09
+0.00
+0.52
+nuclear quantum effects because the atoms involved are
+the lightest. When nuclear quantum effects are included
+the O–H bond weakens and the width of the distribution
+for the lengths of the O-H and hydrogen bonds within the
+interlayer region increases. This result explains our ob-
+servation of cell parameter C being substantially affected
+by NQEs in I, as it is heavily dependent on interactions
+involving O–H groups. By comparison, the other RDFs
+pictured within figure 2 are largely unaffected. Overall
+however, NQEs have a very small effect on the overall
+structure and mainly affect the value of the C parame-
+ter.
+B.
+Vibrational Properties
+We have shown in the previous section that explicit
+treatment of the nuclear quantum has little effect on the
+structural dynamics of these clay systems. Nuclear quan-
+tum effects are much more likely to affect the vibrational
+properties that are acutely affected by subtle changes in
+the strength of the interatomic bonds. In this next sec-
+tion we thus investigate the vibrational spectra of the
+material.
+Classical phonon densities of states (PDOS) were cal-
+culated using revPBE + D3, revPBE + vdW, ClayNN-
+D3 and ClayNN-vdW. PHONOPY91 was used in all these
+calculations with CP2K and the respective NNPs serving
+as external force calculators. Fig. 3 shows the PDOS that
+were obtained from this procedure. Both NN potentials
+produce PDOS that almost exactly match the ab initio
+reference methods within the higher frequency region be-
+tween 3600 to 3800 cm−1. The NNPs thus successfully
+replicate the number and the positions of the vibrational
+O–H modes that are observed with the corresponding
+DFT functional. In the low frequency region (i.e. be-
+low 1200 cm−1) there are larger discrepancies between
+the potentials and the reference DFT methods. This is
+most evident below 800 cm−1. In this region it becomes
+more difficult to distinguish different lattice modes and
+the spectra predicted using the NNPs ‘shift’ away from
+the reference.
+Figure 4 shows the high-frequency region of the clas-
+sical PDOS computed using revPBE + D3 and revPBE
++ vdW in dashed red and dashed blue lines respectively.
+These are presented alongside experimental IR spectra
+obtained at 1.5 K62.
+The complete computed PDOS,
+including the low frequency modes, is available within
+the SI. Neither functional reproduces the experimentally-
+observed O–H stretching vibrations. In experiment there
+are four distinct vibrational modes. The revPBE + D3
+PDOS has only two distinct modes, while the PDOS for
+revPBE + vdW has three.
+We label individual modes within the experimental IR
+within Figure 4 to assign the corresponding theoreti-
+cally obtained modes. Label A points to the inner O–
+H stretching mode around 3620 cm−1. Label B points
+to an anti-phase O–H stretching mode of the interlayer
+O–H groups at 3652 cm−1. Label C points to a higher
+frequency interlayer O–H anti-phase stretching mode at
+3669 cm−1. Finally, label D points to the in-phase inter-
+layer O–H stretching mode located at 3695 cm−1.
+We consider revPBE + D3 first.
+It can be clearly
+seen that there is excellent reproduction of modes A and
+D. Additionally, we observe a slight shoulder toward the
+lower frequency side of mode D. This shoulder could be
+attributed to mode C; however, we find no presence of
+its lower frequency counterpart at B. Due to the lack of
+any peak that could be assigned as mode B and the poor
+position of this shoulder, we cannot assign with absolute
+confidence this shoulder as mode C.
+The reproduction of the high frequency vibrational
+modes of the experimental IR using revPBE + vdW is
+much different. We find that revPBE + vdW accounts
+much better for modes B and C and reproduces them
+with only slight errors in peak positions. Mode A is en-
+
+6
+0
+1
+2
+3
+4
+5
+6
+0
+5
+10
+15
+20
+25
+gH−O(r)
+0
+1
+2
+3
+4
+5
+6
+0
+5
+10
+15
+20
+25
+gH−O(r)
+CL
+NQE
+Experiment
+0
+1
+2
+3
+4
+5
+6
+0
+5
+10
+15
+20
+25
+gSi−O(r)
+0
+1
+2
+3
+4
+5
+6
+0
+5
+10
+15
+20
+25
+gSi−O(r)
+0
+1
+2
+3
+4
+5
+6
+r(˚A)
+0
+5
+10
+15
+gAl−O(r)
+0
+1
+2
+3
+4
+5
+6
+r(˚A)
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+gAl−O(r)
+0.8
+1.0
+1.2
+0
+20
+1.4
+1.6
+1.8
+0
+20
+1.6
+1.8
+2.0
+0
+10
+0.8
+1.0
+1.2
+0
+20
+1.4
+1.6
+1.8
+0
+20
+1.6
+1.8
+2.0
+0
+10
+FIG. 2: Radial distribution functions (RDFs) for O–H (top), Si–O (middle) and Al–O (bottom) elemental pairs.
+The left and right plots correspond to results obtained using ClayNN-D3 and ClayNN-vdW respectively. The
+dashed blue lines indicate the result obtained using classical molecular dynamics simulations while dashed red lines
+shows the result obtained when NQEs are included by using PIMD. The vertical black lines indicate the interatomic
+distance for the corresponding atom pair that is obtained from experiment. The inserts show a zoomed in view of
+the first peak in the RDF.
+tirely neglected, while mode D could be assigned to the
+highest frequency theoretical peak. Similarly to our as-
+signments for revPBE + D3, the position of the theoret-
+ical peak leaves us hesitant to assign this mode as mode
+D.
+Overall, due mainly to the better replication of the
+more intense vibrational modes at 3620 cm−1 and 3695
+cm−1, we conclude that the classical description of the
+vibrational properties of kaolinite are better described
+through the use of revPBE + D3, and therefore, ClayNN-
+
+7
+0
+200
+400
+600
+800
+1000
+1200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+revPBE + D3
+ClayNN-D3
+3600
+3650
+3700
+3750
+0
+200
+400
+600
+800
+1000
+1200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+revPBE + vdW
+ClayNN-vdW
+3600
+3650
+3700
+3750
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ω / cm−1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Relative Intensity
+FIG. 3: Computed phonon densities of states for kaolinite showing the low frequency region (left) and the high
+frequency region (right) for both the reference DFT methods and the trained NNPs that were used within this work.
+The upper panels show revPBE + D3 (solid black line) and ClayNN-D3 (dashed red line) whilst the lower panels
+show revPBE + vdW (solid black line) and ClayNN-vdW (dashed red line). All of the presented spectra have been
+normalised to have unit area.
+D3. This result comes somewhat as a surprise, as one
+would predict that the better reproduction of dispersion
+related structural properties of the system using revPBE
++ vdW would equate to a more accurate replication of
+O–H bond dynamics.
+Our finding that revPBE + D3
+provides a better description than revPBE + vdW when
+considering O–H dynamics is one that is supported by
+previous work76,92.
+Figure 4 indicates that revPBE + D3 does a better job
+of reproducing the experimental IR structure. We there-
+fore used the NN potential that was fitted to the energies
+and forces that were calculated using this functional to
+computed the vibrational density of states (VDOS) at
+temperatures between 10 K and 300 K. Averaged VDOS
+were obtained by performing Fourier transforms of the
+velocity autocorrelation function (VAF). These VAF were
+calculated from 10 individual canonical (NVE) ensem-
+ble simulations that were started from randomly sampled
+frames that were taken from an equilibrated region of the
+corresponding NST trajectories.
+Figure 5 shows some of the VDOS that were obtained
+from our simulations. The full set of results are available
+within the SI. Increasing the temperature broadens the
+peaks in the spectrum. This makes it more difficult to as-
+sign each peak to a specific vibrational mode. However,
+the same problem exists for experiments that are per-
+formed at finite temperature. You would not expect to
+see all the fine structure in the 0 temperature PDOS in a
+300 K experimental IR spectrum. Indeed, the black line
+in figure 6 shows that the four peaks in the experimental
+IR spectrum (see figure 4) broaden when the temperature
+is increased from 1.5 to 300 K. Importantly, however, you
+can still distinguish between the four peaks in the exper-
+imental spectrum at 300 K. In the theoretical spectra, by
+
+8
+3600
+3650
+3700
+3750
+ω / cm−1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Relative Intensity
+A
+B
+C
+D
+revPBE + D3
+revPBE + vdW
+Experimental
+FIG. 4: Computed phonon densities of states for
+kaolinite showing the high frequency region for both
+reference DFT methods. The dashed red line shows the
+result for revPBE + D3, the dashed blue line shows the
+result for revPBE + vdW and the black line shows the
+IR spectra that was obtained from experiments run at
+1.5 K. All three spectra have been normalised to have
+unit area.
+contrast, all the fine structure at high frequency disap-
+pears and there is thus only one high-frequency peak.
+0
+200
+400
+600
+800
+1000
+1200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+3600
+3800
+0 K
+10 K
+50 K
+100 K
+300 K
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ω / cm−1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Relative Intensity
+FIG. 5: Vibrational densities of states for kaolinite
+showing the low frequency region (left) and the high
+frequency region (right) at a range of different
+temperatures. These densities of states were computed
+using ClayNN-D3. The black line shows the 0 K result,
+which was calculating by finding the PDOS. The blue,
+yellow, orange and red lines are then the results
+obtained at 10 K, 50 K, 100 K and 300 K respectively.
+As discussed in section IV A kaolinite contains many
+light atoms. It is therefore important to consider the role
+of nuclear quantum effects. This holds doubly true for the
+dynamic properties, which are more sensitive to factors
+affecting interatomic interactions. Figure 6 shows what
+happens to the spectrum when nuclear quantum effects
+are included. The blue line in this figure is the classical
+result for revPBE + D3 at 300 K that was shown in red
+in figure 5. Meanwhile, the red line shows the spectrum
+that was obtained from PACMD simulations at 300 K.
+You can clearly see that the peak in the high frequency
+spectrum is red shifted when nuclear quantum effects are
+included in the model. This result reflects the weaken-
+ing of O–H bonds within the system when NQEs are
+included, and agrees with previous work which employed
+the revPBE + D3 functional for aqueous systems92. In
+both that work and ours, there was good agreement
+between the computed and experimentally-obtained IR
+spectra when NQEs were not included.
+However, the
+O–H frequencies were red shifted away from the experi-
+mentally values when NQE were modelled explicitly. As
+such, the good fit between low-level theory and experi-
+ment is fortuitous, as any result which does not account
+for the quantum mechanical nature of the nuclei is inher-
+ently flawed. Therefore, using path integral techniques to
+accurately model the NQEs is essential when performing
+these comparisons with experiment that determine the
+level of theory that is required to describe the electronic
+degrees of freedom accurately.
+V.
+CONCLUSION
+In this work, we have fitted neural network potentials
+for kaolinite using data extracted from single point cal-
+culations that were performed using the revPBE + D3
+and revPBE + vdW functionals. We find that revPBE +
+vdW does an excellent job of reproducing the experimen-
+tally observed static properties of the material. However,
+revPBE + D3 is better at reproducing the IR spectrum.
+We also compared the static and dynamic properties
+that are obtained when a fully-quantum-treatment of nu-
+clei is employed with the results that are obtained from
+classical simulations. We find that NQEs do not have
+a substantial impact on the static properties. However,
+there is a substantial change in the dynamic properties
+when NQEs are included.
+Overall, our work further illustrates the rich physical
+insight that can be obtained by comparing simulation
+and experiment. By employing functionals that use dif-
+ferent levels of theory researchers can extract meaningful
+insight into the physics that matter for a particular sys-
+tem. Furthermore, the advances in machine learning that
+have taken place over the last 15 years now mean that it
+is possible to perform these investigations on chemically-
+complex systems such as kaolinite.
+ACKNOWLEDGEMENTS
+S.S. gratefully acknowledges the support of the North-
+ern Ireland Department for the Economy (NI-DfE). We
+also appreciate the provision of any and all computa-
+
+9
+0
+200
+400
+600
+800
+1000
+1200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+3400
+3500
+3600
+3700
+3800
+Classical MD
+PACMD
+Experimental
+ω / cm−1
+Relative Intensity
+FIG. 6: Vibrational densities of states for kaolinite showing the low frequency region (left) and the high frequency
+region (right). These densities of states were computed at 300 K using ClayNN-D3 and both classical nuclei (blue
+line) and path integrals (red line). Experimental data13 (black line) at this temperature was only recorded for the
+high frequency region. We highlight the large shift away from the experimentally obtained O–H peak positions
+within the high frequency domain when we include NQEs.
+tional resources used throughout the creation of this
+work.
+We thank both the UK Materials and Molecu-
+lar Modelling Hub for computational resources, which
+is partially funded by EPSRC (EP/P020194/1) and
+EP/T022213/1), and the Northern Ireland High Perfor-
+mance Computing (NI-HPC) service funded by EPSRC
+(EP/T022175).
+The authors thank Rhys Bunting and
+Ethan Crawford for helpful comments on the manuscript.
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+E. Passaro, C. Johnston, A. Merkys, A. Cepellotti, N. Mounet,
+N. Marzari, B. Kozinsky,
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+M. Parrinello,
+T. E. Markland,
+and D. E.
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+B. A. Helfrecht, P. Juda, S. P. Bienvenue, W. Fang, J. Kessler,
+I. Poltavsky, S. Vandenbrande, J. Wieme, C. Corminboeuf, T. D.
+K¨uhne, D. E. Manolopoulos, T. E. Markland, J. O. Richardson,
+A. Tkatchenko, G. A. Tribello, V. Van Speybroeck, and M. Ce-
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+simulations,” Computer Physics Communications 236, 214–223
+
+12
+(2019).
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+materials science,” Scripta Materialia 108, 1–5 (2015).
+92O. Marˇs´alek and T. E. Markland, “Quantum Dynamics and Spec-
+troscopy of Ab Initio Liquid Water: The Interplay of Nuclear and
+Electronic Quantum Effects,” The Journal of Physical Chemistry
+Letters 8, 1545–1551 (2017).
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+sentation of High-Dimensional Potential-Energy Surfaces,” Phys.
+Rev. Lett. 98, 146401 (2007).
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+ing high-dimensional neural network potentials,” J. Chem. Phys.
+134, 074106 (2011).
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+Kalman filters for training feedforward neural networks,” J.
+Chemometrics 8, 391–407 (1994).
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+statistics–classical dynamics method is best for water?” Faraday
+Discussions 221, 350–366 (2020).
+
+13
+Supplementary Information
+VI.
+GITHUB REPOSITORY
+Exemplary input files are provided for a number of the calculations and simulations performed throughout this
+work. These are available within our github repository. Efforts have been taken to make these files as ‘general’ as
+possible, however we recommend anyone wishing to use these models ‘out of the box’ to first familiarise themselves
+with the I-PI90, LAMMPS89 and N2P279,80 codes.
+VII.
+OVERVIEW OF CLAYNN MODEL
+We first discuss the atomic descriptors that are used to translate the chemical structure of the kaolinite system to
+a machine learnable input. We then follow this with a brief description of the architecture and training/expansion
+procedure of the potentials.
+A.
+Symmetry Functions
+The atomic structure of the kaolinite system is translated to a more ‘machine-learnable’ form of input through the
+use of atom-centered symmetry functions (ACSFs). Three varieties of these functions were used to provide a suitable
+environmental descriptor for all atomic configurations. These functions are given within the literature93,94 as G2, G3
+and G9 type symmetry functions respectively. The form of the cutoff function and the symmetry functions are shown
+here as Eqns 1, 2, 3 and 4 respectively.
+fc(r) = tanh3(1 − r
+rc
+)
+(1)
+G2
+i =
+�
+j̸=i
+e−η(rij−rs)2fc(rij)
+(2)
+G3
+i = 21−ζ �
+j̸=i
+j<k
+(1 + λ cos θijk)ζe−η[(rij−rs)2+(rik−rs)2+(rjk−rs)2]fc(rij)fc(rik)fc(rjk)
+(3)
+G9
+i = 21−ζ �
+j̸=i
+j<k
+(1 + λ cos θijk)ζe−η[(rij−rs)2+(rik−rs)2]fc(rij)fc(rik)
+(4)
+The complete set of symmetry functions we generated is provided within our github repository. We use cutoffs of 8
+and 12 Bohr when creating the symmetry function set, yielding a ‘fine’ description of atomic environments at short
+ranges whilst maintaining distinction over longer distances.
+These functions were produced using an iterative python script (which we provide within our github). In order to
+adequately lower the number of functions used to a more representative set, we utilised the pruning feature within of
+N2P2. This procedure calculated all symmetry function values encountered within the training data and removed any
+functions which didn’t sufficiently contribute to the distinction of atomic environments. The metric used to determine
+this is the range of values spanned by the symmetry function when considering all the training data. This process
+results in a final symmetry function set of 511 functions, a further breakdown of these functions is given in Table. II
+B.
+Architecture and Training
+We use a typical architecture for the elemental neural networks by employing a single input layer with the number
+of nodes determined by the number of symmetry functions for that element. This is then followed by two hidden
+layers, each with 25 nodes, and a final output layer consisting of a single node.
+
+14
+Function
+Element G2 G3 G9 Total
+H
+29 11 116
+156
+O
+39 14 160
+213
+Al
+10
+0
+52
+62
+Si
+15
+4
+61
+80
+TABLE II: Final Symmetry Functions sets assigned to each element broken down by function type.
+We provide the reference free atomic energies, system energies, and atomic forces to the neural network potential
+in atomic units. These can be seen within the respective input files in the github repository. We further sanitize our
+data by adjusting the energies provided within the training data. This is achieved through the subtraction of the
+mean system energy of the dataset from the system energy of every configuration.
+The weights are updated through the use of the Kalman filter95 implemented within N2P2. The training procedure
+is allowed to run for 25 epochs. After these epochs have completed, we select the most accurate epochs weights,
+taking care to ensure that the potential has not been overfit to the training data, and use these as the weights for our
+potential.
+The expansion and improvement of the potentials is achieved through a procedure named query by committee, a
+process first employed by Schran et al83. We trained ten other potentials using the same training data but differing
+randomization seeds. This procedure enables us to distinguish between inaccuracies within the training data and
+regions of the potential energy surface that are inadequately sampled through analysis of the standard deviation of
+the committee predictions. This procedure was repeated until adequately good coverage over all required phase spaces
+was achieved. This procedure was used initially to create a NVT ‘stable’ potential and then repeated for the NST
+ensemble.
+C.
+Validation of ClayNN Potentials
+We validate the models used throughout by the process of query by committee. We computed the RMSE and
+standard deviation of the predictions of the committee of potentials. To this end, we randomly sample and recompute
+500 frames from both the ClayNN-D3 and ClayNN-vdW driven classical NVT and NST simulations run at 300K.
+This provides us with a measure of both static and dynamic accuracy of the potentials. These are presented in Fig.
+7 and Fig. 8 for ClayNN-D3 and ClayNN-vdW respectively. We split these into frames from both the NVT and
+NST simulations to provide clearer distinction between the two ensembles and the regions of phase space which they
+explore. A similar procedure for the quantum mechanical simulations was not completed. This recalculation with the
+baseline DFT functional would incur the same computational expense as running ab initio path integral molecular
+dynamics and ultimately defeat the purpose of using MLIPs.
+As can be seen from both Fig. 7 and Fig. 8, the decrease of accuracy moving from NVT to NST is low. The RMSE
+increases whilst remaining relatively low for both the predicted energy and forces of the kaolinite system studied
+using both potentials. This is reasonably sensible, as the change of ensemble from NVT, which ensured a constant cell
+volume, to the NST ensemble allowed for a much larger region of chemical phase space to be explored. This results
+in slightly larger errors and standard deviation values across both potentials.
+We note the relatively large error bars associated with the energetic predictions for both ClayNN-D3 and ClayNN-
+vdW with NST predictions. We computed the errors as the standard deviation of the committee potentials at 0.36
+meV/atom (54.64%) and 0.35 meV/atom (62.85%) respectively. We attribute this to the disparity of available system
+energies within the reference data. The amount of data available equates to N and 3n(N) points respectively where
+N is the number of frames and n the number of atoms. Ultimately, while the energy predictions are a useful indicator
+of phase space coverage, they provide a limited understanding into the performance of the potential when driving
+molecular dynamics. Regardless of this fact, we present the energy predictions of the potentials as a general ‘sign of
+wellness’ for both NVT and NST simulations at the conditions studied.
+In contrast to the energy predictions, the force predictions of both potentials remains very accurate throughout
+both ensembles. We computed a maximum error of 45.54 meV/˚A and 45.27 meV/˚A corresponding to the ClayNN-D3
+NVT and ClayNN-vdW NST predictions respectively. In addition to this, we report extremely low standard deviation
+values for both potentials for the sampled trajectory at 9.97 meV/˚A (21.89%)and 10.7 meV/˚A (23.64%) respectively.
+This corresponds to well covered regions of the phase space under the conditions we study within this work, and
+allows us to confidently use these potentials as surrogates to DFT level calculations.
+Overall, we observed that throughout all simulations, the potential was extremely accurate when predicting on
+
+15
+structures with cell parameters similar to those that were prevalent within the training data. As such, the energy
+and force errors are extremely low throughout these simulations and we observe very few (¡5) symmetry function
+extrapolation warnings during these simulations. When we consider NST simulations, it is of little surprise that
+deformations away from the cell parameters that dominate the training data incur an increase in the errors associated
+with the predicted energy and forces of the system. In order to correct this, a thorough coverage of the phase space
+accessible to the system when simulating using NST at 300 K would need to be accounted for and shown to the
+potential. We deem this as unnecessary, as ultimately, we see no statistically significant increase in the errors of the
+predicted atomic forces when simulated using the NST ensemble. We therefore conclude that both ClayNN-D3 and
+ClayNN-vdW are suitable for both ensembles in their current form.
+VIII.
+KAOLINITE CHARACTERISATION
+A.
+Cell Parameters
+All reported cell parameters were obtained from geometry/cell optimization calculations using the reference DFT
+functionals or respective NNPs, or obtained as ensemble averages from NST molecular dynamics simulations. DFT
+cell optimization was completed using the 2 x 2 x 2 kaolinite supercell with CP2K while NNP optimization was
+completed using LAMMPS. We provide input files for both of these calculations within our github.
+In addition, we report that during simulations involving the ClayNN-D3 potential with and without NQEs, we
+observed definite warpings of the cell, with extreme deviations of cell parameter C, α and β at prolonged simulation
+times (¿ 1 ns). We attribute the revPBE + D3 functional and it’s description of the interlayer interactions, as this was
+not something we observed when using the ClayNN-vdW potential. When we included NQEs within these simulations,
+this warping occurred on a shorter timescale (¿ 250 ps), and as a result, we attribute this to the description of the
+intermolecular interactions between the kaolinite layers. We note that we did not include any portion of the trajectory
+after this phenomenon was observed within the statistical averages when reporting cell parameters or sampling frames
+for simulations involving dynamical properties.
+B.
+Radial Distribution Functions
+Figures 9 and 10 show the radial distribution functions for all elemental pairs within kaolinite computed using
+ClayNN-D3 and ClayNN-vdW respectively. The dashed blue lines represent the classically obtained results while the
+dashed red lines represent those calculated when we include NQEs. Figure 11 shows the classically obtained RDFs
+for ClayNN-D3 and ClayNN-vdW as the dashed blue line and dashed red line respectively. The main results from
+both Figure 9 and 10 have been discussed within the main text. We plot Figure 11 to more clearly compare the
+differences in modelling provided by the respective functionals. It can be easily seen that both functionals provide a
+similar description of most interactomic interactions within the system. The largest differences are seen in gH–H(r)
+and gH–Si(r), which show a substantial difference in the location of the first maxima for both functions of around 0.1
+- 0.2 ˚A and further differences in the more distant regions of both functions.
+C.
+Vibrational Density of States
+All reported phonon densities of states (PDOS) were obtained through the use of phonopy interfaced with CP2K.
+The workflow that we followed during this process is available on our github, along with a short configuration file we
+used to create the final plots. We attribute the differences between the NNP predicted phonon density of states and
+that computed using DFT as being due to the intrinsic error incurred through the learning procedure.
+Figure 12 shows the vibrational density of states (VDOS) computed using simulations driven by ClayNN-D3 at a
+range of temperatures. These temperatures range from 0 K (as in, the PDOS) to 300 K. A number of these were
+omitted from the figure presented within the main text for clarity. We observe significant degradation of the peak
+positions due to thermal effects upon an increase in temperature of 50 K. This is most clearly seen within the high
+frequency region of the VDOS (the right panel), where clear distinction of the two major peaks is sufficiently blurred
+at 50 K to leave it difficult to assign with confidence.
+When including NQEs in our study of the dynamical properties of kaolinite, we used both thermostatted ring
+polymer molecular dynamics86,87 (TRPMD) and partially adiabatic centroid molecular dynamics88 (PACMD). Results
+obtained using PACMD are presented within the main text. The results obtained using TRPMD suffered from a
+
+16
+well known ‘peak broadening’96 problem and were unsuitable for presentation. Figure 15 shows the differences in
+vibrational spectrum obtained using PACMD and TRPMD as the red and blue lines respectively.
+Due to our conclusion that ClayNN-D3, and by extension, revPBE + D3, provided a better description of the
+vibrational properties of the kaolinite systme, we did not present any results obtained using ClayNN-vdW. Figure 14
+shows the VDOS computed classically and using PACMD in the blue and red lines respectively. These results were
+also obtained from simulations performed at 300 K. We note the relatively good agreement in peak position of the
+classical VDOS obtained using ClayNN-vdW to experimental results (pictured in black within the high frequency
+region). It is useful to reiterate that this result is entirely fortuitous, as the classically obtained results do not consider
+the quantum mechanical nature of the nuclei within the system. The agreement with experiment is therefore tainted
+as a result of this fact. Finally. figure 15 shows the VDOS computed using PACMD and TRPMD as the red and blue
+lines respectively.
+IX.
+MISCELLANEOUS
+A.
+Path Integral Bead Convergence
+To ensure we used a sufficient number of replicas when performing path integral molecular dynamics, we converged
+the number of beads with respect to the total system energy. We simulated the system in the NVT ensemble using
+4, 8, 16 and 32 beads. The difference between 16 and 32 beads was calculated via the difference of the mean total
+energy, which we compute as 0.02 atomic units. As the total energy was computed as the referential energy within
+the ClayNN-D3 potential, we add this back to the values before computation of the percentage difference between the
+values. We present the system energy trajectories for these simulations within Fig 16.
+
+17
+−0.40
+−0.38
+−0.36
+−0.34
+−0.32
+−0.30
+−0.28
+Reference Value (a.u)
+−0.40
+−0.38
+−0.36
+−0.34
+−0.32
+−0.30
+−0.28
+Predicted Value (a.u)
+RMSE = 0.35847 meV/atom
+STD = 0.30056 meV/atom
+Energy Correlation Plot - NvT Subset
+Reference
+Prediction
+−0.38
+−0.36
+−0.34
+−0.32
+−0.30
+−0.28
+−0.26
+Reference Value (a.u)
+−0.40
+−0.38
+−0.36
+−0.34
+−0.32
+−0.30
+−0.28
+−0.26
+Predicted Value (a.u)
+RMSE = 0.66553 meV/atom
+STD = 0.36363 meV/atom
+Energy Correlation Plot - NsT Subset
+−4
+−2
+0
+2
+4
+Reference Value (eV/˚A)
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+Predicted Value (eV/˚A)
+RMSE = 45.54496 meV/˚A
+STD = 9.96860 meV/˚A
+Force Correlation Plot - NvT Subset
+−4
+−2
+0
+2
+Reference Value (eV/˚A)
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+Predicted Value (eV/˚A)
+RMSE = 42.74805 meV/˚A
+STD = 9.60939 meV/˚A
+Force Correlation Plot - NsT Subset
+FIG. 7: Committee validation plots for the revPBE + D3 potential, ClayNN-D3 with red points and black lines
+indicating the mean committee prediction and standard deviation of the committee prediction respectively, with the
+blue lines representing the reference value. Upper plots show energy validation for both NVT (left) and NST (right)
+while lower plots show force validation for both NVT (left) and NST (right).
+
+18
+−0.30 −0.28 −0.26 −0.24 −0.22 −0.20 −0.18
+Reference Value (a.u)
+−0.32
+−0.30
+−0.28
+−0.26
+−0.24
+−0.22
+−0.20
+−0.18
+Predicted Value (a.u)
+RMSE = 0.55826 meV/atom
+STD = 0.24990 meV/atom
+Energy Correlation Plot - NvT Subset
+Reference
+Prediction
+−0.32
+−0.30
+−0.28
+−0.26
+−0.24
+−0.22
+Reference Value (a.u)
+−0.32
+−0.30
+−0.28
+−0.26
+−0.24
+−0.22
+Predicted Value (a.u)
+RMSE = 0.56347 meV/atom
+STD = 0.35414 meV/atom
+Energy Correlation Plot - NsT Subset
+Reference
+Prediction
+−4
+−2
+0
+2
+Reference Value (eV/˚A)
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+Predicted Value (eV/˚A)
+RMSE = 44.73245 meV/˚A
+STD = 10.54304 meV/˚A
+Force Correlation Plot - NvT Subset
+−4
+−2
+0
+2
+Reference Value (eV/˚A)
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+Predicted Value (eV/˚A)
+RMSE = 45.26973 meV/˚A
+STD = 10.70550 meV/˚A
+Force Correlation Plot - NsT Subset
+FIG. 8: Committee validation plots for the revPBE + vdW potential, ClayNN-vdW with red points and black lines
+indicating the mean committee prediction and standard deviation of the committee prediction respectively, with the
+blue lines representing the reference value. Upper plots show energy validation for both NVT (left) and NST (right)
+while lower plots show force validation for both NVT (left) and NST (right).
+
+19
+0
+1
+2
+3
+4
+5
+6
+0.0
+0.5
+1.0
+1.5
+2.0
+gH−H(r)
+0
+1
+2
+3
+4
+5
+6
+0
+5
+10
+15
+gO−Al(r)
+CL
+NQE
+0
+1
+2
+3
+4
+5
+6
+0
+10
+20
+gH−O(r)
+0
+1
+2
+3
+4
+5
+6
+0
+10
+20
+gO−Si(r)
+0
+1
+2
+3
+4
+5
+6
+0
+2
+4
+6
+8
+gH−Al(r)
+0
+1
+2
+3
+4
+5
+6
+0
+5
+10
+15
+gAl−Al(r)
+0
+1
+2
+3
+4
+5
+6
+0.0
+0.5
+1.0
+1.5
+2.0
+gH−Si(r)
+0
+1
+2
+3
+4
+5
+6
+0
+2
+4
+6
+gAl−Si(r)
+0
+1
+2
+3
+4
+5
+6
+r(˚
+A)
+0
+2
+4
+gO−O(r)
+0
+1
+2
+3
+4
+5
+6
+r(˚
+A)
+0
+5
+10
+15
+gSi−Si(r)
+FIG. 9: Radial distribution functions (RDFs) for all elemental pairs within the kaolinite system calculated from
+revPBE + D3 trained neural network driven molecular dynamics. Blue dashed lines show the RDF calculated from
+classical molecular dynamics while the red dashed lines show the RDF calculated when accounting for quantum
+effects using PIMD.
+
+20
+0
+1
+2
+3
+4
+5
+6
+0
+1
+2
+gH−H(r)
+0
+1
+2
+3
+4
+5
+6
+0
+5
+10
+gO−Al(r)
+CL
+NQE
+0
+1
+2
+3
+4
+5
+6
+0
+10
+20
+gH−O(r)
+0
+1
+2
+3
+4
+5
+6
+0
+10
+20
+gO−Si(r)
+0
+1
+2
+3
+4
+5
+6
+0
+2
+4
+6
+8
+gH−Al(r)
+0
+1
+2
+3
+4
+5
+6
+0
+5
+10
+gAl−Al(r)
+0
+1
+2
+3
+4
+5
+6
+0
+1
+2
+gH−Si(r)
+0
+1
+2
+3
+4
+5
+6
+0
+2
+4
+6
+gAl−Si(r)
+0
+1
+2
+3
+4
+5
+6
+r(˚
+A)
+0
+1
+2
+3
+4
+gO−O(r)
+0
+1
+2
+3
+4
+5
+6
+r(˚
+A)
+0
+5
+10
+gSi−Si(r)
+FIG. 10: Radial distribution functions (RDFs) for all elemental pairs within the kaolinite system calculated from
+revPBE + vdW trained neural network driven molecular dynamics. Blue dashed lines show the RDF calculated
+from classical molecular dynamics while the red dashed lines show the RDF calculated when accounting for
+quantum effects using PIMD.
+
+21
+0
+1
+2
+3
+4
+5
+6
+0
+1
+2
+gH−H(r)
+0
+1
+2
+3
+4
+5
+6
+0
+5
+10
+15
+gO−Al(r)
+revPBE + D3
+revPBE + vdW
+0
+1
+2
+3
+4
+5
+6
+0
+10
+20
+gH−O(r)
+0
+1
+2
+3
+4
+5
+6
+0
+10
+20
+gO−Si(r)
+0
+1
+2
+3
+4
+5
+6
+0
+2
+4
+6
+8
+gH−Al(r)
+0
+1
+2
+3
+4
+5
+6
+0
+5
+10
+15
+gAl−Al(r)
+0
+1
+2
+3
+4
+5
+6
+0
+1
+2
+gH−Si(r)
+0
+1
+2
+3
+4
+5
+6
+0
+2
+4
+6
+gAl−Si(r)
+0
+1
+2
+3
+4
+5
+6
+r(˚
+A)
+0
+2
+4
+gO−O(r)
+0
+1
+2
+3
+4
+5
+6
+r(˚
+A)
+0
+5
+10
+15
+gSi−Si(r)
+FIG. 11: Radial distribution functions (RDFs) for all elemental pairs within the kaolinite system calculated using
+both the revPBE + D3 (blue) and revPBE + vdW (red) functionals. All functionals shown here are obtained from
+classical molecular dynamics, in order to isolate the effects of different dispersion corrections when simulating the
+kaolinite system.
+
+22
+0
+200
+400
+600
+800
+1000
+1200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+3500
+3600
+3700
+3800
+3900
+0 K
+10 K
+20 K
+30 K
+40 K
+50 K
+75K
+100 K
+300 K
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ω / cm−1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Relative Intensity
+FIG. 12: Computed VDOS for the various studied temperatures using ClayNN-D3 plotted against the computed
+PDOS. We present the low frequency (< 1200 cm−1) region and high frequency (> 3500 cm−1) on the left and right
+of the figure respectively.
+
+23
+0
+200
+400
+600
+800
+1000
+1200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Relative Intensity
+PACMD
+TRPMD
+3400
+3600
+3800
+ω / cm−1
+FIG. 13: Vibrational density of states computing using ClayNN-D3 within both the TRPMD (blue line) and
+PACMD (red line) formalism of path integral molecular dynamics.
+0
+200
+400
+600
+800
+1000
+1200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Relative Intensity
+3400
+3600
+3800
+ω / cm−1
+Classical MD
+PACMD
+Experimental
+FIG. 14: Vibrational density of states of kaolinite computed using ClayNN-vdW within both classical mechanics
+(blue line) and path integrals (red line). We include the experimentally obtained IR spectra of the high frequency
+region (right panel) to highlight the differences between the computationally obtained result and experiment when
+using this parameterization of ab initio theory.
+
+24
+0
+200
+400
+600
+800
+1000
+1200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Relative Intensity
+PACMD
+TRPMD
+3400
+3600
+3800
+ω / cm−1
+FIG. 15: Vibrational density of states computed using ClayNN-vdW within both the TRPMD (blue line) and
+PACMD (red line) formalism of path integral molecular dynamics.
+10000
+20000
+30000
+40000
+50000
+Trajectory Step
+−0.2
+−0.1
+0.0
+0.1
+0.2
+0.3
+Predicted Energy
+4 replicas
+8 replicas
+16 replicas
+32 replicas
+FIG. 16: Total system energy of NVT simulations separated based on the number of replicas used within the
+simulation. We present these as 4 replica (blue lines), 8 replica (green lines), 16 replica (yellow lines) and 32 replica
+(red lines) simulations. The predicted energy is presented rather than the total energy, as the mean system energy is
+subtracted prior to training.
+

UtE1T4oBgHgl3EQfawTU/content/tmp_files/2301.03166v1.pdf.txt

@@ -0,0 +1,2215 @@
+Improving Energy Saving of One-sided Matrix
+Decompositions on CPU-GPU Heterogeneous Systems
+Jieyang Chen
+[email protected]
+University of Alabama at Birmingham
+Birmingham, Alabama, USA
+Xin Liang
+[email protected]
+University of Kentucky
+Lexington, Kentucky, USA
+Kai Zhao
+[email protected]
+University of Alabama at Birmingham
+Birmingham, Alabama, USA
+Hadi Zamani Sabzi
+[email protected]
+University of California, Riverside
+Riverside, California, USA
+Laxmi Bhuyan
+[email protected]
+University of California, Riverside
+Riverside, California, USA
+Zizhong Chen
+[email protected]
+University of California, Riverside
+Riverside, California, USA
+Abstract
+One-sided dense matrix decompositions (e.g., Cholesky, LU,
+and QR) are the key components in scientific computing in
+many different fields. Although their design has been highly
+optimized for modern processors, they still consume a con-
+siderable amount of energy. As CPU-GPU heterogeneous
+systems are commonly used for matrix decompositions, in
+this work, we aim to further improve the energy saving of one-
+sided matrix decompositions on CPU-GPU heterogeneous
+systems. We first build an Algorithm-Based Fault Tolerance
+protected overclocking technique (ABFT-OC) to enable us
+to exploit reliable overclocking for key matrix decomposi-
+tion operations. Then, we design an energy-saving matrix
+decomposition framework, Bi-directional Slack Reclamation
+(BSR), that can intelligently combine the capability provided
+by ABFT-OC and DVFS to maximize energy saving and
+maintain performance and reliability. Experiments show that
+BSR is able to save up to 11.7% more energy compared with
+the current best energy saving optimization approach with no
+performance degradation and up to 14.1% 𝐸𝑛𝑒𝑟𝑔𝑦×𝐷𝑒𝑙𝑎𝑦2 re-
+duction. Also, BSR enables the Pareto efficient performance-
+energy trade-off, which is able to provide up to 1.43× perfor-
+mance improvement without costing extra energy.
+CCS Concepts: • Hardware → Power and energy; • Com-
+puter systems organization → Dependable and fault-tolerant
+systems and networks.
+Keywords: GPU, matrix decomposition, energy saving, fault
+tolerance
+Permission to make digital or hard copies of all or part of this work for
+personal or classroom use is granted without fee provided that copies are not
+made or distributed for profit or commercial advantage and that copies bear
+this notice and the full citation on the first page. Copyrights for components
+of this work owned by others than ACM must be honored. Abstracting with
+credit is permitted. To copy otherwise, or republish, to post on servers or to
+redistribute to lists, requires prior specific permission and/or a fee. Request
+permissions from [email protected].
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+© 2023 Association for Computing Machinery.
+ACM ISBN 979-8-4007-0015-6/23/02...$15.00
+https://doi.org/10.1145/3572848.3577496
+1
+Introduction
+To meet performance requirements for current mission-critical
+scientific computing, millions of computing cores are equipped
+in modern High Performance Computing (HPC) systems con-
+suming tens of megawatts of power [16]. With the increasing
+need for higher performance, it is anticipated that future HPC
+systems will consist of even more computing cores and con-
+sume more power. As HPC systems are achieving higher
+parallelism, how to achieve high performance and energy ef-
+ficiency while ensuring computing reliability has become a
+critical challenge for scientific computing.
+As the type of processor that contributes the most of the
+computing parallelism in many current and future HPC sys-
+tems, Graphics Processing Units (GPUs), equipped with thou-
+sands of low-power cores, offer high computational power
+and energy efficiency. Many applications and libraries have
+been designed and optimized for GPU accelerators [1, 3, 8, 9,
+13, 25, 34, 36, 42, 43]. Benefiting from the fact that GPUs are
+designed for highly parallelizable computations while CPUs
+are more efficient with serial computations, CPUs and GPUs
+that are linked through fast interconnections [30, 31] are usu-
+ally used together to form heterogeneous systems that can
+efficiently handle a large spectrum of scientific computing
+workloads. Many scientific applications or software begin
+to have an optimized design for CPU-GPU heterogeneous
+systems such as the MAGMA linear algebra library [15].
+One-sided dense matrix decomposition such as Cholesky,
+LU, and QR play a pivotal role in many scientific applica-
+tions. Their state-of-the-art designs for CPU-GPU heteroge-
+neous systems are proposed in [44, 45], and they have been
+highly optimized in the MAGMA library and used as key
+computational kernels by many applications across different
+fields [18, 21, 23, 24, 35, 46, 53].
+Many works have been done to improve the energy effi-
+ciency of linear algebra operations. [7] proposed to use a
+DVFS-based approach to optimize matrix decompositions.
+[2, 51, 52] seek to use reduced core supply voltage to reduce
+the energy consumption of matrix-matrix multiplication op-
+erations. Although much work has been done to improve the
+arXiv:2301.03166v1  [cs.DC]  9 Jan 2023
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Jieyang Chen, Xin Liang, Kai Zhao, Hadi Zamani Sabzi, Laxmi Bhuyan, and Zizhong Chen
+energy saving of matrix decomposition on CPU-GPU hetero-
+geneous systems, it is still desirable to further improve their
+energy saving since matrix decompositions as they still con-
+sume a considerable amount of energy. Improving the energy
+saving of matrix decomposition can lead to more energy-
+efficient scientific computing. However, the major challenge,
+as pointed out in [27, 28, 38, 51, 52], is that aggressive energy-
+saving optimizations can weaken the reliability of the system
+and cause performance degradation, which is unacceptable
+for time-sensitive and mission-critical scientific applications.
+In this work, we aim to further improve the energy saving
+of one-sided matrix decompositions on CPU-GPU hetero-
+geneous systems while maintaining performance and reli-
+ability. We first build an Algorithm-Based Fault Tolerance
+protected overclocking technique (ABFT-OC) to enable us
+to exploit reliable overclocking for key matrix decomposi-
+tion operations. Then, we design an energy-saving matrix
+decomposition framework, Bi-directional Slack Reclamation
+(BSR), that can intelligently combine the capability provided
+by ABFT-OC and Dynamic Voltage and Frequency Scaling
+(DVFS) to maximize energy saving and maintain perfor-
+mance and reliability. Also, BSR enables the Pareto efficient
+performance-energy trade-off. Specifically, our contributions
+are listed as follows:
+• We propose the first adaptive algorithm-based fault tol-
+erance protected overclocking technique (ABFT-OC)
+for matrix decompositions on CPU-GPU heterogeneous
+systems. Overclocking with an optimized voltage guard-
+band can enable us to exploit higher clock frequen-
+cies with higher energy efficiency. However, aggressive
+overclocking can decrease system reliability, so we
+propose to couple ABFT with overclocking to enable
+trustable computation. To reduce fault tolerance over-
+head, we further propose a lightweight adaptive-ABFT
+technique that automatically adjusts its fault tolerance
+strength according to the error rate.
+• Next, based on ABFT-OC, we propose a novel slack-
+based energy saving framework - Bi-directional Slack
+Reclamation (BSR), which aims to exploit slack, pro-
+cessor idle time, to save energy and enable flexible
+Pareto efficient performance-energy trade-off. Differ-
+ent from existing works, BSR reclaims slack in both
+directions using both ABFT-OC and DVFS to save
+more energy and enable performance improvement.
+• We implement our BSR on three key one-sided matrix
+decompositions: Cholesky, LU, and QR. We evaluate
+our implementation on a modern CPU-GPU hetero-
+geneous system with Nvidia GPU. Experiments show
+that BSR is able to save up to 11.7% more energy com-
+pared with the current best energy saving optimiza-
+tion approach with no performance degradation and up
+to 14.1% 𝐸𝑛𝑒𝑟𝑔𝑦 × 𝐷𝑒𝑙𝑎𝑦2 reduction. Also, BSR en-
+ables the Pareto efficient performance-energy trade-off,
+which is able to provide up to 1.43× performance im-
+provement without costing extra energy.
+A22
+A12
+A21
+A11
+A'22
+L11
+U11
+L21
+L12
+(a) Blocked algorithm
+PD(k+1)
+Row Panel
+L
+L
+TMU'(k)
+TMU(k)
+On GPU
+On CPU
+Col. Panel
+Tailing 
+Matrix
+PU'(k)
+L
+PU(k)
+Row Panel
+Col. Panel
+Sync.
+Sync.
+Slack
+Slack
+OR
+…
+…
+(b) State-of-the-art design on CPU-GPU heterogeneous systems
+Figure 1. One iteration of LU decomposition
+(a) Double precision
+(b) Single precision
+Figure 2. Slacks occur when decomposing a 30720 × 30720
+matrix on our heterogeneous system. Block size is optimized
+for performance. Positive values represent slacks on the CPU
+side and negative values represent slacks on the GPU side.
+2
+Related Works and Problem Statement
+In this section, we first introduce the design of state-of-the-art
+matrix decomposition on CPU-GPU heterogeneous systems
+and we focus on discussing their key computing character-
+istics. Then, we review how existing works leverage such
+computing characteristics to optimize for energy efficiency.
+Finally, we formulate our research problem and challenges.
+2.1
+State-of-the-art matrix decompositions
+The state-of-the-art matrix decompositions for CPU-GPU het-
+erogeneous systems use the blocked version matrix decom-
+position algorithms. Blocks, logically divided sub-matrices,
+form Panel and Trailing Matrix. The decomposition process
+begins from the up left corner of the matrix and moves to-
+wards the down right corner iteratively. An illustration of one
+iteration of the LU decomposition is shown in Figure 1(a).
+Each iteration includes three major operations: 1 Panel de-
+composition (PD): 𝐿 · 1 × 𝑈 11 ← 𝐴 · 1; 2 Panel update
+
+5000
+(ms)
+4000
+Slack Length (
+3000
+2000
+1000
+0
+1
+11
+21
+31
+41
+51
+Iteration
+Cholesky
+LU
+QR100
+Slack Length (ms)
+0
+-100
+-200
+-300
+-400
+1 11 21 31 41 51 61 71
+Iteration
+-Cholesky
+一LU
+QRImproving Energy Saving of One-sided Matrix Decompositions on CPU-GPU Heterogeneous Systems
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+(PU): 𝑈 12 ← (𝐿11)−1 × 𝐴12; and 3 Trailing matrix update
+(TMU): 𝐴′22 ← 𝐴22 −𝐿21 ×𝑈 12. Cholesky, LU, and QR de-
+composition all share similar three operations. On CPU-GPU
+heterogeneous systems, the three operations are assigned to
+different processors based on their characteristics. PD is as-
+signed to the CPUs since it is highly sequential. PU and TMU
+are assigned to the GPUs as they are high parallelizable. As il-
+lustrated in Figure 1(b), to overlap the computation on CPUs
+and GPUs, a look-ahead optimization [26] is used that allows
+the partial PU and TMU to be done first (i.e., PU’ and TMU’),
+so that the PD of the next iteration can be done with the rest
+of PU and TMU concurrently. Depending on the computa-
+tional power of the CPU/GPU and the amount of workload
+assigned during decomposition, those concurrent tasks may
+finish at different times, which leads to idle computing cycles
+on the CPU or GPU. The idle is called slack. Figure 2 show
+how slack length can change during Cholesky, LU, and QR
+decompositions on our test platform.
+2.2
+Existing slack-based energy saving
+Matrix decompositions have been designed to maximize their
+usage on highly optimized BLAS-3 GPU kernels, so their
+energy efficiency is inherently high, which leaves limited
+room for further optimization. As for now, the most effective
+class of energy-saving optimizations for matrix decomposi-
+tions on CPU-GPU heterogeneous systems is DVFS-based
+approaches, which aim to exploit different energy-saving tech-
+niques when there are slacks.
+TMU’
+TMU
+DtoH
+HtoD
+On GPU
+On CPU
+Slow down CPU 
+to reclaim slack
+PD
+(b) Single directional slack reclamation (SR) 
+PU'
+PU
+TMU’
+TMU
+DtoH
+HtoD
+On GPU
+On CPU
+Halt: Reduce to 
+lowest power state
+PD
+(a) Race-to-Halt (R2H)
+PU'
+PU
+Race: Execute at 
+highest perf.
+Figure 3. Existing slack-based energy saving
+There are two main strategies for optimizing energy costs:
+Race-to-Halt (R2H) [22, 37, 39] and Slack Reclamation (SR) [7].
+As shown in Figure 3, the main idea of R2H is to timely re-
+duce clock frequency to the minimum as soon as the tasks
+on the non-critical path finish. The processor maintains its
+minimum clock frequency during the slack to save energy.
+This strategy is usually implemented by the hardware or the
+operating system leveraging their workload monitoring ca-
+pabilities. SR saves energy by slowing down the tasks on
+the non-critical path. The reason this strategy can save en-
+ergy is due to the relation between the dynamic power of
+the processor and its clock frequency 𝑃𝑑𝑦𝑛𝑎𝑚𝑖𝑐 ∝ 𝑓 2.4 [17].
+Theoretically, SR is able to save more energy compared with
+R2H [7]. Since the processor’s clock frequency need to be
+adjusted before the execution of each task and the length of
+slack can change as shown in Figure 2, some form of compu-
+tation pattern prediction is necessary. In the start-of-the-art
+SR [7], the authors propose to predict computation patterns
+leveraging algorithmic knowledge in matrix decompositions.
+2.3
+Motivation of further improving energy saving
+Despite a lot of research efforts have been made to improve
+the energy saving of matrix decomposition on CPU-GPU
+heterogeneous systems, it is still desirable to further improve
+their energy saving since matrix decompositions are heavily
+used in many scientific applications. Thus improving the
+energy saving of matrix decomposition can lead to more
+energy-efficient scientific computing.
+2.4
+Challenges of further improving energy saving
+2.4.1
+Performance degradation. DVFS is designed to en-
+able performance-energy trade-off while maintaining pro-
+cessor reliability. So, existing DVFS-based energy-saving
+techniques can only be applied to tasks on the non-critical
+path to avoid negatively impacting the overall performance.
+This has already been extensively exploited by existing works.
+To save even more energy, the only other choice is to apply
+DVFS-based energy-saving techniques to tasks on the critical
+path, however, this will inevitably lead to performance degra-
+dation to the overall decomposition since modern CPU and
+GPU processors tend to have better energy efficiency when
+running at lower clock frequencies.
+2.4.2
+Reliability degradation. Other approaches such as
+processor undervolting can also be used to reduce the energy
+cost of computation. Since it works by decreasing the core
+supply voltage while maintaining the same clock frequencies,
+it can save energy without performance degradation. However,
+they can decrease system reliability [2, 51, 52]. Such reliabil-
+ity degradation can be manifested as hard errors (e.g., process
+or system crash) or SDCs (e.g., incorrect calculation, bit-flips
+in memory cells), which can seriously decrease the reliabil-
+ity of matrix decomposition. Although ABFT has been used
+with undervolting in [2, 51] to improve the energy efficiency
+of matrix-matrix multiplications and ensure computing cor-
+rectness, applying existing ABFT techniques can still bring
+considerable performance overhead. This overhead can be
+especially high for matrix decompositions since the iterative
+computing fashion is prone to error propagation, which needs
+the strongest variant of ABFT, full checksum ABFT [4], to
+provide sufficient protection.
+2.5
+Research questions
+In this work, we try to answer the following research ques-
+tions:
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Jieyang Chen, Xin Liang, Kai Zhao, Hadi Zamani Sabzi, Laxmi Bhuyan, and Zizhong Chen
+RQ:1 How to further improve energy saving of matrix de-
+compositions on CPU-GPU heterogeneous system beyond
+the state-of-the-art works?
+RQ:2 How to maximize energy saving for matrix decompo-
+sition while maintaining both performance and reliability at
+the same time?
+RQ:3 How to enable performance-energy trade-off in matrix
+decomposition?
+Hardware 
+Profiling
+Enhanced 
+Online Slack 
+Prediction
+Online 
+Profiling
+Matrix Decomposition
+BSR
+ABFT-OC
+History 
+Profiling Data
+Slack 
+Prediction 
+Model
+At Offline
+At Runtime
+Information
+Control
+DVFS
+Figure 4. Overview of our energy-saving matrix decomposi-
+tion framework
+3
+Design of Energy-Saving Matrix
+Decomposition
+In this work, we propose to build a matrix decomposition
+framework that maximizes energy saving while maintaining
+both performance and reliability at the same time. Figure
+4 shows the overview of our framework. We first focus on
+enabling reliable computation when overclocking by cou-
+pling ABFT with overclocking - ABFT-OC. To reduce fault
+tolerance overhead, we further propose a lightweight adaptive-
+ABFT technique that automatically adjusts its fault tolerance
+strength according to the error rate. Next, based on ABFT-OC,
+we propose a novel slack-based energy saving optimization
+framework - BSR, which aims to exploit slack, to save en-
+ergy and enable flexible Pareto efficient performance-energy
+trade-off. Different from existing works, BSR reclaims slack
+in both directions using both ABFT-OC and DVFS to save
+more energy and enable performance improvement.
+3.1
+Adaptive Algorithm-Based Fault Tolerance
+Protected Overclocking (ABFT-OC)
+To design a technique that maximize energy saving for ma-
+trix decompositions, we seek hardware energy optimization
+techniques beyond DVFS. DVFS has been extensively used
+for energy saving by both hardware and applications. It op-
+timizes energy efficiency by lowering the core voltage (𝑉𝑑𝑑)
+with the decrease of clock frequency for reducing energy
+consumption. However, lowering frequency can inevitably
+cause performance degradation. Processor voltage guardband
+optimization largely mitigates this issue by allowing lowering
+of the core voltage without decreasing clock frequency or
+overclocking without violating the hardware power limit.
+3.1.1
+Voltage guardband optimization for overclocking.
+In this work, we define overclocking as the processor state
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+2
+2.5
+3
+3.5
+4
+4.5
+5
+5.5
+500
+600
+700
+800
+900
+1000
+1100
+1200
+1300
+1400
+1500
+1600
+1700
+1800
+1900
+2000
+2100
+2200
+Power Reduction Factor (⍺)
+Energy Efficiency 
+(Gflops/Watt) 
+Clock Frequency (MHz)
+Default Guardband
+Optimized Guardband
+Power Reduction Factor
+Fault
+Free
+SDC 
+𝑓!"#$%&!%'
+𝑓(#)*+&,
+Fault Free
+(a) GPU energy efficiency
+(b) GPU SDC error rate
+(c) CPU energy efficiency
+0
+20
+40
+60
+80
+500
+700
+900
+1100
+1300
+1500
+1700
+1900
+2100
+Temperature (C)
+Clock Frequency (MHz)
+Default
+Optimized
+(d)
+Maximum
+sustained
+GPU core temperature
+0
+20
+40
+60
+80
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+Temperature (C)
+Clock Frequency (MHz)
+Default
+Optimized
+(e)
+Maximum
+sustained
+CPU core temperature
+Figure 5. Profiling results of our testing CPU and GPU
+where it sustains at a higher-than-default clock frequencies.
+Figure 5 (a) shows the achievable overclocking frequency
+range and their energy efficiency of our test GPU at different
+clock frequencies after we apply voltage guardband optimiza-
+tion. Please note unlike previous works that were based on
+Windows-based GPU driver [27, 29, 51] where the core volt-
+age can be directly adjusted and monitored, the Linux-based
+GPU driver does not allow us to directly control and monitor
+the GPU core voltage. Even though we find that optimizing
+the voltage guardband of GPU on Linux is still achievable
+through the clock offset command of the NVML API on
+
+0.7
+(Num. of Errors/sec)
+OD Error
+1D Error
+2D Error
+0.6
+0.5
+Error Rate
+0.4
+0.3
+0.2
+0.1
+0
+500
+009
+700
+800
+006
+1000
+1200
+1300
+1400
+0
+1600
+1700
+1800
+1900
+2000
+2100
+2200
+5
+Clock Frequency5
+1
+n Factor (α)
+fautoboost
+foverclock
+Energy Efficiency
+4
+0.9
+Gflops/Watt)
+Fault Free
+3
+Fault Free
+Power Reduction
+0.8
+2
+0.7
+1
+fdef ault.
+0
+0.6
+1000 1500 2000 2500 3000 3500 4000 4500
+Clock Frequency (MHz)
+-Default Guardband
+-Optimized Guardband
+- Power Reduction FactorImproving Energy Saving of One-sided Matrix Decompositions on CPU-GPU Heterogeneous Systems
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Linux-based GPU driver. We omit the details due to the page
+limit. CPU undervolting can be directly achieved on the Linux
+system. We set the offset of the CPU core voltage using a
+third-party tool intel-undervolt. Figure 5 (c) shows
+the CPU energy efficiency before and after we set the opti-
+mized voltage guardband. Please note unlike our testing GPU,
+our testing CPU can achieve overclocking with the default
+guardband, but an optimized guardband can help us achieve
+higher energy efficiency.
+Finding the optimized guardband is done by gradually low-
+ering specific power settings of CPU/GPU to the point where
+energy efficiency is maximized without process or OS crash.
+The whole process can be done in less than 30 minutes and it
+only needs to be done once during software installation. As
+optimized guardband can be workload-dependent, we specif-
+ically use the workload in matrix decomposition i.e., TMU
+on GPU and PD on CPU to find optimized guardband. Also,
+as shown in Figure 5 (b), we observe that setting to extreme
+high clock frequencies for the GPU can weaken the reliability
+of computation e.g., SDCs. So, we propose to incorporate
+fault tolerance with overclocking by designing ABFT-OC.
+3.1.2
+Design of ABFT-OC. Reliable computation is the
+foundation of our optimized matrix decomposition. As over-
+clocking achieved through the use of optimized guardband
+can lead to SDCs, we propose to use ABFT [4–6, 10–12, 14,
+19, 20, 32, 33, 40, 41, 47–50] to handle SDCs during matrix
+decompositions. Since the processor power state is under con-
+trol and the corresponding SDC error rate is known, SDC
+error rate is predictable during matrix decompositions. So
+we propose the first ABFT that can adjust its fault tolerance
+strength and overhead at runtime based on the predicted error
+rate to minimize fault tolerance overhead and ensure correct-
+ness. SDC refers to the kind of error that only causes incorrect
+calculation results without process or system crash. When
+using our optimized guardband, the SDC is caused by insuffi-
+cient core voltage supply when at high clock frequencies. The
+rate of SDC can increase as we increase the clock frequency
+when we apply a optimized guardband at the same time.
+A
+checksum(A)
+A'
+checksum(A')
+OP
+Update checksum
+Encode 
+checksum
+Can detect 
+and correct 
+0D error
+0D error
+A
+checksum(A)
+A'
+checksum(A')
+OP
+Update checksum
+Encode 
+checksum
+Can detect 
+and correct 
+1D error
+1D error
+checksum(A)
+checksum(A')
+(a) Single-side checksum ABFT
+(b) Full checksum ABFT
+Figure 6. ABFT checksum for detecting and correcting SDCs
+in matrix operations
+Depending on where the hardware fault occurs, it may be
+manifested as different kinds of SDC. For example, calcula-
+tion error is usually caused by faults in the logic part of ALU
+or FPU. Memory storage error is usually caused by faults
+(e.g., bit flips) in the storage cells of DRAM, cache, or regis-
+ters. For matrix operations, matrix elements can be repeatedly
+accessed to obtain final results. If an element whose value
+is corrupted gets repeatedly referenced, it may cause error
+propagation. Depending on the cause of the error and the
+computation pattern (i.e., how data is used/reused) of a matrix
+operation, the error pattern can be different. The degrees of
+error propagation [4] can be classified as: 0D, 1D, and 2D.
+0D: a single standalone error with no error propagation; 1D:
+an error propagates to entire/part of one row/column; 2D: an
+error propagates beyond one row/column. So, we distinguish
+different degrees of error propagation in Figure 5.
+Table 1. Theoretical estimation on ABFT fault coverage (FC)
+on the TMU operation of the 5𝑡ℎ, 10𝑡ℎ, and 15𝑡ℎ iteration of
+LU decomposition if we apply different clock frequencies.
+Iter.
+ABFT
+1800MHz
+1900MHz
+2000MHz
+2100MHz
+2200MHz
+5𝑡ℎ
+Single
+Fault-free
+Full Coverage
+99.86%
+97.51%
+96.45%
+Full
+Fault-free
+Full Coverage
+Full Coverage
+Full Coverage
+Full Coverage
+10𝑡ℎ
+Single
+Fault-free
+Full Coverage
+99.94%
+98.92%
+98.46%
+Full
+Fault-free
+Full Coverage
+Full Coverage
+Full Coverage
+Full Coverage
+15𝑡ℎ
+Single
+Fault-free
+Full Coverage
+99.98%
+99.76%
+99.65%
+Full
+Fault-free
+Full Coverage
+Full Coverage
+Full Coverage
+Full Coverage
+Algorithm 1: Adaptive-ABFT strategy
+1 Function ABFT-OC():
+In
+:Desired ABFT fault coverage 𝐹𝐶𝑑𝑒𝑠𝑖𝑟𝑒𝑑
+In
+:Desired GPU clock freq. 𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑
+In
+:Default GPU clock freq. 𝐹𝐺𝑃𝑈
+𝐵𝐴𝑆𝐸
+In
+:Predicted operation execution time 𝑇 ′𝐺𝑃𝑈
+2
+𝑆𝑖𝑛𝑔𝑙𝑒𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘 ← 𝐹𝐴𝐿𝑆𝐸
+3
+𝐹𝑢𝑙𝑙𝐴𝐵𝐹𝑇𝐶ℎ𝑒���𝑘 ← 𝐹𝐴𝐿𝑆𝐸
+4
+while (𝜆𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑,0𝐷 > 0 || 𝜆𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑,1𝐷 > 0 ||
+𝜆𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑,2𝐷 > 0) && !𝑆𝑖𝑛𝑔𝑙𝑒𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘 &&
+!𝐹𝑢𝑙𝑙𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘 do
+5
+𝑇𝐺𝑃𝑈
+𝑝𝑟𝑜 𝑗𝑒𝑐𝑡𝑒𝑑 = 𝑇 ′𝐺𝑃𝑈 ×
+𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑
+𝐹𝐺𝑃𝑈
+𝐵𝐴𝑆𝐸
+6
+if 𝐹𝐶𝑠𝑖𝑛𝑔𝑙𝑒 (𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑,𝑇𝐺𝑃𝑈
+𝑝𝑟𝑜 𝑗𝑒𝑐𝑡𝑒𝑑) ⩾ 𝐹𝐶𝑑𝑒𝑠𝑖𝑟𝑒𝑑
+then
+7
+𝑆𝑖𝑛𝑔𝑙𝑒𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘 = 𝑇𝑅𝑈𝐸
+8
+else if 𝐹𝐶𝑓 𝑢𝑙𝑙 (𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑,𝑇𝐺𝑃𝑈
+𝑝𝑟𝑜 𝑗𝑒𝑐𝑡𝑒𝑑) ⩾ 𝐹𝐶𝑑𝑒𝑠𝑖𝑟𝑒𝑑
+then
+9
+𝐹𝑢𝑙𝑙𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘 = 𝑇𝑅𝑈𝐸
+10
+else
+11
+𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 = 𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 − 100𝑀𝐻𝑧
+12
+end
+13
+end
+14 return 𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑, 𝑆𝑖𝑛𝑔𝑙𝑒𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘, 𝐹𝑢𝑙𝑙𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘
+ABFT is based on the idea that if we encode a certain
+amount of matrix information in checksums before a matrix
+operation and apply the same matrix operation to checksums,
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Jieyang Chen, Xin Liang, Kai Zhao, Hadi Zamani Sabzi, Laxmi Bhuyan, and Zizhong Chen
+the checksum relation would still hold for the resulting matrix.
+By verifying the checksum relations after the operation, we
+can detect and correct errors in the result matrix. Depend-
+ing on how much information is encoded in checksums, the
+fault tolerance strength is different. As shown in Figure 6,
+there are two commonly schemes for checksum encoding: 1
+Single side checksum encodes matrices along either rows or
+columns. Since it only encodes the matrix in one dimension,
+it brings relative lower overhead. However, it can only effi-
+ciently tolerate 0D error pattern. 2 Full checksum encodes
+matrices along both rows and column at the same time. Since
+it encodes matrices in both dimensions, it brings stronger
+protection i.e., both 0D and 1D error patterns. However, it
+also brings higher fault tolerance overhead.
+Given that the fault tolerance strength is limited, we must
+determine suitable ABFT protection according to the error
+rate and limit the clock frequency range to ensure all errors
+can be detected and corrected with a high probability. Other-
+wise, undetected or uncorrected errors would cause serious
+error propagation later, which requires recovery with high
+overhead. In this work, we find that it is useful to estimate the
+probability that a certain kind of ABFT can detect and correct
+all errors given different error rates at different overclocking
+frequencies. In order to do that, we first define an error rate
+function 𝑅 given clock frequency derived from our profiling
+results in Figure 5: 𝜆𝑓 ,𝐸𝑟𝑟𝑇 𝑦𝑝𝑒 = 𝑅(𝑓 , 𝐸𝑟𝑟𝑇𝑦𝑝𝑒) where 𝜆 is
+the error rate of a certain error type (𝐸𝑟𝑟𝑇𝑦𝑝𝑒). The error type
+can be 0D, 1D, or 2D. 𝑓 is the processor clock frequency.
+Assuming the rate is constant for a given clock frequency, we
+treat the distribution of probability errors that occur during a
+period of time as the Poisson distribution. So, the probability
+of having 𝑘 errors in a certain type during a period of time
+𝑇 can be estimated using the Poisson distribution function:
+𝑝 =
+𝑒−𝜆𝑓 ,𝐸𝑟𝑟𝑇 𝑦𝑝𝑒𝑇 (𝜆𝑓 ,𝐸𝑟𝑟𝑇 𝑦𝑝𝑒𝑇)𝑖
+𝑖!
+. Both single-side and full check-
+sum encode the matrix for each matrix block individually.
+They cannot tolerate more than one fault strike to a matrix
+block during one error detection interval (i.e., one iteration of
+matrix decomposition). Assuming the matrix is of size 𝑛 with
+matrix block size 𝑏, single-side checksum ABFT can tolerate
+up to 𝑆 = 𝑛
+𝑏 × 𝑛
+𝑏 0D errors, as long as two 0D errors do not
+strike the same matrix block within one iteration of matrix
+decomposition. Full checksum ABFT can tolerate up to 𝑆 0D
+and 1D errors combined, as long as two 0D/1D errors do not
+strike the same matrix block within one iteration of matrix de-
+composition. Assuming error occurs randomly and uniformly
+in time and space, we provide the theoretical estimation on
+the probability that ABFT can detect and correct all errors in
+one detection interval (i.e. Fault Coverage (FC)).
+𝐹𝐶𝑠𝑖𝑛𝑔𝑙𝑒 (𝑓 ,𝑇) =
+� 𝑆
+∑︁
+𝑘=0
+𝑒−𝜆𝑓 ,0𝐷𝑇 (𝜆𝑓 ,0𝐷𝑇)𝑘
+𝑘!
+𝑘
+�
+𝑖=0
+𝑆 − 𝑖
+𝑆
+�
+𝑒−𝜆𝑓 ,1𝐷𝑇𝑒−𝜆𝑓 ,2𝐷𝑇
+𝐹𝐶𝑓 ����𝑙𝑙 (𝑓 ,𝑇) =
+��
+�
+𝑆
+∑︁
+𝑘=0
+𝑆−𝑘
+∑︁
+𝑗=0
+𝑒−𝜆𝑓 ,0𝐷𝑇 (𝜆𝑓 ,0𝐷𝑇)𝑘
+𝑘!
+𝑒−𝜆𝑓 ,1𝐷𝑇 (𝜆𝑓 ,1𝐷𝑇)𝑗
+𝑗!
+𝑘+𝑗
+�
+𝑖=0
+𝑆 − 𝑖
+𝑆
+��
+�
+𝑒−𝜆𝑓 ,2𝐷𝑇
+Table 1 show the example estimation results based on dif-
+ferent GPU overclocking frequencies and the execution time
+of the TMU operation in three selected iterations of the LU
+decomposition. We define 𝐹𝐶 > 99.9999% as Full Cover-
+age. Having the capability of fault coverage estimation, we
+propose an adaptive-ABFT scheme. Unlike existing ABFT
+works, which enable ABFT during the entire matrix decompo-
+sition process, our adaptive-ABFT only enables ABFT error
+detection and correction when the error rate is above 0. Al-
+gorithm 1 shows the adaptive-ABFT strategy. We first check
+the error rate function in Line 4. If the rate of any kind of
+error is above zero, we check if applying ABFT can provide
+enough fault coverage (Line 5 - 9). We prioritize single-side
+ABFT over full ABFT to lower fault tolerance overhead. If
+none of the ABFT schemes can provide enough fault cover-
+age, we progressively lower the GPU clock frequency (Line
+11) until enough fault coverage is provided. Finally, we return
+the adjusted clock frequency together with flags indicating
+if we need to do a single or full ABFT check. Please note
+ABFT-OC would also work for CPU. We exclusively apply it
+to GPU in our algorithm since SDCs only occur to the GPU
+on our test system.
+3.2
+Bi-directional slack reclamation (BSR)
+TMU’
+TMU
+DtoH
+HtoD
+On GPU
+On CPU
+Slow down CPU using 
+DVFS to reclaim slack
+Speed up GPU using 
+ABFT-OC to reclaim slack
+Performance 
+improvement
+ABFT overhead
+PD
+PU'
+PU
+Figure 7. Bi-directional slack reclamation (BSR)
+The current best energy-saving approach, single directional
+slack reclamation (SR) [7], saves energy by slowing down
+tasks on the non-critical paths via DVFS. This work pro-
+poses a novel Bi-directional slack reclamation (BSR) energy-
+saving technique that reclaims slacks in two directions at
+the same time using both ABFT-OC and DVFS. Specifically,
+BSR reclaims slacks by simultaneously slowing down tasks
+on the non-critical path using DVFS and speeding up tasks on
+the critical path using ABFT-OC. An illustration of BSR is
+shown in Figure 7. Compared with SR, BSR brings three
+major advantages: 1 potential higher energy saving through
+both DVFS and ABFT-OC at the same time; 2 performance
+improvement in addition to energy saving optimization; 3
+enabling performance-energy consumption trade-off.
+3.2.1
+Enhanced Algorithmic-based Slack Prediction. Slack
+prediction is critical for making correct power status adjust-
+ments so that energy saving can be maximized. As BSR en-
+ables more opportunities for slack reclamation, it is more
+critical for it to make accurate slack predictions. The state-of-
+the-art algorithmic slack prediction was first proposed by [7].
+
+Improving Energy Saving of One-sided Matrix Decompositions on CPU-GPU Heterogeneous Systems
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Table 2. Ratios of time complexity of PD, PU, TMU, transfer
+size, and ABFT-related operations between 𝑘𝑡ℎ and 𝑘 + 1𝑡ℎ
+iteration. 𝑛 and 𝑏 are the total size and the block size of the
+input matrix respectively. PU of Cholesky and QR are omitted
+since they do not affect the slack
+Operation
+Computation
+&
+Checksum
+Update
+Data Transfer
+Checksum Verifi-
+cation
+PD-Cho.
+1
+1
+1
+TMU-Cho.
+(1+𝑘)(1−
+𝑏
+𝑛−𝑘𝑏−𝑏 )
+N/A
+1 −
+𝑏
+𝑛−𝑘𝑏−𝑏
+PD-LU
+1 −
+6𝑏
+3𝑛−(3𝑘−1)𝑏
+1 −
+1
+𝑛−𝑘𝑏
+1 −
+1
+𝑛−𝑘𝑏
+PU-LU
+1 −
+𝑏
+𝑛−𝑘𝑏−𝑏
+N/A
+1 −
+𝑏
+𝑛−𝑘𝑏−𝑏
+TMU-LU
+1 −
+2𝑏
+𝑛−𝑘𝑏
+N/A
+1 −
+2𝑏
+𝑛−𝑘𝑏
+PD-QR
+1 −
+𝑏
+6𝑛−(6𝑘+1)𝑏
+1 −
+𝑏
+𝑛−𝑘𝑏−𝑏
+1 −
+𝑏
+𝑛−𝑘𝑏−𝑏
+TMU-QR
+1
+−
+𝑏
+𝑛−𝑘𝑏−𝑏
+−
+𝑏
+𝑛−𝑘𝑏+𝑏
++
+𝑏2
+(𝑛−𝑘𝑏−𝑏) (𝑛−𝑘𝑏+𝑏)
+N/A
+1
+−
+𝑏
+𝑛−𝑘𝑏−𝑏
+−
+𝑏
+𝑛−𝑘𝑏+𝑏
++
+𝑏2
+(𝑛−𝑘𝑏−𝑏) (𝑛−𝑘𝑏+𝑏)
+It mainly works by profiling the tasks in the 1𝑠𝑡 iteration of
+decomposition and using the profiled time together with ratios
+of computational time complexity between 𝑘𝑡ℎ iteration and
+the 1𝑠𝑡 to predict the execution time of tasks in the 𝑘𝑡ℎ itera-
+tion of decomposition. By leveraging algorithmic knowledge
+and profiling results, algorithmic slack prediction can achieve
+much higher prediction accuracy compared with statistical-
+learning-based approaches and hardware-based approaches.
+However, we find that the accuracy of current algorithmic
+slack prediction highly relies on the profiling accuracy of the
+1𝑠𝑡 iteration and the assumption that computational efficiency
+stays constant across different iterations on a given processor.
+As the measurement of the 1𝑠𝑡 iteration can be inaccurate
+(e.g., when it is short) and the computational efficiency of
+tasks can also change considerably throughout the decomposi-
+tion process, all these inaccuracies can accumulate and cause
+large prediction errors in the latter part of the decomposition
+process, which lead to wrong slack reclamation decisions.
+In BSR, we propose an enhanced algorithmic-based slack
+prediction that greatly improves slack prediction accuracy.
+The enhanced algorithmic-based slack prediction rely on the
+profiled execution time of the 𝑝 last neighbor iterations to pre-
+dict the execution time of the current iteration to reduce the
+negative impacts bring by inaccurate profiling and changes
+in computational efficiency since tasks in neighbor iterations
+tend to have similar input sizes and thus similar computa-
+tional efficiencies. Since a closer neighbor has a more ac-
+curate estimation of computational efficiency, we apply dif-
+ferent weights to different profiling results in our enhanced
+algorithmic-based slack prediction. Specifically, the execution
+time of a task in 𝑘𝑡ℎ iteration (𝑇
+′𝑂𝑃
+𝑘
+) is predicted as:
+𝑇
+′𝑂𝑃
+𝑘
+= 𝑤1𝑟𝑂𝑃
+𝑘−1,𝑘𝑇𝑂𝑃
+𝑘−1 + 𝑤2𝑟𝑂𝑃
+𝑘−2,𝑘𝑇𝑂𝑃
+𝑘−21 + ... + 𝑤𝑝𝑟𝑂𝑃
+𝑘−𝑝,𝑘𝑇𝑂𝑃
+𝑘−𝑝
+where 𝑟𝑂𝑃
+𝑗,𝑘 is the ratio of theoretical time complexity of 𝑂𝑃
+between 𝑗𝑡ℎ and 𝑘𝑡ℎ iteration, which can be calculated based
+on the algorithm time complexity and relative change of the
+input sizes of 𝑂𝑃. Table 2 shows the ratios of key components
+of matrix decompositions. We omit the calculation process
+due to the page limit.𝑇𝑂𝑃
+𝑘−𝑖 is the actual profiled execution time
+of 𝑂𝑃 of the 𝑖𝑡ℎ last neighbor. 𝑤1 is the weight we applied to
+the 𝑖𝑡ℎ last neighbor. Through empirical study, we find that
+𝑝 = 4 and 𝑤1 = 1
+2,𝑤2 = 1
+4,𝑤3 = 1
+8,𝑤4 = 1
+8 can help provide
+enough prediction accuracy for energy saving. When ABFT
+is applied, the slack of the 𝑘𝑡ℎ iteration is predicted as:
+𝑠𝑙𝑎𝑐𝑘𝑘 = 𝑇
+′𝑇𝑀𝑈
+𝑘
++𝑇
+′𝑇𝑀𝑈 𝑐ℎ𝑒𝑐𝑘𝑠𝑢𝑚 𝑢𝑝𝑑𝑎𝑡𝑒
+𝑘
++𝑇
+′𝑇𝑀𝑈 𝑐ℎ𝑒𝑐𝑘𝑠𝑢𝑚 𝑣𝑒𝑟 𝑓
+𝑘
+𝑇
+′𝑃𝑈
+𝑘
++𝑇
+′𝑃𝑈 𝑐ℎ𝑒𝑐𝑘𝑠𝑢𝑚 𝑢𝑝𝑑𝑎𝑡𝑒
+𝑘
++𝑇
+′𝑃𝑈 𝑐ℎ𝑒𝑐𝑘𝑠𝑢𝑚 𝑣𝑒𝑟 𝑓
+𝑘
+−𝑇
+′𝑃𝐷
+𝑘
+−𝑇
+′𝑃𝐷 𝑐ℎ𝑒𝑐𝑘𝑠𝑢𝑚 𝑢𝑝𝑑𝑎𝑡𝑒
+𝑘
+−𝑇
+′𝑃𝐷 𝑐ℎ𝑒𝑐𝑘𝑠𝑢𝑚 𝑣𝑒𝑟 𝑓
+𝑘
+−𝑇
+′𝐷𝑎𝑡𝑎 𝑇𝑟𝑎𝑛𝑠𝑓 𝑒𝑟
+𝑘
+−𝑇
+′𝑇𝑟𝑎𝑛𝑠𝑓 𝑒𝑟 𝑐ℎ𝑒𝑐𝑘𝑠𝑢𝑚
+𝑘
+3.2.2
+Bi-directional slack reclamation strategies. Com-
+pared with SR, BSR offers more flexibility by reclaiming
+slacks from both directions, so the fractions of slacks that
+are reclaimed by the two tasks are adjustable, which in turn
+controls the performance-energy efficiency trade-off. So, we
+define reclamation ratio (𝑟) to be the fraction of the slack
+we try to reclaim by speeding up the task on the critical path
+and 1 − 𝑟 to be the fraction we try to reclaim by slowing
+down the task on the non-critical path. Algorithm 2 shows
+our BSR algorithm that makes decisions at the beginning of
+each matrix decomposition iteration. The execution time of
+tasks and slack are predicted in Line 3 - 4 using our enhanced
+algorithmic-based slack prediction. Given reclamation ratio 𝑟,
+we calculate the desired execution time of tasks on CPU and
+GPU in Line 5 - 11. We also consider the overhead of DVFS
+operations in our calculation to minimize the impact on per-
+formance. Line 12 - 15 calculate the desired CPU/GPU clock
+frequencies and limit them within the available frequency
+range. Line 16 - 17 calculates the projected execution time
+if we apply the desired frequencies. Note that the projected
+time may be different from the desired time since desired
+frequencies could be out of the available range. Finally, we
+make decisions on whether or not we adjust CPU/GPU clock
+frequencies in Line 18 - 22. If the projected time suggests
+that it can make a negative impact on the performance, it
+will skip frequency adjustment for this iteration i.e., setting
+AdjustCPU/GPU to FALSE. Note that this does not mean
+we do not reclaim slack of this iteration. Since we still keep
+the adjusted CPU/GPU frequencies from the last iteration,
+the partial of slack can still be reclaimed. This strategy en-
+sures we reclaim most of the slacks while minimizing perfor-
+mance impact. Line 23 invokes our adaptive-ABFT strategy
+for overclocking. Finally, we return the final decisions re-
+garding CPU/GPU clock frequency adjustments and ABFT
+protection strength for the current iteration.
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Jieyang Chen, Xin Liang, Kai Zhao, Hadi Zamani Sabzi, Laxmi Bhuyan, and Zizhong Chen
+Algorithm 2: BSR strategy
+1 Function BSR():
+In
+:reclamation ratio 𝑟
+In
+:iteration 𝑘
+In
+:GPU DVFS latency 𝐿𝐺𝑃𝑈
+In
+:CPU DVFS latency 𝐿𝐶𝑃𝑈
+In
+:Desired ABFT fault coverage 𝐹𝐶𝑑𝑒𝑠𝑖𝑟𝑒𝑑
+2
+Apply optimized guardband for both CPU and
+GPU
+3
+𝑇 ′𝐶𝑃𝑈 ,𝑇 ′𝐺𝑃𝑈 , 𝑇 ′𝐷𝑎𝑡𝑎𝑇𝑟𝑎𝑛𝑠𝑓 𝑒𝑟 ←
+EnhancedAlgorithmicPrediction(𝑘)
+4
+𝑠𝑙𝑎𝑐𝑘𝑘 ←= 𝑇 ′𝐺𝑃𝑈 −𝑇 ′𝐶𝑃𝑈 −𝑇 ′𝐷𝑎𝑡𝑎𝑇𝑟𝑎𝑛𝑠𝑓 𝑒𝑟
+5
+if 𝑠𝑙𝑎𝑐𝑘𝑘 > 0 then
+6
+𝑇𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 ← 𝑇 ′𝐺𝑃𝑈 − (𝑠𝑙𝑎𝑐𝑘𝑘 × 𝑟) − 𝐿𝐺𝑃𝑈
+7
+𝑇𝐶𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 ← 𝑇𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 − 𝐿𝐶𝑃𝑈 −𝑇 ′𝐷𝑎𝑡𝑎𝑇𝑟𝑎𝑛𝑠𝑓 𝑒𝑟
+8
+else
+9
+𝑇𝐶𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 ← 𝑇 ′𝐶𝑃𝑈 − (𝑠𝑙𝑎𝑐𝑘𝑘 × 𝑟) − 𝐿𝐶𝑃𝑈
+10
+𝑇𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 ← 𝑇𝐶𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 − 𝐿𝐺𝑃𝑈 +𝑇 ′𝐷𝑎𝑡𝑎𝑇𝑟𝑎𝑛𝑠𝑓 𝑒𝑟
+11
+end
+12
+𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 ← 𝑅𝑜𝑢𝑛𝑑𝑢𝑝(𝐹𝐺𝑃𝑈
+𝐵𝐴𝑆𝐸 × 𝑇 ′𝐺𝑃𝑈
+𝑇𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 , 100𝑀ℎ𝑧)
+13
+𝐹𝐶𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 ← 𝑅𝑜𝑢𝑛𝑑𝑢𝑝(𝐹𝐶𝑃𝑈
+𝐵𝐴𝑆𝐸 × 𝑇 ′𝐶𝑃𝑈
+𝑇𝐶𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 , 100𝑀ℎ𝑧)
+14
+𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 = 𝐿𝑖𝑚𝑖𝑡𝑇𝑜𝑅𝑎𝑛𝑔𝑒(𝐹𝐺𝑃𝑈
+𝑚𝑖𝑛 , 𝐹𝐺𝑃𝑈
+𝑚𝑎𝑥 )
+15
+𝐹𝐶𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 = 𝐿𝑖𝑚𝑖𝑡𝑇𝑜𝑅𝑎𝑛𝑔𝑒(𝐹𝐶𝑃𝑈
+𝑚𝑖𝑛 , 𝐹𝐶𝑃𝑈
+𝑚𝑎𝑥 )
+16
+𝑇𝐺𝑃𝑈
+𝑝𝑟𝑜 𝑗𝑒𝑐𝑡𝑒𝑑 = 𝑇 ′𝐺𝑃𝑈 ×
+𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑
+𝐹𝐺𝑃𝑈
+𝐵𝐴𝑆𝐸
+17
+𝑇𝐶𝑃𝑈
+𝑝𝑟𝑜 𝑗𝑒𝑐𝑡𝑒𝑑 = 𝑇 ′𝐶𝑃𝑈 ×
+𝐹𝐶𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑
+𝐹𝐶𝑃𝑈
+𝐵𝐴𝑆𝐸
+18
+𝑇𝑚𝑎𝑥 = 𝑚𝑎𝑥(𝑇 ′𝐺𝑃𝑈,𝑇 ′𝐶𝑃𝑈 +𝑇 ′𝐷𝑎𝑡𝑎𝑇𝑟𝑎𝑛𝑠𝑓 𝑒𝑟)
+19
+if 𝑇𝐺𝑃𝑈
+𝑝𝑟𝑜 𝑗𝑒𝑐𝑡𝑒𝑑 > 𝑇𝑚𝑎𝑥 then 𝐴𝑑 𝑗𝑢𝑠𝑡𝐺𝑃𝑈 ← 𝐹𝐴𝐿𝑆𝐸;
+20
+else 𝐴𝑑 𝑗𝑢𝑠𝑡𝐺𝑃𝑈 ← 𝑇𝑅𝑈𝐸;
+21
+if 𝑇𝐶𝑃𝑈
+𝑝𝑟𝑜 𝑗𝑒𝑐𝑡𝑒𝑑 > 𝑇𝑚𝑎𝑥 then 𝐴𝑑 𝑗𝑢𝑠𝑡𝐶𝑃𝑈 ← 𝐹𝐴𝐿𝑆𝐸;
+22
+else 𝐴𝑑 𝑗𝑢𝑠𝑡𝐶𝑃𝑈 ← 𝑇𝑅𝑈 𝐸;
+23
+𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑, 𝑆𝑖𝑛𝑔𝑙𝑒𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘, 𝐹𝑢𝑙𝑙𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘 ←
+ABFT-OC(𝐹𝐶𝑑𝑒𝑠𝑖𝑟𝑒𝑑, 𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑, 𝐹𝐺𝑃𝑈
+𝐵𝐴𝑆𝐸, 𝑇 ′𝐺𝑃𝑈 )
+24 return 𝐴𝑑 𝑗𝑢𝑠𝑡𝐶𝑃𝑈 , 𝐴𝑑 𝑗𝑢𝑠𝑡𝐺𝑃𝑈 , 𝐹𝐶𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑 ,𝐹𝐺𝑃𝑈
+𝑑𝑒𝑠𝑖𝑟𝑒𝑑,
+𝑆𝑖𝑛𝑔𝑙𝑒𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘, 𝐹𝑢𝑙𝑙𝐴𝐵𝐹𝑇𝐶ℎ𝑒𝑐𝑘
+3.2.3
+Theoretical performance improvement and energy
+saving analysis. Next, we provide a theoretical analysis of
+performance improvement and energy saving. With losing
+generality, we assume that the slack on the CPU in the fol-
+lowing discussion for simplification. The performance im-
+provement mainly comes from speeding up the tasks on
+the critical path. So, the performance improvement of iter-
+ation 𝑘 can be simply calculated as: Δ𝑇 = 𝑇𝑜𝑙𝑑
+𝑘
+− 𝑇 𝑛𝑒𝑤
+𝑘
+=
+𝑇𝐺𝑃𝑈
+𝑘
+− (𝑇𝐺𝑃𝑈
+𝑘
+− 𝑠𝑙𝑎𝑐𝑘𝑘 × 𝑟) = 𝑠𝑙𝑎𝑐𝑘𝑘 × 𝑟. This suggests that
+higher 𝑟 leads to higher performance. As for energy consump-
+tion, the theoretical amount of energy saving on the CPU
+when adopting BSR with reclamation ratio 𝑟 in the iteration 𝑘
+can be estimated as:
+Δ𝐸𝐶𝑃𝑈
+𝑘
+= Δ𝐸𝐶𝑃𝑈 _𝑑𝑦𝑛𝑎𝑚𝑖𝑐
+𝑘
++ Δ𝐸𝐶𝑃𝑈 _𝑠𝑡𝑎𝑡𝑖𝑐
+𝑘
+Δ𝐸𝐶𝑃𝑈 _𝑑𝑦𝑛𝑎𝑚𝑖𝑐
+𝑘
+= 𝐸𝐶𝑃𝑈 _𝑑𝑦𝑛𝑎𝑚𝑖𝑐_𝑜𝑙𝑑
+𝑘
+− 𝐸𝐶𝑃𝑈 _𝑑𝑦𝑛𝑎𝑚𝑖𝑐_𝑛𝑒𝑤
+𝑘
+=
+𝑑𝐶𝑃𝑈 𝑃𝐶𝑃𝑈
+𝑡𝑜𝑡𝑎𝑙𝑇𝐶𝑃𝑈
+𝑘
+−
+𝛼𝐶𝑃𝑈
+� 𝑓 𝐶𝑃𝑈 _𝑛𝑒𝑤
+𝑓 𝐶𝑃𝑈 _𝑜𝑙𝑑
+�2.4
+𝑑𝐶𝑃𝑈 𝑃𝐶𝑃𝑈
+𝑡𝑜𝑡𝑎𝑙 (𝑇𝐶𝑃𝑈
+𝑘
++ 𝑠𝑙𝑎𝑐𝑘𝑘 (1 − 𝑟)) =
+𝑑𝐶𝑃𝑈 𝑃𝐶𝑃𝑈
+𝑡𝑜𝑡𝑎𝑙𝑇𝐶𝑃𝑈
+𝑘
+−
+𝛼𝐶𝑃𝑈
+�
+𝑇𝐶𝑃𝑈
+𝑘
+𝑇𝐶𝑃𝑈
+𝑘
++ 𝑠𝑙𝑎𝑐𝑘𝑘 (1 − 𝑟)
+�2.4
+𝑑𝐶𝑃𝑈 𝑃𝐶𝑃𝑈
+𝑡𝑜𝑡𝑎𝑙
+(𝑇𝐶𝑃𝑈
+𝑘
++ 𝑠𝑙𝑎𝑐𝑘𝑘 (1 − 𝑟)) =
+�
+1 − 𝛼𝐶𝑃𝑈
+(𝑇𝐶𝑃𝑈
+𝑘
+)1.4
+(𝑇𝐶𝑃𝑈
+𝑘
++ 𝑠𝑙𝑎𝑐𝑘𝑘 × (1 − 𝑟))1.4
+�
+𝑑𝐶𝑃𝑈 𝑃𝐶𝑃𝑈𝑇𝐶𝑃𝑈
+𝑘
+Δ𝐸𝐶𝑃𝑈 _𝑠𝑡𝑎𝑡𝑖𝑐
+𝑘
+= (𝑇𝐶𝑃𝑈
+𝑘
+− 𝛼𝐶𝑃𝑈 (𝑇𝐶𝑃𝑈
+𝑘
++ 𝑠𝑙𝑎𝑐𝑘𝑘 (1 − 𝑟)))
+(1 − 𝑑𝐶𝑃𝑈 )𝑃𝐶𝑃𝑈
+𝑡𝑜𝑡𝑎𝑙
+Similarly, we can estimate the energy saving on GPUs as
+follows:
+Δ𝐸𝐺𝑃𝑈
+𝑘
+=
+�
+1 − 𝛼𝐺𝑃𝑈
+(𝑇𝐺𝑃𝑈
+𝑘
+)1.4
+(𝑇𝐺𝑃𝑈
+𝑘
+− 𝑠𝑙𝑎𝑐𝑘 × 𝑟)1.4
+�
+𝑑𝐺𝑃𝑈 𝑃𝐺𝑃𝑈
+𝑡𝑜𝑡𝑎𝑙𝑇𝐺𝑃𝑈
+𝑘
++
+(𝑇𝐺𝑃𝑈
+𝑘
+− 𝛼𝐺𝑃𝑈 (𝑇𝐺𝑃𝑈
+𝑘
+− 𝑠𝑙𝑎𝑐𝑘𝑘 × 𝑟))(1 − 𝑑𝐺𝑃𝑈 )𝑃𝐺𝑃𝑈
+𝑡𝑜𝑡𝑎𝑙
+Where 𝛼𝐶𝑃𝑈 /𝐺𝑃𝑈 are total power reduction factors when
+we use optimized guardband of CPU/GPU. We measure that
+in our hardware profiling work Figure 5. For clock frequen-
+cies out of the default range, we use constant values of the last
+measured value to estimate (dashed line). 𝑇𝐶𝑃𝑈 /𝐺𝑃𝑈
+𝑘
+are the
+original task execution time of CPU/GPU. 𝑃𝐶𝑃𝑈 /𝐺𝑃𝑈
+𝑡𝑜𝑡𝑎𝑙
+are the
+total power of CPU/GPU at the default guardband and clock
+frequencies. 𝑑𝐶𝑃𝑈 /𝐺𝑃𝑈 are the ratios of the CPU/GPU dy-
+namic power in the total power consumption. The change of
+CPU/GPU dynamic power is estimated using: 𝑃𝑑𝑦𝑛𝑎𝑚𝑖𝑐 ∝ 𝑓 2.4
+[17]. When the critical path is on the GPU, it is for sure we
+can save energy on the CPU. However, whether or not we can
+save energy on the GPU depends on 𝛼𝐺𝑃𝑈 and 𝑟. Assuming
+power reduction factor 𝛼𝐺𝑃𝑈 is fixed and minimized by apply-
+ing optimized processor guardband, then the reclamation ratio
+𝑟 controls the trade-off between performance improvement
+and energy consumption. Higher 𝑟 leads to higher perfor-
+mance but less energy saving, and vice versa. The highest
+energy saving can be achieved with 𝑟𝑚𝑎𝑥_𝑒𝑛𝑒𝑟𝑔𝑦 = 0 without
+performance improvement. The max 𝑟 that achieves maxi-
+mum without impacting energy efficiency is hard to be solved
+directly. So, we use a numerical approach to solve for 𝑟. By
+solving Δ𝐸𝐶𝑃𝑈
+𝑘
++ Δ𝐸𝐺𝑃𝑈
+𝑘
+= 0 using Newton’s method, we are
+able to get estimated solutions. For example, for decomposi-
+tion with input 30730×30720, the averaged reclamation ratios
+across all iterations are 0.28 for Cholesky, 0.26 for LU, and
+
+Improving Energy Saving of One-sided Matrix Decompositions on CPU-GPU Heterogeneous Systems
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+0.31 for QR, which approximately matches our experimental
+results in Figure 11.
+4
+Experimental Evaluation
+4.1
+Evaluation Methodology
+We compare BSR with two state-of-the-art energy-saving
+approaches R2H and SR together with the original design in
+the MAGMA library.
+• Original: The original matrix decompositions in the
+state-of-the-art MAGMA library. We keep the CPU/GPU
+clock frequency fixed at the default (autoboost dis-
+abled).
+• R2H: The original matrix decompositions in the state-
+of-the-art MAGMA library with CPU/GPU autoboost
+feature enabled. The processor clock frequency is dy-
+namically set according to the workload.
+• SR: The state-of-the-art energy efficient matrix decom-
+positions using single directional slack reclamation [7].
+• BSR: Our proposed matrix decomposition with BSR en-
+ergy efficiency optimization and ABFT-OC. Clock fre-
+quencies can reach greater ranch where SDCs can occur
+but are correctable by ABFT.
+All the above versions are implemented for Cholesky, LU,
+and QR decomposition for double precision inputs with block
+size tuned for performance.
+Table 3. Hardware/System Configuration for Experiments.
+Processor
+Intel Core i7-9700K
+NVIDIA RTX 2080 Ti
+Base Clock
+3.5(↑by0.1)GHz
+1.3(↑by 0.1)GHz
+Overclocking
+3.6-4.5(↑by0.1)GHz
+1.4-2.2(↑by 0.1)GHz
+Memory
+32 GB RAM
+12 GB RAM
+Default guard-
+band
+Vcore offset: 0mV
+Graphics clock offset: 0
+Optimized
+guardband
+Vcore offset: -150mV
+Graphics clock offset: +200
+0%
+10%
+20%
+30%
+40%
+50%
+2
+12
+22
+32
+42
+52
+Slack Prediction Error 
+Iteration
+Profile First Iteration
+Online Calibration
+Figure 8. Slack prediction error of the LU decomposition
+using different approaches
+4.2
+Experimental Environment
+All experiments are performed on a power-aware CPU-GPU
+server. Table 3 lists hardware configuration of the experimen-
+tal platform and system tools used for adjusting CPU/GPU
+guardband/clock frequencies and for measuring the energy
+consumption of CPU and GPU. Limited to the capability of
+our test platform, we only measure the energy consumption of
+1
+11
+21
+31
+41
+51
+Decomposition Iteration
+No FT
+Single-side ABFT
+Full ABFT
+No FT
+Single-ABFT
+Full-ABFT
+Adaptive 
+ABFT 
+Overhead Correct
+0%
+23.28%
+8%
+76.11%
+12%
+100.00%
+4%
+100.00%
+Figure 9. Comparing overhead and correctness when differ-
+ent ABFT scheme is applied in double precision LU decom-
+position with reclamation ratio 𝑟 = 0.25
+the CPU package and GPU device. For accurate measurement
+of energy consumption and stable SDCs error rate at reduced
+guardband, we adjust the external cooling system to stabilize
+the CPU/GPU temperature at 45°C and 55°C respectively.
+From the software perspective, all matrix decomposition ver-
+sions are built with GCC 7.4.0 and CUDA 11.6 with the
+highest optimization flags turned on. NVIDIA cuBLAS 11.1
+and Intel MKL 2020 are used as linear algebra computing ker-
+nels. MKL is configured to use all CPU cores. The operating
+system is Ubuntu 18.04.
+4.3
+Evaluation Results
+4.3.1
+Online slack prediction accuracy comparison. Fig-
+ure 8 shows the relative online prediction error using only the
+first iteration to predict [7] vs. our enhanced slack prediction
+approach proposed in this work. We can see both approaches
+can give less than 10% relative error for the first 2/3 of the
+iterations. However, since [7] only depends on the profiling
+result of the first iteration, the error caused by profiling and
+prediction will accumulate and become significant (about
+11.4% on average) as the decomposition progresses. Our en-
+hanced algorithmic slack prediction uses an online calibration
+approach to effectively avoid error from accumulating and
+reducing relative prediction error to around 4% on average.
+4.3.2
+ABFT overhead and correctness comparison. Fig-
+ure 9 shows the computational overhead and probability of
+computing correctness when different ABFT schemes are
+applied. We use double precision LU decomposition with
+BSR reclamation ratio 𝑟 = 0.25 as an example. The correct-
+ness is estimated by repeating the decomposition 100,000
+times and comparing the results. We observe similar results
+on other types of decompositions. Due to relative short slack
+in the later part of decomposition, higher GPU clock frequen-
+cies are needed, which reach degrees of overclocking that
+can have SDC errors. If we do not apply any fault tolerance,
+only 23.28% of the overall matrix decomposition tests out-
+put correct results. If we apply single-side checksum ABFT,
+it improves the percentage of tests with correct output to
+76.11% since 0D errors can be effectively detected and cor-
+rected. However, 1D error cannot be handled by single-side
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Jieyang Chen, Xin Liang, Kai Zhao, Hadi Zamani Sabzi, Laxmi Bhuyan, and Zizhong Chen
+0
+500
+1000
+1500
+2000
+2500
+3000
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+Org
+R2H
+SR
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+Time (ms)
+PD
+Data Transfer
+TMU+PU
+ABFT Overhead
+Frequency Change Overhead
+BSR
+(a) Time breakdown (2𝑛𝑑 iteration)
+0
+200
+400
+600
+800
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+Org
+R2H
+SR
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+Time (ms)
+PD
+Data Transfer
+TMU+PU
+ABFT Overhead
+Frequency Change Overhead
+BSR
+(b) Time breakdown (50𝑡ℎ iteration)
+-100
+-50
+0
+50
+100
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+Org
+R2H
+SR
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+Energy Saving (J)
+CPU Energy Saving
+GPU Energy Saving
+BSR
+(c) Energy saving breakdown (2𝑛𝑑 iteration)
+-25
+-15
+-5
+5
+15
+25
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+CPU
+GPU
+Org
+R2H
+SR
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+Energy Saving (J)
+CPU Energy Saving
+GPU Energy Saving
+BSR
+(d) Energy saving breakdown (50𝑡ℎ iteration)
+Figure 10. Time and energy saving breakdown of the 2𝑛𝑑 and 50𝑡ℎ iteration of the LU decomposition (Input size: 30720 × 30720).
+Energy saving is compared with the original design. Positive values represent energy saving and negative values represent extra
+energy costs.
+2000
+2500
+3000
+3500
+4000
+4500
+300
+350
+400
+450
+500
+550
+Energy Consumption (J)
+Performance (Gflop/s)
+Original
+R2H
+SR
+BSR (Ours)
+0.05
+0.1
+0.15
+0.2
+0
+0.25
+0.3
+Max Perf. Improv.: 49.9%
+Max Energy Sav.: 30.7%
+(a) Cholesky
+4000
+5000
+6000
+7000
+8000
+300
+350
+400
+450
+500
+550
+Energy Consumption (J)
+Performance (Gflop/s)
+Original
+R2H
+SR
+BSR (Ours)
+0.05
+0.1
+0.15
+0.2
+0
+0.25
+Max Perf. Improv.: 40.1%
+Max Energy Sav.: 28.2%
+(b) LU
+9000
+11000
+13000
+15000
+300
+350
+400
+450
+500
+550
+Energy Consumption (J)
+Performance (Gflop/s)
+Original
+R2H
+SR
+BSR (Ours)
+0.05
+0.1
+0.150.2
+0
+0.25
+Max Perf. Improv.: 43.6%
+0.3
+Max Energy Sav.: 28.8%
+(c) QR
+Figure 11. Pareto efficient performance-energy consumption trade-off enabled by adjusting the reclamation ratio. Input size:
+30720 × 30720 double precision
+0%
+5%
+10%
+15%
+20%
+25%
+30%
+35%
+Cholesky
+LU
+QR
+Energy Saving
+R2H
+SR
+BSR (Ours)
+(a) Energy Saving
+0%
+5%
+10%
+15%
+20%
+25%
+30%
+35%
+Cholesky
+LU
+QR
+R2H
+SR
+BSR (Ours)
+Energy	×	Delay!	Saving
+(b) ED2P Reduction
+Figure 12. Overall energy saving and ED2P Reduction com-
+pared with the original design. Input size: 30720 × 30720.
+0%
+5%
+10%
+15%
+20%
+25%
+30%
+35%
+5K
+10K
+15K
+20K
+25K
+30K
+Energy Saving
+Matrix Size (N*N)
+R2H
+SR
+BSR (Ours)
+Figure 13. Overall energy saving of LU compared with the
+original design with different input matrix sizes
+checksum ABFT. When full checksum ABFT is applied, it
+can ensure all decomposition tests are correct, but it also
+brings 12% overhead. Our adaptive-ABFT can adaptively
+apply necessary levels of fault tolerance to ensure high re-
+liability and low overhead. For example, when we set the
+reclamation ratio 𝑟 = 0.25, the first 41 iterations are running
+at fault-free clock frequencies (1700Mhz), so adaptive-ABFT
+completely disables ABFT for eliminating unnecessary fault
+tolerance overhead. For 42𝑡ℎ − 49𝑡ℎ iteration, the slacks need
+to be reduced by BSR using more aggressive overclocking
+(up to 1900Mhz), so it applies single-side checksum ABFT.
+Finally, it applies full checksum ABFT after 50𝑡ℎ iteration
+since higher clock frequencies are used (up to 2200Mhz). So,
+with adaptive-ABFT, we can still ensure all decomposition
+tests are correct with only 4% fault tolerance overhead.
+4.3.3
+Per iteration performance and energy comparison.
+To understand how each of the different approaches affects
+the performance and energy efficiency of matrix decomposi-
+tions, we show the profiling results of 2𝑛𝑑 and 50𝑡ℎ iteration
+of the LU decomposition in terms of time and energy costs
+breakdown in Figure 10. For the original version, we can see
+the slack occurs on the CPU side for the 2𝑛𝑑 iteration and
+GPU side for the 50𝑡ℎ iteration. For clarity, we refer to the
+case that slack is on the CPU side as C and the case that
+slack is on the GPU side as G in our following discussion.
+For R2H, we observe noticeable energy saving in both C and
+
+Improving Energy Saving of One-sided Matrix Decompositions on CPU-GPU Heterogeneous Systems
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+G due to reduced energy consumption on the CPU side and
+GPU side respectively. For SR, we see slack is fully reclaimed
+in G , but not fully reclaimed in C due to the limited clock
+frequency range on the CPU and longer slack length. For
+BSR, we test different reclamation ratios 𝑟 and mark their
+values under the bars. We set 𝑟 from 0 to a certain value that
+leads to maximum achievable performance. This maximum 𝑟
+is higher for C than G since GPU has greater overclocking
+capabilities than CPU in our system when we apply optimized
+guardband. We can see maximum energy saving is achieved
+when 𝑟 = 0, which is consistent with our previous theoret-
+ical analysis. Maximum performance 𝑟 = 0.25 for C and
+G , which are close to our theoretical estimation. When we
+increase 𝑟, we see an increase in energy consumption for the
+processor on the critical path due to the increase in clock fre-
+quency. For C , we observe a slight increase in energy-saving
+since the slack is long enough for the CPU to always run at
+the lowest clock frequency, and reducing the total execution
+time can save more CPU static energy. We also observe a
+slight decrease in energy saving in G , mainly due to the
+slight increases in clock frequencies. Even though it can still
+save energy since 1) the clock frequencies are low; 2) power
+reduction brings by optimized guardband. Finally, Thanks
+to ABFT-OC, we can exploit higher overclocking frequen-
+cies where we can achieve higher performance and energy
+efficiency in C .
+4.3.4
+Overall energy saving and energy efficiency com-
+parison. Next, we show the overall energy-saving capability
+of different approaches in Figure 12(a). We evaluate all three
+matrix decompositions with an input size of 30720 × 30720.
+All four versions of each type of matrix decomposition pro-
+duce a similar performance. To maximize energy saving the
+reclamation ratio of BSR is set to 0. We can see that compared
+with the state-of-the-art MAGMA library, our BSR is able
+to save energy by 30.7% for Cholesky, 28.2% for LU, and
+28.8% for QR. That is 1.31 × −1.49× more energy saving
+compared with the current state-of-the-art SR energy saving
+approach and 2.03 × −2.20× more energy saving compared
+with R2H. In addition, we use 𝐸𝑛𝑒𝑟𝑔𝑦 × 𝐷𝑒𝑙𝑎𝑦2 (ED2P) to
+measure the energy efficiency of matrix decompositions. As
+shown in Figure 12(b), compared with the original design,
+our BSR is able to reduce ED2P by 29.3%-31.6%. Compared
+with R2H, BSR is able to reduce ED2P by 18.6%-20.7%. Fi-
+nally, compared with SR, BSR is able to reduce ED2P by
+10.8%-14.1%.
+4.3.5
+Overall energy saving on different input sizes. In
+Figure 13, we show the results of applying energy-saving
+approaches on LU decomposition with different input sizes.
+Limited by the page space, we only show the results for LU de-
+composition. Other matrix decompositions behave similarly.
+We can see our BSR is able to stably save energy consumption
+across different input matrix sizes ranging from 5120 × 5120
+and above. Note that it is hard to save energy on smaller ma-
+trices since they either lead to high fault tolerance overhead
+or small slacks that are hard to be reclaimed.
+4.3.6
+Overall Pareto efficient performance-energy con-
+sumption trade-off. Finally, we show the overall Pareto ef-
+ficient performance-energy consumption trade-off enabled
+by adjusting the reclamation ratio in BSR. As shown in Fig-
+ure 11, by adjusting the reclamation ratio to a minimum 0, we
+achieve max energy saving with similar performance to the
+original design. In this case, compared with the original de-
+sign, BSR is able to save energy by 28.2%-30.7%. Compared
+with R2H, BSR is able to save energy by 17.1%-18.9%. Com-
+pared with SR, BSR is able to save energy by 9.6%-11.7%.
+By increasing the reclamation ratio, we are able to adjust the
+performance or energy consumption of matrix decomposi-
+tions. For example, with equal or less energy consumption,
+compared with the original design BSR is enable to improve
+the performance by 1.38×-1.51×. Also, compared with R2H,
+BSR is enable to improve the performance by 1.33×-1.43×.
+In addition, compared with SR, BSR is enable to improve the
+performance by 1.36×-1.43×. Finally, we see the results of
+BSR with different reclamation ratios form a Pareto set such
+that we cannot improve energy saving and performance at the
+same time without reliability degradation.
+5
+Conclusion
+In this work, we focused on further improving the energy sav-
+ing of matrix decompositions on CPU-GPU heterogeneous
+systems beyond existing state-of-the-art works. To achieve
+our goal, we first proposed ABFT-OC, a novel overclocking
+technique that is protected by ABFT to enable reliable com-
+putation for key operations in matrix decompositions when
+overclocking. Next, based on ABFT-OC, we proposed BSR,
+a novel matrix decomposition framework, that aims to maxi-
+mize energy saving while maintaining performance and relia-
+bility. We evaluated BSR on three key matrix decomposition
+algorithms - Cholesky, LU, and QR. Experiments show that
+BSR is able to save up to 11.7% more energy compared with
+the current best energy saving optimization approach with no
+performance degradation and up to 14.1% ED2P reduction.
+Also, BSR enables the Pareto efficient performance-energy
+trade-off, which is able to provide up to 1.43× performance
+improvement without costing extra energy.
+6
+Acknowledgement
+This work was supported by the U.S. Department of Energy,
+Office of Science, Office of Advanced Scientific Computing
+Research, Scientific Discovery through the Advanced Com-
+puting (SciDAC) program under Award Number DESC0022209.
+The research was also partly supported by NSF Grant 1907401.
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Jieyang Chen, Xin Liang, Kai Zhao, Hadi Zamani Sabzi, Laxmi Bhuyan, and Zizhong Chen
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+

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+First-principles Investigation of Electrides Derived from Sodalites
+Byungkyun Kang1 and Qiang Zhu1, ∗
+1Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA
+(Dated: January 10, 2023)
+Recently, the electride materials, with excess anionic electrons confined in their empty space, have
+received a growing attention due to their promising applications in catalysis, nonlinear optics and
+spin-electronics. However, the utilization of electride materials is limited by their thermal instability.
+Here we introduce an alternative way to achieve the localized anionic electron states via the removal
+of high symmetric Wyckoff sites of anions from the existing sodalite compounds. Using four halide
+sodalites as the parental structures, our simulation reveals that the materials after the removal of
+anionic halide sites exhibit typical electride behaviors that are characterized by the existence of
+localized electronic states near the Fermi level. Compared to most previously studied electrides,
+these materials are expected to be more thermally stable due to the complex structural framework
+and thus promising for practical applications.
+Among them, Na4(AlSiO4)3 manifests magnetic
+electronic structure. We demonstrate that this magnetism originates from a highly localized excess
+electron state surrounded by electronpositive alkaline cations. Our results suggest Na4(AlSiO4)3
+could be a promising spintronics component, thus encouraging further experimental study.
+I.
+INTRODUCTION
+Electides represent a unique class of materials in which
+the excess electrons are trapped inside crystal cavities
+and serve as the anions1–5.
+Localization of electrons
+provides the early examples of quantum confinement3.
+These trapped electrons are usually loosely bound and
+float around Fermi level to form unique interstitial energy
+bands6, giving rise to interesting transport and magnetic
+behaviors.
+Therefore, electrides have attracted grow-
+ing attention for various material applications.
+While
+the first crystalline organic electride was made by Dye
+and coworkers in 19837, the use of organic electrides
+for practical applications was limited by their thermal
+instability3.
+In 2003, Hosono and coworkers reported
+the synthesis of the first thermally stable electride based
+on the inorganic mineral mayenite (12CaO·7Al2O3)8.
+The resulting electride Ca6Al7O16 (C12A7:2e−), with
+the excess electrons confined in the zero-dimension (0D)
+cages, exhibit excellent thermal stability and low reac-
+tivity with air, opening a new chapter in electride syn-
+thesis and utilization. The discovery of C12A7:2e− has
+stimulated many new efforts to search for other inor-
+ganic electrides.
+Recently, a number of new electrides
+with electrons confined in different geometries, including
+2D (Ca2N9, Y2C10), 1D (Y5Si311, Sr5P312, Mn5Si313,
+La8Sr2(SiO4)614), and 0D (YH215), have been increas-
+ingly investigated in experiment. In the mean time, many
+new materials with improved functionality were proposed
+theoretically thanks to the recent advances in computer
+simulation16–26.
+Despite the growing number of reports on identify-
+ing new electride materials, most of them are only sta-
+ble at low temperatures and very sensitive to air and
+moisture. To date, C12A7:2e− remains one of the only
+room-temperature stable electrides from practical appli-
+cation perspective, with the exception of several recent
+reports27–29. Therefore, it is necessary to examine the
+uniqueness of C12A7:2e− from the viewpoint of struc-
+tural stability. The original composition of mayenite can
+be represented as Ca12Al14O33, which is an air-stable
+neutral material at ambient temperature and atmosphere
+with a melting point of 1415 ◦C4. Its crystal structure
+has a cubic body centered unit cell, consisting of four
+positively charged [Ca6Al7O16]+ as the main framework
+with the following Wyckoff sites: Ca (24d), Al1 (16c),
+Al2 (12a), O1 (48e), O2 (16c)30. As shown in Fig. 1a,
+this framework constitutes twelve cages with a free inner
+space of 4-6 ˚A in diameter in the unit cell.
+In addi-
+tion, there exist another type of oxygen atoms (O3) that
+are statistically distributed inside centers of the twelve
+cages, which can be regarded as the partial occupancy of
+the 12b Wyckoff site with a ratio of 1
+6. Inside the cage,
+each O3 site is collinearly coordinated to two Ca atoms
+with a distance of 2.77 ˚A, which is a bit higher than 2.40
+˚A in the rocksalt CaO compound.
+The resulting two
+oxygen may be regarded as the counteranion (O2−) to
+the giant framework cation. Due to the long Ca-O dis-
+tance, it is expected that O2− anions are loosely bound
+to the cages8. When the material is thermally treated
+in a reductive atmosphere, part of the loosely bonded
+O2− ions may escape as gaseous O2 from the lattice and
+leave electrons in the cages, considered as the electron
+captured oxygen vacancy, or F-center. With the proper
+reductive condition, all of the loosely bonded O2− anions
+can be removed from the lattice. In this case, the rel-
+atively free electrons play the role of anions and result
+in a high conductivity. In the meantime, the unpaired
+feature of these free electrons also generate nontrivial
+magnetic8 and superconducting properties31. Thanks to
+its excellent stability, C12A7:2e− has been used for am-
+monia synthesis32–34 and as an electron-injection barrier
+material35.
+Inspired by the synthesis of C12A7:2e−, we proceed
+to check the feasibility of designing new electrides from
+the existing minerals sharing structural features similar
+to mayenite that allow the localization of anionic elec-
+trons.
+Clearly, the most important characteristic of a
+arXiv:2301.02716v1  [cond-mat.mtrl-sci]  6 Jan 2023
+
+2
+FIG. 1.
+Two representative porous structures consisting of anions at the center of the cages. (a) mayenite Ca12Al14O33 and
+(b) sodalite Na4(AlSiO4)3Cl. Both structures are featured by the presence of anions (partially) occupying the Wyckoff site
+around the center of cages. The left panels highlight the overall distribution of cages, while the right panels shows the chemical
+environment of each representative cage. In each cage, the coordination environment of the center anion is also displayed by
+the red dash lines. For clarity, only 6 out of 12 cages are shown for the mayenite structure.
+good precursor is the presence of anions occupying the
+high-symmetry Wyckoff site.
+In the entire materials
+database, there exist many compounds meeting such a
+requirement. In this work, we focus on a common type
+of sodalite minerals consisting of the halide anions at
+the high symmetry Wyckoff sites (see Fig. 1b). Simi-
+lar to mayenite, the sodalite has a high melting point of
+1079 ◦C36. Its crystal structure also contains the porous
+framework with two cages centering around the corner
+and body center.
+The halide anions (X−) occupy the
+cage centers at the 2a Wyckoff site, exhibiting a tetrag-
+onal coordination with the neighboring alkaline metals
+M + (e.g., Na, K). The resulting M-X distance is about
+the same bond length as that in the typical ionic com-
+pound. For instance, the Na-Cl contact in sodalite is 2.76
+˚A, which is close to 2.79 ˚A in the rocksalt NaCl com-
+pound. Due to the large dimeter size, it is possible to
+form many diffusion pathway if the X anion is removed
+from the M4 tetrahedron to realize the electride state.
+Using these materials as the precursor, we immediately
+found that the so called black sodalite Na4(AlSiO4)3 has
+been found in our recent screening work25. In this work,
+we checked several thermal-stable sodalite compounds
+and derived the likely crystalline electride structures via
+rigorous structural analysis. Based on these structures,
+we further performed a thorough computational investi-
+gation on their electronic and magnetic properties based
+on first-principles calculations. Our results suggest that
+the electrides derived from sodalite are likely to achieve
+similar properties as found in C12A7:2e−. We hope this
+work will encourage more investigations of porous zeo-
+lites as the precursor to synthesize more thermal stable
+electrides in the future.
+II.
+COMPUTATIONAL METHODS
+In this work, we selected four materials for detailed
+study,
+including
+Na4(AlSiO4)3Cl,
+Na4(AlSiO4)3Br,
+Na4(AlGeO4)3I and Na4(GaSiO4)3I. For each of them,
+we first created the new structures by removing the halide
+anions in the unit cell. The structures are subsequently
+relaxed to obtain the optimum cell parameters.
+The
+structural stability was then checked by phonon calcu-
+lation. Whenever the imaginary frequencies were found,
+we followed the largest soft mode to obtain a new struc-
+ture with a subgroup symmetry37.
+We used the projector augmented wave (PAW) method
+with the plane-wave code VASP38–40 to carry out all
+calculations with the framework of density functional
+theory (DFT). The Generalized Gradient Approxima-
+tion (GGA) with Perdew, Burke and Ernzerhof (PBE)
+functional41 was adopted.
+We have used a 3×3×3 Γ-
+centered k-point grid for geometric relaxation of unit
+cell, which contain two formula units.
+The cutoff en-
+ergy for all calculations is 520 eV, and forces are con-
+verged within 0.03 eV/˚A. Using the optimized structure,
+we constructed a 2×2×2 supercell (368 atoms) for the
+F-center defect (with the halide anion vacancy) calcu-
+lation, in which a single Γ point was used for sampling
+
+(a) Mayenite
+Space group: I-43d
+Number of cages: 12
+Occupancy ratio: 1/6
+Diameter: ~4-6 nm
+Contacts: 2 collinear cations
+(b) Sodalite
+Space group: P-43n
+Number of cages: 2
+Occupancy ratio: 1
+Diameter: ~10 nm
+Contacts: 4 tetragonal cations3
+FIG. 2.
+The optimized crystal structures of four sodalite materials.
+the Brillouin zone. Phonon bands were calculated using
+the force constants, which were calculated by the finite
+displacement methods using the Phonopy code42.
+III.
+RESULTS AND DISCUSSIONS
+A.
+Pristine Sodalites
+Sodalite is a common structural type for many nature-
+occurring minerals.
+The typical crystal structure pos-
+sesses a cubic symmetry in space group P¯43n (218). As
+shown in Fig. 2, each unit cell contain two formula units.
+According to X-ray diffraction analysis43–45, the min-
+eral Na4(AlSiO4)3Cl is characterized by a β-cage made
+of four-membered (Al, Si)O4 rings on the (100) plane
+linked together in six-membered rings parallel to (100)
+plane.
+This results in a highly symmetric framework
+with just five atoms in the asymmetric unit. Other ma-
+terials, including Na4(AlSiO4)3Br, Na4(AlGeO4)3I and
+Na4(GaSiO4)3I, show the same packing behavior except
+a slight difference in terms of lattice parameters.
+TABLE I. The comparison of cell parameters and band gaps
+for sodalite materials between simulation and experimental
+values.
+System
+Cell Parameters (˚A) Band Gap (eV)
+DFT
+Expt.
+DFT
+Expt.
+Na4(AlSiO4)3Cl
+8.973
+8.87646
+4.69
+6.147
+Na4(AlSiO4)3Br
+9.031
+8.93246
+4.61
+5.947
+Na4(AlGeO4)3I
+9.287
+9.17448
+3.39
+Na4(GaSiO4)3I
+9.198
+9.07448
+3.59
+Table I summarizes the calculated lattice parameters,
+which agree well with the experimental reports. Using
+the dispersion corrected calculations with the DFT-D3
+method49, we found that the relaxed lattice parameter of
+Na4(AlSiO4)3Cl is 8.897 ˚A, which is approximately 0.1 ˚A
+smaller than the value with the standard PBE functional.
+This indicates a minor effect of dispersion correction in
+the sodalites. Our results are close to other DFT results
+in which the calculated lattice parameters are 8.72150,
+8.9151, and 8.78 ˚A52.
+Fig. 3 shows the band structures and density of states
+(DOS) of all four sodalite materials.
+Clearly, none of
+their electronic structures show spin polarization in their
+band structures.
+This means that the electron energy
+levels are not affected by the spin of the electron. In ad-
+dition, all four compounds exhibit direct band gaps at the
+Γ point. The calculated band gaps are also summarized
+in Table I. For Na4(AlSiO4)3Cl, we obtained a gap of 4.69
+eV, similar to other DFT studies (e.g., 4.39 eV53,54 and
+5.0 eV55). All of these values are smaller than the exper-
+imentally reported 6.1 eV47. This is expected since DFT
+systematically underestimates the energy gap.
+Similar
+trends were also observed in other compounds.
+From the DOS analysis, the computed valence bands
+near Fermi level in all four sodalite structures are very
+flat valence bands, with the contribution of O-p or-
+bitals.
+On the contrary, their conduction bands are
+rather dispersive, which are very similar to that of
+mayenite56. For Na4(AlSiO4)3Cl, its valence band max-
+imum (VBM) consists of large O-p and small Cl-p or-
+bitals, while Na4(AlSiO4)3Br has a VBM comprised of
+large Br and small O orbitals. This is consistent with
+the fact that the Br is more electronegative than Cl.
+For Na4(AlGeO4)3I and Na4(GaSiO4)3I, their highest
+valance bands are mainly comprised of I orbitals and are
+separated from the other valence bands. It can be ex-
+plained by the lowest electronegativity of iodine among
+the halides (I, Br, Cl), which leads to less hybridization
+with O-p and a higher energy level.
+
+Na
+Al
+Ga
+Si
+Ge
+C1
+Br
+Na4(A1Si04)3C1
+Na4(A1SiO4)3Br
+Na4(A1Ge04)3I
+Na4(GaSiO4)3I4
+FIG. 3.
+The electronic band structures and atomic projected density of states of (a) Na4(AlSiO4)3Cl, (b) Na4(AlSiO4)3Br,
+(c) Na4(AlGeO4)3I, and (d) Na4(GaSiO4)3I. In each band structure, blue solid lines represent spin up bands and red dashed
+lines represent spin down bands. In the plots of density of states, two different scales of density of states are used: one below
+3 eV and one above 3 eV.
+B.
+The Structures of Crystalline Electrides
+Prior to investigation of electride, an electron trapped
+at the site of an anion vacancy in ionic crystals is known
+as an F-center in solid state physics. F-center is a point
+defect and its concentration is so small that interaction
+between them is negligible. On the other hand, electrides
+may be regarded as materials having a stoichiometric F-
+center, and the electron concentration in the electride is
+high enough to allow them to interact with each other4.
+In the past, most works attempted to derive the crystal
+structure by directly removing the anions from the par-
+ent crystal57,58. However, such treatment failed to take
+into account the likely structural changes due to complete
+removal of anions (instead of low-concentration point de-
+fects). Therefore, it is important to examine the resulting
+crystalline electride in a more rigorous manner.
+As we discussed earlier, we start by removing the
+halide anions from the parental crystal structures. Since
+Na4(AlSiO4)3Cl and Na4(AlSiO4)3Br share the same
+framework, this lead to three candidate structures for
+further consideration.
+After the structural relaxation,
+we found that there exist notable imaginary frequencies
+in the computed phonon spectrum of Na4(AlSiO4)3 and
+Na4(AlGeO4)3. As shown in the left panel of Fig.
+4a
+for Na4(AlSiO4)3, the most negative phonon frequency
+is located at the Γ point. Therefore, We derived a new
+structure by displacing atoms along the eigenvector of
+the soft mode (see the middle panel of Fig. 4a) with
+TABLE II. The simulated crystallographic data for three crys-
+talline electrides.
+Na4(AlSiO4)3,
+Space group P23,
+a = 8.98 ˚A
+Atomic coordinates
+Na1
+4e
+0.72
+0.72
+0.72
+Na2
+4e
+0.17
+0.17
+0.17
+Al
+6g
+0.75
+0.00
+0.50
+Si
+6h
+0.25
+0.50
+0.00
+O1
+12j
+0.36
+0.05
+0.65
+O2
+12j
+0.86
+0.85
+0.45
+Na4(AlGeO4)3,
+Space group P23,
+a = 9.28 ˚A
+Atomic coordinates
+Na1
+4e
+0.18
+0.18
+0.18
+Na2
+4e
+0.70
+0.70
+0.70
+Al
+6g
+0.25
+0.00
+0.50
+Ge
+6h
+0.75
+0.50
+0.00
+O1
+12j
+0.65
+0.93
+0.65
+O2
+12j
+0.85
+0.85
+0.43
+Na4(GaSiO4)3,
+Space group P¯43n,
+a = 9.09 ˚A
+Atomic coordinates
+Na
+8e
+0.18
+0.18
+0.18
+Ga
+6d
+0.25
+0.00
+0.50
+Si
+6c
+0.25
+0.50
+0.00
+O
+24i
+0.57
+0.87
+0.15
+the maximum magnitude of 0.2 ˚A on the Na atoms.
+Then, the new structure was relaxed again. We repeated
+this process until the absolute minimum of the poten-
+
+(a)Na4(A1Si04)3Cl
+(b) Na4(A1SiO4)3Br
+0
+0.05
+0.1
+0
+0.1
+0.2
+81
+N
+6
+6
+(eV)
+(eV)
+4
+4
+Na
+Al
+NaAl
+2
+spin up
+2
+$pin up
+Si
+C1
+Si
+Br
+spin down
+$pin dbwn
+.0
+0
+0
+0
+2
+-21
+T
+X
+M
+r
+R
+X/M
+RO
+5
+10
+X
+M
+R
+X/M
+RO
+10
+20
+DOs (eV-l)
+(c)Na4(A1GeO4)3I
+(d) Na4(GaSiO4)3I
+0
+0.1
+0.2
+0
+0.2
+0.4
+81
+6
+6
+(eV)
+4
+4
+Na -
+-Al
+Na -
+-Ga
+2
+E
+2
+spin up
+Ge
+$pin up
+Si
+-1
+spin down
+-0
+$pin dbwn
+0
+0
+0
+MM
+MM
+2
+-2层
+X
+M
+R
+X/M
+RO
+10
+T
+5
+X
+T
+M
+r
+R
+X/M
+RO
+5
+10
+DOS (eV-l)
+DOS (eV-l)5
+FIG. 4.
+The computed phonon spectrum for three different electrides (a) Na4(AlSiO4)3, (b) Na4(AlGeO4)3, and (c)
+Na4(GaSiO4)3. In (a), the left panel shows phonon dispersion curves of relaxed Na4(AlSiO4)3, which has a soft mode shown in
+the middle panel. The right panel shows phonon dispersion curves of relaxed Na4(AlSiO4)3 after applying atomic displacements
+according to the depicted soft mode.
+tial energy surface was reached and no imaginary fre-
+quency phonon occurred. The similar method has been
+used by the authors in a previous study37. After these
+treatments, we obtained new structures with the energy
+drops of 71 and 4 meV per formula unit for Na4(AlSiO4)3
+and Na4(AlGeO4)3, respectively.
+On the other hand,
+Na4(GaSiO4)3 retains the same space group symmetry
+as found in the original sodalite form.
+Due to the soft modes, the original Na (8e) and O
+(24i) sites in space group P¯43n were split into the 4e
+and 12j sites in space group P23. Therefore, the final
+stable structure for these two electrides has a subgroup
+symmetry P23 (195). Table II summarizes the crystal-
+lographic information for all three structures. Compared
+to Table I, one can clearly found that the cell parameters
+nearly remains the same regardless of either the symme-
+try change or atomic removal.
+C.
+Electronic Properties
+Next, we investigated the electronic properties of the
+resulting three crystalline electrides. In particular, the
+Cl removed electride Na4(AlSiO4)3 corresponds to the
+previously investigated black sodalite, in which each pe-
+riodic β-cage contains one Na43+ cluster and its color is
+black due to a metallic band structure59–61. As shown
+in Fig. 5, the three structures manifest electride bands
+inside the gap of the parent sodalite. The partial charge
+density obtained in the energy range −2 < E − EF < 0
+is centered inside the cage after the removal of halide an-
+ions, indicating that the excess electrons are localized at
+the crystal cages and forming the electride bands. There
+are sizable contributions of oxygen orbitals in the charge
+density, giving rise to relatively dispersed electride bands
+in comparison to the valence bands of the parent com-
+pounds. Therefore, the dispersion of the electride bands
+can be attributed to the hybridization between localized
+excess electrons in Na4 tetrahedron and oxygen orbitals.
+Recently, Stoliaroff et al.
+have suggested that high
+concentrations of chlorine vacancies can be created by
+F-center point defects using DFT modeling62.
+It was
+found that ground and excited states of the singly occu-
+pied molecular orbital and the triply degenerate lowest
+unoccupied molecular orbital for the F-center63. It was
+observed that the F-center is an electron trapped at an
+halide anion vacancy in alkali halides64.
+The trapped
+electron is in a set of quantized states similar to a hy-
+drogen atom: the ground state is the atomic 1s state,
+and the first excited state is composed of the 2s and
+
+P43n Na4(A1SiO4)3
+P23 Na4(A1SiO4)3
+(a)
+Soft Mode
+30
+30
+Frequency (THz)
+Frequency (THz)
+20
+20
+10
+0
+0
+r
+X
+M
+R
+X
+X
+X
+M
+R
+M
+R
+M
+R
+P23 Na4(A1GeO4)3
+P43n Na4(GaSiO4)3
+(b)
+(c)
+30
+30
+Frequency (THz)
+Frequency (THz)
+20
+20
+10
+10
+0E
+ob
+X
+M
+r
+R
+X
+M
+R
+X
+M
+R
+X
+M
+R6
+FIG. 5.
+The electronic band structures and isosurfaces of partial charge density in the range of -2 < E - EF < 0 eV of (a)
+ferromagnetic Na4(AlSiO4)3, (b) antiferromagnetic Na4(AlSiO4)3, (c) Na4(AlGeO4)3, and (d) Na4(GaSiO4)3. In each band
+structure, blue solid lines represent spin up bands and red dashed lines represent spin down bands.
+FIG. 6.
+The computed wavefunctions for pristine Na4(AlSiO4)3Cl, F-center of Cl vacancy in the supercell of Na4(AlSiO4)3Cl,
+and electride states in Na4(AlSiO4)3. F and E denote the F-center and electride state, respectively. In the left panel, the
+magnified views of O-p and O-s wavefunctions are displayed to aid the visual analysis. Three gray lines denote states that are
+triply degenerate. In the right panel, wavefunctions which consist of electride bands are marked by black boxes.
+
+(a) FM Na4(A1SiO4)3
+(b) AFM Na4(A1SiO4)3
+4
+4
+2
+(eV)
+(eV)
+E
+EI
+spin|up
+spin |up
+-2
+ spindown
+spin down
+.4
+-4
+0
+X
+M
+R
+X/M
+X
+M
+R
+R
+X/M
+R
+(c) Na4(A1GeO4)3
+(d) Na4(GaSiO4)3
+2
+2
+(eV)
+(eV)
+EI
+spin up
+spin up
+ spin down
+spin down
+-4
+.4
+6
+X
+M
+R
+X/M
+R
+M
+I
+R
+X/M
+RC1 Vacancy
+Na4(A1Si04)3
+Na4(A1Si04)3C1
+F-center
+E-s+
+(Si,Al,O)-s,p
+F-p+O-p
+Energy
+E-s+
+(Si,Al,O)-s
+E-p+O-p
+O-s+C1-s
+F-s+(O,Si,C1)-s
+CBM
+F-p+O-s
+E-p+O-s,p
+O-p
+VBM
+E-s+O-p
+F-s+O-s
+C1-p+O-1
+CBM
+E-s+O-s
+F-s+O-p
+O-p
+VBM7
+2p states65. However, it is shown that hybridization be-
+tween metal and electride orbitals leads to large splitting
+between electride associated states66. Electride associ-
+ated states are split into two when hybridization between
+metal and electride orbitals occurs.
+Otto et al.
+pro-
+posed that the particle-in-a-box state is better described
+as a particle in a box hybridized with framework oxy-
+gen atoms for the black sodalite Na4(AlSiO4)367. In this
+model, the particle is confined to a box defined by the
+oxygen atoms, and the electron density is spread more
+uniformly throughout the structure. These results sug-
+gest that the Cl removed electride can be described by
+periodic F-center and it may keep dual nature of local-
+ized and itinerant electrons. However, all aforementioned
+studies on the black sodalite were based on the dynam-
+ically unstable P¯43n Na4(AlSiO4)3. Thus, we need to
+investigate the co-operative localization and hybridiza-
+tion in the ground state P23 Na4(AlSiO4)3 electride.
+Fig. 6 displays the computed wavefunctions for pristine
+P¯43n Na4(AlSiO4)3Cl, F-center of Cl vacancy in the su-
+percell structure, and crystalline P23 Na4(AlSiO4)3. In
+the pristine structure (see the left panel of Fig.
+6), it
+shows that the wavefunction of VBM is comprised of O-p
+while the CBM is comprised of [O-s + Cl-s]. The valence
+band right below the VBM is comprised of [Cl-p + O-p],
+in which the wavefunction manifests a spherical shape
+inside of Na4 tetrahedron at the vicinity of Cl. This is a
+typical for an ionic bonding in compound like NaCl.
+As shown in the middle panel of Fig. 6, the introduc-
+tion of Cl vacancy gives rise to the formation of a defect
+F-center. Like the atomic orbital, the F center exhibits a
+set of quantized states similar to a hydrogen atom. The
+first F-center associated state appears between the VBM
+and CBM. In comparison to the spherical [Cl-p + O-p]
+state in the pristine structure, this state is mainly com-
+prised of localized F-s state in the Na4 tetrahedron and
+a few O-p states centered at the O atoms. Around the
+CBM, we observed the hybridization of [F-s + O-p] as
+the first unoccupied state. At the second lowest conduc-
+tion bands, we also found triply degenerate hybridization
+of F-p and O-s, where a spherical wavefunction inside
+Na4 tetrahedron contains an empty region at the center.
+Similar to F-s, the linear combination of F-p also has a
+spherical shape, but it has an inner hollow due to the
+requirement of orthogonality.
+Furthermore, the higher
+energy conduction bands consist of [F-s + (O, Si, Cl)-s]
+and another triply degenerate [F-p + O-p].
+Finally, the crystalline electride states (as shown in the
+right panel of Fig. 6) have qualitatively the same proper-
+ties with F-center. Their electride states exhibit distinct
+local s- or p-like features similar to F-s and F-p. These E-
+s and E-p states also have notable hybridization with the
+surrounding atomic orbitals. Due to the hybridization,
+the band width becomes larger, suggesting a notable ex-
+cess electron hopping through β frames in the crystalline
+electride67.
+D.
+Magnetism of Na4(AlSiO4)3
+Of the three proposed electrides, Na4(AlSiO4)3 man-
+ifests ferromagnetic (FM) electronic structure. Blake et
+al. first showed that the antiferromagnetic (AFM) tran-
+sition is associated with the incompletely doped regions
+of the sample61. But no such transition was observed in
+the fully doped sample. However, a later experiment by
+Heinmaa and Lippmass reported the AFM phase with a
+Neel temperature of TN = 54 K for heavily sodium-doped
+sodalite Na4(AlSiO4)3 via nuclear magnetic resonance
+spectroscopy study68.
+This experiment was also sup-
+ported by following theoretical calculations. Sankey et
+al. reported the magnetic ground state of Na4(AlSiO4)3
+(i.e. fully doped case) by conducting local-spin-density
+approximation (LSDA) simulations67.
+They showed a
+lower energy of AFM ordering was favored over the FM
+ordering. Our calculations also reveal that AFM config-
+uration has a lower energy of 33 meV per formula unit
+than FM configuration. As shown in Fig. 5b, the AFM
+spin ordering opens a gap, which is an indirect gap of
+0.45 eV from R to X point, which is consistent with the
+previous study on the P¯43n (Na, K)4(AlSiO4)358.
+In this work, we also investigated the origin of mag-
+netization within FM configuration to compare existing
+ferromagnetic electrides Y2C17 and Ca5Ga2N422.
+The
+calculated total energy of spin-polarized Na4(AlSiO4)3 is
+lower by 42 meV per formula unit than the non-spin-
+polarized system.
+The total magnetization moment is
+0.97 µB per formula unit. These values are between ∼ 15
+meV and 0.38 µB for a weak itinerant ferromagnetic elec-
+tride Y2C17, and 65 meV and 1.25 µB for Ca5Ga2N422.
+Similar to the previous study22, we found that the
+strong magnetization of Na4(AlSiO4)3 is due to excess
+electron localization inside the Na4 tetrahedron.
+As
+shown in Fig. 7a, increasing lattice parameter (length
+of Na4 tetrahedron edges) from 8.98 ˚A (4.51 ˚A) to 9.28
+˚A (5.71 ˚A) results in less excess electron confinement
+in the larger Na4 tetrahedron making the magnetic mo-
+ments disappear.
+The electron localization can be de-
+termined by the electron localization function (ELF),
+which is a measure of the likelihood of finding an elec-
+tron in the neighborhood space of a reference electron
+located at a given point69. The ELF value is a dimen-
+sionless index.
+ELF = 1 corresponding to perfect lo-
+calization and ELF =
+1
+2 corresponding to the electron
+gas. As shown in Fig. 7c, the maximum ELF value at
+the cage center for Na4(AlSiO4)3 is 0.99 when a=8.98
+˚A, which is higher than 0.94 when a=9.28 ˚A. For the
+reference, the corresponding maximum ELF values are
+0.85 for Na4(AlGeO4)3, and 0.96 for Na4(GaSiO4)3. The
+trend of ELF values is also consistent with the results on
+the band width. The calculated electride spin up band
+widths are 0.72 eV for Na4(AlSiO4)3 with a=8.98 ˚A),
+1.19 eV for Na4(AlSiO4)3 with a=9.28 ˚A, 1.39 eV for
+Na4(AlGeO4)3, and 1.20 eV for Na4(GaSiO4)3. Clearly,
+these localizations are strongly associate with the elec-
+tride band widths.
+
+8
+FIG. 7.
+The calculated electronic structures of electrides. (a) The total density of states and wavefunctions of fully occupied
+band at Γ point for Na4(AlSiO4)3 with two different lattice parameters. (b) Atomic (Ge,Si) projected density of states of
+Na4(AlGeO4)3, Na4(AlSiO4)3 (a=8.98 ˚A), and Na4(AlSiO4)3 (a=9.28 ˚A). The spin up density of states are illustrated. The
+bottom panel shows isosurfaces of partial charge density in the range of -2 < E - EF < 0 eV for Na4(AlSiO4)3 (a=8.98 ˚A)
+and Na4(AlGeO4)3. (c) The electron localization functional maps at (100) plane of Na4(AlSiO4)3 (a=8.98 ˚A), Na4(AlSiO4)3
+(a=9.28 ˚A), Na4(AlGeO4)3, and Na4(GaSiO4)3.
+For itinerant ferromagnets, magnetism can be under-
+stood by the Stoner criterion of D(EF)I > 1, where
+D(EF) is density of state at the Fermi level and I is
+the exchange parameter. As shown in Fig. 7a, the sharp
+D(EF) of Na4(AlSiO4)3 (a=8.98 ˚A) suggests the mag-
+netism is raised from the Stoner-type instability.
+We
+showed the sharp D(EF), which associates with the small
+electride band width, is due to localization of excess elec-
+tron. Then, a question is raised for the non-magnetism
+of Na4(AlSiO4)3 and Na4(GaSiO4)3.
+We checked the
+possible magnetism of Na4(AlGeO4)3 with smaller lat-
+tice parameters to try to localize the excess electron
+and increase D(EF). All attempts resulted in non-spin-
+polarized electronic structures. This can be attributed
+to the higher electronegativity of Ge than Si. As shown
+in Fig. 7b, owing to the electronegativity, the Ge’s pro-
+jected DOS (PDOS) in Na4(AlGeO4)3 is greater than
+the Si’s PDOS in Na4(GaSiO4)3 below the Fermi level,
+while there is no notable difference in the Si’s PDOS in
+the two different lattices. The large metal cation’s elec-
+tronegativity leads to increasing hybridization66, which
+results in less localization of excess electrons and a large
+band width. This hinders spin polarization of the excess
+electrons.
+Therefore, there is no magnetism in either
+Na4(AlSiO4)3 or Na4(GaSiO4)3.
+IV.
+SUMMARY AND CONCLUSIONS
+In sum, we present an approach to realize the elec-
+tride material via the crystallographic symmetry analy-
+sis. Inspired by the synthesis of C12A7:2e−, we proposed
+to make potentially stable electrides from thermally sta-
+ble inorganic compounds that possess the high symmetry
+Wyckoff site being occupied by anions. Using the well-
+known halide sodalites as the examples, we found that
+the materials after the removal of anionic halide anions
+exhibit typical electride behaviors that are characterized
+by the existence of localized electronic states near the
+Fermi level. Among them, Na4(AlSiO4)3 manifests anti-
+ferromagnetic electronic structure. We demonstrate that
+this magnetism originate from a highly localized excess
+electron state surrounded by electronpositive alkaline
+cations.
+While some compounds (e.g., Na4(AlSiO4)3)
+have been investigated previously due to interest in F-
+center research57,58,61,67,68, our work provided more rig-
+orous phonon analysis to derive the truly stable ground
+state structures of crystalline electrides.
+Compared to
+most previously studied electrides, these materials are
+derived from the sodalite materials with high melting
+points. Therefore, they are expected to be more ther-
+mally stable due to the complex structural framework.
+Thanks to the improved stability, these predicted ma-
+terials are more amenable for fabrication and practical
+applications. In the present work, we only investigated
+several halide sodalite to demonstrate the proof of con-
+cept. The extension to other sodalite type materials with
+different anions is the subject of future work. In addi-
+tion, our work suggests the connection between porous
+zeolite and electride since the existence of crystalline ze-
+olite can naturally provide the crystal cavity to accom-
+modate the excess electrons.
+While efforts on making
+zeolite electrides have been attempted in the past70,71,
+the extraordinary diversity in zeolite structures72 war-
+rants many opportunities for future exploration.
+
+(a)
+(b)
+(c)
+15
+0.021
+Na4(A1Si04)3
+Na4(A1SiO4)3
+Ge:Na4(A1GeO4)3
+Si:Na4(A1Si04)3i
+spin up
+10
+a= 8.98 A
+a= 9.28 A
+1.0
+5
+0
+soa
+0.01
+DOS 
+5
+ 8.98 A
+a=
+ Na4(GaSiO4)3
+Na4(A1Ge04)3
+ELI
+10
+a= 9.28 A
+T
+spin down
+15
+0.00+
+-0.5
+0.0
+1.0
+1.5
+-1.5
+-1.0
+0.5
+-1.0
+-0.5
+0.0
+0.5
+E-E, (eV)
+E-Er (eV)
+a: 8.98 A
+a: 9.28 A
+Na4(A1SiO4)3
+Na4(A1GeO4)3
+0.0
+Na-Na: 4.51 A
+Na-Na: 5.71 A9
+ACKNOWLEDGMENTS
+This research was sponsored by the U.S. Department
+of Energy, Office of Science, Office of Basic Energy Sci-
+ences, Theoretical Condensed Matter Physics program
+and the DOE Established Program to Stimulate Com-
+petitive Research under Award Number DE-SC0021970.
+The computing resources are provided by ACCESS (TG-
+DMR180040) and National Energy Research Scientific
+Computing Center (NERSC).
+∗ [email protected]
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+MNRAS 000, 1–14 (2015)
+Preprint 11 January 2023
+Compiled using MNRAS LATEX style file v3.0
+The eROSITA Final Equatorial-Depth Survey (eFEDS) - Splashback
+radius of X-ray galaxy clusters using galaxies from HSC survey
+Divya Rana1★, Surhud More1,2†, Hironao Miyatake2,3,4, Sebastian Grandis5,6, Matthias Klein6,
+Esra Bulbul7, I-Non Chiu8,9,10, Satoshi Miyazaki11, Neta Bahcall12
+1 Inter University Centre for Astronomy and Astrophysics, Ganeshkhind, Pune 411007, India
+2 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, 5-1-5, Kashiwanoha, 2778583, Japan
+3 Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University, Nagoya, 464-8602, Japan
+4 Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan
+5 Institut für Astro- und Teilchenphysik, Universität Innsbruck, Technikerstr. 25/8, 6020 Innsbruck, Austria
+6 Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679, Munich, Germany
+7 Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße 1, D-85748 Garching, Germany
+8 Department of Physics, National Cheng Kung University, 70101 Tainan, Taiwan
+9 Department of Astronomy, School of Physics and Astronomy, and Shanghai Key Laboratory for Particle Physics and Cosmology,
+Shanghai Jiao Tong University, Shanghai 200240, China
+10 Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China
+11 Subaru Telescope, National Astronomical Observatory of Japan, 650, N Aohoku Place, Hilo, HI 96720 USA
+12 Department of Astrophysical Sciences, Peyton Hall, 4 Ivy Lane, Princeton University, Princeton, NJ 08544
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We present the splashback radius measurements around the SRG/eROSITA eFEDS X-ray selected galaxy clusters by cross-
+correlating them with HSC S19A photometric galaxies. The X-ray selection is expected to be less affected by systematics
+related to projection that affects optical cluster finder algorithms. We use a nearly volume-limited X-ray cluster sample selected
+in 0.5-2.0 keV band having luminosity 𝐿𝑋 > 1043.5ergs−1h−2 within the redshift 𝑧 < 0.75 and obtain measurements of the
+projected cross-correlation with a signal-to-noise of 17.43. We model our measurements to infer a three-dimensional profile and
+find that the steepest slope is sharper than −3 and associate the location with the splashback radius. We infer the value of the 3D
+splashback radius 𝑟sp = 1.45+0.30
+−0.26 h−1Mpc. We also measure the weak lensing signal of the galaxy clusters and obtain halo mass
+log[𝑀200m/h−1M⊙] = 14.52 ± 0.06 using the HSC-S16A shape catalogue data at the median redshift 𝑧 = 0.46 of our cluster
+sample. We compare our 𝑟sp values with the spherical overdensity boundary 𝑟200m = 1.75 ± 0.08 h−1Mpc based on the halo mass
+which is consistent within 1.2𝜎 with the ΛCDM predictions. Our constraints on the splashback radius, although broad, are the
+best measurements thus far obtained for an X-ray selected galaxy cluster sample.
+Key words: keyword1 – keyword2 – keyword3
+1 INTRODUCTION
+The gravitational collapse of high-density peaks in the initial matter
+distribution results in the formation of virialized massive dark matter
+halos in the Universe. The most massive of these halos host clusters
+of galaxies that we see today (see Kravtsov & Borgani 2012; Walker
+et al. 2019; Vogelsberger et al. 2020, for a recent review). The matter
+distribution in the dark matter halos is driven by the profile of the
+initial density peak from which it forms. There have been extensive
+theoretical and numerical studies to understand the structure of the
+dark matter halos (e.g. Gunn & Gott 1972; Fillmore & Goldreich
+1984; Bertschinger 1985; Navarro et al. 1997; Moore et al. 1999).
+Since the seminal study of (Navarro et al. 1997), it has been well
+known that dark matter halos follow self-similar density distribution
+★ E-mail: [email protected]
+† E-mail: [email protected]
+in their inner regions, known as the Navarro-Frenk-White (NFW)
+profile.
+On the other hand, the outskirts of massive dark matter halos
+have received attention in the last few years and both theoretical and
+observational aspects of it remain a subject of active research. Diemer
+& Kravtsov (2014) showed that the matter distribution at the outskirts
+of stacked dark matter halo profiles differs from extrapolations of the
+NFW profile. In the outskirts of these halos, the density distribution
+shows logarithmic slopes (𝑑 log 𝜌/𝑑 log 𝑟) which are much steeper
+than the asymptotic value of −3 expected from the NFW profile.
+The location of the steepest slope also is dependent upon the mass
+accretion rate of these dark matter halos. Adhikari et al. (2014)
+explicitly showed with a phase space analysis that these locations
+correspond to the position where recently accreted particles reach
+the apocenters of the orbits for the first time.
+The resultant sharp drop in the density profile at this location is
+reminiscent of the last density caustic predicted in the model of sec-
+© 2015 The Authors
+arXiv:2301.03626v1  [astro-ph.CO]  9 Jan 2023
+
+2
+Rana et al.
+ondary infall around a spherically symmetric overdensity (Fillmore
+& Goldreich 1984; Bertschinger 1985; Shi 2016). Adhikari et al.
+(2014) termed this location the "splashback radius" as after reaching
+the apocenters, the particles are expected to splash back in to the
+halo. Subsequently, the splashback radius has been proposed as a
+physical boundary of the dark matter halo and found to be primarily
+dependent on the mass of the halo, its accretion rate and redshift
+(More et al. 2015; Diemer et al. 2017). These results triggered an in-
+terest in various aspects of splashback radius studies in simulations
+(see e.g Mansfield et al. 2017; Okumura et al. 2018; Fong et al.
+2018; Mansfield & Kravtsov 2020; Sugiura et al. 2020; Xhakaj et al.
+2020; Diemer 2020; Deason et al. 2021; O’Neil et al. 2022a), for
+different dark matter and dark energy theories (Adhikari et al. 2018;
+Contigiani et al. 2019a; Banerjee et al. 2020) and galaxy evolution
+scenarios in clusters (Dacunha et al. 2022).
+Observational investigations of the splashback radius have fo-
+cussed on galaxy clusters as the density drops at the splashback
+radius are expected to be quite significant due to their higher current
+accretion rates. Secondly on cluster scales it is easier to select iso-
+lated halos than on galaxy and group scales. More et al. (2016) used
+the cross-correlation of the SDSS redMaPPer clusters (Rykoff et al.
+2014) with SDSS photometric galaxies to present the first detection
+of the splashback radius. They detected a steepening of the projected
+galaxy number density profile as observed by (Diemer & Kravtsov
+2014) in numerical simulations. However, the inferred location of
+the splashback radius was found to be about 20 percent smaller than
+expected from the dark matter simulations based on ΛCDM model
+(More et al. 2015). The robustness of these results to effects such
+as the miscentering of clusters and to priors were demonstrated in
+Baxter et al. (2017). Busch & White (2017) argued that selection
+effects induced by optical clusters could affect the cross-correlation
+measurements and could therefore be important in understanding the
+origin of this difference. Using clusters selected by a mock redMaP-
+Per algorithm run on a simulation, Sunayama & More (2019) showed
+that these differences could indeed arise from such optical selection
+effects. Such selection effects are quite sensitive to the background
+subtraction scheme employed while running the cluster finding al-
+gorithm. Murata et al. (2020) explored clusters selected with the
+CAMIRA cluster finder (Oguri 2014) using data from the Subaru
+Hyper Suprime-Cam survey (Aihara et al. 2018a) which use a local
+background subtraction scheme. They obtained measurements of the
+splashback radius which were consistent with ΛCDM prediction.
+Such issues related to the projection effects in optically selected
+galaxy clusters can be avoided by using clusters selected using the
+Sunyaev Zeldovich (SZ) effect (Sunyaev & Zeldovich 1970, 1980).
+Zürcher & More (2019) used galaxy clusters selected from the Planck
+SZ survey and cross-correlated them with galaxies detected in the
+Pan-STARRS. Shin et al. (2019) used SZ clusters selected from
+the Atacama Cosmology Telescope (ACT) Polarimeter (Hilton et al.
+2018) and the South Pole Telescope (SPT, Bleem et al. 2015) SZ
+survey along with galaxy catalog data from the Dark Energy Survey
+(DES, The Dark Energy Survey Collaboration 2005). They use both
+the galaxy number density and weak lensing profiles to obtain the
+constraints on the splashback radius and found the results consistent
+with the ΛCDM predictions, albeit with a larger errorbar which does
+not entirely preclude the initial results from optically selected clus-
+ters. Similarly, Shin et al. (2021) show consistency of the measured
+splashback radius with expectations from ΛCDM using a larger clus-
+ter catalogue with improved precision, yet not with errors that could
+rival those obtained with optical clusters (More et al. 2016; Baxter
+et al. 2017; Chang et al. 2018; Murata et al. 2020). Furthermore,
+studies along the same lines have used galaxy clusters identified us-
+ing the X-rays emitted by the bremsstrahlung emission from ICM
+(Umetsu & Diemer 2017; Contigiani et al. 2019b; Bianconi et al.
+2021) and found similar results.
+In this study, we use galaxy clusters selected based on their X-
+ray emission and cross-correlate their positions with those of optical
+galaxies (More et al. 2016; Baxter et al. 2017; Chang et al. 2018;
+Murata et al. 2020). Previous studies with X-ray clusters (Umetsu &
+Diemer 2017; Contigiani et al. 2019b; Bianconi et al. 2021) are based
+on either a limited number of galaxy clusters due to the shallower
+depth of the survey or their limited area coverage. Instead, our goal is
+to use the X-ray galaxy clusters from the extended ROentgen Survey
+with an Imaging Telescope Array (eROSITA, Merloni et al. 2012;
+Predehl et al. 2021) on board the SRG mission, a highly sensitive
+space-based X-ray telescope launched in July 2019. The eROSITA
+mission aims to conduct an all sky survey in X-rays which will
+yield a catalog of ∼ 100000 X-ray galaxy clusters by the end of
+four years of operation (Borm et al. 2014). Before observing the
+entire sky, eROSITA first collected data from a smaller equatorial
+field of approximately 140 sq. deg. at its planned depth to test its
+performance. The X-ray galaxy cluster data provided by the eROSITA
+final equatorial depth survey (eFEDS) has been extensively studied
+for cluster cosmology (see for e.g. Sanders et al. 2022; Ramos-Ceja
+et al. 2022; Ghirardini et al. 2022; Chiu et al. 2022b,a; Bulbul et al.
+2022; Bahar et al. 2022; Klein et al. 2022). As a pilot study we will
+use the eFEDS X-ray galaxy clusters and obtain constraints on the
+location of the splashback radius by measuring the number density
+profiles of galaxies correlated with these clusters.
+Inference of the splashback radius using measurements of the
+galaxy number density profile is possible if galaxies act as test par-
+ticles in the cluster potential and dynamically are distributed similar
+to dark matter particles. Galaxies residing in massive subhaloes are
+expected to be affected by dynamical friction (Chandrasekhar 1943),
+which slows down their motion around the cluster centre and de-
+creases the radius at which they reach their first apocenters. This
+effect on the orbit could potentially bias the measurements of the
+splashback radius and has been seen in simulations (More et al.
+2016; O’Neil et al. 2022b) and with observational claims (Adhikari
+et al. 2014, cf. More et al. 2016). Dynamical friction can be min-
+imized by using the faintest of galaxies residing in low-mass sub-
+haloes that are affected very little by dynamical friction. The Hyper
+Suprime Cam (HSC) survey provides photometric galaxies down to
+an i-band magnitude of 26 with excellent seeing conditions (Aihara
+et al. 2018a). We use galaxies from the HSC S19A internal data re-
+lease (Aihara et al. 2022) and cross-correlate them with X-ray galaxy
+clusters selected from eFEDS. We constrain the splashback radius
+and compare its magnitude with the commonly used spherical over-
+density boundary 𝑅200m. For halo mass estimates, we use the galaxy
+shape catalogue data from HSC S16A (Mandelbaum et al. 2018a) to
+measure the weak lensing signal around our clusters.
+We describe the different data catalogs we use in Section 2, while
+the measurement and the modelling techniques are described in Sec-
+tion 3. In Section 4 we present the main results and compare them
+with earlier works. We then summarize our findings in Section 5.
+Throughout the work, we use flat ΛCDM cosmological model with
+matter density Ωm = 0.27, baryon density Ωb = 0.049, power law
+index of the initial power spectrum 𝑛s = 0.95, variance of density
+fluctuation 𝜎8 = 0.81, temperature of the cosmic microwave back-
+ground TCMB = 2.726 K and the Hubble parameter ℎ = 0.7 as
+out fiducial cosmological model. The symbol 𝑟 represents the three-
+dimensional while 𝑅 represents the projected two-dimensional radial
+distance from the cluster center. We use the halo mass definition of
+𝑀200m and corresponding halo boundary 𝑅200m as the radius en-
+MNRAS 000, 1–14 (2015)
+
+Splashback radius around X-ray galaxy clusters
+3
+closing the matter density 200 times the present matter density of the
+universe and the log used is the logarithm at base ten.
+2 DATA
+2.1 X-Ray Cluster Catalogue
+The eROSITA (Merloni et al. 2012; Predehl et al. 2021) is a seven
+telescope module capable of detecting X-rays onboard the Russian-
+German Spectrum-Roentgen-Gamma (SRG) satellite (Sunyaev et al.
+2021) orbiting around the Lagrange point L2. It provides a field of
+view of ≈ 1deg2 with excellent imaging quality. The on-axis energy
+resolution is ≈ 18′′ at 1.48 keV and ≈ 26′′ average angular resolu-
+tion over the full field of view (Predehl et al. 2021). The eROSITA
+final equatorial depth survey (eFEDS) is a small field having an
+area of 140 deg2 with a vignetted corrected average exposure time
+of ∼ 1.3ks carried out at a depth similar to the depth to be achieved
+after eROSITA All-Sky Survey is complete. Thus it works as a per-
+formance verification phase for the whole survey.
+The data for the eFEDS comprises of observations in the energy
+band 0.2-2.3 keV. The data were collected during the first quarter
+of Nov 2019. This data was processed using the eROSITA Standard
+Analysis Software System (eSASS) pipeline for source detections
+(Brunner et al. 2022). The candidate source detection for the X-ray
+clusters provides a detection likelihood Ldet and an extent likelihood
+Lext for each source. In order to select galaxy clusters we apply
+quality cuts of Ldet > 5 and Lext > 6. The values of these thresh-
+olds were calibrated using simulations which demonstrate that these
+cuts tend to give a secure sample with about 80 percent genuine
+clusters (Comparat et al. 2020). The application of the quality cuts
+results in a sample of 542 X-ray galaxy clusters (for more detail,
+see Brunner et al. 2022; Liu et al. 2022). A number of these galaxy
+clusters were optically confirmed using the Multi-Component Match
+filter (MCMF) algorithm (Klein et al. 2018, 2019). The MCMF uses
+galaxies from both HSC S20A and the DESI Legacy Imaging Sur-
+veys (Dey et al. 2019) to provide redshifts estimate 𝑧cl, richness 𝜆cl
+and secondary peak contamination fraction 𝑓cont. The contamina-
+tion fraction provides a quantitative measure for the probability of
+the chance superposition in the optical. As suggested in Klein et al.
+(2022), we use a cut of 𝑓cont < 0.2 which selects an X-ray clus-
+ter sample with minimal contamination as deduced from the optical
+confirmation. In fact due to the luminosity cut we use for selecting
+the X-ray clusters, a vast majority of them (98 out of 109) have a con-
+tamination in the optical confirmation at a level less than 5% (i.e.,
+𝑓cont < 0.05). The full description of MCMF optimal confirmation
+analysis for our dataset is given in Klein et al. (2022).
+In the current work, we use X-ray cluster data from the pub-
+licly released eFEDS catalogue version 2.1 as described in Klein
+et al. (2022). The catalogue provides the sky position of each of
+the galaxy clusters in the columns RA_CORR, DEC_CORR, the
+redshifts and its associated error in the columns Z_BEST_COMB,
+SIGMA_Z_BEST_COMB, along with the contamination fraction in
+the column F_CONT_BEST_COMB. Similar, we use the columns
+ML_FLUX, ML_FLUX_ERR for the flux and its errors in the 0.5-2.0
+keV band. We select those clusters which have less than 20 percent
+relative error on both the redshift and flux measurements.
+Finally, we also convert the flux values to luminosities 𝐿X =
+4𝜋𝑑2
+l (𝑧) 𝑓X, where 𝑑l(𝑧) is the luminosity distance at redshift 𝑧 for
+our fiducial cosmological model. We select galaxy clusters that sat-
+isfy a threshold of 𝐿X > 1043.5erg s−1h−2 and have redshift 𝑧 < 0.75
+as shown in Figure 1. The solid red line shows our selection which
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+z
+1041
+1042
+1043
+1044
+LX(erg s−1h−2)
+Figure 1. Sample selection: The above figure represents the scatter plot
+between X-ray luminosity 𝐿𝑋 and redshift 𝑧 for our eFEDS galaxy cluster
+sample using clusters residing in the HSC S19A footprint with 1𝜎 errors in
+the y direction. The solid red line represents a roughly volume-limited sample
+selection at luminosity threshold 𝐿𝑋 > 1043.5erg s−1h−2 within the redshift
+of 𝑧 < 0.75 and providing 109 X-ray galaxy clusters for our analysis (without
+the k-correction). The eROSITA eFEDS field has roughly 90 deg2 masked
+overlapping region with HSC S19A.
+corresponds to 109 X-ray galaxy clusters that we use for our analysis.
+These selection cuts help us to prune out less massive clusters at low
+redshift while giving us a sample that looks approximately volume
+limited. We note that we have not applied any k and 𝑛H corrections
+to keep our sample selection simple, where 𝑛H is the galactic neutral
+hydrogen column density. However, we have also checked by using
+the corrected values for luminosities given by Liu et al. (2022) and
+our sample changes by at most 10 percent.
+We will also use a subsample of 45 X-ray clusters that reside within
+the footprint that encompasses the HSC S16a internal data release.
+This subsample will be used to carry out a weak lensing analysis with
+the help of the first year shape catalog from the HSC survey. In order
+to explore any systematic effects, we use 100 times more random
+points in the same field constructed using the sensitivity map from
+the eFEDS region. We assign redshifts to these randoms by drawing
+with replacement from the parent redshift distribution of our cluster
+sample.
+2.2 Galaxy Catalogues
+We use a galaxy catalogue from the Subaru Hyper Suprime-Cam
+(HSC) optical imaging survey carried out using the wide-field
+(1.77 deg2) Hyper Suprime-Cam instrument (Komiyama et al. 2018;
+Miyazaki et al. 2018). The Subaru is a 8 meter class telescope situ-
+ated at the summit of Maunakea in Hawai’i where the median seeing
+is close to ∼ 0.6”. The HSC survey collaboration has concluded a
+wide angle imaging survey of 1200 sq. deg. in the 𝑢𝑔𝑟𝑖𝑧𝑦 bands at
+a depth of 𝑖 = 26 under the Subaru Strategic Program (SSP, Aihara
+et al. 2018a).
+We use galaxies from an area ∼ 90 deg2 in the GAMA09H field
+MNRAS 000, 1–14 (2015)
+
+4
+Rana et al.
+from the internal data release S19A of the HSC survey. Following
+Murata et al. (2020), we use fluxes from the forced photometry table,
+and use objects brighter than z-band cmodel magnitude (galactic
+extinction corrected) 𝑚𝑧 < 24.5. We select only those objects that
+have flux errors less than 20 percent and restrict ourselves to sources
+that are extended in the z-band (z_extendedness_value != 0). We
+note that given the redshifts of our clusters, the the 4000Å never
+enters the 𝑧 band, which avoids any non-uniformities in the sample
+selection.
+In order to avoid selecting galaxies from regions which do not
+have sufficient depth, we use galaxies which have been observed in
+a minimum number of exposures in the HSC survey. Such a selec-
+tion is made by using the columns [gr]_inputcount_value≥2
+and [izy]_inputcount_value≥4. We also apply various flags to
+remove galaxies affected by bad pixels and remove duplicates due to
+overlaps in the area processed by the software pipeline. In particular
+we use
+• z_deblend_skipped = False
+• z_cmodel_flag_badcentroid = False
+• z_cmodel_flag = False
+• z_pixelflags_edge = False
+• z_pixelflags_interpolatedcenter = False
+• z_pixelflags_saturatedcenter = False
+• z_pixelflags_crcenter = False
+• z_pixelflags_bad = False
+• z_pixelflags_suspectcenter = False
+• z_pixelflags_clipped = False
+• z_detect_isprimary = True
+• z_sdsscentroid_flag = False
+In addition we also use flags from the mask table in order to reduce
+the impact of bright stars in the HSC survey - the ghost mask, the halo
+mask and the blooming mask as described in Aihara et al. (2022).
+These conditions can be summarized as:
+• z_mask_brightstar_ghost = False
+• z_mask_brightstar_blooming = False
+• z_mask_brightstar_halo = False
+We also infer the halo masses for our cluster sample using weak
+gravitational lensing. For this purpose, we use the shape information
+of galaxies from the incremental data release S16A that includes the
+first year shape catalogue (Aihara et al. 2019). This data release is has
+slightly more data compared to the first public data release (Aihara
+et al. 2018b) of the HSC survey. The entire S16A shape catalog
+spans six different fields - HECTOMAP, GAMA09H, WIDE12H,
+GAMA15H, XMM, and VVDS covering a total area of 136.9 deg2
+and has an effective galaxy number density of 21.5 arcmin−2 with a
+median redshift of 0.8.
+We use data from the GAMA09H field as it overlaps with the
+eFEDS cluster catalogue in terms of sky area. The shape catalogue
+provides the sky positions for galaxies, their estimated shapes and
+their corresponding calibrations. The shapes of galaxies are rep-
+resented by two ellipticities (𝑒1, 𝑒2) = (𝑒 cos 2𝜙, 𝑒 sin 2𝜙) with
+𝑒 = (𝑎2 − 𝑏2)/(𝑎2 + 𝑏2), where a and b are the semimajor and
+semiminor axis of the galaxies (Bernstein & Jarvis 2002) with 𝜙
+as the position angle of major axis in equatorial coordinate system.
+These shapes are estimated using the re-Gaussianization technique
+(Hirata & Seljak 2003), which has been extensively studied using
+data from the SDSS survey (Mandelbaum et al. 2005; Reyes et al.
+2012; Mandelbaum et al. 2013).
+The shape catalog also provides additive bias (𝑐1, 𝑐2), multiplica-
+tive bias 𝑚 corrections, the rms value for intrinsic ellipticities 𝑒rms
+and the measurement error 𝜎𝑒 for each galaxy. This additional cal-
+ibration data were obtained by image simulations performed using
+an open-source software package - GALSIM (Rowe et al. 2015) that
+mimic observation conditions for the HSC survey (Mandelbaum et al.
+2018b). While computing the weak lensing signal, we assign weights
+𝑤𝑠 to each galaxies using their 𝑒rms and 𝜎𝑒 as 𝑤𝑠 = (𝑒2rms +𝜎2𝑒)−1 as
+is described in more detail in Mandelbaum et al. (2018b). The shape
+catalogue already includes various quality cuts, primary amongst
+which is an i-band magnitude limit of 𝑖 < 24.5 as suggested in Man-
+delbaum et al. (2018a) for studies related to weak lensing cosmology.
+The HSC survey provides full probability distribution function 𝑝(𝑧)
+for each galaxy in the shape catalogue for each of the six different
+methods (Tanaka et al. 2018). We use the 𝑝(𝑧) computed using the
+classical template fitting code - Mizuki (Tanaka 2015) for the weak
+lensing signal measurements along with a selection cut on the 𝑝(𝑧)
+to extract a secured sample of background galaxies as described in
+Section 3.1. We also checked our results using other photometric
+redshift estimation methods and found no significant change in the
+inferred values for the model parameters.
+3 MEASUREMENTS AND MODELLING
+Here we will describe the methods used for measurements and mod-
+elling of the weak lensing profiles and the cluster galaxy cross-
+correlations for our sample.
+3.1 Weak Lensing Profile
+The weak gravitational lensing imprints a coherent tangential dis-
+tortion pattern in the shapes of background source galaxies due to
+matter present in and around foreground lens galaxies (see Kilbinger
+2015; Mandelbaum 2018, for a recent review). These distortions can
+be measured as a function of projected cluster-centric comoving ra-
+dial distance 𝑅 and are related with the projected matter distribution
+in the foreground lens such that
+¯Σ(𝑅) − ⟨Σ(𝑅)⟩ = Σcrit𝛾𝑡
+(1)
+where the ¯Σ(𝑅) − ⟨Σ(𝑅)⟩ = ΔΣ(𝑅) is known as excess surface
+density (ESD). The quantity ¯Σ(𝑅) =
+∫ 𝑅
+0 Σ(𝑅′)2𝜋𝑅′ 𝑑𝑅′/𝜋𝑅2 is the
+average surface matter density within a projected distance 𝑅 while
+⟨Σ(𝑅)⟩ is the azimuthally averaged surface matter density at the
+same distance. The quantity 𝛾𝑡 denotes the average tangential shear,
+and Σcrit denotes the critical surface density, a geometric factor that
+quantifies the lensing efficiency of a given lens-source pair, such that,
+Σcrit =
+𝑐2
+4𝜋𝐺
+𝐷a(𝑧s)
+(1 + 𝑧l)2𝐷a(𝑧l)𝐷a(𝑧l, 𝑧s) .
+(2)
+Here the quantities 𝐷a(𝑧s), 𝐷a(𝑧l) and 𝐷a(𝑧l, 𝑧s) are the angular
+diameter distances between us and the source at redshift 𝑧s, us and the
+lens at redshift 𝑧l and between a given lens-source pair, respectively.
+The (1 + 𝑧l)2 factor in the denominator is related to our use of
+comoving coordinates (Mandelbaum et al. 2006).
+Following Mandelbaum et al. (2018a), we compute the ESD
+ΔΣ(𝑅𝑖) for our lensing sample using the HSC background source
+galaxies at ten logarithmic cluster centric projected comoving radial
+MNRAS 000, 1–14 (2015)
+
+Splashback radius around X-ray galaxy clusters
+5
+distance bins 𝑅𝑖 ranging from 0.2 to 2 h−1Mpc, such that
+ΔΣ(𝑅𝑖) =
+1
+1 + 𝑚
+� �
+ls∈𝑅𝑖 𝑤ls𝑒𝑡,ls⟨Σ−1
+crit⟩−1
+2R �
+ls∈𝑅𝑖 𝑤ls
+�
+−
+1
+1 + 𝑚
+� �
+ls∈𝑅𝑖 𝑤ls𝑐𝑡,ls⟨Σ−1
+crit⟩−1
+�
+ls∈𝑅𝑖 𝑤ls
+�
+.
+(3)
+Here 𝑒𝑡,ls and 𝑐𝑡,ls are the components of the ellipticities (𝑒1, 𝑒2) and
+additive bias (𝑐1, 𝑐2) in the tangential direction to the line joining
+the lens and the source. The summation is over all the lens-source
+pair ls having separation of 𝑖th radial bin 𝑅𝑖. We also use minimum
+variance weights such that 𝑤ls = 𝑤s⟨Σ−1
+crit⟩2. The quantity ⟨Σ−1
+crit⟩ is
+the average of the inverse surface critical density over the probability
+distribution of source redshift 𝑝(𝑧s) given by
+⟨Σ−1
+crit⟩ = 4𝜋𝐺(1 + 𝑧l)2
+𝑐2
+∫ ∞
+𝑧l
+𝐷a(𝑧l)𝐷a(𝑧l, 𝑧s)
+𝐷a(𝑧s)
+𝑝(𝑧s)𝑑𝑧s .
+(4)
+The 1 + 𝑚 factor in the denominator of Eq. 3 corresponds to the
+multiplicative bias with 𝑚 = �
+ls∈𝑅𝑖 𝑤ls𝑚s/�
+ls∈𝑅𝑖 𝑤ls and R is the
+shear responsitivity which corrects for the response of ellipticities to
+an applied value of shear and is computed using 𝑒rms (Bernstein &
+Jarvis 2002) as,
+R = 1 −
+�
+ls∈𝑅𝑖 𝑤ls𝑒2
+rms,ls
+�
+ls∈𝑅𝑖 𝑤ls
+.
+(5)
+In order to have reduce contamination of the lensing signal by
+foreground galaxies or galaxies correlated with our clusters, we only
+use sources which satisfy
+∫ ∞
+𝑧max+𝑧diff
+𝑝(𝑧s)𝑑𝑧s > 0.99 .
+(6)
+We use 𝑧max = 0.75 as the maximum redshift of our lensing sam-
+ple and 𝑧diff = 0.2 as an additional offset for better selection of
+the background. We also use galaxies with a photo-z quality cut of
+photo_z_risk_best_value < 0.5. Further, we also apply a mul-
+tiplicative bias 𝑚sel related to the selection of galaxies above the
+resolution threshold (𝑅2 ≥ 0.3) which is used to select source galax-
+ies during the construction of the shape catalog (see Mandelbaum
+et al. 2018b). This selection bias is related to the probability density
+𝑝(𝑅2) at the threshold 𝑅2 = 0.3 such that 𝑚sel = 𝐴𝑝(𝑅2 = 0.3) with
+𝐴 = 0.00865. The probability 𝑝(𝑅2 = 0.3) is computed using the
+lens-source weights 𝑤ls for individual radial bin 𝑅𝑖.
+We apply the redshift selection cuts on source galaxies to select
+the background galaxy population. Any residual galaxies left in our
+source population that are associated with the cluster could poten-
+tially dilute the weak lensing signal. We can correct for such a dilution
+by multiplying the signal by a boost factor 𝐶(𝑅𝑖) (for example Hirata
+et al. 2004; Mandelbaum et al. 2005; Miyatake et al. 2015) which is
+the ratio between weighted lens-source pair counts to the weighted
+random-source pair counts in a given radial bin 𝑅𝑖 such that,
+𝐶(𝑅𝑖) =
+𝑁r
+�
+ls∈𝑅𝑖 𝑤ls
+𝑁l
+�
+rs∈𝑅𝑖 𝑤rs
+.
+(7)
+Here the quantities 𝑁r and 𝑁l are the number of random points and
+lenses in our sample, respectively. The weight 𝑤ls and 𝑤rs correspond
+to lens-source and random-source pairs. We use 100 different random
+realizations in our work such that they have the same number as lenses
+within the same sky coverage, and also satisfy the same star mask,
+and by construction, have the same redshift distribution as our lensing
+sample. We find that the boost parameters are mostly consistent with
+unity demonstrating the utility of our redshift cuts, so we neglect
+them in this study. We refer to Appendix A for more discussion on
+boost parameter measurements for our lensing sample.
+Along with the correction for the dilution of ESD signal using the
+boost parameters, we also check for any systematic bias due to the
+use of photometric redshifts for the source galaxies while computing
+the critical surface density. Following eqn. 5 in Mandelbaum et al.
+(2008), we compute the magnitude of such biases 𝑏(𝑧𝑙) for a lens at
+redshift 𝑧𝑙 using
+ΔΣ
+ΔΣ𝑡 = 1 + 𝑏(𝑧l) =
+�
+s 𝑤ls⟨Σ−1
+crit,ls⟩−1 �
+Σ𝑡
+crit,ls
+�−1
+�
+s 𝑤ls
+.
+(8)
+Here the quantities with superscript 𝑡 represent the true values of
+the corresponding quantities and the sum runs over all the source
+galaxies. Ideally the bias needs to be estimated using spectroscopic
+redshifts for a representative subsample of source galaxies. However,
+given the depth of the HSC survey, a reasonably large survey with
+such spectroscopic redshift is not available. Therefore, we use ro-
+bust estimates of photometric redshifts from the COSMOS-30 band
+photometry (Ilbert et al. 2009) and assume it to be a much more
+realistic estimate of the redshifts of our source galaxies. We also in-
+clude weights for each source galaxy 𝑤som that match the colour and
+magnitudes distribution of COSMOS-photoz galaxies to our source
+galaxy sample. These 𝑤som are included in the 𝑤ls while doing the
+computations as done in previous studies (for e.g. Nakajima et al.
+2012; Miyatake et al. 2019; Murata et al. 2019). We then use eqn.
+23 from Nakajima et al. (2012) to assign appropriate weights for
+our lenses to compute the average bias for the ESD signal. For our
+sample, we obtain three percent bias on average, which is negligible
+given the statistical uncertainty in the signal measurements.
+We also subtract the ESD signal around random points from the
+signal computed from our lensing sample to correct for any scale
+dependent systematics (for more details refer to Sheldon et al. 2004;
+Mandelbaum et al. 2005; Singh et al. 2017). We compute the ran-
+dom signal by averaging over ESD measurements from 100 different
+realizations of randoms.
+The presence of the same source galaxy at different radial bins
+for different galaxy clusters can give rise to a covariance between
+the ESD measurements in the different radial bins. We quantify this
+covariance by randomly rotating the shapes of our source galaxies
+which washes away the shear signal imparted on them and allows us
+to estimate the covariance due to shape noise. In our study, we utilize
+200 different random rotations and use the ESD measurements in
+each case to estimate the covariance, as discussed in Appendix B. As
+we model signal upto 1h−1Mpc we expect shapenoise to dominate
+over large scale structure contribution on these length scales. So, we
+use the shape noise covariance for our analysis and infer the halo
+masses for our lensing sample.
+We model the dark matter distribution around clusters as a NFW
+density profile 𝜌nfw(𝑟) at a three-dimensional radius 𝑟.
+𝜌nfw(𝑟) =
+𝛿c𝜌m
+�
+𝑟
+𝑟s
+� �
+1 + 𝑟
+𝑟s
+�2 ,
+(9)
+𝛿c = 200
+3
+𝑐3
+ln(1 + 𝑐) −
+𝑐
+1+𝑐
+.
+(10)
+Here 𝑟s = 𝑟200m/𝑐 is the scale radius of the galo with 𝑟200m as the
+three dimensional radius of a sphere enclosing the average density
+of 200 times the present matter density 𝜌m of the Universe,
+𝑟200m =
+� 3𝑀200m
+4𝜋200𝜌m
+�1/3
+.
+(11)
+MNRAS 000, 1–14 (2015)
+
+6
+Rana et al.
+So given the model we just need two parameters Θ = (𝑀200m, 𝑐)
+the halo mass 𝑀200m and the concentration parameter 𝑐 to predict
+the ESD profile ΔΣ(𝑅). In practice, we use the analytical form for
+ΔΣ(𝑅) given by eqn. 136-140 in Schneider (2006) for the case of the
+NFW profile.
+In principle, previous works have also added a point mass contri-
+bution for the baryonic component of the central galaxy of the cluster
+(e.g., Kobayashi et al. 2015). However for the scales of our interest,
+the dark matter contribution is the dominant component. We also do
+not split the dark matter contribution into 1-halo and 2-halo terms (for
+more details, see Hikage et al. 2013; Miyatake et al. 2016) or use an
+off-centering kernel (Johnston et al. 2007) in order to account for the
+possible misidentification of the cluster centre. Given the scales we fit
+and the statistical errors on our mass estimates, none of these effects
+would cause a significant difference to our conclusions. Furthermore
+we also do not carry out a halo occupation distribution (HOD) based
+modelling (e.g., Seljak 2000; Cooray & Sheth 2002; van den Bosch
+et al. 2013) as the quantity of our interest is the average halo mass
+rather than the entire distribution of the halo masses. At the high mass
+end corresponding to our galaxy clusters, the distribution is expected
+to be peaked due to the presence of the exponential tail of the mass
+function. Due to all these reason we limit our modelling to the simple
+NFW profile based modelling scheme as described above. In future,
+we expect the splashback radius measurements using X-ray clusters
+to improve further, which will require a more detailed analysis of the
+weak lensing signal of these galaxy clusters.
+3.2 Galaxy Number Density
+We follow the method used in More et al. (2016) for the measurement
+of the cluster galaxy cross-correlation measurements. We use the
+Davis-Peebles estimator (Davis & Peebles 1983) to compute the
+projected cross-correlation between our X-ray clusters with the HSC
+photometric galaxies. The projected cross-correlation 𝜉2D(𝑅) at a
+projected comoving radius 𝑅 is given by
+𝜉2D(𝑅) = 𝐷c𝐷g
+𝑅c𝐷g
+− 1 ,
+(12)
+where 𝐷c𝐷g are the pair counts between clusters-galaxies and 𝑅c𝐷g
+are the pair counts between cluster randoms-galaxy at a comoving
+projected separation of 𝑅. The number counts for the random points
+are normalized to account for the difference in the number of clusters
+and that of randoms. We measure the signal over nine logarithmically
+spaced comoving projected radial bins from 0.1 to 2.8 h−1Mpc. We
+use 40 times more randoms than the number of galaxy clusters in our
+sample.
+The flux limit of the photometric survey can affect the galaxy distri-
+bution around the galaxy clusters by detecting many fainter galaxies
+around the clusters closer to us and can bias our measurements. To
+remove such biases, we use the photometric galaxies having z-band
+absolute magnitude cut 𝑀𝑧 −5 log ℎ < −19.36, which corresponds to
+an apparent magnitude limit of 𝑚𝑧 = 24.0 at redshift 𝑧 = 0.75 which
+ensures the population of galaxies has similar intrinsic properties
+around each cluster in our sample even though they are at different
+redshifts. We estimate the covariance of our measurements using 20
+jackknife (Miller 1974) regions that have an approximately square
+shape and an area of 5 deg2. The side of the squares corresponds to
+47 h−1Mpc at the median redshift 𝑧 = 0.46 for our sample and is
+larger than the radial range for signal measurement, justifying our
+choice of jackknife region size. We have shown the corresponding
+covariance in Appendix B.
+We parameterize the density profile with the functional form sug-
+gested by Diemer & Kravtsov (2014) in order to model the cross
+correlation signal as is standard practice in the literature (More et al.
+2016; Baxter et al. 2017; Chang et al. 2018; Nishizawa et al. 2018;
+Zürcher & More 2019; Shin et al. 2019; Murata et al. 2020; Shin
+et al. 2021). The density profile 𝜉3D(𝑟) consists of an inner Einasto
+profile (Einasto 1965) 𝜌in(𝑟) and an outer powerlaw profile 𝜌out(𝑟)
+connected by a smooth transistion function 𝑓trans(𝑟) at a three di-
+mensional radial distance of 𝑟,
+𝜉3D(𝑟) = 𝜌in(𝑟) 𝑓trans(𝑟) + 𝜌out(𝑟) ,
+(13)
+𝜌in(𝑟) = 𝜌s exp
+�
+− 2
+𝛼
+�� 𝑟
+𝑟s
+� 𝛼
+− 1
+��
+,
+(14)
+𝜌out(𝑟) = 𝜌o
+� 𝑟
+𝑟out
+�−𝑠e
+,
+(15)
+𝑓trans(𝑟) =
+�
+1 +
+� 𝑟
+𝑟t
+�𝛽�−𝛾/𝛽
+.
+(16)
+We compute the two-dimensional cross-correlation profile 𝜉2D(𝑅)
+at the projected radius 𝑅 by integrating the three-dimensional profile
+𝜉3D(𝑟) along the line of sight direction 𝜋,
+𝜉2D(𝑅) =
+1
+𝑅max
+∫ 𝑅max
+0
+𝜉3D
+�√︃
+(𝑅2 + 𝜋2)
+�
+𝑑𝜋 .
+(17)
+Following More et al. (2016) we fix the maximum projected length
+of 𝑅max = 40ℎ−1Mpc. Our modelling scheme eqn. 17 comprises
+of nine parameters - 𝜌s, 𝛼, 𝑟s, 𝜌o, 𝑟out, 𝑠e, 𝛽, 𝑟t, 𝛾. As 𝜌o and 𝑟out are
+degenerate with each other, we fix 𝑟out = 1.5ℎ−1Mpc which gives us
+a total of eight parameters Θ = (𝜌s, 𝛼, 𝑟s, 𝜌o, 𝑠e, 𝛽, 𝑟t, 𝛾) modelling
+scheme.
+We use the Bayesian analysis to get the posterior probability for
+our model parameters Θ given the data D with priors 𝑃(𝜃) on the
+parameters as given in Table 1. We use a flat prior on most model
+parameters and Gaussian priors on log 𝛼, log 𝛽 and log 𝛾. The Gaus-
+sian priors for log 𝛼, log 𝛽 and log 𝛾 are similar to those commonly
+used in literature (for eg., More et al. 2016; Shin et al. 2019; Mu-
+rata et al. 2020) for splashback radius studies as they are motivated
+from simulations (Gao et al. 2008; Diemer & Kravtsov 2014) and
+are characteristic of typical cluster scale halos.
+From the Bayes theorem, we can write the posterior 𝑃(Θ|D) as
+𝑃(Θ|D) ∝ 𝑃(D|Θ)𝑃(Θ) ,
+(18)
+where 𝑃(D|Θ) is the likelihood of the data given model parameters,
+and we are using a Gaussian likelihood given by
+𝑃(D|Θ) ∝ exp
+�
+− 𝜒2(Θ)
+2
+�
+(19)
+with 𝜒2(Θ) = [D −M(Θ)]𝑇 𝐶−1[D −M(Θ)], D is the data vector
+with M(Θ) as the model prediction vector given the parameter Θ.
+The noise in the covariance matrix can result in a bias when inverted
+to obtain the 𝜒2. We follow eqn. 17 in Hartlap et al. (2007) in
+order to account for this bias. We use the affine invariant Monte
+Carlo Markov Chain (MCMC Goodman & Weare 2010) sampler
+as provided by the python package emcee (Foreman-Mackey et al.
+2013) to infer the posterior distribution for our model parameters
+given the measurements. In our work, we run separate MCMC to
+infer parameter posteriors for the weak lensing and cross-correlation
+measurements for our cluster sample and the convergence of the
+chains are assured by visual inspections.
+MNRAS 000, 1–14 (2015)
+
+Splashback radius around X-ray galaxy clusters
+7
+Model Parameters
+Parameter
+Prior
+log 𝜌s
+Flat[-3, 5]
+log 𝛼
+Gauss(log(0.2), 0.6)
+log 𝑟s
+Flat[log(0.1), log(5.0)]
+log 𝜌o
+Flat[-1.5, 1.5]
+𝑠𝑒
+Flat[0.1, 4]
+log 𝑟t
+Flat[log(0.5), log(1.6)]
+log 𝛽
+Gauss(log(6.0), 0.2)
+log 𝛾
+Gauss(log(4.0), 0.2)
+log[𝑀200m/h−1M⊙]
+Flat[12, 16]
+𝑐
+Flat[0, 20]
+Table 1. Parameter priors: The table provides the prior distributions used for
+running the MCMC chains on our model parameters. The Flat[a, b] denotes
+uniform prior in the interval (a, b), and Gauss(𝜇,𝜎) is the Gaussian priors with
+𝜇 mean and 𝜎 standard deviation. The parameters log 𝜌s, log 𝛼, log 𝑟s mod-
+els the inner Einasto profile with 𝜌in(𝑟); 𝑠𝑒, log 𝜌o for outer profile 𝜌out(𝑟)
+and log 𝑟t, log 𝛽, log 𝛾 for the transition function 𝑓trans(𝑟) and are used to
+model the galaxy number density profile. The halo mass log[𝑀200m/h−1M⊙]
+and concentration 𝑐 are for the NFW profile-based modelling of the weak
+lensing measurements for halo mass calibrations.
+4 RESULTS AND DISCUSSIONS
+In this section, we describe our results for the weak lensing and pro-
+jected cluster-galaxy cross-correlation on the X-ray cluster sample.
+The weak lensing measurements allows us to infer the average halo
+mass of our sample, while the cross-correlation analysis will allow
+us to infer the location of the splashback radius in three dimensions.
+We will combine the two to present constraints on 𝑟sp/𝑟200m.
+4.1 Halo Mass for Galaxy Clusters
+We follow the methodology described in Sec 3.1 and measure the
+weak lensing signal ΔΣ(𝑅) around our X-ray clusters in ten projected
+logarithmically-spaced radial bins from the X-ray cluster centre with
+comoving distances in the range of [0.1, 2.0] h−1Mpc. We then model
+the signals using a simple NFW profile and infer the corresponding
+halo mass 𝑀200m and use it to obtain the spherical overdensity size
+estimate 𝑟200m.
+In the top panel of Figure 2, the blue data points represent our
+measurements ΔΣ and the errors are a result of shape noise. Our
+measurements have a total signal-to-noise of 17.93. The weak lensing
+signal shows the expected behavior and decreases as a function of the
+projected radial distance to the cluster centre. The best fit model is
+shown by the solid red line and corresponds to a reduced chi-squared
+𝜒2
+red = 5.6/8.02, where the degrees of freedom have been computed
+using eq. 29 in Raveri & Hu (2019). The blue shaded region indicates
+the 68 percent of the model predictions with 𝜒2 closest to the best fit.
+It shows that our model is a good description of our measurements.
+The bottom panel of Figure 2 shows the results of systematic tests
+for our signal measurements. The blue points with errors show the
+cross-component ΔΣ× of the signal. This signal is consistent with
+zero shown as the dashed grey line with a 𝜒2 of 5.26 for 10 degrees of
+freedom and a corresponding p-value of 0.86. The weak gravitational
+lensing is a result of the correlated dark matter distribution around our
+sample of lenses. Therefore, we expect a null signal when we measure
+the same around random points albeit any systematics. Similar to the
+ΔΣ×, we also obtain a null signal ΔΣrand for the random points
+residing in the survey region where the error is computed as the
+101
+102
+∆Σ[hM⊙pc−2]
+SNR = 17.93
+χ2/dof = 5.60/8.02 = 0.70
+Data
+100
+R[h−1Mpc]
+−20
+0
+20
+R · ∆Σ[106M⊙pc−1]
+∆Σ×
+∆Σrand
+Figure 2. Weak lensing signals: In the top panel, the blue data points corre-
+spond to our weak lensing measurements with shape noise errors for a nearly
+volume-limited selected X-ray cluster sample (see Figure 1) with signal-to-
+noise given in the bottom right corner. The solid red line denotes the best
+fit model prediction with 𝜒2
+red ≡ 𝜒2/dof value given in the upper right cor-
+ner. The blue-shaded region represents the 68 percentile around the median
+model predictions, and Table 2 presents the constrained model parameter
+values. The bottom panel shows the systematic checks for our weak lensing
+signal ΔΣ measurements. ΔΣ× denotes the cross component of weak lensing
+signal with shape noise errors, and ΔΣrand shows the measurements around the
+random points with errors over mean computed using the random realisations.
+The grey dashed zero line shows the expected null signal.
+scatter from 100 random realizations. These two systematic checks
+show that our signal measurements are robust.
+In Figure 3, the light and dark shaded blue regions show the 68 and
+95 percent credible regions in the inference of the halo mass and its
+concentration parameter. We obtain 13 percent constraints on the halo
+mass, i.e., log[𝑀200m/h−1M⊙] = 14.52±0.06 and approximately 20
+percent on the concentration parameter 𝑐 = 4.49+0.90
+−0.79. The ΛCDM
+expectation for the concentration at the halo mass of our interest
+computed using the concentration-mass relation by Diemer & Joyce
+(2019) from the package COLOSSUS (Diemer 2018) is equal to 4.83
+which is consistent with our inference. We have also cross-checked
+our measured halo masses with those made using a joint calibration
+of weak lensing and cluster abundance for our X-ray cluster sample
+(Chiu et al. 2022a)1 and found log[𝑀200m/h−1M⊙] = 14.58 , which
+is consistent with our measurements. In principle, the weak lensing
+measured halo mass may be different from the true halo mass due to
+1 This study uses 𝑀500c as the halo mass definitions and we use an
+concentration-mass relation by Diemer & Joyce (2019) in COLOSSUS to
+convert the masses into the definition used in our work 𝑀200m.
+MNRAS 000, 1–14 (2015)
+
+8
+Rana et al.
+14.40
+14.55
+14.70
+log M200m
+4
+6
+8
+c
+4
+6
+8
+c
+Figure 3. Mass profile fits posteriors: The above contour plot shows the
+degeneracy between our cluster halo mass log 𝑀200m and concentration pa-
+rameter 𝑐, respectively. The light and dark blue shaded contours are 68 and
+95 confidence levels.
+various projection effects as well as triaxiality of the halo (see e.g.,
+Becker & Kravtsov 2011). In the case of HSC WL of eFEDS clusters,
+this bias consistent with 0, and is known to about 3-6 percent (Grandis
+et al. 2021; Chiu et al. 2022a,b). It is thus negligible compared to our
+statistical mass uncertainty of 13 percent.
+Based on our inferred halo mass 𝑀200m, we derive the corre-
+sponding value of the traditional halo boundary 𝑟200m = 1.75 ±
+0.08 h−1Mpc. In Section 4.2, we will use this constraint on 𝑟200m
+and compare it to our inferred value of the splashback radius.
+4.2 Splashback radius of X-ray Clusters
+We follow the methodology described in Section 3.2 to measure the
+cross-correlation signal between our eFEDS X-ray cluster sample
+and the HSC S19A galaxies in nine logarithmically spaced comoving
+projected radial bins in the range [0.1, 2.8] h−1Mpc away from the
+X-ray centre of our clusters. We model these measurements using
+Eq. 17 for the projected cross-correlation profile 𝜉2D(𝑅) and infer
+the radial location for the splashback feature 𝑅sp. We also inferred
+the corresponding three-dimensional cross-correlation profile 𝜉3D(𝑟)
+and the value of the splashback radius 𝑟sp.
+In Figure 4, the blue data points with errors correspond to our mea-
+surements of the cross-correlation signal. The errors were obtained
+using the jackknife technique. The cross-correlation signal 𝜉2D(𝑅)
+is detected with a signal-to-noise ratio of 17.43. As expected the pro-
+jected number of galaxies correlated with the cluster centre decrease
+as we move further away from the cluster centre. The solid red line
+shows the best fit model and corresponds to a reduced chi-square
+𝜒2
+red = 1.12 and computed using the effective degrees of freedom
+dofeff = 2.71 (see eqn. 29 in Raveri & Hu (2019)). The dark blue
+shaded regions show the 68 percent credible regions which show that
+Parameters
+Constraints
+log 𝜌s∗
+0.78+0.55
+−0.67
+log 𝛼
+−0.71+0.33
+−0.30
+log 𝑟s∗
+0.09+0.40
+−0.32
+log 𝜌o∗
+−0.85+0.57
+−0.45
+𝑠e
+1.69+0.75
+−0.85
+log 𝑟t
+0.00+0.08
+−0.08
+log 𝛽
+0.78+0.06
+−0.06
+log 𝛾
+0.60+0.06
+−0.06
+𝑅sp/h−1Mpc
+1.19+0.30
+−0.22
+𝑟sp/h−1Mpc
+1.45+0.30
+−0.26
+𝑑 log 𝜉3D
+𝑑 log 𝑟 |𝑟=𝑟sp
+−5.05+0.88
+−0.73
+log[𝑀200m/h−1M⊙]
+14.52+0.06
+−0.06
+𝑐
+4.49+0.90
+−0.79
+𝑟200m/h−1Mpc
+1.75+0.08
+−0.08
+𝜒2
+wl/dofeff
+5.6/8.02
+𝜒2sp/dofeff
+3.03/2.71
+Table 2. Parameter constraints: The table provides the median values for the
+model parameter constraints with errors based on the 16 and 84 percentile of
+the posterior distribution obtained from modelling the measurements for our
+eFEDS X-ray galaxy cluster sample. The first eight rows present the parameter
+constraints for fitting the galaxy number density profiles. The two-dimensional
+𝑅sp and three-dimensional 𝑟sp splashback radius estimates along with the
+logarithmic slope values at 𝑟sp as given in rows 9-11. Row 12-14 presents
+the parameter values from the weak lensing analysis and the corresponding
+spherical overdensity size 𝑟200m value. The last two rows present the best fit
+𝜒2 values for the weak lensing 𝜒2
+wl and galaxy density profile 𝜒2sp with the
+associated effective degrees of freedom dofeff (based on eqn. 29 of (Raveri
+& Hu 2019)). The asterisk sign indicates the parameter constraints which
+are sensitive to the choice of the priors. But as we can see from Figure. C1
+log 𝜌s and log 𝑟s are degenerate among each other but do not correlate with
+the inferred value of the splashback radius. For log 𝜌o see appendix C.
+the measurements are in reasonable agreement with the expectations
+from the model. In the top left and right panel, the blue vertical dashed
+line and the shaded region shows the median of the inferred location
+for the projected splashback radius 𝑅sp along with the 68 percent
+credible interval. Furthermore, the orange dotted vertical line with
+the shaded region shows the inferred value for the boundary of the
+halo in the traditional sense - 𝑟200m inferred from our weak lensing
+analysis as described in Section 4.1.
+The location of the projected splashback radius 𝑅sp was estimated
+using the minima of the logarithmic slope 𝑑 log 𝜉2D/𝑑 log 𝑅 profile
+for the model predicted 𝜉2D. The top right panel in Figure 4 shows
+the 𝑑 log 𝜉2D/𝑑 log 𝑅 profile and the dark blue shaded regions repre-
+sent the 68 percent credible region around the median. In Table 2 we
+present the median values for the model parameters and the location
+of the splashback radius along with errors computed from the 16-th
+and 84-th percentiles of the corresponding posterior distributions.
+The corresponding two-dimensional parameter posterior plots show-
+ing the correlation among different model parameters is presented in
+Appendix C. In our analysis, we obtain a 25 percent constraint on the
+𝑅sp with a median at 1.19 h−1Mpc. The quantity 𝑅sp corresponds to
+location of the steepest slope of the projected correlation function,
+and is expected to be at a location which is smaller than the location
+of the steepest slope in three dimensional distance from the cluster.
+MNRAS 000, 1–14 (2015)
+
+Splashback radius around X-ray galaxy clusters
+9
+R[h−1Mpc]
+10−3
+10−2
+10−1
+100
+101
+102
+ξ2D(R)
+SNR = 17.43
+Data
+χ2/dof = 3.03/2.71 = 1.12
+Rsp
+R[h−1Mpc]
+−6
+−5
+−4
+−3
+−2
+−1
+0
+d log ξ2D/d log R
+100
+r[h−1Mpc]
+10−2
+10−1
+100
+101
+102
+103
+104
+ξ3D(r)
+rsp
+r200m
+100
+r[h−1Mpc]
+−6
+−5
+−4
+−3
+−2
+−1
+0
+d log ξ3D/d log r
+Figure 4. Galaxy number density profile: The blue data points in the top left panel show our measurements for cross-correlation 𝜉2D(𝑟) between the eFEDS X-ray
+galaxy cluster and HSC S19A optical galaxies with signal-to-noise given in the bottom left corner. The solid red line represents our best fit model predictions
+with 𝜒2
+red ≡ 𝜒2/dof value given in the top left corner, and the dark blue shaded regions denoted the 68 percentile around the median model predictions. The top
+right panel shows the inferred logarithmic derivative of model predictions for projected cross-correlation profile 𝜉2D(𝑟). In the top row, The blue dashed vertical
+line represents the median value for the inferred two-dimensional splashback radius 𝑅sp with a light blue 68 percentile shaded error region around it. Similarly,
+the bottom row shows the corresponding three-dimensional 𝜉3D(𝑟) profile and the associated logarithmic derivative. In the bottom row, The blue dotted dashed
+vertical line with a light blue shaded region around it indicates our constraints on the three-dimensional splashback radius 𝑟sp estimates. In all the panels, the
+orange dotted vertical line with a shaded region shows the weak lensing calibrated three-dimensional spherical overdensity size 𝑟200m using the source galaxies
+from the HSC S16A shape catalogue. The solid black vertical line is the expectation value for 𝑟sp from ΛCDM prediction (More et al. 2015). We provide the
+model parameter constraints with the corresponding splashback radius and spherical overdensity sizes in Table 2.
+MNRAS 000, 1–14 (2015)
+
+10
+Rana et al.
+0
+1
+2
+3
+Rsp[h−1Mpc]
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+1
+2
+3
+rsp[h−1Mpc]
+0.0
+0.5
+1.0
+1.5
+2.0
+r200m
+−9
+−7
+−5
+−3
+−1
+d log ξ3D
+d log r |r=rsp
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Figure 5. Splashback radius posteriors: The plot in the left and middle panels represents the distributions of the two-dimensional 𝑅sp and three-dimensional 𝑟sp
+splashback radius values. The right panel shows the distribution of the logarithmic slope for the three-dimensional 𝜉3D(𝑟) galaxy number density profile at 𝑟sp.
+The orange dotted line with a shaded region in each panel shows the weak lensing constraints on the spherical overdensity size 𝑟200m. The solid black vertical
+line in the right panel denotes the expectation from the standard NFW profile.
+Therefore, we also infer the three-dimensional splashback radius
+𝑟sp for our cluster sample using our model. In Figure 4 the bottom left
+panel shows the three dimensional cross-correlation profile 𝜉3D(𝑟)
+inferred from the model fits to the measurements of 𝜉2D(𝑅), while
+the bottom right panel shows the corresponding logarithmic slope
+𝑑 log 𝜉3D/𝑑 log 𝑟. Here, the blue vertical dotted dash line with a light
+blue shaded region around it represents the location of splashback
+radius 𝑟sp. From our analysis we obtain 𝑟sp = 1.45 ± 0.30 h−1Mpc.
+We note that the median values of 𝑟sp and 𝑅sp differ by roughly 20
+percent as expected in cluster scale halos (see e.g., More et al. 2016).
+Following Baxter et al. (2017), in Figure 5 we show the posterior
+distribution of the splashback radius in 2-d (𝑅sp), and 3-d (𝑟sp) along
+with the slope 𝑑 log 𝜉3D/𝑑 log 𝑟 at the location of 𝑟 = 𝑟sp. The value
+of the logarithmic slope we obtain −5.05 ± 0.88 is steeper than −3
+at more than 2𝜎 (see Table 2). The black solid vertical line denotes
+the logarithmic slope of −3 reached asymptotically on large scales
+for the NFW halos without the presence of the two-halo term. This
+shows that the location of the minimum corresponds to the splashback
+feature for our cluster sample associated with the steepest slope. We
+note that our slope value is consistent with those obtained in the
+X-ray study by Contigiani et al. (2019b) and Murata et al. (2020)
+study using optically selected clusters and also with values seen
+in the literature (More et al. 2016; Baxter et al. 2017; Umetsu &
+Diemer 2017; Chang et al. 2018; Zürcher & More 2019; Shin et al.
+2019; Bianconi et al. 2021; Shin et al. 2021), especially given the
+large errors. These difference in the value of slope could possibly
+arise from the variations in the halo mass accretion rate in different
+samples, which correlates well with the sharpness in the splashback
+radius feature (Diemer & Kravtsov 2014) and more investigation will
+be done in the future works.
+We compare the values we infer for 𝑟sp with the expectations from
+ΛCDM model. We estimate the approximate location for𝑟sp using the
+weak lensing calibrated halo mass estimate 𝑀200m = 1014.52 h−1M⊙
+at the median redshift of 𝑧 = 0.46 for our sample. We use the 𝑀200m
+and 𝑧 to computing the peak height 𝜈200m ≡ 𝛿𝑐/𝜎(𝑀200m)/𝐷(𝑧).
+We then use the 𝜈200m as an input parameter in the fitting re-
+lations given by eqn. 7 of More et al. (2015) and obtain 𝑟sp =
+1.80±0.07 h−1Mpc as the ΛCDM prediction. This estimate is shown
+by the solid black vertical line in Figure 4, which is consistent with
+our results within ∼ 1.2𝜎. We further calculated the halo mass acc-
+retation rate Γ = 5.86 ± 3.78 for our sample using eqn 5 in More
+et al. (2015). The large error on the Γ value is related to the broader
+constraints on the splashback feature 𝑟sp.
+In the current analysis, we only work with a single z-band absolute
+magnitude cut 𝑀𝑧 − 5 log ℎ < −19.36 which corresponds to an
+apparent magnitude limit of 𝑚𝑧 = 24.0 at redshift 𝑧 = 0.75. Note that
+the knee of the Schechter function is brighter in the z-band than in the
+i-band by about 0.3 magnitudes (Blanton et al. 2001) . Our magnitude
+cut in the z-band is therefore even fainter compared to the magnitude
+at the knee of the Schechter function than galaxies used in More et al.
+(2016) who use an i-band magnitude cut of 𝑀𝑖 − 5 log ℎ < −19.43.
+Such cuts are thus expected to minimize the effects of dynamical
+friction (Adhikari et al. 2016). In principle, we could use various
+magnitude limits to check for the effects of the dynamical friction as
+commonly done in literature (for, e.g. More et al. 2016; Murata et al.
+2019). However, the current sample size yields fairly low signal-
+to-noise cross-correlations when restricted to brighter samples. In
+future work, we intend to pursue such measurements.
+Finally, in Figure 6 we compare our results with those in the lit-
+erature in the 𝜈200m − 𝑟sp/𝑟200m plane. Given that this relation is
+fairly independent of redshift (More et al. 2015), this allows for a
+uniform comparison of results in the literature. The grey dashed line
+represents the expectations from ΛCDM using the parameterization
+given by eqn. 7 in More et al. (2015) and calibrated from numerical
+simulations. Round data points are used to present the constraints
+from optically selected clusters, triangular data points for those ob-
+tained using clusters identified using the SZ effect, and square data
+points for those obtained using X-ray selected clusters. In the leg-
+end, we have also labelled the name of the data used along with
+the method used in the corresponding investigation - galaxy number
+density profile (GP), weak lensing (WL) and stacked luminosity pro-
+file (LP). The black square point shows our constraint based on the
+galaxy number density profile around eFEDS X-ray clusters from this
+work, which yields 𝑟sp/𝑟200m ≈ 0.83 ± 0.17. This results is a 20 per-
+MNRAS 000, 1–14 (2015)
+
+Splashback radius around X-ray galaxy clusters
+11
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+ν200m
+0.5
+1.0
+1.5
+2.0
+rsp/r200m
+More+15
+SDSS; More+16,Baxter+17:GP
+CLASH; Umetsu+17:WL+SL
+DES; Chang+18:GP
+DES; Chang+18:WL
+Planck; Z¨urcher+19:GP
+CCCP; Contigiani+19:WL
+SPT; Shin+19:GP
+ACT; Shin+19:GP
+HSC; Murata+20:GP
+LoCuSS; Bianconi+21:LP
+ACT; Shin+21:WL
+ACT; Shin+21:GP
+eFEDS; This Work:GP
+Figure 6. Comparison with other studies: In the above figure, we compare our
+results with the existing works in the literature. The x-axis represents the peak
+height 𝜈200m and the y-axis represents the ratio between splashback radius 𝑟sp
+and the spherical overdensity 𝑟200m radius estimates. The round data points
+show the values computed using the optically selected clusters, the triangular
+points are for the analysis using SZ selected clusters, and the squared points
+are for the X-ray selected clusters. In the legends, we label each datapoint
+using their survey name, reference and the analysis method used - the weak
+lensing (WL), strong lensing(SL), galaxy number density (GP) or stacked
+Luminosity profile (LP). The dashed grey line shows the expectation from the
+ΛCDM cosmology as a function of 𝜈200m computed using the fitting function
+given by More et al. (2015). The black datapoint with errors represents our
+results from this study using the eFEDS X-ray clusters and shows marginal
+consistency (1.2𝜎) with the ΛCDM predictions.
+cent constraint and yields results consistent ≈ 1.2𝜎 with the ΛCDM
+prediction. It represents an improvement over the weak-and-strong
+lensing based study on CLASH X-ray cluster sample by Umetsu &
+Diemer (2017) which provided a lower limit for 𝑟sp/𝑟200m > 0.89.
+Similarly, our analysis also puts nearly 25 percent tighter bounds than
+the one obtained with another weak lensing result using the CCCP
+X-ray clusters (Contigiani et al. 2019b).
+We find that our constraints are similar to those obtained by galaxy
+number density studies using SZ clusters by Zürcher & More (2019);
+Shin et al. (2019), the weak lensing-based analysis in Shin et al.
+(2021) using ACT SZ clusters and the stacked luminosity profile
+results in LoCuSS X-ray clusters (Bianconi et al. 2021). The preci-
+sion with which we determine the splashback radius is worse when
+compared to galaxy density profile studies using SZ selected clusters
+(Shin et al. 2021) or optically selected clusters (More et al. 2016;
+Baxter et al. 2017; Chang et al. 2018; Murata et al. 2020), albeit the
+latter are likely affected by systematics related to projection effects.
+This is expected to get better as our sample size gets larger.
+5 CONCLUSIONS
+The splashback radius which denotes the physical boundary of dark
+matter halos depends upon the current accretion mass rate. Various
+studies in the literature have attempted to constrain the location of this
+radius in observations. In this work, we studied the splashback radius
+around the eROSITA eFEDS X-ray clusters by cross-correlating them
+with the optical photometric galaxies. Our use of X-ray selected
+clusters avoids systematics related to projection effects which have
+affected the measurement of the splashback radius from optically
+selected clusters. We select X-ray clusters having luminosities above a
+threshold 𝐿𝑋 > 1043.5erg s−1h−2 within the redshift 𝑧 < 0.75, which
+provides us a nearly volume limited sample. Finally, we compared our
+inferred value of the splashback radius with the standard spherical
+overdensity radius 𝑟200m calibrated using weak gravitational lensing.
+We briefly summarize our major findings below :
+• We measure the stacked weak lensing signal around our clus-
+ter sample using the background galaxies from HSC S16A shape
+catalogue data and obtain measurements with a signal-to-noise of
+17.93. We then use a simple NFW profile to model the stacked
+signal and infer a halo mass log[𝑀200m/h−1M⊙] = 14.52 ± 0.06
+which corresponds to an spherical overdensity size of 𝑟200m =
+1.75 ± 0.08 h−1Mpc.
+• We measure the projected cross-correlation with an signal-to-
+noise of 17.43 for our X-ray galaxy clusters using the optical galax-
+ies from HSC S19A data having z-band absolute magnitude cut
+𝑀𝑧 −5 log ℎ < −19.36. We model these measurements by projecting
+the three-dimensional functional form given by Diemer & Kravtsov
+(2014) to infer the location of the steepest slope and associate it with
+the splashback radius.
+• We find the value for the 3D steepest slope to be −5+0.88
+−0.73,
+more than 2𝜎 away from the asymptotic −3 value in the case of the
+standard NFW profile which provides evidence for the presence of a
+splashback feature.
+• Our analysis corresponds to ∼ 25 percent constraint on the
+projected 𝑅sp = 1.19+0.30
+−0.22 h−1Mpc and ∼ 20 percent constraint on
+the three dimensional 𝑟sp = 1.45+0.30
+−0.26 h−1Mpc splashback radius.
+These values are in the range as expected commonly for the massive
+halos in numerical simulations (Diemer & Kravtsov 2014; More et al.
+2015).
+• Our constraints on 𝑟sp are broadly comparable to spherical over-
+density estimates 𝑟200m = 1.75 ± 0.08 h−1Mpc and are marginally
+consistent (≈ 1.2𝜎) with the expectation from numerical simulation
+for the weak lensing calibrated halo mass at the median redshift
+𝑧 = 0.46 for our cluster sample. The results from our analysis sig-
+nificantly improve the errors on the splashback radius measurement
+based on X-ray selected galaxy clusters.
+• We infer a halo mass accretion rate Γ = 5.86 ± 3.78 with large
+errors, where the error is dominated by the error on the inferred
+splashback radius 𝑟sp.
+In our analysis, we use X-ray centres of the galaxy clusters for the
+weak lensing and cross-correlation signal measurements. We have
+checked the dependence of our results on the choice of the centre
+by using the optically confirmed galaxy as the centre and find that
+our results are robust to this choice. Given the differences between
+X-ray and optical centres - 92.6+44.3
+−35.1kpc (Seppi et al. 2022), we have
+also explore effects of miscentering on our conclusions. We have re-
+analyzed both the weak lensing and cross-correlation measurements
+after removing signals in radial bins below 200 h−1kpc and found no
+significant change in our results.
+We also want to point out that in the present analysis, we only
+MNRAS 000, 1–14 (2015)
+
+12
+Rana et al.
+use one z-band absolute limit on the S19A optical galaxies for cross-
+correlation measurements. We choose the faintest possible limit such
+that we can get less noisy measurements on large scales. In future
+studies with a larger cluster sample, we would also like to test for
+possible dynamical friction effects.
+Our results provide meaningful constraints on the splashback ra-
+dius of galaxy clusters, although with large statistical errors. How-
+ever, in the near future, we expect better constraints by using all-sky
+cluster catalogues from the eRASS (Merloni et al. 2012). The statis-
+tical precision of these measurements will allow comparisons with
+expectations from the hydrodynamic simulations like IllustrisTNG
+(Weinberger et al. 2017; Pillepich et al. 2018) and thorough investi-
+gations of any differences between the observations and theoretical
+predictions.
+ACKNOWLEDGEMENTS
+We are grateful to Navin Chaurasiya, Amit Kumar, Moun Meenakshi,
+Preetish K. Mishra, Ayan Mitra, Masahiro Takada and Keiichi
+Umetsu for discussions and their insightful comments on the earlier
+version of the manuscript. DR thank the University Grants Com-
+mission (UGC) of India, for providing financial support as a senior
+research fellow. We acknowledge the use of the high performance
+computing facility - Pegasus at IUCAA. This work was supported
+in part by the World Premier International Research Center Initia-
+tive (WPI Initiative), MEXT, Japan, and JSPS KAKENHI Grant Nos.
+20H01932 and 21H05456. E.B. acknowledges financial support from
+the European Research Council (ERC) Consolidator Grant under the
+European Union’s Horizon 2020 research and innovation programme
+(grant agreement CoG DarkQuest No 101002585).
+The Hyper Suprime-Cam (HSC) collaboration includes the astro-
+nomical communities of Japan and Taiwan, and Princeton University.
+The HSC instrumentation and software were developed by the Na-
+tional Astronomical Observatory of Japan (NAOJ), the Kavli Institute
+for the Physics and Mathematics of the Universe (Kavli IPMU), the
+University of Tokyo, the High Energy Accelerator Research Orga-
+nization (KEK), the Academia Sinica Institute for Astronomy and
+Astrophysics in Taiwan (ASIAA), and Princeton University. Fund-
+ing was contributed by the FIRST program from the Japanese Cab-
+inet Office, the Ministry of Education, Culture, Sports, Science and
+Technology (MEXT), the Japan Society for the Promotion of Science
+(JSPS), Japan Science and Technology Agency (JST), the Toray Sci-
+ence Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton
+University.
+This paper is based in part on data collected at the Subaru Tele-
+scope and retrieved from the HSC data archive system, which is
+operated by Subaru Telescope and Astronomy Data Center (ADC) at
+NAOJ. Data analysis was in part carried out with the cooperation of
+Center for Computational Astrophysics (CfCA), NAOJ.
+This work is based on data from eROSITA, the soft X-ray instru-
+ment aboard SRG, a joint Russian-German science mission supported
+by the Russian Space Agency (Roskosmos), in the interests of the
+Russian Academy of Sciences represented by its Space Research
+Institute (IKI), and the Deutsches Zentrum für Luft- und Raum-
+fahrt (DLR). The SRG spacecraft was built by Lavochkin Associa-
+tion (NPOL) and its subcontractors, and is operated by NPOL with
+support from the Max Planck Institute for Extraterrestrial Physics
+(MPE). The development and construction of the eROSITA X-ray
+instrument was led by MPE, with contributions from the Dr. Karl
+Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg),
+the University of Hamburg Observatory, the Leibniz Institute for
+Astrophysics Potsdam (AIP), and the Institute for Astronomy and
+Astrophysics of the University of Tübingen, with the support of DLR
+and the Max Planck Society. The Argelander Institute for Astronomy
+of the University of Bonn and the Ludwig Maximilians Universität
+Munich also participated in the science preparation for eROSITA.
+DATA AVAILABILITY
+This work uses publically available catalogue data accessible through
+survey websites. The eROSITA eFEDS X-ray cluster catalogue
+is available at https://erosita.mpe.mpg.de/edr/index.php and Subaru
+HSC dataset can be found at https://hsc-release.mtk.nao.ac.jp/doc/.
+The
+measurements
+for
+cross-correlation
+and
+weak
+lens-
+ing
+analysis
+are
+available
+on
+https://github.com/divyarana-
+cosmo/rsp_efeds_hsc_s19a.
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+APPENDIX A: BOOST PARAMETERS
+As discussed in Section 3.1, we estimate the boost parameters using
+eqn. 7 to quantify the dilution in the weak lensing signal due to
+the systematics in the photometric redshift 𝑝(𝑧) distribution in the
+source galaxies. In Figure A1, the blue data points represent the
+boost parameters from our weak lensing signal with 1𝜎 error bars
+from random realizations. We found that they are mostly consistent
+with the unity line apart from the third datapoint, which differs by
+more than 2𝜎. We test the impact of the third data point on our
+inferred halo mass. We reran our signal fitting after removing the
+third data point and found no significant change in the inferred value
+of the halo mass.
+APPENDIX B: COVARIANCES
+Figure B1 and Figure B2 show the correlation coefficient 𝑟ij for the
+corresponding covariances of the weak lensing and galaxy density
+profile measurements. We estimate 𝑟ij from the covariance 𝐶 and
+given by
+𝑟ij =
+𝐶ij
+√︁𝐶ii𝐶jj
+(B1)
+where subscript ij represents the ith and jth radial bins. We use
+the shape noise covariance for the weak lensing profile and area
+jackknifes to compute covariance for the galaxy number density
+profile. We describe the details in Sec 3.
+MNRAS 000, 1–14 (2015)
+
+14
+Rana et al.
+100
+R[h−1Mpc]
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+C(R)
+Data
+Figure A1. Boost parameters: The blue data points denotes the boost param-
+eter 𝐶(𝑅) for our weak lensing measurement ΔΣ and are estimated using
+eqn. 7 with y errors computed from the scatter in random realizations. The
+plot shows consistency of boost parameter with grey dashed unity line.
+0.12
+0.16
+0.21
+0.29
+0.39
+0.53
+0.71
+0.96
+1.29
+1.74
+R[h−1Mpc]
+0.12
+0.16
+0.21
+0.29
+0.39
+0.53
+0.71
+0.96
+1.29
+1.74
+R[h−1Mpc]
+−1.00
+−0.75
+−0.50
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+rij
+Figure B1. Weak lensing profile covariance: The above plot shows the cor-
+relation coefficient 𝑟ij matrix for the 𝑖th and 𝑗th radial bin of the weak lensing
+signal measurements ΔΣ given in Sec 3.1 and estimated by shape noise using
+200 different random rotations of background source galaxies with x-y axis
+showing our radial binning for signal measurements.
+APPENDIX C: DEGENERACIES IN THE MODEL
+PARAMETERS
+Figure C1 shows the correlations between the parameters used for
+modelling the cluster-galaxy cross-correlation measurements. The
+parametric dependence is given by the function form in eqn. 14, 15
+and 16. We note that the posterior sample for log 𝜌o pile up at the
+lower edge of the prior range. We tested our results with broader
+0.12
+0.18
+0.27
+0.39
+0.58
+0.86
+1.27
+1.88
+2.78
+R[h−1Mpc]
+0.12
+0.18
+0.27
+0.39
+0.58
+0.86
+1.27
+1.88
+2.78
+R[h−1Mpc]
+−1.00
+−0.75
+−0.50
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+rij
+Figure B2. Cluster-galaxy cross correlation covariance: The above plot
+present the correlation coefficient 𝑟ij matrix for the 𝑖th and 𝑗th radial bin
+of our galaxy number density profile measurements 𝜉2D given in Sec 3.2 and
+estimated using 20 jackknife region with each having roughly 5 deg2 area.The
+x and y axis denotes our radial binning for signal measurements.
+priors and found some values for 𝑅sp to be at the largest radial bins.
+These values arise from the poor signal measurements at large scales
+to get a well-constrained outer profile. In such cases, the model tries
+to fit the whole measurement without the outer profile by using a
+smaller amplitude log 𝜌o giving rise to a spurious secondary peak in
+the 𝑅sp posterior.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–14 (2015)
+
+Splashback radius around X-ray galaxy clusters
+15
+−1.6
+−0.8
+0.0
+0.8
+log α
+−0.8
+−0.4
+0.0
+0.4
+log rs
+−1.0
+−0.5
+0.0
+log ρo
+0.8
+1.6
+2.4
+3.2
+se
+−0.15
+0.00
+0.15
+log rt
+0.60
+0.75
+0.90
+log β
+0.45
+0.60
+0.75
+log γ
+0.6
+1.2
+1.8
+2.4
+Rsp
+−0.8
+0.0
+0.8
+1.6
+2.4
+log ρs
+0.5
+1.0
+1.5
+2.0
+2.5
+rsp
+−1.6
+−0.8
+0.0
+0.8
+log α
+−0.8
+−0.4
+0.0
+0.4
+log rs
+−1.0
+−0.5
+0.0
+log ρo
+0.8
+1.6
+2.4
+3.2
+se
+−0.15
+0.00
+0.15
+log rt
+0.60
+0.75
+0.90
+log β
+0.45
+0.60
+0.75
+log γ
+0.6
+1.2
+1.8
+2.4
+Rsp
+0.5
+1.0
+1.5
+2.0
+2.5
+rsp
+Figure C1. Splashback parameter posteriors: The above corner plot shows the degeneracies among the model parameters described in Sec 3.2 for the projected
+cluster-galaxy cross-correlation measurements.
+MNRAS 000, 1–14 (2015)
+

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@@ -0,0 +1,2080 @@
+arXiv:2301.00873v1  [hep-ph]  2 Jan 2023
+The production cross section of heavy quarks in e−p interaction at the NLO
+approximation
+S. Zarrin, S. Dadfar, and M. Sayahi
+Department of Physics, University of Sistan and Baluchestan, Zahedan, Iran
+We present the production cross section of heavy quarks σc¯c, σb¯b and σt¯t at the next-to-leading
+order in the electron-proton interaction by using the quarks and gluon distribution functions at
+the initial scale Q2
+0.
+To do this, we use a fitted form of the heavy quark coefficient functions
+for deep-inelastic lepton-hadron scattering to obtain the structure functions of heavy quarks.
+Then, we calculate the reduced cross section of heavy quarks by using the structure functions and
+subsequently present the single differential and the integrated cross section of heavy quarks at the
+center-of-mass energies of √s = 319 GeV , 1.3 T eV and 3.5 T eV in the electron-proton collision.
+The obtained numerical results of the cross section of the charm and beauty quarks are compared
+with the HERA data, which is a combination from the results of the H1 and ZEUS detectors,
+and with the predictions from H1PDF, MSTW2008 and MSRT03. Furthermore, we present the
+production cross section of top quark as a direct prediction from our calculations.
+PACS numbers: 13.60.Hb, 13.85.Lg, 14.65.Dw, 14.65.Fy, 14.65.Ha
+I.
+INTRODUCTION
+The study of the heavy quarks production is one of the
+most important subjects of research at the present and
+future colliders and the test of quantum chromodynamics
+(QCD). These quarks can be generated in the hadron-
+hadron, photon-hadron, electron-positron and lepton-
+hadron interactions.
+The production of heavy quarks
+are studied in two different prescriptions in the frame-
+work of QCD analyses. The first framework (the region
+Q2 = m2
+q) is the so-called variable-flavour number scheme
+(VFNS) [1]. In this scheme, the heavy quarks contribu-
+tions are described by a parton density and treated as a
+massless quark in the hadron. In the ‘massless’ scheme,
+the dominant contribution at the leading order (LO) ap-
+proximation is due to the quark parton model (QPM)
+process and at the next-to-leading order (NLO) approx-
+imation the contributions of the photon-gluon fusion
+(PGF) and QCD Compton processes are also considered.
+In the second framework (the region Q2 ≫ m2
+q), heavy
+quarks are treated as a massive quark and their contri-
+butions are given by the fixed flavour number scheme
+(FFNS)[2, 3]. In this scheme (the ‘massive’ scheme), the
+dominant LO process is the PGF and the NLO diagrams
+are of order α2
+s.
+At HERA, at the LO approximation, the PGF is the
+dominant contribution for the heavy quarks production
+in electron-proton interaction e− + p → q¯q + e− + X.
+In this process, by the interaction of a virtual photon
+emitted by the incoming electron with a gluon from the
+proton, is formed a heavy quark-antiquark pair.
+The
+HERA data show that the production of heavy quarks is
+sensitive to the gluon distribution (which the minimum
+momentum fraction of gluon xg in photoproduction to
+generate a heavy quark pair is arranged such that xtt
+g >
+xbb
+g
+> xcc
+g ) and also is dependent on the mass of these
+quarks. Therefore, the calculations of the heavy quarks
+structure functions are dependent on the squared energy
+scale µ2 [4–8].
+The measurements of the open charm (c) cross section
+in DIS at HERA have mainly been exclusive for D or D∗
+meson production [9–13]. The measurement of the open
+beauty (b) cross section is challenging since b events con-
+tain only a small fraction (typically < 5%) of the total
+cross section.
+The b cross section has been measured
+in DIS (Q2 > 2GeV 2) by ZEUS [14] and in photopro-
+duction (Q2 < 1GeV 2 and 0.1 < y < 0.8) by H1 [15]
+and ZEUS [16], using the transverse momentum distribu-
+tion of muons relative to the b jet in semi-muonic decays.
+Moreover, in Ref. [5], the production of c and b quarks
+in ep interactions has been studied with the ZEUS de-
+tector at HERA for exchanged four-momentum squared
+5 < Q2 < 1000GeV 2 using an integrated luminosity of
+354pb−1. Also, measurements of the c and b contribu-
+tions to the inclusive proton structure function F2 have
+been recently presented in deep inelastic scattering (DIS)
+at HERA, using information from the H1 vertex detector,
+for values of the negative square of the four momentum
+of the exchanged boson Q2 > 150GeV 2 and of inelastic-
+ity 0.1 < y < 0.7 [17]. In this region, the inclusive c and
+b cross sections have been found σc¯c = 373 ± 39 ± 47 pb
+and σb¯b = 55.4 ± 8.7 ± 12.0 pb, respectively and the data
+show that a fraction of ∼ 18% (∼ 3%) of DIS events con-
+tain c (b) quark. Furthermore, inclusive c and b cross
+sections have been measured in e−p and e+p neutral
+current collisions at HERA in the kinematic region of
+5 < Q2 < 2000GeV 2 and 0.0002 < x < 0.05 which x is
+the Bjorken scaling variable. In which the e−p center-
+of-mass energy (CME) is √s = 319GeV , with a proton
+beam energy of Ep = 920GeV and electron beam energy
+of Ee = 27.6GeV [4].
+In high energy processes, the contribution of heavy
+quarks in the proton structure functions will be stud-
+ied in projects such as the Large Hadron electron Col-
+lider (LHeC) and the Future Circular Collider electron-
+hadron (FCC-eh) which operate at high enough ener-
+gies to observe new phenomenon [18–23]. At the LHeC
+
+2
+project, the possibility of colliding an electron beam from
+a new accelerator with the existing LHC proton is inves-
+tigated. In this project, the e−p CME is planned to reach
+√s = 1.3T eV [18, 22]. Beyond the LHeC, the next gener-
+ation ep collider (the FCC-eh project) is an ideal environ-
+ment to increase center-of-mass energy. In the proposed
+FCC-eh program, the distribution of heavy quarks will
+be examined at √s = 3.5T eV [23].
+Recently, the theoretical predictions of inclusive heavy
+quark production have been presented within the VFNS
+scheme at the next-to-next-to-leading order approxima-
+tion (NNLO) [24].
+The predictions for the c and b
+cross sections have been obtained from fits [25] to the
+HERA inclusive F2 data based on CCFM evolution [26].
+Also, the production of heavy quarks in the FFNS ap-
+proach have been predicted according to the LO PGF
+off-shell matrix elements convoluted with the CCFM kT -
+unintegrated gluon density of the proton [25]. In Refs.
+[27–32], the connection between the gluon distribution
+and the structure functions of heavy quarks (c and b)
+has been theoretically shown at small x. Moreover, in
+Refs. [32–34] the authors present the necessary condi-
+tions for predicting the top structure function F t¯t
+2 with
+respect to the different predictions for the behavior of the
+gluon at low x and high Q2 values. Besides these predic-
+tions, various successful phenomenological methods have
+presented to obtain the c and b structure functions and
+the ratios of Rc¯c and Rb¯b [28, 32, 35]. The importance
+of these studies, along with the t-quark density, can be
+explored at future circular collider energies and may lead
+us to new physics in the future [36, 37].
+At small x, both the H1 and ZEUS detectors have mea-
+sured the charm and beauty components of the proton
+structure function from the measurement of the inclu-
+sive heavy quark cross sections after applying small cor-
+rections to the longitudinal structure function of heavy
+quarks at low and moderate inelasticity. But, in the re-
+gion of high inelasticity, this function may has a signifi-
+cant effect on the heavy quark production cross section.
+The heavy quark deferential cross section is written in
+terms of the heavy quark structure functions as:
+d2σq¯q
+dxdQ2 = 2πα2
+xQ2
+�
+Y+F q¯q
+2 (x, Q2) − y2F q¯q
+L (x, Q2)
+�
+= 2πα2
+xQ2 Y+σq¯q
+r (x, Q2),
+(1)
+where Y+ = 1 + (1 − y)2 and y = Q2/(xs) is the inelas-
+ticity variable in which s and Q2 are the CME squared
+and the photon virtuality, respectively. The heavy quark
+structure functions F q¯q
+2 (x, Q2) and F q¯q
+L (x, Q2) with re-
+spect to the behavior of the gluon density are given by:
+F q¯q
+k,g(x, Q2, m2
+q) = xe2
+H
+� zmax
+x
+Hk,g(z, ξ)g(x
+z , µ2)dz
+z , (2)
+where µ = (Q2+4m2
+q)1/2 is the default common value for
+the factorization and renormalization scales, zmax = Q2
+µ2
+and ξ =
+Q2
+m2q .
+In general, the heavy quark coefficient
+functions of Hk,g(z, ξ) (with k = 2, L) are expanded in
+αs as follows:
+Hk,g(z, ξ) =
+∞
+�
+i=1
+�αs(µ2)
+4π
+�i
+H(i)
+k,g(z, ξ),
+k = 2, L,
+(3)
+where the heavy quark coefficient functions at the LO
+and NLO approximations, H(1)
+k,g and H(2)
+k,g, are as follows:
+H(1)
+k,g(z, ξ) = ξ
+πz c(0)
+k,g(η, ξ),
+H(2)
+k,g(z, ξ) = 16πξ
+z
+�
+c(1)
+k,g(η, ξ) + ¯c(1)
+k,g(η, ξ) ln
+� µ2
+m2q
+��
+,
+(4)
+where the coefficient functions c(0)
+k,g(η, ξ) have been given
+in Ref. [3] and the coefficients c(1)
+k,g and ¯c(1)
+k,g are rather
+lengthy, and not published in print and they are only
+available as computer codes [3]. In Ref. [38], the analytic
+form of the heavy quark coefficient functions have been
+presented for deep-inelastic lepton-hadron scattering in
+the kinematical regime Q2 ≫ m2
+q in which Q2 and m2
+q
+stand for the masses squared of the virtual photon and
+heavy quark, respectively.
+This paper is organized as follows. In the next sec-
+tion, the theoretical formalism is presented, including the
+structure functions of heavy quarks. In section III, we
+present a detailed numerical analysis. In the last section,
+we summarize our main conclusions and remarks.
+II.
+THEORETICAL FORMALISM
+In this paper, we use the obtained heavy quark struc-
+ture functions F q¯q
+2
+and F q¯q
+L
+at the LO and NLO ap-
+proximations in Ref.
+[32].
+These functions have been
+presented by using the heavy quark coefficient functions
+in Ref. [32, 38] obtained by using the presented heavy
+quark coefficient functions (in the kinematical regime
+Q2 ≫ m2
+q) in Ref.
+[38] and a set control coefficients
+which have been obtained by using the heavy quark struc-
+ture functions from HERA [5, 15, 17, 39–43], LHeC [44],
+and other works such as Ref. [45] (for c and b quarks)
+and Ref. [31] (for t quark). Then, we present the inte-
+grated and differential cross sections for heavy quarks in
+DIS and compare our numerical results with HREA data
+[4, 5, 14, 17] and with the results from the MSTW2008
+[46], MSRT03 [47] and H1PDF [48].
+Let us first present a brief summary of our previous
+work. The presented heavy quark coefficient functions in
+Ref. [38] at the regimes of Q2 ≥ m2
+q and Q2 ≤ m2
+q do not
+provide appropriate and acceptable results and the ob-
+tained structure functions by these coefficient functions
+are very ascendant, especially for the charm quark. To
+control this growth, by using HERA data and the re-
+sults from HERA [5, 15, 17, 39–43], LHeC [44], and other
+
+3
+Table (1): The fixed numbers d and e in the control
+coefficients of the heavy quark coefficient functions.
+d
+e
+mc
+1.007324
+−11.9235 × 10−4
+mb
+1.003669
+−2.35842 × 10−4
+mt
+1.000011
+−8.93697 × 10−8
+works such as Ref. [45] (for c and b quarks) and Refs. [31]
+(for t quarks), a series of control coefficients have been
+presented that are only a function of Q2. By multiplying
+these coefficients by the obtained coefficient functions in
+Ref. [38], they have given acceptable results. The gen-
+eral form of these control coefficients is as b − exp(cQ2),
+which d and e are fixed numbers and are shown in Table
+(1) for heavy quarks.
+To solve the Eq. (2), it is necessary to obtain the gluon
+density function, for this aim, we used the DGLAP evo-
+lution equations and the Laplace transform method as
+Ref.
+[49–51] by considering this fact that the Laplace
+transform of the convolution factors is simply the ordi-
+nary product of the Laplace transform of the factors. The
+coupled DGLAP integral-differential equations are as fol-
+lows [52]:
+∂Fs(x, Q2)
+∂ ln Q2
+= αs(Q2)
+2π
+�
+Pqq(x, Q2) ⊗ Fs(x, Q2)
++2nfPqg(x, Q2) ⊗ G(x, Q2)
+�
+,
+(5)
+∂G(x, Q2)
+∂ ln Q2
+= αs(Q2)
+2π
+�
+Pgq(x, Q2) ⊗ Fs(x, Q2)
++Pgg(x, Q2) ⊗ G(x, Q2)
+�
+,
+(6)
+where αs is the running strong coupling constant and
+Pab(x)’s are the Altarelli-Parisi splitting functions.
+In
+the above equations, the symbol ⊗ represents the con-
+volution integral which is defined as f(x) ⊗ h(x) =
+� 1
+x f(y)h(x/y)dy/y.
+To convert Eqs.
+(5) and (6) into
+Laplace space, we inserted the variables x = exp(−v),
+y = exp(−w) and τ(Q2, Q2
+0) =
+1
+4π
+� Q2
+Q2
+0 αs(Q′2)d ln(Q′2)
+into the DGLAP evolution equations.
+By using the
+Laplace transform method, one can turn the convolution
+equations at the LO and NLO approximations from v-
+space into s-space, and then solve them straightforwardly
+in s-space as:
+f (i)(s, Q2) = k(i)
+ff(s, τ)f (i)(s, Q2
+0) + k(i)
+fg(s, τ)g(i)(s, Q2
+0),
+(7)
+g(i)(s, Q2) = k(i)
+gf (s, τ)f (i)(s, Q2
+0) + k(i)
+gg (s, τ)g(i)(s, Q2
+0),
+(8)
+with i = LO or NLO. The functions of f(s, Q2
+0) and
+g(s, Q2
+0) are the singlet and gluon distribution functions
+at initial scale τ = 0 and L[ ˆH(v, τ), v, s] = h(s, τ). In
+Eqs. (7) and (8), the kernels of kij(s, u)’s at LO and NLO
+approximations can be found [51, 53, 54] . Since the ob-
+tained gluon distribution function in the above equation
+(8) is in Laplace space s and its exact solution is not pos-
+sible through analytical techniques, so its inverse Laplace
+transform must be computed numerically [53, 55]. To ob-
+tain the heavy quarks structure functions in terms of the
+distribution functions at the initial scale, we turned Eq.
+(2) to the Laplace space s.
+To this aim, the variable
+z = x/y and transformation x → xe(ln 1/a) (where a is
+larger than one) are used, therefore, Eq. (2) is obtained
+as follows:
+F q¯q
+k,g(xe(ln 1/a), Q2, m2
+q) = e2
+H
+� 1
+x
+G(y, µ2)dy
+y
+×Ck,g(xe(ln 1/a)
+y
+, ξ),
+k = 2, L,
+(9)
+where G(y, Q2) = yg(y, Q2) and Ck,g(x) = xHk,g(x). By
+using the variables x = exp(−v), y = exp(−w), one can
+rewrite the above equation as:
+ˆF q¯q
+k,g(v − ln 1/a, Q2, m2
+q) = e2
+H
+� v
+0
+ˆG(w, µ2)
+× ˆCk,g(v − w − ln 1/a, ξ)dw,
+k = 2, L,
+(10)
+Using the Laplace transform method, we could turn the
+above equation from v-space into s-space as follows:
+f q¯q
+k,g(s, Q2, m2
+q) = e2
+Hg(s, µ2)hk,g(s, ξ),
+k = 2, L, (11)
+where hk,g(s, ξ) = L[ ˆC(v − ln 1/a, ξ)], v, s] at the LO ap-
+proximation have been given in Ref. [32]. To obtain the
+heavy quarks components F q¯q
+2
+and F q¯q
+L
+of the structure
+functions in Laplace space at the LO and NLO approxi-
+mations, the obtained gluon distribution function in Eq.
+(8) are inserted into Eq. (11). But before that, Q2 must
+be replaced by µ2. With these descriptions, one can write
+the structure functions as follows:
+f q¯q
+k,g(s, Q2, m2
+q) = e2
+H
+�
+jgf(s, µ2)f(s, Q2
+0)
++jgg(s, µ2)g(s, Q2
+0)
+�
+,
+k = 2, L,
+(12)
+where
+jgf(s, µ2) = hk,g(s, ξ)kfg(s, τ(µ2, Q2
+0))/as,
+
+4
+jgg(s, µ2) = hk,g(s, ξ)kgg(s, τ(µ2, Q2
+0))/as.
+(13)
+Finally, using the Laplace inverse transform, we could
+obtain these structure functions in the usual space x as
+follows;
+F q¯q
+k,g(x, Q2, m2
+q) = e2
+H
+�
+Jgf(x, µ2) ⊗ Fs(x, Q2
+0)
++Jgg(x, µ2) ⊗ G(x, Q2
+0)
+�
+,
+k = 2, L
+(14)
+where Jgf(x, µ2)
+=
+L−1[jgf(s, µ2), s, v]|v=ln(1/x) and
+Jgg(x, µ2) = L−1[jgg(s, µ2), s, v]|v=ln(1/x). It should be
+noted that, to obtain the heavy quarks structure func-
+tions in Eq. (14), it is requires only a knowledge of the
+singlet Fs(x) and gluon G(x) distribution functions at
+the starting value Q2
+0.
+III.
+NUMERICAL RESULTS
+Now, we present our numerical results of the produc-
+tion cross section of heavy quarks in the e−p interaction
+at the LO and NLO approximations obtained by using
+Eq. (1) and the DGLAP evolution equations. In order
+to present more detailed discussions on our findings, the
+numerical results for the structure functions and the pro-
+duction cross section of heavy quarks are compared with
+HREA data [4, 5, 9, 14, 17, 41–43] and with the results
+from the MSTW2008 [46], MSRT03 [47] and H1PDF
+[48].
+To extract numerical results, we use the pub-
+lished MSTW2008 [46] initial starting functions Fs(x)
+and G(x). It should be said that, we have considered the
+uncertainties due to the running charm, beauty and top
+(t) quark masses mc = 1.29+0.077
+−0.053 GeV , mb = 4.049+0.138
+−0.118
+GeV [6] and mt = 173.5+3.9
+−3.8 GeV [56] where the uncer-
+tainties are obtained through adding the experimental fit,
+model and parameterization uncertainties in quadrature.
+Figures (1) and (2) indicate the numerical results of the
+c quark structure functions (F c¯c
+2 (x, Q2) and F c¯c
+L (x, Q2))
+at Q2 = 11, 60, 130 and 500 GeV 2 and the b quark
+structure functions (F b¯b
+2 (x, Q2) and F b¯b
+L (x, Q2)) at Q2 =
+12, 60, 200 and 650 GeV 2. These results are presented
+at the LO and NLO approximations and compared with
+those presented by ZEUS [41–43] and H1 [9, 13, 17] collid-
+ers and with the results from the MSTW2008 predictions
+[46]. In these figures, since nowhere is presented data on
+the longitudinal structure function of heavy quarks, we
+only compare F c¯c
+2 (x, Q2) and F b¯b
+2 (x, Q2) with those pre-
+sented by ZEUS and H1 colliders . As can be seen, our
+numerical results at the NLO approximation are closer
+to the experimental data than the results at the LO ap-
+proximation.
+In figure (3) as a comparison, we show our numerical
+results of the c quark reduced cross section and ZEUS [5]
+data as a function of x. The numerical results of this cross
+section are showed at the LO and NLO approximation
+at Q2 = 6.5, 12, 30, 80, 160 and 600 GeV 2.
+It can be
+concluded that in the low-energy scale, our results are
+very close to ZEUS data and show that the presented
+control coefficients are suitable. We also compare the c
+quark reduced cross section at the NLO approximation in
+Q2 = 5, 12, 60, 200, 650 and 2000 GeV 2 with H1 [4] data
+and with the results from the MSTW2008 predictions
+[46] in figure (4).
+Figure (5) indicates our numerical results of the b
+quark reduced cross section at the LO and NLO approx-
+imations in Q2 = 6.5, 12, 30, 80, 160 and 600 GeV 2 com-
+pared with ZEUS [5] data. Moreover, in figure (6), we
+compare the b quark reduced cross section at the NLO
+approximation with H1 [4] data and with the results from
+the MSTW2008 predictions [46].
+In figure (7), we present the numerical results of the t
+quark reduced cross section at the LO and NLO approxi-
+mations at Q2 = 6.5, 12, 30, 80, 160 and 600 GeV 2. Here,
+we must state that the t quark longitudinal structure
+function at the LO and NLO approximations are very
+small relative to the structure function F t¯t
+2 (x, Q2), so we
+can almost consider σt¯t
+r equal to F t¯t
+2 (x, Q2). At the NLO
+approximation in the energy interval 6.5 < Q2 < 600
+GeV 2, the t quark longitudinal structure function is zero.
+In figure (8), it is presented a comparison between the
+reduced cross section of heavy quarks at Q2 = 1000, 5000
+and 10000 GeV 2. In this figure at Q2 = 1000GeV 2 and
+minimum x, the ratios of σb¯b
+r /σc¯c
+r
+and σt¯t
+r /σc¯c
+r
+are ap-
+proximately 0.158 and 0.005, respectively and at Q2 =
+10000GeV 2 and minimum x, these ratios are approxi-
+mately 0.225 and 0.047, respectively. These results show
+that with the increase of energy, the production cross
+section of t quark grows more than the production cross
+section of b quark.
+In order to assess the significance of the theoretical
+uncertainty at the LO and NLO approximations, we in-
+dicate the Q2 dependence of the single differential cross
+section of heavy quarks dσqq/dQ2 at the LO and NLO ap-
+proximations in figure (9). In this figure, the differential
+cross section of c, b and t quarks are presented at the cen-
+ter of mass energies of √s = 319GeV , √s = 1.3T eV and
+√s = 3.5T eV in e−p interaction at 0.1 < y < 0.7. We
+also show the single differential cross section of the c and
+b quarks as a function of Q2 at the CME of √s = 319GeV
+and 0.02 < y < 0.7 at the NLO approximation in figure
+(10). The data have been given together with their sta-
+tistical and systematic uncertainties (not including the
+error on the integrated luminosity). Moreover, the single
+differential cross sections of c and b quarks as a function
+of x at the CME of √s = 319GeV and 0.02 < y < 0.7 at
+the NLO approximation are presented in figure (11).
+The integrated cross sections are also compared with
+H1 [30] and ZEUS [14] data and with the predictions
+from NLO QCD. The integrated cross sections for c and
+b quarks have been respectively presented 373 ± 39 ± 47
+pb and 55.4 ± 8.7 ± 12.0 pb by the H1 vertex detector
+for values of photon virtuality Q2 > 150GeV 2 and of
+
+5
+inelasticity 0.1 < y < 0.7. Our numerical result of this
+cross section at the LO and NLO approximations for the c
+quark at the e−p CME of √s = 319 GeV and at Q2 > 150
+GeV 2 and inelasticity 0.1 < y < 0.7 are 320 ± 7 pb and
+332 ± 3 pb, respectively and for the b quark they are
+32.9 ± 1.2 pb and 36.7 ± 1.0 pb. These results are pre-
+sented in Table (2) and compared with the VFNS pre-
+dictions from MRST03 [47] and the H1PDF 2000 fit [48]
+and with results from the FFNS scheme, which was used
+for the QCD predictions in Ref. [14]. In Ref. [14], the re-
+sults have been obtained by using the parton distribution
+function set CTEQ5F3 [57]. In addition to these results,
+we show the integrated cross sections for c and b quarks
+at the e−p CME of √s = 1.3T eV and √s = 3.5T eV . It
+should be noted that the integrated cross sections for c
+and b quarks at √s = 319GeV at the NLO approxima-
+tion are larger than those at the LO approximation but at
+√s = 1.3T eV and √s = 3.5T eV the results are inverse.
+Furthermore, we obtain and present the integrated cross
+sections for t quark at √s = 1.3T eV and √s = 3.5T eV
+in Table (2) as a prediction from our calculations. This
+cross section at √s = 319GeV for values of photon vir-
+tuality Q2 > 150GeV 2 and of inelasticity 0.1 < y < 0.7
+is zero.
+All of the results clearly show that the extraction
+procedure provides correct behaviors of the structure
+functions and the production cross section of the heavy
+quraks at the LO and NLO approximations. Moreover,
+it should be noted that the NLO corrections are small
+for values of high x , but at low x region these correc-
+tions have many effects on the results especially at low
+Q2.
+Furthermore, they often allow one to reduce the
+uncertainties of the predicted results, as one can see by
+comparing the bands in almost all of the plots presented
+in the figures.
+IV.
+CONCLUSION
+In conclusion, we have presented the production cross
+section of heavy quarks (σc¯c, σb¯b and σt¯t) and the sin-
+gle differential cross sections (dσqq/dQ2 and dσqq/dx) of
+them by utilizing the heavy quarks structure functions
+F q¯q
+2
+and F q¯q
+L
+obtained by Dokshitzer-Gribov-Lipatov-
+Altarelli-Parisi evolution equations and a suitable fit for
+the heavy quarks coefficient functions at the NLO ap-
+proximation. Indeed, we have shown that the Laplace
+transform method is the suitable and alternative scheme
+to solve the DGLAP evolution equations and Eq. (2).
+It should be noted that, the obtained equations are gen-
+eral and require only a knowledge of the parton distri-
+bution functions Fs(x), G(x) at the starting value Q2
+0.
+The comparisons have shown that our numerical results
+of the charm and beauty production cross section are in
+agreement with the H1 and ZEUS data well within er-
+rors.
+Also, in this paper, we have compared the pro-
+duction cross sections at the center-of-mass energy of
+√s = 319GeV and at 0.1 < y < 0.7 with the ex-
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Q
+2
+=500G
+e
+V
+2
+F(x,Q
+2
+)
+x
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Q
+2
+=11G
+e
+V
+2
+ F
+2
+c c   
+(NLO
+)
+ F
+L
+c c
+ (NLO
+)
+ F
+2
+c c  
+(MST
+W-NLO
+)
+ H1
+ Z
+EUS
+F
+(x,Q
+2
+)
+x
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+Q
+2
+=130G
+e
+V
+2
+F
+(x,Q
+2
+)
+x
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Q
+2
+=60G
+e
+V
+2
+F
+(x,Q
+2
+)
+x
+Figure (1): The charm quark structure functions F c¯c
+2(L)
+compared with data from H1 [9, 13, 17], ZEUS [41–43], and
+MSTW2008 [46] at the NLO approximation.
+perimented results by H1 PDF 2000 and MSRT03 and
+MSTW2008.
+Also, we have obtained the production
+cross sections of heavy quarks at the-center-of mass en-
+ergies of √s = 1.3T eV and √s = 3.5T eV and considered
+the uncertainties due to the running charm, beauty and
+top quark masses. In addition, we have presented the
+production cross section of the quark top at center-of-
+mass energies of √s = 1.3T eV and √s = 3.5T eV .
+
+-
+-1
+-1.1
+1
+[!6
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+Q
+2
+=650G
+e
+V
+2
+F
+(x,Q
+2
+)
+x
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+-0.002
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+Q
+2
+=12G
+e
+V
+2
+ F
+2
+bb  
+(NLO
+)
+ F
+L
+bb
+ (NLO
+)
+ F
+2
+bb 
+(MST
+W-NLO
+)
+ H1
+F
+(x,Q
+2
+)
+x
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Q
+2
+=200G
+e
+V
+2
+F
+(x,Q
+2
+)
+x
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+Q
+2
+=60G
+e
+V
+2
+F
+(x,Q
+2
+)
+x
+Figure (2): The beauty quark structure functions F b¯b
+2(L)
+compared with data from H1 [17] and MSTW2008 [46] at
+the NLO approximation.
+1E-4
+0.001
+0.01
+0.1
+1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Q
+2
+=80G
+e
+V
+2
+s
+r
+c c
+(x,Q
+2
+)
+x
+1E-4
+0.001
+0.01
+0.1
+1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Q
+2
+=600G
+e
+V
+2
+s
+r
+c c
+(x,Q
+2
+)
+x
+1E-4
+0.001
+0.01
+0.1
+1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Q
+2
+=160G
+e
+V
+2
+s
+r
+c c
+(x,Q
+2
+)
+x
+1E-4
+0.001
+0.01
+0.1
+1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Q
+2
+=6.5G
+e
+V
+2
+ NLO
+ LO
+ Z
+EUS
+s
+r
+c c
+(x,Q
+2
+)
+x
+1E-4
+0.001
+0.01
+0.1
+1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Q
+2
+=30G
+e
+V
+2
+s
+r
+c c
+(x,Q
+2
+)
+x
+1E-4
+0.001
+0.01
+0.1
+1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Q
+2
+=12G
+e
+V
+2
+s
+r
+c c
+(x,Q
+2
+)
+x
+Figure (3): The reduced charm quark cross section as a
+function of x for six different values of Q2 at the LO and
+NLO approximations compared with the ZEUS data [5]. The
+error bars represent the statistical, systematic (not including
+the error on the integrated luminosity) and extrapolation
+uncertainties added in quadrature. The shaded areas are the
+uncertainties due to the running quark mass.
+1E
+-4
+0.001
+0.01
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Q
+2
+=
+120G
+eV
+2
+s
+r
+cc
+(x,Q
+2
+)
+x
+0.001
+0.01
+0.1
+Q
+2
+=
+300G
+eV
+2
+x
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+ N
+LO
+ M
+S
+T
+W2008
+ H
+1
+s
+r
+cc
+(x,Q
+2
+)
+Q
+2
+=
+12G
+eV
+2
+0.001
+0.01
+0.1
+Q
+2
+=
+2000G
+eV
+2
+x
+Q
+2
+=
+60G
+eV
+2
+Figure (4): The reduced charm quark cross section as a
+function of x at the NLO approximation compared with the
+H1 data [4] and the results from the MSTW2008 predictions
+[46].
+
+1
+-
+!
+--r
+1
+11111
+17
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+Q
+2
+=80G
+e
+V
+2
+s
+r
+bb
+(x,Q
+2
+)
+x
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+Q
+2
+=600G
+e
+V
+2
+s
+r
+bb
+(x,Q
+2
+)
+x
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+Q
+2
+=160G
+e
+V
+2
+s
+r
+bb
+(x,Q
+2
+)
+x
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+Q
+2
+=6.5G
+e
+V
+2
+ NLO
+ LO
+ Z
+EUS
+s
+r
+bb
+(x,Q
+2
+)
+x
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+Q
+2
+=30G
+e
+V
+2
+s
+r
+bb
+(x,Q
+2
+)
+x
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+Q
+2
+=12G
+e
+V
+2
+s
+r
+bb
+(x,Q
+2
+)
+x
+Figure (5): The reduced beauty quark cross section. For
+more details, see the caption of figure (3).
+Q
+2
+=
+60G
+eV
+2
+0.001
+0.01
+0.1
+Q
+2
+=
+2000G
+eV
+2
+x
+1E
+-4
+0.001
+0.01
+0.1
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+Q
+2
+=
+200G
+eV
+2
+s
+r
+bb
+(x,Q
+2
+)
+x
+0.001
+0.01
+0.1
+Q
+2
+=
+650G
+eV
+2
+x
+Q
+2
+=
+12G
+eV
+2
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+ N
+LO
+ M
+S
+T
+W2008
+ H
+1
+s
+r
+bb
+(x,Q
+2
+)
+Q
+2
+=
+5G
+eV
+2
+Figure (6): The reduced beauty quark cross section as a
+function of x at the NLO approximation compared with the
+H1 data [4] and the results from the MSTW2008 predictions
+[46].
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+10
+-4
+10
+-3
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Q
+2
+=80G
+e
+V
+2
+s
+r
+tt
+(x,Q
+2
+)�10
+-4
+x
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+10
+-4
+10
+-3
+0
+5
+10
+15
+20
+25
+30
+35
+40
+ NLO
+ LO
+s
+r
+tt
+(x,Q
+2
+)�10
+-4
+x
+Q
+2
+=600G
+e
+V
+2
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+10
+-4
+10
+-3
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Q
+2
+=160G
+e
+V
+2
+s
+r
+tt
+(x,Q
+2
+)�10
+-4
+x
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+10
+-4
+0
+2
+4
+6
+8
+10
+12
+Q
+2
+=6.5G
+e
+V
+2
+s
+r
+tt
+(x,Q
+2
+)�10
+-5
+x
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+10
+-4
+0
+2
+4
+6
+8
+10
+12
+Q
+2
+=30G
+e
+V
+2
+s
+r
+tt
+(x,Q
+2
+)�10
+-5
+x
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+10
+-4
+0
+2
+4
+6
+8
+10
+12
+Q
+2
+=12G
+e
+V
+2
+s
+r
+tt
+(x,Q
+2
+)�10
+-5
+x
+Figure (7): The reduced top quark cross section as a
+function of x for six different values of Q2 at the LO and
+NLO approximations.
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+10
+-3
+10
+-2
+10
+-1
+10
+0
+10
+1
+10
+2
+10
+3
+ s
+r
+cc
+ (1000G
+eV
+2
+)
+ s
+r
+bb
+ (1000G
+eV
+2
+)
+ s
+r
+tt
+  (1000G
+eV
+2
+)
+ s
+r
+cc
+ (5000G
+eV
+2
+)
+ s
+r
+bb
+ (5000G
+eV
+2
+)
+ s
+r
+tt
+ (5000G
+eV
+2
+)
+ s
+r
+cc
+ (10000G
+eV
+2
+)
+ s
+r
+bb
+(10000G
+eV
+2
+)
+ s
+r
+tt
+  (10000G
+eV
+2
+)
+s
+r
+qq
+(x,Q
+2
+)
+x
+Figure (8): A comparison between the reduced cross section
+of heavy quarks at large values of Q2 at the NLO
+approximation.
+
+11
+-
+-
+-
+-008
+Figure (9): The results of the differential cross section of the
+c, b and t quarks as a function of Q2 at the center-of-mass
+energies of √s = 319GeV , √s = 1.3T eV and √s = 3.5T eV
+at the LO and NLO approximations.
+Figure (10): The results of the differential cross section of
+the c and b quarks as a function of Q2 at the center-of-mass
+energy of √s = 319GeV at the LO and NLO approximations
+compared with ZEUS data [5].
+Figure (11): The results of the differential cross section of
+the c and b quarks as a function of x at the center of mass
+energy of √s = 319GeV at the NLO approximation
+compared with ZEUS data [5].
+Table (2): The integrated cross sections of the c, b and
+t quarks which are integrated over the range Q2 > 150
+GeV 2 and 0.1 < y < 0.7. Our numerical results for the
+c and b quark cross sections are compared with the H1
+[17] and ZEUS [14] data and with the results from
+H1PDF [48] and MSRT03 [47].
+√s = 319GeV
+√s = 1.3TeV
+√s = 3.5TeV
+σc¯c (pb)
+LO
+320 ± 7
+2147 ± 42
+6225 ± 115
+NLO
+332 ± 3
+1791 ± 25
+4449 ± 71
+H1
+373 ± 39 ± 47
+− − −
+− − −
+H1PDF
+455
+− − −
+− − −
+ZEUS
+419
+− − −
+− − −
+MRST03 426
+− − −
+− − −
+σb¯b (pb)
+LO
+32.9 ± 1.2
+248.4 ± 7.3
+751.9 ± 22.9
+NLO
+36.7 ± 1.0
+217.2 ± 4.7
+560.6 ± 12.9
+H1
+55.4 ± 8.7 ± 12.0 − − −
+− − −
+H1PDF
+52
+− − −
+− − −
+ZEUS
+37
+− − −
+− − −
+MRST03 47
+− − −
+− − −
+σt¯t (fb)
+LO
+− − −
+4.18 ± 0.56
+141.1 ± 11.1
+NLO
+− − −
+4.78 ± 0.58
+143.5 ± 11.5
+
+109
+108
+107
+106
+(qd)xp/op
+105
+104
+ZEUS
+(CC)
+103
+NLO
+(CC)
+(bb)
+102
+ZEUS
+(bb)
+NLO
+101
+100
+1E-4
+0.001
+0.01
+0.1
+X104
+102
+LO
+NLO
+ZEUS
+103
+101
+docc/dQ?(pb/GeV2)
+dob/dQ2(pb/GeV2)
+102
+100
+101
+10-1
+100
+10-2
+10-1
+10-2
+10-3
+101
+102
+103
+101
+102
+103
+Q?(GeV2)
+Q?(GeV2)101
+100
+10-5
+(319 GeV)
+LO
+NLO
+(319 GeV)
+LO
+(1.3 TeV)
+100
+NLO (1.3 TeV)
+10-1
+10-6
+LO
+(3.5 TeV)
+NLO (3.5 TeV)
+10-1
+10-2
+10-7
+dob/dQ2(nb/GeV2)
+dot/dQ2(nb/GeV2)
+10-2
+10-3
+10-8
+10-3
+104
+10-9
+104
+10-5
+10-10
+10-5
+106
+10-11
+10-6
+10-7
+10-7
+10-12
+101
+102
+103
+104
+101
+102
+103
+104
+101
+102
+103
+104
+105
+Q(GeV2)
+Q?(GeV2)
+Q?(GeV2)9
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+

XNE2T4oBgHgl3EQfYgch/content/tmp_files/2301.03854v1.pdf.txt

@@ -0,0 +1,871 @@
+Frascati Physics Series Vol. 74 (2022)
+ISBN: 978-88-86409-76-6
+Frontier Objects in Astrophysics and Particle Physics
+September 25- October 1, 2022
+CONSTRAINTS ON DARK MATTER FROM REIONIZATION
+Marco Castellano
+INAF - Osservatorio Astronomico di Roma,
+via di Frascati 33, 00078 Monte Porzio Catone (RM), Italy
+Nicola Menci
+INAF - Osservatorio Astronomico di Roma,
+via di Frascati 33, 00078 Monte Porzio Catone (RM), Italy
+Massimiliano Romanello
+INAF - Osservatorio Astronomico di Roma,
+via di Frascati 33, 00078 Monte Porzio Catone (RM), Italy;
+Dipartimento di Fisica e Astronomia - Alma Mater Studiorum Universit`a di Bologna,
+via Piero Gobetti 93/2, I-40129 Bologna, Italy;
+INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna,
+via Piero Gobetti 93/3, I-40129 Bologna, Italy
+Abstract
+This conference proceedings paper provides a short summary of the constraints pre-
+sented in Menci et al. 2016, 2017 [1, 2] on the mass of thermal WDM candidates,
+and of the results presented in Romanello et al. 2021 [3] on how Reionization sce-
+narios are affected by early galaxy formation in WDM cosmologies. The abundance
+of galaxies in the epoch of reionization (z >6) is dependent on fundamental cosmo-
+logical parameters, most importantly on the properties of dark matter, such that it can
+be used as a powerful cosmological probe. Here we show how the number density of
+primordial galaxies allows to constrain the mass of thermal WDM candidates, and the
+constraints that will be made possible by future JWST observations. We then inves-
+tigate how the Reionization process is affected by early galaxy formation in different
+cosmological scenarios. We use a semi-analytic model with suppressed initial power
+spectra to obtain the UV Luminosity Function in thermal Warm Dark Matter and ster-
+ile neutrino cosmologies. For each cosmology, we find an upper limit to fixed fesc,
+which guarantees the completion of the process at z < 6.7.
+arXiv:2301.03854v1  [astro-ph.CO]  10 Jan 2023
+
+1
+Introduction
+The Epoch of Reionization (EoR) marked a fundamental phase transition in the history
+of the Universe, during which the Intergalactic Medium (IGM) became transparent to
+UV photons. The most recent observations indicate a late-Reionization scenario [4–
+6], with the end of the EoR at z ≈ 6, but the exact contribution from different ionizing
+sources and the exact timeline and topology of Reionization are still unknown.
+From a theoretical point-of-view, cosmic Reionization depends on non-linear
+and non-local phenomena, in which the physics of galaxy formation couples with the
+physics of gravity and radiation transport. The first process is determined by both
+baryonic physics and poorly known feedback effects, but also by the initial power
+spectrum of density fluctuations: in fact, dark matter produces the potential wells in
+which baryonic perturbation undergo an accelerated growth. Therefore, the study of
+Reionization is strongly related to the comprehension of cosmological framework in
+which cosmic structures form and grow.
+The currently most acknowledged cosmological model is the ΛCDM model. It
+is based on the contribution of the cosmological constant Λ (≈69%) and Cold Dark
+Matter (≈26%) and provides a coherent large-scale description of the Universe with
+respect to the available data. The ΛCDM model postulates the existence of Dark
+Matter in a “cold” version, i.e., composed by Weakly Interacting Massive Particles
+(WIMPs) with mX > 0.1 GeV or condensates of light axions, with mX ≈ 10−5–10−1
+eV.
+However, there are some possible tensions related to observations at galactic and
+sub-galactic scales, of the order of kpc. Furthermore, the lack of detection of CDM
+candidates has suggested the possibility to investigate on alternative cosmological sce-
+narios, based on the existence of Warm Dark Matter particles, with mass of the order
+of keV. While in the ΛCDM model, due to the high mass particles, all the cosmolog-
+ical density perturbations can become gravitationally unstable, in a WDM scenario,
+depending on the value of mX, only perturbations above the kpc scale can collapse,
+producing shallower density profiles and a smaller number of low-mass halos. This,
+in the context of the hierarchical growth of the cosmic structures, implies a reduction
+in the number of faint galaxies and a delay in their formation [1, 7].
+In WDM cosmologies, the simplest approach is to consider particles that behave
+as “thermal relics”, resulting from the freeze-out of DM species initially in thermal
+equilibrium with the early Universe, e.g. [8, 9].
+A possible alternative is offered by sterile neutrinos (SN) or right-handed neutri-
+
+nos, which are particles predicted in the context of Standard Model extensions. Since
+they are produced out-of-equilibrium, from the oscillations of active neutrinos, they
+are characterized by a non-thermal power spectrum, which depends both on mass and
+on sin(2θ), where θ is the mixing angle [10].
+In the present paper we first summarise the results presented in Menci et al.
+2016,2017 [1, 2] where stringent constraints on DM models with suppressed power
+spectra by have been derived by comparing the maximum number density of DM
+halos φ expected at redshift z = 6 to the observed number density φobs of galaxies at
+the same redshift in the HFF. We then summarise the results presented in Romanello
+et al. 2021 [3] on how Reionization scenarios are affected by early galaxy formation in
+WDM cosmologies. We have used the theoretical model by Menci et al. (2018) [11],
+where the collapse history of dark matter halos is modelled through the Extended
+Press-Schechter (EPS) formalism and baryonic processes taking place in each halo
+are included through physically motivated analytical recipes.
+2
+Method
+2.1
+Warm Dark Matter thermal relics
+The simplest alternative to CDM is provided by Warm Dark Matter models assuming
+DM to be the result from the freeze-out of particles with mass in the keV range ini-
+tially in thermal equilibrium in the early Universe. In these models, the population of
+low-mass galaxies is characterized by lower abundances and shallower central density
+profiles compared to Cold Dark Matter (CDM) due to the dissipation of small-scale
+density perturbations produced by the free-streaming of the lighter and faster DM par-
+ticles. In this case, the mass of the DM particle completely determines the suppression
+of the density power spectrum compared to the CDM case
+The computation of the halo mass function for the WDM scenario is based on
+the standard procedure described and tested against N-body simulations. The differ-
+ential halo mass function (per unit log M) based on the extended Press & Schechter
+approach [e.g. 12] reads:
+d φ
+d logM = 1
+6
+ρ
+M f(ν) d log σ2
+d logr .
+(1)
+Here ν ≡ δ2
+c(t)/σ2 depends on the linearly extrapolated density for collapse in the
+spherical model δc = 1.686/D(t) and D(t) is the growth factor of DM perturbations.
+A spherical collapse model for which f(ν) = √2ν/π exp(−ν/2) is assumed.
+
+The key quantity entering Eq. 1 is the variance of the linear power spectrum
+P(k) of DM perturbations (in terms of the wave-number k = 2π/r). Its dependence on
+the spatial scale r of perturbations is:
+d log σ2
+d log r = −
+1
+2 π2 σ2(r)
+P(1/r)
+r3
+.
+(2)
+In WDM scenarios the spectrum PWDM is suppressed with respect to the CDM
+case PCDM below a characteristic scale depending on the mass mX of the WDM
+particles. In the case of relic thermalized particles, the suppression factor can be
+parametrized as in Bode et al. 2001 [13]:
+TWDM(k) = PWDM(k)
+PCDM(k) =
+�
+1 + (α k)2 µ�−10/µ .
+(3)
+where µ = 1.12 and the quantity α is linked to the WDM free-streaming scale:
+α = 0.049
+� ΩX
+0.25
+�0.11 � mX
+keV
+�−1.11 � h
+0.7
+�1.22 h−1
+Mpc,
+(4)
+where mX is the WDM particle mass, ΩX is the WDM density parameter (ΩX) and h
+the Hubble constant in units of 100 km/s/Mpc.
+The mass function is computed through Eq. 1 after substituting Eq. 2, with a
+power spectrum P(k) = PWDM(k) determined by the WDM particle mass mX after Eqs.
+3 and 4. A half-mode wavenumber is defined, as the khm at which the transfer function
+TWDM(k) is equal to 1/2 [13, 14]. Correspondingly, a half-mode mass Mhm can also
+be defined:
+Mhm = 4π
+3 ρm
+�
+πϵ(2µ/5 − 1)−1/2µ�3 .
+(5)
+2.2
+Semi-Analytic Model
+To investigate the interplay between WDM scenarios and reionization history, we use
+the semi-analytic model developed by Menci et al. (2018), to which we refer for fur-
+ther informations [11]. The model retraces the collapse of dark matter halos through a
+Monte Carlo procedure on the basis of the merging history given by EPS formalism,
+at 0 < z < 10 [11]. In this framework, the DM structures formation is determined by
+the power spectrum: the WDM P(k) is computed by the suppression of the CDM one,
+due to the particles free streaming at kpc scale, as described in the previous section.
+Conversely, for sterile neutrino based cosmological scenarios, we refer to Mhm
+from Lovell et al. (2020), obtained comparing CDM and WDM simulations performed
+
+within the same cosmic volume and in which the parameterization of the WDM halo
+mass function is given by Rfit [15]:
+Rfit = nWDM
+nCDM
+=
+������1 +
+�
+α Mhm
+Mhalo
+������
+γ
+,
+(6)
+where nCDM and nWDM are the differential halo mass functions and Mhalo is the halo-
+mass. The numerical value of α, β and γ coefficients changes if we consider central
+(α = 2.3, β = 0.8, γ = −1.0) or satellite halos (α = 4.2, β = 2.5, γ = −0.2) [15].
+We perform our analysis with five different sterile neutrino models, with a mass
+of 7.0 keV, labelled according to the lepton asymmetry number (L6), which is indicated
+in the last part of the name. For example, L6 = 120 is named LA120, L6 = 8 is
+named LA8 and so on. Among them, the models LA9, LA10 and LA11 are based on
+decaying-particles that are compatible with the X-ray 3.55 keV emission line observed
+in galaxy clusters [15].
+The semi-analytic model associates a galactic luminosity to each halo, depend-
+ing on cooling process and merging history. The gas in the halo, initially set to have a
+density given by the universal baryon fraction and to be at the virial temperature, cools
+due to atomic processes and settles into a rotationally supported disk. Then, the cooled
+gas is gradually converted into stars, with a SFR given by:
+˙M∗ = Mgas
+τ∗ , according to
+the Schmidt-Kennicut law with a gas conversion time scale τ∗ = qτd, proportional
+to the dynamical time scale τd through the free parameter q [11]. Moreover, galaxy
+interactions occurring in the same host halo may induce the sudden conversion of a
+fraction f of cold gas into stars on a short time-scale given by the duration of the in-
+teraction [11]. Feedback phenomena due to supernovae, AGNs and photoionization
+are also included, as described by Menci et al. (2018) [11]. Finally, the luminosity
+produced by the stellar populations is computed by assuming a Salpeter IMF [11]. In
+our analysis, we integrate the rest-frame UV (∼1400 Å) dust-corrected LF between
+the limits Mlim
+UV = [−25, −12], in order to obtain the corresponding luminosity density:
+ρUV =
+� Mlim
+UV
+dMUV
+dN
+dMUV
+RfitLUV,
+(7)
+which is dominated by the contribution of systems with MUV ≥ −20 (see Section 3.2).
+The number density of UV photons that actively participate to hydrogen ioniza-
+tion process is obtained by multiplying for two quantities [16]:
+˙Nion = fescξionρUV.
+(8)
+
+The ionizing photon production efficiency (ξion) is expressed in Hz/erg units and
+it describes how efficiently is possible to get UV ionizing photons from an UV con-
+tinuum radiation field. Finally, the escape fraction fesc converts the intrinsic ionizing
+emissivity ˙Nion,intrinsic = ξionρUV into an effective one. It is defined as the fraction of
+ionizing photons that can escape from the source galaxy instead of being reabsorbed
+inside it and which therefore actively participates in the ionization of the IGM.
+In our study, we model the Reionization history with different values of fesc.
+Fixed escape fraction is useful to broadly characterize the Reionization history, al-
+though a universal value for fesc is highly unrealistic.
+Nevertheless, the investigation of the degenerate quantities fescξion, which drive
+the Reionization process, can yield to interesting upper limits to the escape fraction.
+Once obtained ˙Nion, the equation that accounts for ionization and recombination,
+which regulates the evolution of the hydrogen filling fraction QHII is:
+˙QHII =
+˙Nion
+¯nH
+− QHII
+trec
+,
+(9)
+where the comoving hydrogen mean density is computed as ¯nH ≈ 2×10−7(Ωbh2/0.022)
+cm−3 and the recombination time-scale is trec ≈ 3.2 Gyr [(1 + z)/7]−3C−1
+HII [17]. We
+consider case B of recombination, in which electrons fallen to the ground level gen-
+erate ionizing photons that are re-absorbed by the optically thick IGM, having no
+consequences on the overall ionization balance. We treat the evolution of the clump-
+ing factor CHII with redshift, due to the effect of UVB generated by Reionization,
+according to, e.g., Haardt et al. 2012 [18]:
+CHII = 1 + 43z−1.71.
+(10)
+After the reconstruction of the Reionization history, we use the redshift evolu-
+tion of the filling fraction to compute the integral:
+τes(z) = cσT ¯nH
+� z
+0
+QHII(z′)(1 + z′)2 �
+1 + ηY
+4X
+�
+H−1(z′)dz′,
+(11)
+in which helium is singly-ionized (η = 1) at z > 4 and doubly-ionized (η = 2) at z < 4.
+Then, the electron scattering optical depth has been compared with observational con-
+straints on τes obtained, from CMB anisotropy, by Planck and WMAP.
+
+Figure 1: Adapted from Menci et al. 2016 [1]: the cumulative mass functions com-
+puted at z = 6 for different values of the WDM particle mass mX from 1 to 3 keV
+(bottom to top). The shaded areas correspond to the observed number density of HFF
+galaxies within 1-σ, 2-σ, and 3-σ confidence levels.
+3
+Results
+3.1
+Constraints on thermal WDM from the abundance of high-redshift galaxies
+We compare the halo number densities in WDM cosmologies to the observed number
+density φobs of galaxies derived by integrating the galaxy luminosity function (LF) at
+z = 6 by [19] down to the faintest bin MUV = −12.5. Constraints on DM models are
+simply put by requiring that observed galaxies cannot outnumber their host DM halos
+(φ ≥ φobs). The reference luminosity function has been estimated from objects in the
+Abell 2744 and MACS 0416 cluster fields, selected on the basis of their photometric
+redshift.
+In Fig. 1 we show the cumulative mass function φ(> M) at z = 6 for different
+assumed WDM particle masses. All the mass functions saturate to a maximum number
+density φmX ≈ φ(Mhm). This is compared with the observed number density φobs of
+galaxies with MUV ≤ −12.5. The condition φobs ≤ φmX yields mX ≳ 2.9 keV at 1-
+σ level, mX ≥ 2.4 keV at 2-σ level, and mX ≥ 2.1 keV at 3-σ level. In Fig. 2 we
+
+Figure 2: Constraints on the abundance of dark-matter halos derived from the galaxy
+LF at different redshifts (black arrows). The corresponding limits on the mass of
+WDM thermal relics are shown as colored contours. JWST measurements reaching
+z > 9 lensed galaxies down to MUV ≤ −12.5 (red arrows) can significantly strengthen
+currents constraints if no cut-off of the LF will be found.
+show the constraints on the thermal relic WDM particle mass from the abundance
+of galaxies in available observations at z < 7, including the quoted constraint from
+the HFF observations at z ∼ 6, compared to what could be achieved by hypothetical
+JWST observations of strongly lensed galaxies at z > 9 (red arrows). We derive the
+estimate by extrapolating to MUV ≤ −12.5 the UV LF at z = 9 − 16 recently published
+by Harikane et al. 2022 [20]. If JWST will confirm a steep faint-end of the UV LF
+at extremely high-redshifts, thermal relic particles with masses mX < 5 keV will be
+ruled out by galaxy abundance measurements alone.
+3.2
+The reionization history in WDM scenarios
+Here we investigate the unfolding of the reionization epoch in WDM cosmologies,
+summarising the cited work by Romanello et al. 2021 [3].
+
+Relic Particle Mass (keV)
+0.30
+1.00
+2.00
+4.00
+6.00
+8.00
+10.00
+3
+log Φ/Mpc
+-2
+3
+5
+10
+15
+ZFigure 3: Adapted from Romanello et al. 2021 [3]. The multiple panels show the inte-
+grated photons ratio rphot(< MUV
+lim ), where ˙Nion,tot is computed by integrating Equation
+(7) between intrinsic Msup
+UV = −12 and Minf
+UV = −25. We compare with CDM two ther-
+mal WDM cosmologies (WDM3 is an intermediate case between WDM4 and WDM2),
+and three sterile neutrino cosmologies (here LA10 is the only representative scenario
+for radiatively decay Dark Matter, which is compatible with the 3.5 keV emission line
+observed in galaxy clusters).
+In Figure 3 we plot the integrated ionizing photons ratio:
+rphot(< MUV
+lim ) =
+˙Nion(MUV < Mlim
+UV)
+˙Nion,tot
+(12)
+in which we compute ˙Nion,tot, using Equation (7) between intrinsic Msup
+UV = −12 and
+Min f
+UV = −25, while the numerator is obtained by varying the upper limit of the integral
+from −24 to −12, including so the photons from progressively dimmer sources, until
+the unity is reached.
+From Figure 3 we can identify two important features, through which we can
+understand the role and the different contribution of faint and bright galaxies during
+EoR. The first is the increasing of the relative contribution of the brightest systems
+
+1.0
+LA8
+CDM
+LA10
+CDM
+Muv<-24
+CDM
+Muv<-22
+Muv<-24
+Muv<-24
+0.8
+Muv<-20
+Muv<-22
+Muv<=22
+>
+ Muv<-18
+Muv<-20
+Muv<-18
+Muv<-18
+Muv<-16
+Muv<-14
+Muv<-16
+Muv<-16
+0.6
+Muv<-12
+Muv<-14
+Muv<-14
+Muv<-12
+Muv<-12
+0.4
+0.2
+0.0
+1.0
+LA120
+WDM4
+WDM2
+CDM
+CDM
+CDM
+Muv<-24
+Muv<-24
+Muv<-24
+0.8
+Muv<-22
+Muv<-22
+Muv<=22
+Muv<-20
+Muv<-20
+>
+Muv<-18
+Muv<-18
+Muv<-18
+Muv<-16
+Muv<-16
+Muv<-16
+0.6
+Muv<-14
+Muv<-14
+Muv<-14
+Muv<-12
+Muv<-12
+Muv<-12
+0.4
+0.2
+0.0
+6
+7
+6
+7
+6
+7
+5
+5
+8
+5
+redshiftFigure 4: Adapted from Romanello et al. 2021 [3]. The product ξion fesc required to
+ionize the IGM at z = 6.7, for a set of different cosmologies. The dot size increases
+with Mhm; lighter colours refer to initial condition QHII(z = 10) = 0.2, while darker
+colours are for QHII(z = 9) = 0.0.
+(respectively with intrinsic Msup
+UV < −24, Msup
+UV < −22 and Msup
+UV < −20) with the age
+of universe. In the ΛCDM model, rphot(< −22) passes from 2.8% at z = 8, to 10%
+at z = 6.3. In parallel, for Msup
+UV = −20 we have a raise from 31% at z = 8 to 44% at
+z = 6.3. We can interpret this trend in the light of the hierarchical growth of cosmic
+structures: merging phenomena between galaxies give origin to more massive and
+brighter structures, increasing their overall contribution. However, the role of faint
+galaxies in the Reionization process is still predominant.
+The second issue to be highlighted derives from a comparison between differ-
+ent cosmological scenarios, which reveals that WDM models present a relative ˙Nion
+higher than the CDM ones. Again, the reason resides in the effect of free-streaming,
+which determines a suppression in the number density of the faint-galaxies and so a
+decreasing in their relative contribution for each Msup
+UV . The difference between cos-
+mologies is summarized in the half-mode mass and is not negligible: if we compare
+CDM with LA8 and WDM4, at z = 8 it values ≈ 1–2% , respectively for Msup
+UV = −20
+and −18, but it increases to 8−10% for WDM2 and LA120. Finally, we noted that the
+continue (WDM) and the dashed (CDM) lines in Figure 3 approach each other with
+
+24.25
+24.20
+24.15
+24.10
+24.05
+24.00
+23.95
+WDM2
+WDM3
+WDM4time; for example, at z = 5 the differences between CDM and WDM2-LA120 reduce
+respectively to 4–6%. Again, we can interpret this result by looking at the evolution
+of the UV LFs with z.
+The current analysis is based on intrinsic UV luminosity and it is independent
+from the dust extinction, which is summarized in the escape fraction value: in fact
+fesc appears only as a multiplicative constant, so it simplifies in the ratio between
+˙Nion. Conversely, if we consider other escape fraction dependencies, we could expect
+a more various behaviour.
+3.2.1
+Implications on fesc
+The evolution of the filling fraction with cosmic time depends also on the initial con-
+dition for Equation (9). In particular, we choose two extreme possibilities, which are
+motivated both with model available in literature and with an observational point of
+view.
+The first has QHII(z = 10) = 0.2.
+This assumption agrees with the 68%
+credibility interval modelled on the marginalized distribution of the neutral fraction
+(1 − QHII), from the SFR histories and the Planck constraints on τes, from Robertson
+et al. (2015) [26]. Similarly, it is coherent with the range of QHII allowed for the
+model by Bouwens et al. (2015), where Reionization is complete between z = 5.9 and
+z = 6.5 [27]. As a second possibility, we choose QHII(z = 9) = 0.0, which is preferred
+by the two hydrogen neutral fraction measurements performed by Mason et al. (2019)
+and Hoag et al. (2018)[4, 5]. All the others are intermediate cases.
+For each of the two initial conditions we compute the number density of ion-
+izing photons ˙Nion with different combinations of ξion fesc, exploring the effect of the
+parameters degeneracy on the reheating of IGM. Particles free-streaming has conse-
+quences on galaxy formation, determining a lack of faint-galaxies which alters the UV
+LF, with a general reduction in the UV luminosity density in models with a high Mhm.
+Thus, we obtain a delay in the IGM ionizing process, with respect to CDM.
+In Figure 4, we show log(ξion fesc) in CDM, sterile neutrinos and thermal WDM
+cosmologies. Due to the great uncertainty on fesc, we searched for the ξion fesc values
+that ensure the completion of Reionization at z = 6.7. We note that log(ξion fesc) in-
+creases with Mhm: a larger escape fraction and/or UV photons production efficiency
+are needed to complete the Reionization process in WDM scenarios. However, the
+quantity ξion is better constrained than fesc, so we assume from the literature a fidu-
+cial value of log(ξion/(erg−1Hz)) = 25.2 [16, 27], as expected from a low metallicity
+
+Figure 5: Adapted from Romanello et al. 2021 [3]. Left: evolution of the filling frac-
+tion QHII, for sterile neutrino and thermal WDM models, with log(ξion/(erg−1Hz)) =
+25.2. The two upper panels have initial condition QHII(z = 10) = 0.2 and fesc = 0.05.
+The two lower panels are plotted with QHII(z = 9) = 0.0 and fesc = 0.06. The cyan
+shaded region indicates our fiducial late-Reionization redshift interval, 5.8 < z < 6.7.
+The upward triangle labelled Lyα LF includes results by Konno et al. (2014), Konno
+et al. (2017) and Zheng et al. (2017) [21–23]. Right: electron scattering optical
+depth for different models, compared with measurements from Planck and WMAP [6,
+24, 25].
+
+1.0
+0.10
+CDM
+LA8
+LA9
+0.8
+0.08
+LA10
+CDM
+ LA11
+_A8
+LA120
+A9
+0.6
+0.06
+WMAP Year 9
+A10
+ Planck 2015
+A11
+Planck 2018
+A120
+0.04
+0.4
+Mason19
+Q
+Mason18
+*
+Hoag19
+0.02
+Mesinger15
+0.2
+Schenker14
+Lyα LF
+4
+Ouchi17
+0.00
+0.0
+4
+0
+2 
+10
+6
+8
+10
+8
+7
+4 
+5
+Z
+1.0
+0.10
+CDM
+WDM4
+WDM3
+0.8
+0.08
+WDM2
+WMAP Year 9
+ Planck 2015
+CDM
+0.6
+Planck 2018
+0.06
+XX
+WDM4
+HO
+WDM3
+T
+WDM2
+0.04
+0.4
+O
+Mason19
+★
+Mason18
+Hoag19
+Mesinger15
+0.02
+0.2
+Schenker14
+4
+Lyα LF
+Ouchi17
+0.00
+0.0
+10
+0
+2
+4
+6
+10
+5
+6
+8
+9
+4
+7
+N
+Z
+1.0
+CDM
+0.10
+CDM
+LA8
+LA8
+LA9
+LA9
+0.8
+LA10
+LA10
+0.08
+LA11
+LA11
+LA120
+LA120
+Mason19
+O1
+0.6
+0.06
+WMAP Year 9
++
+Mason18
+IHO
+ Planck 2015
+Hoag19
+人
+Planck 2018
+Mesinger15
+0.04
+0.4
+Schenker14
+Lyα LF
+Ouchi17
+0.02
+0.2 
+0.00
+0.05
+10
+0
+8
+9
+2
+9
+4
+6
+7
+5
+3 
+4
+5
+6
+8
+NE
+Z
+1.0
+CDM
+CDM
+0.10
+WDM4
+WDM4
+WDM3
+ WDM3
+0.8
+WDM2
+0.08
+WDM2
+Mason19
+O
+ WMAP Year 9
+Mason18
+ Planck 2015
+Hoag19
+★
+0.6
+0.06
+ Planck 2018
+Mesinger15
+T
+Schenker14
+Lyα LF
+0.4
+0.04
+Ouchi17
+0.02
+0.2
+0.00
+0.0
+10
+4
+5
+6
+8
+9
+7
+0
+2
+3
+8
+1
+5
+Z
+Zsingle-star population. This value is coherent with the Salpeter IMF assumed in the
+semi-analytic model [26]. We did not investigate the variation of ξion with redshift and
+MUV, which we have considered negligible with respect to changes in escape fraction.
+Similarly, we have neglected the variation with galaxy age. These hypotheses allow
+us to set an upper limit to fesc for each different WDM particle and boundary condi-
+tion. In general, models that start from Q = 0 need a higher fesc value to ionize the
+IGM within the same z range. For this reason they are more inclusive and result in a
+weaker constraint to the admitted escape fraction. If fesc > f sup
+esc , Reionization process
+is completed outside the fiducial redshift interval.
+The evolution of the filling fraction QHII in the various cosmological models
+and for different assumptions is summarised in Fig. 5.
+4
+Conclusions
+In this paper we have first summarised the results presented in Menci et al. 2016, 2017
+[1, 2] putting stringent constraints on DM models with suppressed power spectrs. The
+comparison of the predicted maximum number density of DM halos φ to the observed
+number density φobs provide robust constraints through the simple condition that ob-
+served galaxies cannot outnumber their host DM halos (φ ≥ φobs). Remarkably, these
+constraints are conservative, and independent of the modeling of baryonic physics in
+low-mass galaxies. The mass of WDM thermal relic candidates is constrained to be
+mX ≥ 2.9 keV at 1σ confidence level, and mX ≥ 2.4 keV at 2 − σ level. by have
+been derived by comparing the maximum number density of DM halos φ expected at
+redshift z = 6 to the observed number density φobs of galaxies at the same redshift in
+the HFF.
+We have then summarised the results presented in [3] on how Reionization sce-
+narios are affected by early galaxy formation in WDM cosmologies. We have used the
+semi-analytic model described by Menci et al. (2018) [11], to produce the UV LF in a
+ΛCDM framework. We have tested some ΛWDM cosmologies, in which the contri-
+bution of the faint galaxies is suppressed: in particular, we have focused on five sterile
+neutrino models presented in Lovell et al. (2020) [15]), and three thermal WDM mod-
+els with mX = 2–3–4 keV. In both cases, we have found that a higher Mhm leads to
+a general delay in the Reionization process. In CDM cosmology, merging between
+galaxies determines the rise of the intrinsic MUV < −20 systems relative contribution
+to the ionizing photons budget, from ≈30% to ≈45% between 6.3 < z < 8. In the
+WDM case, the particles free-streaming yields to a shift towards brighter sources and
+
+rphot(MUV < −20) undergoes a further 1–10% growth, depending on cosmology; We
+found that in WDM cosmologies a higher fescξion is required, in order to complete the
+Reionization process at the same redshift.
+The most important limits to our analysis are related to observational uncertain-
+ties. We expect significant advances thanks to improved constraints on the UV LF at
+very high-redshift made possible by forthcoming JWST surveys.
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+Konno, A. et al. SILVERRUSH. IV. Lyα luminosity functions at z = 5.7 and 6.6
+studied with ≈ 1300 Lyα emitters on the 14–21 deg2 sky. Publications of the
+Astronomical Society of Japan 70. S16. issn: 0004-6264. https://doi.org/
+10.1093/pasj/psx131 (Dec. 2017).
+23.
+Zheng, Z.-Y. et al. First Results from the Lyman Alpha Galaxies in the Epoch of
+Reionization (LAGER) Survey: Cosmological Reionization at z ∼ 7. The Astro-
+physical Journal 842, L22. https://doi.org/10.3847/2041-8213/aa794f
+(June 2017).
+24.
+Planck Collaboration. Planck 2015 results - XIII. Cosmological parameters. A&A
+594, A13. https://doi.org/10.1051/0004-6361/201525830 (2016).
+25.
+Hinshaw, G. et al. Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP)
+Observations: Cosmological Parameter Results. The Astrophysical Journal Sup-
+plement Series 208, 19. https://doi.org/10.1088/0067-0049/208/2/19
+(Sept. 2013).
+26.
+Robertson, B. E., Ellis, R. S., Furlanetto, S. R. & Dunlop, J. S. Cosmic Reion-
+ization and Early Star-Forming Galaxies: a Joint Analysis of New Constraints
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+L19. https://doi.org/10.1088/2041-8205/802/2/l19 (Apr. 2015).
+27.
+Bouwens, R. J. et al. Reionization After Planck: the Derived Growth of the Cos-
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+0004-637x/811/2/140 (Sept. 2015).
+

ZNE5T4oBgHgl3EQfdw-l/content/tmp_files/2301.05614v1.pdf.txt

@@ -0,0 +1,1108 @@
+Wrapping dynamics and full uptake conditions for nonspherical active nanoparticles
+Ke Xiao∗
+Wenzhou Institute, University of Chinese Academy of Sciences,
+Wenzhou 325016, People’s Republic of China and
+Department of Physics, College of Physical Science and Technology,
+Xiamen University, Xiamen 361005, People’s Republic of China
+Rui Ma and Chen-Xu Wu†
+Fujian Provincial Key Lab for Soft Functional Materials Research,
+Research Institute for Biomimetics and Soft Matter,
+Department of Physics, College of Physical Science and Technology,
+Xiamen University, Xiamen 361005, People’s Republic of China
+(Dated: 16th January, 2023)
+The cellular uptake of self-propelled nanoparticles (NPs) or viruses, usually nonspherical, by cell
+membrane is crucial in may biological processes. In this study, using Onsager variational principle,
+we obtain a general wrapping equation for nonspherical self-propelled nanoparticles. Two analytical
+critical conditions are theoretically derived, one for the continuous full uptake of prolate particles
+and the other for snapthrough full wrapping of oblate particles. They capture considerably well the
+full uptake critical boundaries in the phase diagrams constructed in terms of active force, aspect
+ratio, adhesion energy density, and membrane tension based on numerical calculations. It is found
+that enhancing activity (active force), reducing effective dynamic viscosity, increasing adhesion
+energy density, and decreasing membrane tension, can significantly improve the wrapping efficiency
+for the self-propelled particles. These results elucidate some of the previous specific investigations
+conclusively and may offer novel possibilities for designing an effective active NP-based vehicle for
+controlled drug delivery.
+The lipid bilayer plasma membrane, a physical barrier
+defining organelles of cells and plenty of their surrounding
+environment, plays a crucial role for a spectrum of bio-
+logical processes [1]. Examples range from the transduc-
+tion of biochemical signal and the intake of nutrients [2]
+to budding and fission [3, 4], and endocytosis of viruses,
+pathogens, and particles [5, 6]. The engulfing of a parti-
+cle or virus (pathogen) by a plasma membrane is a widely
+encountered phenomenon in endocytosis processes, in-
+cluding inter- and intracellular transport [7], delivering
+therapeutic agents enveloped by nanoparticles into tu-
+mor cells [8–10], and virus infection [11–13]. Especially,
+cellular uptake, which involves the interaction between
+cell membrane and nanoparticles (NPs) or viruses, is an
+essential step for a wide range of healthy and disease-
+related processes [12].
+Over the past two decades, considerable efforts using
+experiment, theoretical modeling, and numerical sim-
+ulation have been devoted to characterizing how the
+physical parameters, including the particle size [14–20],
+shape [21–23], elastic properties of the particle [24–28],
+ligand/receptor density [29–31], as well as the mechan-
+ical properties of the membrane [32, 33], affect the in-
+vading behaviors. Though the cellular uptake of passive
+particles via endocytic process has been studied exten-
+sively, little work has been done on the active entry of
+self-propelled bacterial pathogens. To name a few exam-
+ples, it has been found that some cytosolic bacteria such
+as Rickettsia rickettsii are able to produce active force to
+facilitate their mobility by forming actin tails [34], and
+Listeria monocytogenes can generate active force to push
+out a tube-like protuberance from the plasma membrane
+by hijacking the actin polymerization-depolymerization
+apparatus of their host [35–39]. How the active force of
+these self-propelled agents affects the engulfing dynamics
+at the cell membrane remains to be elucidated.
+Recently, many model systems by using lipid vesicles to
+encapsulate natural swimmers (Escherichia coli bacteria,
+Bacillus subtilis bacteria, etc.) or artificial microswim-
+mers (synthetic Janus particles) have been developed to
+study the active membrane behaviors in vitro [40–45].
+Such systems are out-of-equilibrium and hence give rise
+to many intriguing behaviors, such as membrane fluctu-
+ations and large deformations [40, 41], shape transfor-
+mations [42–44], and even deformation of lipid vesicles
+into flagellated swimmers [45]. Therefore, in biology, the
+specific interactions between vesicles and bacteria or arti-
+ficial self-propelled particles plays a key role in designing
+active matter systems [46]. Besides, the wrapping dy-
+namics of particles by cell membranes, which is important
+for understanding the cellular uptake, has been carried
+out experimentally, theoretically, and numerically. In the
+limit of low membrane tension and weak reversible ad-
+hesion, Spanke et.
+al. [47] experimentally investigated
+how the spontaneous wrapping dynamics of micron-sized
+particles by giant unilamellar vesicles changes with the
+adhesion energy.
+By combining computer simulations
+and theoretical analysis, the cellular uptake of active
+particles in the absence of membrane tension was stud-
+ied [48], and the deterministic and stochastic uptake dy-
+arXiv:2301.05614v1  [cond-mat.soft]  13 Jan 2023
+
+2
+namics of passive nanoparticles with different geometries
+were also reported [21].
+Understanding such effect of
+forces and membrane properties (adhesion energy den-
+sity and membrane tension) on the dynamics of cellular
+uptake are critical to designing efficient strategies for po-
+tential biomedical applications, including drug/gene de-
+livery [49–51], cell operation and manipulation [52, 53],
+and bioimaging/sensing [8, 54].
+In addition, in reality
+many pathogens and viruses are nonspherical [55], such
+as egg-shaped malaria parasite [56] and cylindrical Liste-
+ria monocytogenes, indicating the significance of probing
+the wrapping dynamics of nonspherical particles for the
+entry of certain pathogens into cells.
+To better understand the cellular uptake dynamics of
+a nonspherical self-propelled nanoparticle by a plasma
+membrane, a detailed and comprehensive investigation
+of how the wrapping time depends on the active force,
+the particle’s aspect ratio, the viscosity, and the mem-
+brane properties (adhesion energy density and membrane
+tension) is needed. Here in this paper, the wrapping dy-
+namics is studied by employing the Onsager variational
+principle [57, 58] for out-of-equilibrium systems. Our re-
+sults show that active force, low effective dynamic vis-
+cosity, strong adhesion force, and small membrane ten-
+sion play a positive role in the wrapping time, indicating
+that it can be manipulated by changing the activity and
+the aspect ratio of the particles, the viscosity, and the
+properties of the membrane such as adhesion energy den-
+sity and membrane tension. The physical insights gained
+from this work could clarify the mechanism of wrapping
+dynamics of nonspherical active nanoparticles.
+We model a self-propelled invader as an axis-symmetric
+ellipsoid (prolate or oblate spheroid) with its principle ro-
+tational axis orthogonal to the flat membrane, as shown
+in Fig. 1, where a and b denote the semi-axes perpendicu-
+Figure 1. (Color online) Schematic depicting wrapping phases
+of an active particle from nonwrapping to partial wrapping
+and to full uptake.
+lar to and along the principle rotational axis, respectively.
+The geometry of the particle is parameterized by the as-
+pect ratio e = b/a, with e > 1 for a prolate ellipsoid,
+and e < 1 for an oblate one. In practice, it is conve-
+nient to write the area element in terms of polar angle:
+dA = 2πa2 sin θ
+�
+cos2 θ + e2 sin2 θ dθ. To model the up-
+take dynamics of an active particle, we have to write
+down the total free energy of the system. Following the
+classical Canham-Helfrich continuum model [16, 59, 60],
+such an energy is given by
+Etot =
+�
+Amem
+κ
+2 (2H)2dA + σ∆A −
+�
+Aad
+ω dA − fZ,
+(1)
+where the elastic energy of the membrane, the adhesion
+energy between the particle and the membrane, and the
+work done by the active particle are taken into account.
+Here the first term, with κ the bending rigidity and H
+the local mean curvature, denotes the bending energy of
+the membrane
+Ebend =
+� α
+0
+πκe2 sin θ[2 + (e2 − 1) sin2 θ]2
+[1 + (e2 − 1) sin2 θ]3 ×
+�
+cos2 θ + e2 sin2 θ dθ,
+(2)
+an integral over the contact area between the membrane
+and the particle.
+The second term of Eq. (1) is con-
+tributed by the surface tension:
+Eten =
+� α
+0
+2πσa2 sin θ
+�
+1 −
+cos θ
+�
+cos2 θ + e2 sin2 θ
+�
+×
+�
+cos2 θ + e2 sin2 θ dθ.
+(3)
+The third term of Eq. (1) representing the gain in adhe-
+sive energy, characterized by a negative adhesive energy
+−ω per unit area, can be written as an integral over the
+wrapping area. The last term of Eq. (1) arises from work
+done by the active force f acting on the particle, which
+is calculated as Ef = −fae(1 − cosα).
+As the particle is being engulfed by the membrane,
+it has been found that it is the friction force near the
+membrane-particle contact line with its circumference
+given by L(α) = 2πa sin α that largely dissipates the
+energy [47]. In the limit of low Reynolds number, the
+dissipation function reads
+Φ = πηa3 sin α(cos2 θ + e2 sin2 θ) ˙α2,
+(4)
+where η is the effective dynamic viscosity with a typical
+order of 1 Pa·s. In order to obtain the equation governing
+the wrapping dynamics of the active particle, first of all
+we construct a Rayleighian R = ˙Etot + Φ, with ˙Etot the
+time derivative of the free energy of the system given by
+˙Etot =
+�κe2
+a2
+[2 + (e2 − 1) sin2 α]2
+[1 + (e2 − 1) sin2 α]3 +
+2σ
+�
+1 −
+cos α
+�
+cos2 α + e2 sin2 α
+�
+− 2ω
+�
+×
+πa2 sin α
+�
+cos2 α + e2 sin2 α ˙α − fae sin α ˙α,
+(5)
+
+bi
+a
+a3
+Figure 2. (Color online) The engulfing angle α/π and the wrapping velocity v of an active particle with different aspect ratios
+(a) e = 0.75, (b) e = 1, and (c) e = 1.5, where the bending rigidity, the adhesion energy density, the tension modulus, and the
+effective dynamic viscosity are set as 25 kBT, 0.044 mJ/m2, 0.9 × 10−5 N/m, and 1 Pa · s, respectively.
+and Φ the energy dissipation function.
+Minimizing R
+with respect to ˙α following the Onsager variational prin-
+ciple, i.e., ∂R/∂ ˙α = 0, we obtain the cellular uptake
+dynamics equation
+˙α =
+1
+ηa
+�
+cos2 α + e2 sin2 α
+�
+ω +
+fe
+2πa
+�
+cos2 α + e2 sin2 α
+− κe2
+2a2
+[2 + (e2 − 1) sin2 α]2
+[1 + (e2 − 1) sin2 α]3
+− σ
+�
+1 −
+cos α
+�
+cos2 α + e2 sin2 α
+��
+,
+(6)
+for a nonspherical active particle. For spherical particles
+a = b = R, the above equation reduces to Eq. (2) in
+Ref. [21] if f = 0.
+A detailed theoretical analysis of Eq. (6) shows that
+there exist two types of critical conditions for a full up-
+take to occur. One is governed by ˙α
+��
+α=π= 0, or
+ω +
+f
+2πae − 2κ
+a2 e2 − 2σ = 0,
+(7)
+corresponding to a second-order wrapping transition for
+prolate particles and spherical particles. For a spherical
+particle, Eq. (7) reduces to a critical radius Rc = [−f +
+�
+f 2 + 32π2κ(ω − 2σ)]/[4π(ω − 2σ)] when ω ̸= 2σ. The
+other is given by ˙α
+��
+α=π/2+B/(2C)= 0 with B = σ/e, and
+C = 1 − e2
+2e2
+�
+σ − ω − f
+πa
+�
++ κ(3e2 + 7)
+4a2e6
+,
+leading to a critical condition
+ω +
+f
+2πa − κ(1 + e2)2
+2a2e4
+− σ − B2
+4C = 0,
+(8)
+for a first-order wrapping transition from a partial wrap-
+ping to a full uptake.
+If the surface tension satisfies
+(2e2 − 1)σ ≪ κ(3e2 + 7)/(a2e4) − 2(1 − e2)[ω + f/(πa)],
+the above condition reduces to
+ω +
+f
+2πa − κ(1 + e2)2
+2a2e4
+− σ = 0.
+(9)
+To investigate the wrapping dynamics of the active
+particle, we first solve Eq. (6) numerically and analyze
+the evolution of the wrapping angle α/π. For an oblate
+spheroid, the engulfing angle (black curves) as shown in
+Fig. 2(a) exhibits a plateau around α = π/2, indicat-
+ing that there exists a stable partial wrapping state at
+this point.
+This can be reflected by its corresponding
+low wrapping velocity (blue curve) as the oblate parti-
+cle has to cross over an energy barrier before it is fully
+wrapped, corresponding to a first-order wrapping tran-
+sition. Figures 2(b) and 2(c) show the similar wrapping
+behavior with a monotonic increase of wrapping angle to
+a completely wrapped state (α/π = 1) for both the pro-
+late spheroid and the spherical particle. The wrapping
+velocity for the spherical particle shows a significant and
+monotonic drop off after the particle is initially inter-
+nalized at a fast speed, until it slows down and is fully
+wrapped by the fluidic film [see Fig. 2(b)]. While for the
+prolate spheroid, the engulfing speed increases against
+time in the beginning, followed by a dramatic decrease,
+and finally levels off to zero, terminating at a full wrap-
+ping state (α/π = 1) [see Fig. 2(c)].
+To systematically study how the active force and the
+aspect ratio of the particle, the effective dynamic vis-
+cosity, and the membrane properties affect the uptake
+dynamics of the active particle at cell membrane, we nu-
+merically calculate the wrapping time by changing these
+parameter values. Figure 3 shows the wrapping time as
+a function of particle size for different active forces and
+dynamic viscosities. For fixed f and η, the curves of up-
+take time share a similar feature that with the increase of
+the particle size, tup drops very quickly before it bounces
+back mildly.
+Meanwhile, the full wrapping regimes in
+the parameter space are also apparently broadened with
+the increase of the active force and the effective dynamic
+viscosity. Figures 3(a) and 3(b) compare the resulting
+wrapping times of the oblate and the spherical particle
+for different values of active force. For the oblate particle,
+we adjust the length of the symmetry axis a by keeping
+
+1.0
+1.0
+(b)
+1.0
+(a)
+0.20
+0.30
+c
+0.16
+α/元
+0.8
+0.8
+0.8
+0.25
+V
+α/元
+0.15
+/ms)
+(rad/ms)
+α/元
+0.12
+(rad/ms)
+0.6 
+0.20
+0.6
+0.6
+(rad/
+0.15
+e=b/a=1
+0.10
+e=b/a=1.5
+0.08
+0.4F
+0.4
+0.4
+e=b/a=0.75
+a=90 nm
+V (
+a=134 nm
+0.10
+a=108 nm
+0.05
+0.2
+0.2
+0.2
+0.04
+0.05
+0.0
+0.00
+0.0
+0.00
+0.0
+0.00
+0
+20
+40
+60
+80
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+100
+t (ms)
+t (ms)
+t (ms)4
+100
+1000
+20
+40
+60
+80
+100
+1000
+20
+40
+60
+80
+200
+400
+600
+800 1000
+20
+40
+60
+80
+100
+120
+100
+1000
+20
+40
+60
+80
+100
+120
+tup (ms)
+a (nm)
+ f=0 pN  
+ f=1 pN  
+ f=3 pN  
+ f=5 pN  
+ f=8 pN  
+ f=10 pN
+e=b/a=0.75
+tup (ms)
+a (nm)
+ f=0 pN  
+ f=1 pN  
+ f=3 pN  
+ f=5 pN  
+ f=8 pN  
+ f=10 pN
+e=b/a=1
+tup (ms)
+a (nm)
+ � =0.5 Pa·s  
+ � =0.75 Pa·s
+ � =1 Pa·s     
+ � =1.25 Pa·s
+ � =1.5 Pa·s  
+e=b/a=0.75
+(d)
+(c)
+(b)
+tup (ms)
+a (nm)
+ � =0.5 Pa⋅s  
+ � =0.75 Pa⋅s
+ � =1 Pa⋅s     
+ � =1.25 Pa⋅s
+ � =1.5 Pa⋅s  
+e=b/a=1
+(a)
+Figure 3. (Color online) Wrapping times as functions of parti-
+cle size for (a) oblate particle, and (b) spherical particle with
+different active forces.
+Uptake time as a function of parti-
+cle size for (c) oblate particle, and (d) spherical particle with
+different effective dynamic viscosities. Parameter values (if
+not varied) are fixed at κ = 25 kBT, ω = 0.044 mJ/m2,
+σ = 0.9 × 10−5 N/m, and f = 0 pN, η = 1 Pa · s.
+the aspect ratio unchanged. Both ellipsoid and spher-
+ical particles share a similar behavior that there exists
+a critical particle size beyond which a full uptake oc-
+curs. Interestingly, the calculation results demonstrate
+that particles with larger active forces are uptaken faster
+than those with smaller forces, indicating that particle
+activity facilitates the uptake process. This conclusion is
+in line with the simulation predictions in Ref. [48], where
+the authors claimed that the uptake efficiency can be en-
+hanced with the increase of Péclet number quantifying
+the strength of active force.
+In order to probe how effectively the dynamic viscosity
+affect the wrapping dynamics, we plot the wrapping time
+for an oblate particle and a spherical particle against par-
+ticle size with different effective dynamic viscosities, as
+shown in Figs. 3(c) and 3(d), respectively. It is found
+that decreasing the effective dynamic viscosity η clearly
+decreases the threshold particle size for the occurrence of
+the complete uptake. This is due to the fact that large
+viscosity increases the friction, which correspondingly in-
+hibits the uptake process and hence leads to longer up-
+take time. Therefore, to achieve faster (slower) wrapping
+process, enhancing (weakening) the particle activity and
+reducing (raising) the effective dynamic viscosity might
+be an effective option.
+Pathogens
+and
+viruses
+come
+in
+many
+different
+shapes [55], but the most frequent occurrence is ellip-
+soid. Therefore, here we focus our discussion on ellipsoid
+with its shape characterized by aspect ratio.
+Further-
+more, to gain more insight into the interrelated influence
+of the active force f and the particle aspect ratio e on
+the wrapping time, we construct a phase diagram for the
+wrapping time in the f − e space under the condition
+of fixed particle volume, as shown in Fig. 4.
+On one
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+0
+2
+4
+6
+8
+10
+ Eq.(7)
+ Eq.(9)
+No uptake
+No uptake
+Full uptake
+f (pN)
+e
+10
+15
+20
+25
+30
+35
+40
+45
+tup (ms)
+Figure 4. (Color online) A two-dimensional phase diagram on
+the (f −e) plane characterizes the interrelated effects of active
+force and particle aspect ratio on the uptake time of wrap-
+ping process, where the bending rigidity, the adhesion energy
+density, the membrane tension, and the effective dynamic vis-
+cosity are set as 25 kBT, 0.044 mJ/m2, 0.9 × 10−5 N/m, and
+1 Pa · s, respectively. The volume of the particle is fixed at
+V = 4πea3/3 = 4πR3
+0/3 with R0 = 100 nm.
+hand, a comparison of the wrapping time between the
+prolate ellipsoidal particle (e > 1) and the spherical par-
+ticle (e = 1) shows that an prolate ellipsoidal particle is
+taken up slower than a spherical one. However, upon de-
+creasing the aspect ratio, the uptake time for the active
+oblate ellipsoidal particles (e < 1) displays a nonmono-
+tonic feature by decreasing to a minimum value first and
+then bouncing back gradually, in stark contrast to the
+monotonic dependence for active prolate particles with
+an aspect ratio of e > 1. On the other hand, Figure 4
+also verifies that enhancing the activity gives rise to the
+wrapping efficiency (with a decrease of tup). Therefore,
+the wrapping efficiency can be regulated by tuning the
+aspect ratio and the activity of particles. To determine
+the boundaries separating the full uptake and no uptake
+regimes, we plotted two boundary curves based on Eq. (7)
+[see the dash curve in Fig. 4] and Eq. (9) [see the dot curve
+in Fig. 4], respectively, which match the numerical cal-
+culations very well. The consistence once again indicates
+that there exist two ways for a nonspherical particle to
+reach full wrapping, i.e. a continuous uptake for prolate
+particles and a snapthrough uptake for oblate particles.
+Such a conclusion is in agreement with the simulations
+done by Khosravanizadeh et. al [61], who demonstrated
+that the oblate ellipsoidal particles exhibit discontinuous
+wrapping phase transition from partial wrapping to full
+wrapping during the uptake process, while the prolate el-
+lipsoidal particles show a continuous wrapping transition
+
+5
+behavior.
+In order to gain more insights into the effects of the
+membrane properties on the wrapping time, we explore
+the wrapping dynamics of an active oblate particle, a pro-
+late ellipsoidal one, and a spherical one with different ad-
+hesion energy densities and membrane tensions, as shown
+in Fig. 5. The colored contour maps of tup on the f − ω
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0
+2
+4
+6
+8
+10
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0
+2
+4
+6
+8
+10
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0
+2
+4
+6
+8
+10
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0
+2
+4
+6
+8
+10
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0
+2
+4
+6
+8
+10
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+0
+2
+4
+6
+8
+10
+Full uptake
+ Eq.(9)
+e=0.75
+No uptake
+f (pN)
+�  (mJ·m
+-2)
+3
+11
+19
+27
+35
+43
+tup (ms)
+ Eq.(8)
+e=0.75
+No uptake
+Full uptake
+f (pN)
+�  (mN·m
+-1)
+8
+18
+28
+38
+48
+58
+68
+78
+tup (ms)
+ Eq.(7)
+e=1
+No uptake
+Full uptake
+f (pN)
+�  (mJ·m
+-2)
+tup (ms)
+2
+12
+22
+32
+42
+52
+(f)
+(e)
+(d)
+(c)
+(b)
+ Eq.(7)
+e=1
+No uptake
+Full uptake
+f (pN)
+�  (mN·m
+-1)
+7
+17
+27
+37
+47
+57
+67
+77
+tup (ms)
+(a)
+Full uptake
+ Eq.(7)
+e=1.5
+No uptake
+f (pN)
+�  (mJ·m
+-2)
+4
+6
+8
+10
+12
+14
+16
+18
+tup (ms)
+ Eq.(7)
+e=1.5
+No uptake
+Full uptake
+f (pN)
+�  (mN·m
+-1)
+11
+14
+17
+20
+23
+26
+29
+tup (ms)
+Figure 5. (Color online) Wrapping time tup as a function of f
+and ω for (a) oblate ellipsoidal, (b) spherical, and (c) prolate
+ellipsoidal particles at equal particle volume. Uptake times
+as a function of active force f and membrane tension σ for
+(d) oblate ellipsoidal, (e) spherical, and (f) prolate ellipsoidal
+particles at equal particle volume. Parameter values (if not
+varied) are fixed at κ = 25 kBT, ω = 0.044 mJ/m2, σ =
+0.9 × 10−5 N/m, and η = 1 Pa · s. The volume of the particle
+is fixed at V = 4πea3/3 = 4πR3
+0/3 with R0 = 100 nm.
+[see Figs. 5(a), 5(b), and 5(c)] and f − σ [see Figs. 5(d),
+5(e), and 5(f)] planes show that given an aspect ratio and
+a fixed active force, a higher (lower) wrapping efficiency
+can be achieved under a stronger (weaker) adhesion force
+or a lower (higher) membrane tension, i.e. higher adhe-
+sion and looser membrane leads to faster wrapping. Such
+a conclusion can be supported by experimental observa-
+tions [47] reported recently that higher adhesion leads to
+faster wrapping, and simulation results in Ref. [21] that
+the uptake time is strongly decreased for higher adhesion
+and looser membrane. Here it is important to note that
+the boundaries of the full uptake (read lines) can be very
+well captured by the critical conditions Eqs (7), (8) and
+(9) theoretically derived using our model.
+Finally we discuss the influence of the membrane prop-
+erties and the aspect ratio of particle to the wrapping
+process. Figures 6(a) and 6(b) show the two-dimensional
+phase diagrams of wrapping time in the parametric
+planes of ω − e and σ − e.
+What is striking is that,
+for the oblate ellipsoidal particles, the uptake time non-
+monotonically depends on e within the scale of ω or σ,
+in comparison to the monotonic dependence of the pro-
+late ellipsoidal particles. Tuning the particle’s aspect ra-
+tio to an optimal value enables one to obtain a minimal
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+0.04
+0.06
+0.08
+0.10
+0.6
+0.8
+1.0
+1.2
+1.4
+0.000
+0.003
+0.006
+0.009
+0.012
+No uptake
+No uptake
+Full uptake
+ Eq.(7)
+ Eq.(9)
+�  (mJ·m
+-2)
+e
+4
+10
+16
+22
+28
+34
+40
+tup (ms)
+ Eq.(7)
+ Eq.(8)
+ Eq.(9)
+(b)
+Full uptake
+No uptake
+No uptake
+�  (mN·m
+-1)
+e
+12
+17
+22
+27
+32
+37
+42
+47
+tup (ms)
+(a)
+Figure 6.
+(Color online) Two-dimensional wrapping time
+phase diagrams in the projection planes of (a) ω and e and
+(b) σ and e, where the bending rigidity, the adhesion energy
+density, the membrane tension, the active force and the ef-
+fective dynamic viscosity (if not varied) are set as 25 kBT,
+0.044 mJ/m2, 0.9 × 10−5 N/m, f = 0 pN and 1 Pa · s, respec-
+tively. The volume of the particle is fixed at V = 4πea3/3 =
+4πR3
+0/3 with R0 = 100 nm.
+wrapping time. In addition, Figure 6 also confirms that
+the uptake efficiency can be improved (abated) by in-
+creasing (decreasing) the adhesion energy density, and
+reducing (raising) the membrane tension. The dash and
+solid curves based analytical results Eq. (7) and Eq. (8)
+coincide with the boundaries obtained from numerical
+calculations. It has been shown, by providing a stochas-
+tic model to study the kinetics of particle wrapping by a
+vesicle, that increasing the attraction strength between
+the particle and vesicle causes the improvement of uptake
+rate [62]. This is again a result in agreement with our
+present conclusions.
+Consequently, we argue that the wrapping time de-
+creases with the increase of the active force; the higher
+the viscosity, the longer the wrapping time; strong ad-
+hesion and low tension improve the wrapping efficiency.
+The reason stems from that the wrapping process is
+largely controlled by the competition among the three
+types of energy: the elastic energy (consisting of bending
+energy and tension energy), the adhesive energy, and the
+work done by the active force. Reducing wrapping time
+can be realized based on the condition that the adhesion
+energy and the work done by the active force is sufficient
+to overcome the energy barrier, namely, the sum of the
+elastic energy and the viscous dissipation. In the presence
+of active force, the work done by the active force reduces
+the free energy and as a result change the uptake force.
+The positive correlation between active force and uptake
+force gives rise to the positive dependence of wrapping
+time on the active force. According to Eq. (6), the ad-
+hesion energy driving the wrapping process is positively
+
+6
+proportional to the adhesion energy density and nega-
+tively proportional to the membrane tension. As a re-
+sult, increasing the adhesion energy density corresponds
+to increasing the driving force for wrapping, decreasing
+the membrane tension decreases the energy penalty for
+uptake, indicating a decreases of the wrapping time. As
+for the effective dynamic viscosity, an increase of it means
+that the induced energy dissipation requires more adhe-
+sion energy and work done by the active force to com-
+pensate, which leads to an increase of wrapping time.
+In summary, we propose a theoretical model to investi-
+gate the wrapping dynamics of a nonspherical active par-
+ticle by a lipid plasma membrane, by taking into account
+the influence of the active force, the particle shape, the
+effective dynamic viscosity, and the membrane properties
+(including the adhesion energy density and the mem-
+brane tension).
+The wrapping equation for the active
+particle, which quantitatively couples the elastic defor-
+mation of the membrane, the work done by the active
+particle, and the energy dissipation, is derived by using
+Onsager variational principle.
+Two critical conditions,
+one for the continuous full uptake of prolate particles and
+the other for snapthrough full wrapping of oblate parti-
+cles, are obtained theoretically. Our results reveal that
+enhancing activity (active force), reducing effective dy-
+namic viscosity, increasing adhesion energy density, and
+decreasing membrane tension, can significantly improve
+the wrapping efficiency for the self-propelled particles.
+Intriguingly, with an increase of aspect ratio, the wrap-
+ping time for oblate ellipsoidal particles exhibits a non-
+monotonic dependence, in stark contrast to the mono-
+tonic dependence for prolate ellipsoidal particles. There-
+fore, the wrapping time can be manipulated by changing
+the activity and the aspect ratio of the particles, the ef-
+fective dynamic viscosity, and the properties of the mem-
+brane, such as adhesion energy density and membrane
+tension. All these findings may not only shed light on
+the influence of the activity and the aspect ratio of par-
+ticles, the viscosity, and the properties of membrane on
+the dynamic behaviors of wrapping process, but also pro-
+vide guidelines to improve the efficiency of active particle-
+based drug delivery systems.
+ACKNOWLEDGMENTS
+We acknowledge financial support from National
+Natural Science Foundation of China under Grant
+Nos.12147142, 11974292, 12174323, and 1200040838, and
+111 project B16029.
+∗ [email protected]
+† [email protected]
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+

aNA0T4oBgHgl3EQfF__0/content/tmp_files/2301.02042v1.pdf.txt

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+arXiv:2301.02042v1  [cs.IT]  5 Jan 2023
+1
+Improved Gilbert-Varshamov bounds for hopping
+cyclic codes and optical orthogonal codes
+Chenyang Zhang, Chong Shangguan, and Gennian Ge
+Abstract
+Hopping cyclic codes (HCCs) are (non-linear) cyclic codes with the additional property that the n cyclic shifts of
+every given codeword are all distinct, where n is the code length. Constant weight binary hopping cyclic codes are also
+known as optical orthogonal codes (OOCs). HCCs and OOCs have various practical applications and have been studied
+extensively over the years.
+The main concern of this paper is to present improved Gilbert-Varshamov type lower bounds for these codes, when
+the minimum distance is bounded below by a linear factor of the code length. For HCCs, we improve the previously best
+known lower bound of Niu, Xing, and Yuan by a linear factor of the code length. For OOCs, we improve the previously
+best known lower bound of Chung, Salehi, and Wei, and Yang and Fuja by a quadratic factor of the code length. As
+by-products, we also provide improved lower bounds for frequency hopping sequences sets and error-correcting weakly
+mutually uncorrelated codes. Our proofs are based on tools from probability theory and graph theory, in particular the
+McDiarmid’s inequality on the concentration of Lipschitz functions and the independence number of locally sparse graphs.
+Keywords. Gilbert-Varshamov bound; non-linear cyclic codes; hopping cyclic codes; optical orthogonal codes;
+frequency hopping sequences sets; error-correcting weakly mutually uncorrelated codes
+I. INTRODUCTION
+Given integers q, n and d, estimating the maximum size of q-ary codes of length n and minimum distance d is a
+fundamental problem in coding theory. The Gilbert–Varshamov bound (GV bound for short) is a classic lower bound
+on the size of codes. For fixed q, n → ∞, and d bounded below by a linear factor of n, improving upon the GV bound
+substantially is a well-known difficult task. In this paper we will present improved GV-type bounds for several classes
+of non-linear cyclic codes.
+To move forward let us begin with some needed definitions. For a positive integer q, let [q] = {0, 1, . . . , q − 1}.
+A vector x ∈ [q]n is denoted by x = (x1, . . . , xn). For two vectors x, y ∈ [q]n, the Hamming distance d(x, y) is the
+number of coordinates where they differ, namely, d(x, y) = |{1 ≤ i ≤ n : xi ̸= yi}|. A code of length n and alphabet
+size q is a subset of [q]n, whose elements are called codewords. The minimum distance d(C) of a code C ⊆ [q]n is
+defined to be d(C) := min{d(x, y) : x, y ∈ C, x ̸= y}. A q-ary code with length n, size M, and minimum distance d
+is denoted as an (n, M, d)q-code.
+Next, let us briefly recall some known GV-type bounds.
+A. GV-type bounds
+a) Generic codes.: For an integer 1 ≤ t ≤ n and a vector x ∈ [q]n, let B(x, t) := {y ∈ [q]n : d(x, y) ≤ t}
+denote the Hamming ball of radius t centered at x. Given a radius t, it can be easily seen that for every x ∈ [q]n, the
+volume of the Hamming ball B(x, t) is independent of x, which satisfies that
+|B(x, t)| =
+t
+�
+i=0
+�n
+i
+�
+(q − 1)i =: V olq(n, t).
+This project is supported by the National Key Research and Development Program of China under Grant Nos. 2020YFA0712100 and
+2018YFA0704703, the National Natural Science Foundation of China under Grant Nos. 11971325, 12231014, and 12101364, the Natural Science
+Foundation of Shandong Province under Grant No. ZR2021QA005, and the Beijing Scholars Program.
+C. Zhang is with the Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao 266237, China (e-mail:
+[email protected]).
+C. Shangguan is with the Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao 266237, China, and also
+with the Frontiers Science Center for Nonlinear Expectations, Ministry of Education, Qingdao 266237, China (e-mail: [email protected]).
+G. Ge is with the School of Mathematical Sciences, Capital Normal University, Beijing 100048, China (e-mail: [email protected]).
+
+2
+The GV bound, proved independently by Gilbert [7] and Varshamov [22], states that there exist (n, M, d)q-codes
+with
+M ≥
+qn
+V olq(n, d − 1).
+(1)
+For binary codes and d/n ≤ 0.499, Jiang and Vardy [11] improved the lower bound in (1) by a linear factor of n. Based
+on their work, Vu and Wu [23] showed that for every q ≥ 2 and τ ≤ d/n ≤ 1 − 1/q − ǫ, where τ, ǫ are some absolute
+reals, there exist (n, M, d)q-codes with
+M = Ω
+�
+nqn
+V olq(n, d − 1)
+�
+.
+(2)
+b) Non-linear cyclic codes.: For every x = (x1, . . . , xn) ∈ [q]n and 0 ≤ i ≤ n − 1, we call the vector
+πi(x) := (xi+1, xi+2, . . . , xi+n)1 the i-th cyclic shift of x. Let C(x) := {πi(x) : 0 ≤ i ≤ n − 1} denote the set
+consisting of all cyclic shifts of x. Note that C(x) is viewed as a multi-set. A code C ⊆ [q]n is said to be a cyclic code
+if for every x ∈ C, C(x) ⊆ C. In the literature, there are a number of works devoting to the constructions of linear
+cyclic codes with minimum distance ranging from a constant to a sublinear function of n, say, O( n
+ln n) (see, e.g. [19,
+Chapter 8]). However, when minimum distance is bounded below by a linear factor of n, understanding whether there
+exist asymptotically good linear cyclic codes is a longstanding open question in coding theory (see [6], [8]).
+For non-linear cyclic codes, the question above has been solved only recently. Haviv, Langberg, Schwartz, and
+Yaakobi [8] proved that for every prime code length, there exist binary cyclic codes asymptotically attaining the GV
+bound. Later, Niu, Xing, and Yuan [16] showed that for every q and n there exist (n, M, d)q-cyclic codes with
+M ≥ qn(1 − n2e− ǫ2(√n−2)
+2
+)
+V olq(n, d − 1) − 1 .
+(3)
+c) Constant weight codes.: For a vector x ∈ {0, 1}n, the weight wt(x) is defined to be the number of its
+non-zero coordinates, namely, wt(x) = |{1 ≤ i ≤ n : xi ̸= 0}|. A code is said to be a constant weight code, if all
+codewords of it have the same weight. An (n, M, d)2-code with constant weight w is denoted as an (n, M, d; w)-code,
+where we omitted the q = 2 in the subscript. For x ∈ {0, 1}n, wt(x) = w, and 0 ≤ t ≤ 2w, let B(x, t; w) := {y ∈
+{0, 1}n : d(x, y) ≤ t, wt(y) = w} denote the constant weight Hamming ball of radius t centered at x. Given t and w,
+it is known that for every x ∈ {0, 1}n with wt(x) = w, the volume of B(x, t; w) is independent of x, which satisfies
+that
+|B(x, t; w)| =
+⌊t/2⌋
+�
+i=0
+�w
+i
+��n − w
+i
+�
+=: V ol(n, t; w).
+Levenshtein [14] proved a GV-type bound for constant weight codes, showing that there exist (n, M, d; w)-codes
+with
+M ≥
+�n
+w
+�
+V ol(n, d − 1; w).
+(4)
+Similarly to the improvement of (2) upon (1), Kim, Liu, and Tran [12] improved (4) by a linear factor of n, showing
+that for τ ≤ d/n ≤ (1 − ǫ)p(1 − p), where τ, ǫ, p are some absolute reals, there exist (n, M, d; pn)-codes with
+M = Ω
+�
+n
+� n
+pn
+�
+V ol(n, d − 1; pn)
+�
+.
+(5)
+d) Constant weight non-linear cyclic codes.: In the literature, there is also a GV-type lower bound for constant
+weight non-linear cyclic codes. More precisely, Chung, Salehi, and Wei [1], [3] and Yang and Fuja [24] implicitly showed
+that there exist (n, M, d; w)-cyclic codes with
+M ≥
+�n
+w
+�
+− f(n, w, d)
+n · V ol(n, d − 1; w),
+(6)
+where f(n, w, d) is some function of n, w, d (see [1, Theorem 2] and [24, Theorem 3] for details).
+1Throughout this paper, the addition in the subscripts of x is calculated modulo n.
+
+3
+Given the discussion above, it is natural to ask whether one can improve (3) and (6) by a linear or polynomial
+factor of n, similarly to the improvements of (2) and (5) made upon (1) and (4). This is one of the motivating questions
+of this paper. Note that the method which proves (2) and (5) cannot be applied directly to improve (3) and (6), as one
+has to take into account the property of cyclic codes. Based on some probabilistic and graph theoretic tools and some
+ideas from [11], [12], [16], [23], we will answer the above question in a strong sense in Theorems II.1 and II.2 below.
+The main concern of this paper is to present improved GV-type bounds for two special classes of non-linear cyclic
+codes, namely, hopping cyclic codes and its constant weight counterpart, optical orthogonal codes. As by-products,
+we also provide improved lower bounds for frequency hopping sequences sets and error-correcting weakly mutually
+uncorrelated codes as well.
+Next, we will introduce these codes and our results in the detail.
+II. MAIN RESULTS
+A. Hopping cyclic codes
+Hopping cyclic codes are (non-linear) cyclic codes with the additional property that the n cyclic shifts of every
+given codeword are all distinct. Formally speaking, a code C ⊆ [q]n is called a hopping cyclic code (HCC for short) if
+for every x ∈ C, C(x) ⊆ C, and moreover C(x) consists of n distinct elements. We will call a hopping cyclic code
+C ⊆ [q]n an (n, M, d)q-HCC, if it is itself an (n, M, d)q-code. HCCs were originally designed to construct frequency
+hopping sequences sets (see [4], [16] and Section III-A below), but have found their own interest as an intriguing class
+of codes. The reader is referred to [16] for more background on HCCs.
+For any absolute constant ǫ > 0 and d < (n − 2√n)(1 − 1/q − ǫ), Niu, Xing, and Yuan (see Theorem III.5 in [16])
+showed that there exist (n, M, d)q-HCCs with M bounded below by (3).
+Our first main result improves the above GV-type bound for HCCs (and hence for non-linear cyclic codes) by
+asymptotically a linear factor of n.
+Theorem II.1. For positive integers n, q, d and absolute reals τ, ǫ ∈ (0, 1 − 1/q) satisfying τ ≤ d/n ≤ 1 − 1/q − ǫ,
+there exist (n, M, d)q-HCCs with
+M ≥
+cnqn
+V olq(n, d − 1),
+where c is a positive real depending only on τ, ǫ, q.
+B. Optical orthogonal codes
+Constant weight binary hopping cyclic codes are also known as optical orthogonal codes (OOCs for short). OOCs
+enable a large number of asynchronous users to transmit information efficiently and reliably. They have been widely
+used in various practical scenarios like code-division multiple-access systems and spread spectrum communication. A
+large number of existing papers were devoted to the constructions of OOCs, see for example [1], [2], [9], [13], [17],
+[24].
+We will denote an (n, M, d)2-HCC with constant weight w by an (n, M, d; w)-OOC, where we omitted the q = 2
+in the subscript. Chung, Salehi, and Wei [1], [3] and Yang and Fuja [24] showed that there exist (n, M, d; w)-OOCs
+with M bounded below by (6).
+Our second main result improves the above GV-type bound for OOCs (and hence for constant weight non-linear
+cyclic codes) by asymptotically a quadratic factor of n.
+Theorem II.2. For positive integers n, q, d and absolute reals p ∈ (0, 1), ǫ ∈ (0, 1/10), τ ∈ (0, p(1 − p)) satisfying
+pn ∈ Z, τ ≤ d/n ≤ (1 − ǫ)p(1 − p), there exist (n, M, d; pn)-OOCs with
+M ≥
+cn
+� n
+pn
+�
+V ol(n, d − 1; pn),
+where c is a positive real depending only on p, ǫ, τ.
+
+4
+C. Sketch of the proofs and comparison with related works
+Since the work of Jiang and Vardy [11], it is now well-known that one can lower-bound the size of a code via
+the lower bound of the independence number of a locally sparse graph defined approximately. Loosely speaking, let
+G be a graph with vertex set [q]n, where two vertices (vectors) x, y ∈ [q]n are connected by an edge if and only if
+d(x, y) ≤ d − 1. Then, an independent set I of G corresponds to an (n, |I|, d)q-code. In fact, there is a one-to-one
+correspondence between independent sets of G and codes in [q]n with minimum distance at least d. Therefore, given
+the minimum distance d, finding a large code in [q]n is equivalent to finding a large independent set in G.
+Since for every x ∈ [q]n, there are exactly V olq(n, d − 1) − 1 vectors y ∈ [q]n \ {x} such that d(x, y) ≤ d − 1,
+every vertex of G is connected to exactly V olq(n, d − 1) − 1 vertices in [q]n. So, one can construct an independent set
+in G of size
+qn
+V olq(n,d−1), or equivalently, an (n,
+qn
+V olq(n,d−1), d)q-code, by greedily picking vertices and throwing their
+neighbors. This gives the GV bound (1).
+Jiang and Vardy [11] showed that the graph G defined above is in fact “locally sparse” (this will be clarified later
+in Section IV-A). Therefore, there is a more economic way than the greedy algorithm to find a large independent set in
+G (see Lemma IV.1 below). This leads to their improvement on the GV bound of generic codes (1) by a linear factor of
+n. Kim, Liu, and Tran [12] used a similar idea in their proof of (5), which improves the GV bound of constant weight
+codes (4) by a linear factor of n.
+The high level idea in the proofs of Theorems II.1 and II.2 will follow the above framework. However, as the codes
+considered in this paper need not only to have large minimum distance but also to be hopping cyclic, the corresponding
+graphs are not as neat as the graph G described above.
+The proofs of Theorems II.1 and II.2 can be summarized as follows:
+Step 1. define an appropriate graph GHCC (resp. GOOC) whose independent sets correspond to HCCs (resp. OOCs);
+Step 2. show that GHCC (resp. GOOC) has sufficiently many vertices;
+Step 3. show that GHCC (resp. GOOC) is in fact locally sparse;
+Step 4. use known lower bound on the independence number of locally sparse graphs to show that GHCC (resp. GOOC)
+has sufficiently large independent sets, and hence there exist sufficiently large HCCs (resp. OOCs).
+Next, we would like to compare our work with some previous papers [11], [12], [16], [23]. On one hand, in
+[11], [12], [23], G and its constant weight counterpart were used to prove improved GV-type bounds for generic codes
+and constant weight codes, respectively. GHCC (resp. GOOC) defined in this paper is quite different from G (resp. its
+constant weight counterpart). To be more precise, for x ∈ [q]n let
+d(x) = min{dH(πi(x), πj(x)) : 0 ≤ i < j ≤ n − 1} = min{dH(x, πi(x)) : 1 ≤ i ≤ n − 1}
+(7)
+denote the minimum distance d(C(x)). By definition, C ⊆ [q]n is an HCC only if for every x ∈ C, d(x) ≥ 1 and
+moreover, C is an (n, M, d)q-HCC only if for every x ∈ C, d(x) ≥ d. Therefore, compared with G whose vertex set
+is [q]n, in the vertex set of GHCC we have to rule out all vectors x with d(x) < d. In fact, the vertex set of GHCC is
+not vectors in [q]n but some well-defined subsets of vectors in [q]n. Hence, to execute Steps 1 and 2, we have to show
+that for many vectors x ∈ [q]n, d(x) is quite large. Similar discussion works for GOOC. The details can be found in
+Sections V and VI below.
+On the other hand, to show that GHCC and GOOC are locally sparse, we will use some well-developed tools that
+were continually improved in [11], [12], [23]. In particular, we will use two upper bounds on the intersection volume
+of Hamming balls, which were proved in [12] to show that G and its constant weight counterpart are locally sparse.
+Lastly, let us compare our work with [16]. Although [16] is not graph theoretic, an important step in their proof of
+(3) was also to show that for qn − o(qn) vectors x ∈ [q]n, d(x) is quite large (see [16, Lemma III.4]). To do so, they
+used some standard concentration inequality for martingales. In this paper, we used the McDiarmid’s inequality instead,
+which makes our proof easier and gives a better control (upper bound) on the lower order term o(qn) (see Remark V.2
+below).
+A summary of previous mentioned GV-type bounds is presented in Table I.
+The rest of this paper is organized as follows. In Section III we will mention two applications of Theorem II.1,
+which gives new lower bounds for frequency hopping sequences sets and error-correcting weakly mutually uncorrelated
+codes. In Section IV we will collect the tools that are used in Steps 1-4. In Section IV-A we will formally define locally
+sparse graphs (this will be used in Step 1) and state a lower bound on its independence number (this will be used in Step
+4, see Lemma IV.1 below). In Section IV-B we will introduce the McDiarmid’s inequality (this will be used in Step 2,
+see Lemma IV.2 below). In Section IV-C we will state two upper bounds on the intersection volume of Hamming balls
+
+5
+TABLE I: A summary of GV-type bounds
+GV-type lower bounds of M
+Improved GV-type lower bounds of M
+Generic (n, M, d)q-codes
+qn
+V olq(n,d−1) (1) ( see [7], [22])
+Ω
+�
+nqn
+V olq(n,d−1)
+�
+(2) (see [11], [23])
+Non-linear (n, M, d)q-cyclic codes
+(1−o(1))qn
+V olq(n,d−1) (3) (see [9], [16])
+cnqn
+V olq(n,d−1) Theorem II.1 (this paper)
+Constant weight (n, M, d; pn)-codes
+� n
+pn
+�
+V ol(n,d−1;pn) (4) ( see [14])
+Ω
+�
+n
+� n
+pn
+�
+V ol(n,d−1;pn)
+�
+(5) (see [12])
+Constant weight non-linear (n, M, d; pn)-cyclic codes
+�n
+w
+�
+−f(n,d,w)
+n·V ol(n,d−1;w) (6) (see [1], [3], [24])
+cn
+� n
+pn
+�
+V ol(n,d−1;pn) Theorem II.2 (this paper)
+(this will be used in Step 3, see Lemmas IV.3 and IV.4 below). The proofs of Theorems II.1 and II.2 will be presented
+in Sections V and VI respectively. We will conclude this paper in Section VII.
+III. APPLICATIONS OF THE MAIN RESULTS
+A. Frequency hopping sequences
+In this subsection, we will present an application of Theorem II.1 to frequency hopping sequences. We will need
+the following easy lemma.
+Lemma III.1.
+(i) For every x, y ∈ [q]n, x ∈ C(y) if and only if y ∈ C(x); moreover, C(x) = C(y) if and only if
+x ∈ C(y).
+(ii) Let C be an (n, M, d)q-HCC. Then n | M, and C can be partitioned into a pairwise disjoint union of M/n C(x)’s,
+where x ∈ C.
+Proof. Note that (i) follows straightforwardly from the definition of C(x). To prove (ii), it is not hard to check that the
+relation ∼ on the set C defined by x ∼ y if C(x) = C(y) is an equivalence relation. By (i) and the definition of an
+HCC, each equivalent class equals to C(x) for some x ∈ C, and hence consists of n distinct vectors. Therefore, (ii)
+follows from the fact that the equivalent classes given by ∼ form a partition of C.
+Let C be an (n, M, d)q-HCC. Given the lemma above, consider the partition of C formed by the equivalent classes
+given by ∼. If we choose a representative element from each of these equivalent classes, then these representatives
+form a set of frequency hopping sequences (FHSs for short). An FHS set obtained by an (n, M, d)q-HCC in the above
+manner is denoted as an (n, M/n, n − d)q-FHS, which is a set F ⊆ [q]n with |F| = M/n such that
+• for every x ∈ F and 0 < i ≤ n − 1, Hx,x(i) := n − dH(x, πi(x)) ≤ n − d;
+• for every distinct x, y ∈ F and 0 ≤ i ≤ n − 1, Hx,y(i) := n − dH(x, πi(y)) ≤ n − d;
+where Hx,x(i) and Hx,y(i) are known as the auto- and cross-Hamming correlation functions at time delay i (see [4],
+[5], [16]). FHSs were designed for the transmission of radio signals and have been studied extensively. In fact, the work
+of Niu et al. [16] was motivated by the study of FHSs.
+Note that one can also define FHSs directly using the Hamming correlation functions. For the sake of saving space,
+we will not formally state such a definition. It was observed in [4], [16] that the two definitions of FHSs are equivalent,
+since there is a one-to-one correspondence between HCCs and FHSs. Based on such a correspondence and (3), Niu et
+al. (see [16, Corollary III.6]) showed that for any absolute constant ǫ > 0 and λ ≥ n − (n − 2√n)(1 − 1/q − ǫ) there
+exist (n, M, λ)q-FHSs with
+M ≥
+qn(1 − n2e− ǫ2(√n−2)
+2
+)
+n(V olq(n, n − λ − 1) − 1).
+(8)
+The following result improves (8) by a linear factor of n.
+Proposition III.2. For positive integers n, q, λ and absolute reals τ, ǫ ∈ (0, 1 − 1/q) satisfying 1/q + ǫ ≤ λ/n ≤ 1 − τ,
+there exist (n, M, λ)q-FHSs with
+M ≥
+cqn
+V olq(n, n − λ − 1),
+
+6
+where c is a positive real depending only on τ, ǫ, q.
+Proof. This is an easy consequence of Lemma III.1 (ii), Theorem II.1, and the definition of FHSs.
+B. Error-correcting weakly mutually uncorrelated codes
+For 1 ≤ κ ≤ n, a code C ∈ [q]n is said to be a κ-weakly mutually uncorrelated code (WMUC for short) if for
+all κ ≤ ℓ ≤ n − 1, no proper prefix of length ℓ of a codeword in C appears as a suffix of another codeword in C,
+including itself. Formally speaking, C is a κ-WMUC if for every x, y ∈ C (possibly identical) and every κ ≤ ℓ ≤ n−1,
+(x1, . . . , xℓ) ̸= (yn−ℓ+1, . . . , yn). A WMUC is said to be error-correcting if it has some sufficiently large minimum
+distance.
+WMUCs and error-correcting WMUCs were introduced by Yazdi, Kiah, Gabrys, and Milenkovic [20] as a technique
+that allows random access to encoded DNA strands in DNA-based storage systems. The reader is referred to [20] for
+more background.
+The next result shows that HCCs have the bonus of being also WMUC.
+Proposition III.3. For positive integers n, q, κ and absolute reals τ, ǫ ∈ (0, 1 − 1/q) satisfying 1/q + ǫ ≤ κ/n ≤ 1 − τ
+there exist κ-WMU (n, M, n − κ + 1)q-codes with
+M ≥
+cqn
+V olq(n, n − κ),
+where c is a positive real depending only on τ, ǫ, q.
+Proof. By Theorem II.1, under the assumption of the proposition, there exists an (n, M ′, n − κ + 1)q-HCC C′ ⊆ [q]n
+with M ′ ≥
+cnqn
+V olq(n,n−κ), where c is a positive real depending only on τ, ǫ, q. By Lemma III.1 (ii), C′ can be partitioned
+into a pairwise disjoint union of M ′/n C(x)’s. Just as the proof of Proposition III.2, let C be a subcode of C′ formed
+by picking exactly one codeword from every C(x) in the partition. Then, C is an (n, M, n − κ + 1)q-code with
+M ≥
+cqn
+V olq(n,n−κ).
+It remains to show that C is also κ-WMU. Assume to the contrary that there exist x, y ∈ C (possibly identical)
+such that for some κ ≤ ℓ ≤ n−1, the prefix (x1, . . . , xℓ) is identical to the suffix (yn−ℓ+1, . . . , yn). Note that {x, y} ⊆
+C ⊆ C′. As C is an HCC, πn−ℓ(y) = (yn−ℓ+1, . . . , yn, y1, . . . , yn−ℓ) ∈ C′. As dH(x, πn−ℓ(y)) ≤ n − ℓ ≤ n − κ,
+which is strictly less than the minimum distance of C′, we must have x = πn−ℓ(y). It implies that {x, y} ⊆ C(x) and
+|C ∩ C(x)| ≥ 2, a contradiction.
+IV. COLLECTION OF SOME USEFUL LEMMAS
+In this section, we will collect the tools that are used in this paper.
+A. Locally sparse graphs and their independent sets
+In this subsection, we will introduce locally sparse graphs and a lower bound on their independence number. A
+graph G = (V, E) consists of a vertex set V and an edge set E, where V is a finite set and E is a family of 2-subsets
+of V . Two vertices u, v ∈ V are said to be connected by an edge if {u, v} ∈ E. Two connected vertices are called
+neighbors. The set of neighbors of a vertex u in G is denoted by NG(u). The degree of a vertex u ∈ V in G, denoted
+by degG(u), is the size of its neighborhood, i.e., degG(u) = |NG(u)|. Note that we will omit the subscript G in NG(u)
+and degG(u) when the underlying graph G is clear from the context. The maximum degree of a graph is the maximum
+degree among all of its vertices. For a subset S ⊆ V , the set of edges induced by S consists of all edges with both
+endpoints in S, namely the edges {{u, v} ∈ E : u, v ∈ S}. A graph is called locally sparse if for every vertex, its
+neighborhood induces only a limit number of edges.
+Given a graph G = (V, E), a subset I ⊆ V is called an independent set of G if every two vertices in I are not
+connected. The independence number of G, denoted by α(G), is the size of the maximum independent set of G. Using
+the greedy algorithm mentioned in Section II-C, it is not hard to see that every graph with maximum degree D has an
+independent set of size at least
+|V |
+D+1. Lemma IV.1 below, which is a direct consequence of [18, Corollary 1], presents
+a better lower bound for the independence number of locally sparse graphs.
+
+7
+Lemma IV.1.
+[18, Corollary 1] Let G be a graph on n vertices with maximum degree at most D, where D → ∞ as
+n → ∞. Suppose that for every vertex of G, its neighborhood induces at most D2
+K edges, where 1 ≤ K ≤ D2 + 1. Then
+α(G) ≥ (1 − o(1)) · |V |
+D · ln(min{D, K}),
+where o(1) → 0 as K → ∞.
+B. McDiarmid’s inequality
+We will make use of the powerful McDiarmid’s inequality, which, roughly speaking, states that a Lipschitz function
+of random variables is concentrated around its expectation.
+Lemma IV.2. [15] (McDiarmid’s inequality). Given sets X1, . . . , Xn, let X1, . . . , Xn be independent random variables
+taking values in X1, . . . , Xn respectively. Let f : X1×· · ·×Xn → R be a mapping. If there exist constants c1, . . . , cn ∈ R
+such that for each 1 ≤ i ≤ n,
+sup
+x′
+i∈Xi,x1∈X1,...,xn∈Xn
+|f(x1, . . . , xi, . . . , xn) − f(x1, . . . , x′
+i, . . . , xn)| ≤ ci,
+(i.e., changing the value of the i-th coordinate xi change the absolute value of f by at most ci), then for any t > 0,
+Pr[f(X1, X2, . . . , Xn) − E[f(X1, X2, . . . , Xn)] ≤ −t] ≤ exp
+�
+−
+2t2
+�n
+i=1 c2
+i
+�
+.
+C. The intersection volume of two Hamming balls
+We will need the following estimation on the intersection volume of two Hamming balls.
+Lemma IV.3.
+[12, Lemma 4.2] For positive integers n, q, d and two absolute reals τ, ǫ ∈ (0, 1 − 1/q) satisfying
+τ ≤ d/n ≤ 1 − 1/q − ǫ, we have that
+V olq(n, d − τn/2)
+V olq(n, d − 1)
+≤ e−Ωτ,q(1)·n.
+Moreover, for every x, y ∈ [q]n,
+|B(x, d − 1) ∩ B(y, d − 1)|
+V olq(n, d − 1)
+= e−Ωτ,q(1)·d(x,y).
+Similar upper bounds can also be proved for the intersection volume of two constant weight Hamming balls.
+Lemma IV.4.
+[12, Lemma 4.3] Given positive integers n, d, and reals p ∈ (0, 1), ǫ ∈ (0, 1/10), τ ∈ (0, p(1 − p))
+satisfying τ ≤ d/n ≤ (1 − ǫ)p(1 − p). Then
+V ol(n, d − τn/2; pn)
+V ol(n, d − 1; pn)
+≤ 2e−Ωǫ,τ (1)n.
+Moreover, for every x, y ∈ {0, 1}n with Hamming weight pn,
+|B(x, d − 1; pn) ∩ B(y, d − 1; pn)|
+V ol(n, d − 1; pn)
+= 2e−Ωǫ(1)·(d(x,y)+d(x,y)2/(d−1)).
+V. PROOF OF THEOREM II.1
+Given two reals τ, ǫ ∈ (0, 1 − 1/q), the goal of Theorem II.1 is to show the existence of large (n, M, d)q-HCCs
+whenever τ ≤ d/n ≤ 1 − 1/q − ǫ. The proof of Theorem II.1 will follow the steps outlined in Section II-C. First of
+all, note that an (n, M, d)q-code C is an (n, M, d)q-HCC only if for every x ∈ C, d(x) ≥ d (recall (7)). Therefore, to
+construct large HCCs, we need to show the existence of a large set A ⊆ [q]n such that for each x ∈ A, d(x) ≥ d. Such
+a set A in fact exists, as shown by the next lemma.
+Lemma V.1. Let n, q be positive integers and ǫ ∈ (0, 1 − 1/q) be a real. Let
+A = {x ∈ [q]n : d(x) > n(1 − 1/q − ǫ)}.
+
+8
+Then
+|A| ≥ qn
+�
+1 − (n − 1) exp(−ǫ2n
+2 )
+�
+.
+Proof. Let X = (X1, . . . , Xn) be a uniformly chosen random element of [q]n. To prove the lemma, it is enough to
+show that
+Pr[d(X) ≤ n(1 − 1/q − ǫ)] ≤ (n − 1) exp(−ǫ2n
+2 ).
+(9)
+Recall that d(X) = min{d(X, πi(X)) : 1 ≤ i ≤ n − 1}, where πi(X) = (Xi+1, . . . , Xi+n). By the union bound,
+Pr[d(X) ≤ n(1 − 1/q − ǫ)] = Pr[∃ 1 ≤ i ≤ n − 1 s.t. d(X, πi(X)) ≤ n(1 − 1/q − ǫ)]
+≤
+n−1
+�
+i=1
+Pr[d(X, πi(X)) ≤ n(1 − 1/q − ǫ)].
+According to the discussion above, to prove (9) it suffices to show that for every fixed 1 ≤ i ≤ n − 1,
+Pr[d(X, πi(X)) ≤ n(1 − 1/q − ǫ)] ≤ exp(−ǫ2n
+2 ).
+Note that choosing X = (X1, . . . , Xn) ∈ [q]n uniformly at random is equivalent to choosing X1, . . . , Xn ∈ [q] uniformly
+and independently at random. Let
+δ : [q] × [q] → {0, 1}
+denote the Kronecker function such that for a, b ∈ [q], δ(a, b) = 1 if a = b and δ(a, b) = 0 if a ̸= b. It is straightforward
+to check by definition that for every x ∈ [q]n,
+d(x, πi(x)) = n −
+n
+�
+j=1
+δ(xj, xj+i).
+(10)
+We will apply Lemma IV.2 with f(x1, . . . , xn) := d(x, πi(x)). Observe that for every 1 ≤ j ≤ n, changing the
+value of xj could only change the values of δ(xj−i, xj) and δ(xj, xj+i) in the right hand side of (10), which in turn
+changes the absolute value of d(x, πi(x)) and hence f by at most 2. In other words, for every 1 ≤ j ≤ n,
+sup
+x′
+j,x1,...,xn∈[q]
+|f(x1, . . . , xj, . . . , xn) − f(x1, . . . , x′
+j, . . . , xn)| ≤ 2.
+(11)
+Moreover, by the linearity of expectation, it is easy to see that for every 1 ≤ i ≤ n−1 and 1 ≤ j ≤ n, E(δ(Xj, Xj+i)) =
+1/q, which implies that E[f(X1, . . . , Xn) = n(1 − 1/q). Applying Lemma IV.2 to f, we obtain
+Pr [d(X, πi(X)) − n(1 − 1/q) ≤ −ǫn] ≤ exp(−ǫ2n
+2 ).
+Plugging the above inequality to the aforementioned union bound proves (9), and hence Lemma V.1.
+Remark V.2. Using a standard concentration inequality for martingales, [16, Lemma III.4] proved a slightly weaker
+result
+|{x ∈ [q]n : d(x) > (n − 2√n)(1 − 1/q − ǫ)}| ≥ qn
+�
+1 − n2e− ǫ2(√n−2)
+2
+�
+.
+Remark V.3. It is not hard to check by definition that A is an HCC with minimum distance, say 1. Therefore, similarly
+to the proof of Lemma III.1 (ii), one can show that with the equivalence relation ∼, A can be partitioned into a family
+V of equivalent classes such that every equivalent class consists of exactly n distinct vectors of [q]n. It follows by
+Lemma V.1 that
+|V | = |A|
+n ≥ 1
+n · qn
+�
+1 − (n − 1) exp(−ǫ2n
+2 )
+�
+.
+(12)
+Moreover, by the definition of A we have that for every C(x) ∈ V ,
+d(x) > n(1 − 1/q − ǫ),
+(13)
+
+9
+and for every distinct C(x), C(y) ∈ V ,
+C(x) ∩ C(y) = ∅.
+(14)
+To prove Theorem II.1, we will construct an auxiliary graph on the vertex set V , and then apply Lemma IV.1 to
+this graph, as detailed below.
+Lemma V.4. With the notation of Theorem II.1, fixing some d ∈ [τn, n(1 − 1/q − ǫ)], let GHCC = (V, E) be a graph
+constructed as below, where V is the family of equivalent classes that partition A, and two distinct vertices (or equivalent
+classes) C(x), C(y) ∈ V form an edge in E if and only if
+d(C(x), C(y)) = min{d(xi, yj) : 0 ≤ i, j ≤ n − 1} ≤ d − 1.
+Then the following holds:
+(i) for every independent set I ⊆ V , the set of vectors C := �
+C(x)∈I C(x) forms an (n, n|I|, d)q-HCC;
+(ii) the maximum degree of GHCC is at most D := V olq(n, d − 1);
+(iii) for every vertex of GHCC, its neighborhood induces at most D2
+K edges, where K = eΩτ,q(1)·n.
+Proof. To prove (i), note first that for every C(x) ∈ V , C(x) consists of n distinct vectors, which implies that C is
+indeed an HCC. Moreover, it follows by (14) that |C| = n|I|. Lastly, to prove that d(C) ≥ d, let c1, c2 ∈ C be two
+distinct codewords. On one hand, if {c1, c2} ⊆ C(x) for some C(x) ∈ I ⊆ V , then by (13) we have
+d(c1, c2) ≥ d(x) > n(1 − 1/q − ǫ) ≥ d.
+On the other hand, if c1 ∈ C(x) and c2 ∈ C(y) for distinct C(x), C(y) ∈ I, then as I is an independence set in G,
+we have
+d(c1, c2) ≥ d(C(x), C(y)) ≥ d,
+completing the proof of (i).
+To prove (ii), note that for every C(x) ∈ V ,
+deg(C(x)) = |{C(y) ∈ V \ {C(x)} : d(C(x), C(y)) ≤ d − 1}|
+= |{C(y) ∈ V \ {C(x)} :
+min
+0≤i≤n−1 d(x, πi(y)) ≤ d − 1}|
+= |{C(y) ∈ V \ {C(x)} : ∃ i ∈ [n] s.t. πi(y) ∈ B(x, d − 1)}|
+≤ |B(x, d − 1)| = D,
+as needed.
+To prove (iii), fixing a vertex C(x) ∈ V , let Γ denote the subgraph induced by the neighborhood of C(x). Partition
+V (Γ) = S ∪ T , where
+S = {C(y) ∈ V (Γ) : d(C(x), C(y)) ≤ d − τn/2}
+and
+T = {C(y) ∈ V (Γ) : d − τn/2 < d(C(x), C(y)) ≤ d − 1}.
+We have the following claim.
+Claim V.5. There exists some K = eΩτ,q(1)·n such that |S| ≤ D/K, and for every vertex C(y) ∈ T , degΓ(C(y)) ≤
+D/K.
+Given the correctness of the claim, it follows that
+|E(Γ)| = 1
+2
+
+ �
+C(y)∈S
+degΓ(C(y)) +
+�
+C(y)∈T
+degΓ(C(y))
+
+
+≤ 1
+2
+�
+|S| · D + |T | · D
+K
+�
+≤ D2
+K ,
+completing the proof of (iii).
+It remains to prove Claim V.5.
+
+10
+Proof of Claim V.5. To prove the first inequality, observe that
+|S| = |{C(y) ∈ V (Γ) : d(C(x), C(y)) ≤ d − τn/2}|
+= |{C(y) ∈ V (Γ) :
+min
+0≤i≤n−1 d(x, πi(y)) ≤ d − τn/2}|
+= |{C(y) ∈ V (Γ) : ∃ i ∈ [n] s.t. πi(y) ∈ B(x, d − τn/2)}|
+≤ |B(x, d − τn/2)| = V olq(n, d − τn/2).
+Therefore,
+|S|
+D ≤ V olq(n, d − τn/2)
+V olq(n, d − 1)
+≤ e−Ωτ,q(1)·n,
+(15)
+where the last inequality follows by the first equation of Lemma IV.3.
+To prove the second inequality, note that for every C(y) ∈ T , we have
+NΓ(C(y)) = N(C(x)) ∩ N(C(y))
+= {C(v) ∈ V \ {C(x), C(y)} : d(C(v), C(x)) ≤ d − 1, d(C(v), C(y)) ≤ d − 1}
+= {C(v) ∈ V \ {C(x), C(y)} :
+min
+0≤i≤n−1 d(v, πi(x)) ≤ d − 1,
+min
+0≤j≤n−1 d(v, πj(y)) ≤ d − 1}
+= {C(v) ∈ V \ {C(x), C(y)} : ∃ 0 ≤ i, j ≤ n − 1 s.t. v ∈ B(πi(x), d − 1) ∩ B(πj(y), d − 1)},
+⊆
+�
+C(v) ∈ V : v ∈
+�
+0≤i,j≤n−1
+B(πi(x), d − 1) ∩ B(πj(y), d − 1)
+�
+,
+which implies that
+|NΓ(C(y))| ≤
+�
+0≤i,j≤n−1
+|B(πi(x), d − 1) ∩ B(πj(y), d − 1)|.
+It thus follows that
+degΓ(C(y))
+D
+=
+|NΓ(C(y))|
+V olq(n, d − 1) ≤
+�
+0≤i,j≤n−1 |B(πi(x), d − 1) ∩ B(πj(y), d − 1)|
+V olq(n, d − 1)
+=
+�
+0≤i,j≤n−1
+B(πi(x), d − 1) ∩ B(πj(y), d − 1)
+V olq(n, d − 1)
+≤
+�
+0≤i,j≤n−1
+e−Ωτ,q(1)·d(πi(x),πj(y))
+≤ n2 · e−Ωτ,q(1)·n = e−Ωτ,q(1)·n,
+(16)
+where the last two inequalities follows from the second equation of Lemma IV.3 and the assumption that d(πi(x), πj(y)) ≥
+d(C(x), C(y)) ≥ d ≥ τn.
+The claim follows straightforwardly by combining (15) and (16).
+We proceed to present the proof of Theorem II.1.
+Proof of Theorem II.1. Let GHCC = (V, E) be the graph defined in Lemma V.4. Applying Lemma IV.1 in concert with
+(12) and Lemma V.4 (ii), (iii) yields that
+α(GHCC) ≥ (1 − o(1)) · |V |
+D · ln(min{D, K})
+≥ (1 − o(1)) · qn
+n ·
+1
+V olq(n, d − 1) · ln(min{V olq(n, d − 1), eΩτ,q(n)})
+≥
+cqn
+V olq(n, d − 1),
+where c is a constant independent of n. Therefore, according to Lemma V.4 (i), there exist (n, M, d)q-HCCs with
+M ≥ n · α(GHCC), completing the proof of the theorem.
+
+11
+VI. PROOF OF THEOREM II.2
+Given reals p ∈ (0, 1), ǫ ∈ (0, 1/10), τ ∈ (0, p(1 − p), the aim of Theorem II.2 is to show the existence of large
+(n, M, d; pn)-OOCs whenever τ ≤ d/n ≤ (1 − ǫ)p(1 − p). For pn ∈ Z+, let
+�[n]
+pn
+�
+= {x ∈ {0, 1}n : wt(x) = pn}. Note
+that an (n, M, d; pn)-code C is an (n, M, d; pn)-OOC only if for every x ∈ C, d(x) ≥ d. Similarly to the discussion
+at the beginning of Section V, to construct large OOCs we need to show the existence of a large set B ⊆
+�[n]
+pn
+�
+such
+that for each x ∈ B, d(x) ≥ d. The proof of this result is in the spirit similar to that of Lemma V.1 but technically a
+bit more involved.
+Lemma VI.1. Let ǫ > 0, 0 < p < 1 be reals and n be a positive integer. Let
+B = {x ∈
+�[n]
+pn
+�
+: d(x) > (1 − ǫ)np(1 − p)}.
+Then
+|B| ≥
+� n
+pn
+� �
+1 − n3/2 exp
+�
+− Ωp,ǫ(1) · n
+2
+��
+.
+Proof. Let X = (X1, . . . , Xn) ∈ {0, 1}n be a random vector, where X1, . . . , Xn are n independent Bernoulli random
+variables such that for 1 ≤ i ≤ n, Pr[Xi = 1] = p and Pr[Xi = 0] = 1 − p. For the ease of notation, let µ = np(1 − p).
+By the law of conditional probability,
+Pr[d(X) ≤ (1 − ǫ)µ|wt(x) = pn] = Pr[d(X) ≤ (1 − ǫ)µ ∧ wt(x) = pn]
+Pr[wt(x) = pn]
+=
+� n
+pn
+�
+− |B|
+� n
+pn
+�
+.
+To prove the lemma, it is enough to show that
+Pr[d(X) ≤ (1 − ǫ)µ|wt(x) = pn] ≤ n3/2 exp
+�
+− Ωp,ǫ(1) · n
+2
+�
+.
+Recall that d(X) = min{1 ≤ i ≤ n − 1 : d(X, πi(X))}, where πi(X) = (Xi+1, . . . , Xi). Then,
+Pr[d(X) ≤ (1 − ǫ)µ|wt(X) = pn]
+= Pr[∃ 1 ≤ i ≤ n − 1 s.t. d(X, πi(X)) ≤ (1 − ǫ)µ|wt(X) = pn]
+≤
+n−1
+�
+i=1
+Pr[d(X, πi(X)) ≤ (1 − ǫ)µ|wt(X) = pn]
+=
+n−1
+�
+i=1
+Pr[d(X, πi(X)) ≤ (1 − ǫ)µ ∧ wt(x) = pn]
+Pr[wt(x) = pn]
+≤
+n−1
+�
+i=1
+Pr[d(X, πi(X)) ≤ (1 − ǫ)µ]
+Pr[wt(X) = pn]
+≤
+�n−1
+�
+i=1
+Pr[d(X, πi(X)) ≤ (1 − ǫ)µ]
+�
+·
+�
+2πnp(1 − p) · ℓ(n),
+with ℓ(n) = exp(−
+1
+12n+1 +
+1
+12pn +
+1
+12(1−p)n), where the first inequality follows by the union bound, and the last
+inequality follows from the fact that
+Pr[wt(X) = pn] =
+� n
+pn
+�
+ppn(1 − p)(1−p)n
+and Stirling’s approximation
+� n
+pn
+�
+ppn(1 − p)(1−p)n ≥
+1
+�
+2πnp(1 − p)
+· exp(
+1
+12n + 1 −
+1
+12pn −
+1
+12(1 − p)n).
+According to the discussion above, to prove the lemma it suffices to show that for every fixed 1 ≤ i ≤ n − 1,
+Pr[d(X, πi(X)) ≤ (1 − ǫ)µ] ≤ exp(−Ωp,ǫ(1) · n
+2
+).
+
+12
+Similarly to the proof of Lemma V.1, we will apply Lemma IV.2 with f(x1, . . . , xn) := d(x, πi(x)) = n−�n
+j=1 δ(xj, xj+i).
+As for every i ̸= j, E[δ(Xi, Xj)] = p2 + (1 − p)2, by the linearity of expectation we have that E[f(X1, X2, . . . , Xn)] =
+2np(1 − p) = 2µ. Given (11), applying Lemma IV.2 to f yields that
+Pr [d(X, πi(X)) − 2µ ≤ −(1 + ǫ)µ] ≤ exp
+�
+− (1 + ǫ)2p2(1 − p)2n
+2
+�
+= exp
+�
+− Ωp,ǫ(1) · n
+2
+�
+,
+and then we have arrived at the desired conclusion.
+Similarly to the discussion in Remark V.3, the set B obtained by Lemma VI.1 can be partitioned into a family V
+of pairwise disjoint equivalent classes, where
+|V | = |B|
+n ≥ 1
+n ·
+� n
+pn
+� �
+1 − n3/2 exp(−Ωp,ǫ(1)n
+2
+)
+�
+.
+(17)
+To prove Theorem II.2, we will construct an auxiliary graph on the vertex set V , as detailed below.
+Lemma VI.2. Given d ∈ [τn, (1 − ǫ)np(1 − p)], let GOOC = (V, E) be a graph constructed as below, where V is the
+family of equivalent classes that partition B, and two distinct vertices C(x), C(y) ∈ V are connected if and only if
+d(C(x), C(y)) ≤ d − 1. Then the following holds:
+(i) for every independent set I ⊆ V , the set of vectors C := �
+C(x)∈I C(x) forms an (n, n|I|, d; pn)-OOC;
+(ii) the maximum degree of GOOC is at most D := V ol(n, d − 1; pn);
+(iii) for every vertex of GOOC, its neighborhood induces at most D2
+K edges, where K = eΩτ,ǫ(1)·n.
+As the proof of Lemma VI.2 is very similar to that of Lemma V.4, we will put it in Appendix.
+Next we present the proof of Theorem II.2.
+Proof of Theorem II.2. Let GOOC = (V, E) be the graph defined in Lemma VI.2. Then, applying Lemma IV.1 in concert
+with (17) and Lemma VI.2 (ii), (iii) yields that
+α(GOOC) ≥ (1 − o(1)) · |V |
+D · ln(min{D, K})
+≥ (1 − o(1)) · 1
+n ·
+� n
+pn
+�
+·
+1
+V ol(n, d − 1; pn) · ln(min{V ol(n, d − 1; pn), eΩτ,ǫ(n)})
+≥
+c
+� n
+pn
+�
+V ol(n, d − 1; pn),
+where c is a constant independent of n. Therefore, according to Lemma VI.2 (i), there exist (n, M, d; pn)-OOCs with
+M ≥ n · α(GOOC), completing the proof of the theorem.
+VII. CONCLUSIONS
+In this paper we present improved GV-type bounds for hopping cyclic codes and optical orthogonal codes, which in
+turn give improved GV-type bounds for non-linear cyclic codes and non-linear constant weight cyclic codes, respectively.
+Several interesting problems remain open.
+Question VII.1. It is known that for prime power q ≥ 49, there exist a class of codes, called algebraic geometry codes,
+that can significantly improve the GV bound (1) by an exponential power exp(Ω(n)) ( see [10], [21]). Does such an
+improvement hold also for (non-linear) cyclic codes? What about constant weight cyclic codes?
+Question VII.2. All of the results mentioned in this paper are non-constructive. Can we explicitly construct a class of
+asymptotically good (non-linear) cyclic codes? Or more precisely, can we present Zyablov-type bounds for (non-linear)
+cyclic codes? What about constant weight cyclic codes?
+ACKNOWLEDGEMENTS
+The first two authors would like to thank Prof. Yanxun Chang for an inspiring talk on constant weight cyclic codes.
+
+13
+REFERENCES
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+vol. 35, no. 3, pp. 595–604, 1989.
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+pp. 3999–4005, 2013.
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+IEEE Trans. Inform. Theory, vol. 55, no. 7, pp. 3297–3304, 2009.
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+no. 7, pp. 3605–3612, 2010.
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+Amer. Math. Soc., Providence, RI, 2015, vol. 634, pp. 79–99.
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+[8] I. Haviv, M. Langberg, M. Schwartz, and E. Yaakobi, “Non-linear cyclic codes that attain the gilbert-varshamov bound,” in 2017 IEEE
+International Symposium on Information Theory (ISIT), 2017, pp. 586–588.
+[9] Haviv, Ishay and Langberg, Michael and Schwartz, Moshe and Yaakobi, Eitan, “Non-linear cyclic codes that attain the gilbert-varshamov bound,”
+in 2017 IEEE International Symposium on Information Theory (ISIT), 2017, pp. 586–588.
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+Netherlands: Elsevier, 1998, pp. 871–961.
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+arXiv preprint arXiv:2112.11274 2021.
+[13] L. Lan, Y. Chang, and L. Wang, “Cyclic constant-weight codes: upper bounds and new optimal constructions,” IEEE Trans. Inform. Theory,
+vol. 62, no. 11, pp. 6328–6341, 2016.
+[14] V. I. Levenshtein, “Upper-bound estimates for fixed-weight codes,” Problemy Peredachi Informatsii, vol. 7, pp. 3–12, 1971.
+[15] C. McDiarmid, On the method of bounded differences, ser. London Mathematical Society Lecture Note Series.
+Cambridge University Press,
+1989, pp. 148–188.
+[16] X. Niu, C. Xing, and C. Yuan, “Asymptotic Gilbert-Varshamov bound on frequency hopping sequences,” IEEE Trans. Inform. Theory, vol. 66,
+no. 2, pp. 1213–1218, 2020.
+[17] R. Pan and Y. Chang, “Combinatorial constructions for maximum optical orthogonal signature pattern codes,” Discrete Math., vol. 313, no. 24,
+pp. 2918–2931, 2013.
+[18] F. Pirot and E. Hurley, “Colouring locally sparse graphs with the first moment method,” arXiv preprint arXiv:2109.15215v3, 2021.
+[19] R. Roth, Introduction to Coding Theory.
+Cambridge University Press, 2006.
+[20] S. M. H. Tabatabaei Yazdi, H. M. Kiah, R. Gabrys, and O. Milenkovic, “Mutually uncorrelated primers for dna-based data storage,” IEEE Trans.
+Inform. Theory, vol. 64, no. 9, pp. 6283–6296, 2018.
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+no. 1, pp. 96–106, 1995.
+VIII. APPENDIX: PROOF OF LEMMA VI.2
+Proof of Lemma VI.2. Note that given the proofs of the first two items of Lemma V.4, Lemma VI.2 (i) and (ii) can be
+proved easily, so we omit both proofs for clarity. To prove (iii), let Γ, S, and T be defined as in the proof of Lemma V.4.
+Then, it suffices to show that Claim V.5 holds also under the setting of Lemma VI.2.
+Claim VIII.1 (restatement of Claim V.5). There exists some K = eΩτ,ǫ,p(1)·n such that |S| ≤ D/K, and for every
+vertex C(y) ∈ T , degΓ(C(y)) ≤ D/K.
+To prove the first part of the claim, note that
+|S|
+D = |{C(y) ∈ V (Γ) : d(C(x), C(y)) ≤ d − τn/2}|
+D
+≤ V ol(n, d − τn/2; pn)
+V ol(n, d − 1; pn)
+≤ 2e−Ωτ,ǫ(1)n,
+(18)
+where the first inequality follows from the same technique that proves the first inequality in (15), and the second
+inequality follows by the first equation of Lemma IV.4.
+As for the second part of the claim, using the same strategy that proves (16) one can show that for every C(y) ∈ T ,
+
+14
+degΓ(C(y))
+D
+=
+|NΓ(C(y))|
+V ol(n, d − 1; pn) ≤
+�
+0≤i,j≤n−1 |B(πi(x), d − 1) ∩ B(πj(y), d − 1)|
+V ol(n, d − 1; pn)
+=
+�
+0≤i,j≤n−1
+B(πi(x), d − 1; pn) ∩ B(πj(y), d − 1)
+V olq(n, d − 1; pn)
+≤
+�
+0≤i,j≤n−1
+e−Ωǫ(1)·(d(πi(x),πj(y))+d(πi(x),πj(y))2/(d−1))
+≤ n2 · e−Ωτ,ǫ,p(1)·n = e−Ωτ,ǫ,p(1)·n,
+(19)
+where the last two inequalities follows from the second equation of Lemma IV.4 and the assumption that d(πi(x), πj(y)) ≥
+d(C(x), C(y)) ≥ d ≥ τn.
+The claim follows straightforwardly by combining (18) and (19).
+

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+SAFEMYRIDES: Application of Decentralized Control Edge-Computing to 
+Ridesharing Monitoring Services 
+ 
+Samaa Elnagar1*, Manoj A. Thomas2, Kweku-Muata Osei-Bryson3 
+1Howard University, 2University Of Sydney, 3Virginia Commonwealth University 
+ 
+ 
+
+ 
+Abstract 
+Edge computing is changing the face of many industries and services. Common edge computing models offload 
+computing which is prone to security risks and privacy violation. However, advances in deep learning enabled Internet 
+of Things (IoTs) to take decisions and run cognitive tasks locally. This research introduces a decentralized-control 
+edge model where most computation and decisions are moved to the IoT level. The model aims at decreasing 
+communication to the edge which in return enhances efficiency and decreases latency. The model also avoids data 
+transfer which raises security and privacy risks. To examine the model, we developed SAFEMYRIDES, a scene-aware 
+ridesharing monitoring system where smart phones are detecting violations at the runtime. Current real-time 
+monitoring systems are costly and require continuous network connectivity. The system uses optimized deep learning 
+that run locally on IoTs to detect violations in ridesharing and record violation incidences. The system would enhance 
+safety and security in ridesharing without violating privacy.  
+ 
+Keywords: Decentralized control; Edge Computing; Deep Learning; Internet of Things; Ridesharing; Monitoring, Privacy. 
+Highlights 
+• 
+A shift is needed towards decentralized control for IoTs to decrease computational costs and maintain privacy.   
+• 
+Technologies such as Deep learning for IoTs and decision support could be used to detect ridesharing violations 
+locally on smartphones. 
+• 
+Monitoring ridesharing sessions is technically and socially challenging. 
+• 
+The system doesn’t record the entire ridesharing session, but it only records violation incidents. 
+• 
+The system enhances the ridesharing services by ensuring privacy, responsiveness, accuracy, and efficiency.   
+1 
+Introduction 
+On the one hand edge computing led to huge reduction in communication cost and speedup computation (T. Wang, 
+2020). However, with tens of IoTs trying to connect single edge node, computation bottlenecks and delays are still 
+problems. Moreover, edge computing didn’t fully address security and privacy issues (T. Wang, 2020). On the other 
+hand, little research has focused on moving and computation and decentralizing decisions to IoTs (Elnagar & Thomas, 
+2020). What inspire decentralized edge-models that smartphones' specifications as the Internet of Things (IoT) are 
+continuously enhanced while the cost is fairly decreasing (El Khaddar & Boulmalf, 2017). Moreover, optimized deep 
+learning models could run on the limited processing, batteries, and memory of IoTs to provide stand-alone decision-
+making and cognitive tasks (Elnagar & Thomas, 2020). Decentralizing control and federating the computation at the 
+IoT level will minimize security, privacy and latency risks that emerge from transferring data to other computation 
+nodes (Li & Taeihagh). 
+One of the IoT applications that can’t contemplate privacy and decision latency is the ridesharing service where safety 
+is a significant concern (Feeney, 2015). Some countries in Europe and developing parts of the world report ridesharing 
+to be comparatively unsafe (Chaudhry, El-Amine, & Shakshuki, 2018), which suggests the dire need of imposing 
+additional safety measures to protect both the driver and the passenger (Hong, 2017).  From the social and governance 
+aspects, recording the entire rideshare session violates passengers’ privacy (Anderson, 2016; Z. Lee, Chan, Balaji, & 
+Chong, 2016). Technical challenges include storing and processing ridesharing monitoring sessions, and latency in 
+detecting violations. Moreover, cellular communication in developing countries, where crime rates are higher, is 
+limited (Dillahunt, Kameswaran, Li, & Rosenblat, 2017). In attempts to solve the latency and cloud dependency issues, 
+edge computing-based monitoring solutions focused on increasing the transmission bandwidth, optimizing workloads, 
+and increasing the number of edge processing nodes (T. Wang, 2020). However, these solutions not only necessitate 
+costly special infrastructure but also impose security and privacy risks. In high traffic, the latency issue would be 
+doubled due to the increased processing workload dedicated to each edge unit (Meshram, Choudhary, & Velaga, 2020; 
+Zhang, Wang, Lu, Liu, & Shi, 2019).  Therefore, edge computing-based solutions partially solve the latency issue, but 
+the privacy, and network dependency issues are not entirely solved.  
+This research proposes a decentralized-control edge model that aims to fill the efficiency, responsiveness, and privacy 
+gaps in current edge computing systems by moving most of the computation and decision making to the IoT level. 
+With application to ridesharing monitoring, the model minimizes cost represented in network dependency, costly 
+infrastructure, or sacrificing latency. So, we introduce a local decentralized scene-aware monitoring system for 
+ridesharing services using the driver’s smartphones. To ensure passenger privacy, the system record only violation 
+incidences instead of recording the entire rideshare session. The system applies state-of-the-art deep learning 
+optimized models to smartphones to detect ridesharing violations. To ensure security, The system encrypts the detected 
+
+violations before submitting them to ridesharing agents. The implementation of the system exhibits superior detection 
+latency among current solutions.  
+ 
+So, the broad research problem is shifting the gear towards decentralized edge computing and empowering the decision 
+making of IoTs. The narrow research focus is implementing SAFEMYRIDES, a decentralized control to in ridesharing 
+monitoring to address the safety concerns in ridesharing sessions.  
+SAFEMYRIDES is motivated by the lack of real-time monitoring systems in ridesharing, the high cost (connectivity 
+cost, infrastructure cost, and processing cost), and the preach of privacy of the current solutions (continuous 
+monitoring of passengers). The system also could be applied in developing countries where high computational 
+resources and mobile internet connections are limited.  
+In the following sections, we review prior research on ridesharing monitoring. Then, an overview of ridesharing 
+violations, followed by the methodology and description of the proposed system. The architecture of the system is 
+described and evaluated experimentally and using illustrative scenarios. Finally, future work and limitations are 
+provided. 
+2 
+Literature Review 
+2.1 
+ Current Research in Edge Computing 
+ 
+Current edge computing research is dedicated to improving media processing and computation latency and accuracy 
+on the edge computational units. Video Analytics in Public Safety (VAPS) is one of the most successful edge 
+computing applications (Zhang et al., 2019). However,  due to the high real-time requirements and communication 
+overhead, it is hard to implement VAPS at scale or in developing countries. Long, Cao, Jiang, and Zhang (2017) 
+developed a cooperative video processing scheme using an edge computing framework to enable cooperative 
+processing on resource-abundant mobile devices for delay-sensitive multimedia IoT tasks.  In the research, several 
+edge nodes cooperate for enhanced media task preprocessing performance. Yi et al. (2017) presents the LAVEA 
+system, an edge computing platform, which offloads computation between clients and edge nodes, and collaborates 
+with nearby edge nodes. The research aims to provide low-latency video analytics at places closer to the users to 
+minimize the response time. 
+(L. Wang, Zhang, Li, Zhong, & Shi, 2019) designed MobileEdge, a three-task offloading system that shares computing 
+tasks to nearby on-road computational units. The system target achieving optimal task scheduling for collaborative 
+computing. The system used Tensorflow lite (TF lite) that is used in our research. The results show that MobileEdge 
+significantly reduces the application response latency. However, on-road computational units exist only in certain 
+areas in developed countries, Moreover, collaborative computing with other computational units is prone to serious 
+security threats and privacy violations. Simaraly,  DeepDecision, a mobile deep learning framework for edge video 
+analytics, was developed  (Ran, Chen, Zhu, Liu, & Chen, 2018). The framework ties together computationally weak 
+front-end devices with more powerful back-end helpers to allow deep learning to choose local or remote execution 
+and determine an optimal offload strategy. Yet, the system assumes a well-defined shared computation infrastructure.  
+2.2 
+Previous Ridesharing Monitoring Systems 
+ 
+Monitoring ridesharing sessions has been a controversial issue. According to Ulrich Beck's concept of risk society 
+(Beck, Lash, & Wynne, 1992), surveillance risks mitigate the safety avails offered. Therefore, using monitoring 
+systems in ridesharing would enhance safety and consequently increase user satisfaction. Y. S. Lee (2018) conducted 
+a study on Lateral Surveillance in Singapore. They found that contrary to the belief that individuals are always 
+skeptical about surveillance, the respondents, on average, reported a positive perception of lateral surveillance. 
+Previous ridesharing monitoring systems include the Safe-ShareRide system developed by L. Liu, Zhang, Qiao, and 
+Shi (2018). The system can detect violations happening in the vehicle by detecting a predefined list of seeking help 
+phrases. The system uses speech recognition for verbal help detection. Then, the system starts recording videos and 
+analyze them on the nearest computational points. However, depending on the seeking help phrases for violation 
+detection is not typical in most cases, especially for passengers under influence.  
+Kumar, Mukherjee, and Singh (2017) used smartphones for monitoring roads through sensing the road surface and 
+conditions from a moving vehicle using fuzzy logic-based road surface roughness classification. The system collects 
+accelerations of users; processes the information and communication them to a central server. Then, a georeferenced 
+database is updated based on collected information and visualization on Google® maps. To obtain a broader picture 
+of the current status of ridesharing monitoring, Table 1 provides a comparison between previous systems and the 
+proposed system in terms of methodology and service quality issues. 
+
+ 
+Table 1: A Comparison Between Pervious Edge computing Systems and The Proposed System. 
+Research 
+Methodology 
+Computation Level 
+Latency 
+Privacy 
+Efficiency 
+Kumar et al. (2017) 
+Fuzzy logic-based 
+Central Server 
+- 
+- 
+yes 
+Ran et al. (2018) 
+Mobile 
+deep 
+learning, 
+Local 
+or 
+remote 
+execution 
+yes 
+- 
+yes 
+Long et al. (2017), Yi 
+et al. (2017), Zhang et 
+al. (2019) 
+Edge computing 
+Nearby edge nodes, or 
+Shared IoT 
+yes 
+- 
+- 
+L. Liu et al. (2018) 
+Speech 
+recognition 
+for 
+verbal 
+help 
+detection 
+At 
+IoT 
+(speech 
+recognition), 
+edge 
+unit (video) 
+yes 
+- 
+- 
+Proposed Approach 
+Tensorflow 
+lite, 
+Optimized 
+deep 
+learning 
+At 
+the 
+IoT 
+(smartphone) level 
+yes 
+yes 
+yes 
+2.3 
+Violations in Ridesharing 
+According to most ridesharing companies, there is an agreeable code of conduct to be followed by both passengers 
+and drivers. A common code of conduct includes (Armant & Brown, 2020; BILL): Avoid sitting next to the driver on 
+the passenger seat. Avoiding any physical interaction with the driver. Physical interaction with the driver includes 
+violent actions, seductions, and arguments. Moreover, the driver cannot sit next to the passenger in the back seats 
+(Macmurdo, 2015). A Driver must wear a seatbelt all the time. The code of conduct also illegalizes the use and hold 
+of weapons during ridesharing sessions. In addition, the code of conduct condemns any sexual conduct between the 
+driver and passenger (Lyft, 2019). In the design of the ridesharing monitoring system, we trained the system's deep 
+learning model to detect these types of violations. 
+3 
+Conceptual Framework 
+The gap in literature could be summarized that current edge solutions are oriented towards optimizing the computation 
+and media processing on the cloud or nearby edge nodes (Cheng, Bao, Zarifis, & Mou, 2019; Z. Lee et al., 2016). 
+Also, there is an inclination towards using edge analytics for media processing where multiple edge units are 
+collaborating to decrease the latency.  On the other hand, solutions that share the computation are inefficient requiring 
+costly infrastructure that is hard to be implemented in developing countries and rural areas. In addition sharing 
+computation is violating the privacy and security of user’s data (Cheng, Su, Luo & Cai, 2021). Finally, none of the 
+edge solutions are trying to move computation locally to the IoT level to decrease network reliance and ensure privacy. 
+On the other hand, the ridesharing monitoring issues discussed in the literature section could be summarized as the 
+lack of privacy, safety, and late response. These issues have affected customer satisfaction and trust. Most edge 
+computing research avoided moving computation to the IoT level because of the limited computational resources of 
+IoT. Therefore, ensuring system efficiency in terms of battery, memory, and storage consumption in our research is 
+crucial. So, the proposed model aims at enhancing efficiency, (Functionality), privacy, and responsiveness as key 
+factors to enhance IoTs applications performance such as ridesharing monitoring services. The description of each 
+factor is discussed below: 
+Privacy: gauges the degree to which the IoT application protects customer information and only with their consent, 
+disseminate it to other business activities (Hizam & Ahmed, 2020). Since mobile devices are accessible to sensitive 
+information including biometric features, privacy has been an essential factor in assessing electronic applications 
+(Wolfinbarger & Gilly, 2003; Zeithaml, Parasuraman, & Malhotra, 2002). In SAFEMYRIDES, the ridesharing session 
+is recorded based on passenger consent and only violation incidence is recorded. Moreover, recorded media (photos 
+or videos) will be encrypted before they are saved. 
+Efficiency: measures the resources and ease with which a system is accessed and used (Zeithaml et al., 2002). The 
+decentralized-control edge approach aims to consume fewer resources in terms of less network communication that 
+consume less battery and data transfer cost. The approach also uses DL optimization techniques such as quantization 
+and compression of DL networks. Quantization maximize memory savings during the inference process, and 
+compression of DL networks decreases the size of the network to few megabytes, makes it efficient to be used in low 
+
+storage devices (Jacob et al., 2018). Moreover, efficiency will be evaluated by the resources consumed such as 
+memory, storage, and battery consumed. 
+Responsiveness: is one of the most important service quality dimensions for human-computer-interaction (HCI) (Ho 
+& Lee, 2007). Responsiveness is how the system delivers the results of an operation to users in a timely and organized 
+manner. Latency is directly correlated with responsiveness, the delayed is the response, the less responsiveness in the 
+system. Therefore, the proposed approach will be evaluating the average latency as a measure of Responsiveness. 
+Accuracy: there are different definitions of accuracy. In the context of SAFEMYRIDES, accuracy is defined as the 
+ability of the system to detect violations correctly out of all violation incidences. The confusion matrix of precision 
+and recall is also a principal part of calculating accuracy. 
+4 
+Methodology 
+In this research, we follow a design science approach  (Hevner, March, Park, & Ram, 2004). The decentralized-control 
+edge computing approach is considered a model artifact, and the ridesharing monitoring system (SAFEMYRIDES) is 
+considered a framework artifact (Peffers, Rothenberger, Tuunanen, & Vaezi, 2012). The framework artifact is 
+implemented to show functionality and test performance. The system is evaluated as instantiation using synthetic or 
+real-world situations to evaluate the detection accuracy, latency, privacy, and consumed resources. So, the evaluation 
+is performed using controlled experiments or the study of the artifact in a controlled environment for qualities. Next, 
+an overview of decentralized-control edge computing approach is presented.  
+4.1 
+Decentralized Control Edge Model 
+Edge computing was built upon the concept of computation offloading where end devices and IoTs perform data 
+collection and sensing (Huda & Moh, 2022) as shown in the right side of Fig.1. Complex processing and real time 
+analysis are performed on the edge units or on the cloud. However, running machine learning and decision-making 
+algorithms on IoTs is now an attainable task fueled by the increase in computational resources are significant decrease 
+in computational cost (Mocnej et al., 2021). Complex computation tasks such as deep learning inference could run on 
+IoTs with regular consumption of resources.  The decentralized control edge model is proposed on the left side of 
+fig.1. The model limits the computation on the edge or the cloud side to limit traffic and computational overheads. 
+ 
+The model moves most of computation and decision-making to the end device level to allow decentralized control 
+(task onloading). The role of edge units and/or cloud platforms in the decentralized-control model is archiving/logging 
+events of the IoT. So, IoTs connect edge units or the cloud platforms for saving logs or upgrading the processing of 
+tasks. The model not only decreases network traffic between end devices and edge units but also, allow a real-time 
+response to IoT sensed events. Thus, this model focuses on IoTs Task Onloading and IoT-edge communication 
+reduction. However, since deep learning models are complex and resources exhaustive, deep learning tasks should be 
+optimized first before they could be applied to IoTs as discussed in the next sections 
+ 
+Fig. 1: Decentralized Control Edge Model 
+ 
+
+TOOLBOX
+EDGECOMPUTING
+Data Processing
+Buffering Optimization
+吧
+[)
+Basic Analytics
+Cloud Platform
+[…]
+Cloud Platform
+Cloud
+Cloud Data
+Cloud Machine
+Functions
+Center
+Learning
+Edg
+c-c-m
+Event Processlng
+Data Cashing
+Decision Malking
+Cloud Media
+Cloud
+Cloud Storage
+Database Server
+Cloud Controller
+Edge
+Manager
+Functions
+Seve Logs,
+Jpgrade Processing
+Upgrade Actlons
+Real time Analytics
+Cloud Media
+Database Server
+Edge
+Process Darta, Events
+Collect Data, Events
+Take Decisions
+Transmit Data
+1
+Processed
+Processed
+Transmit
+Date
+sehso
+vents
+loTthing car
+loT thing camera loT thing bicycleRoad Device
+T
+loT thing car
+loT thing camera loT thing bicycle Road Device 
+4.2 
+Deep Learning Networks Optimization 
+DL models are known for their high computational cost and complexity, so they mostly run on servers on the cloud 
+(Wu et al., 2018). On the other hand, end devices (e.g., IoTs) and edge devices (e.g., routers) are often battery-powered, 
+have limited memory, and processing to store and process data. Applying DL models to IoT is challenging; DLNs 
+must be optimized and compressed to fit IoTs limited computational sources. In addition, the optimization of the IoTs 
+themselves is also necessary in terms of memory and hardware optimization (Synced, 2017). 
+Compression is one of the main optimization techniques that aim to reduce the massive size of DL networks. One of 
+the popular compression methods is the Pruning technique that eliminates the connections between neurons to directly 
+reduce the feature map width and shrink the network size. Quantization is another optimization technique that aims at 
+compacting the number of bits required to store the DLN weights, usually from 64 bit to 8 bits (Han, & Dally, 2015). 
+4.3 
+SAFEMYRIDES Design 
+The proposed system is an on-site decision support system for monitoring ridesharing vehicles, where the driver's 
+smartphone can detect the violation and decides the proper action. We trained a Mobile SSD network to detect 
+violation incidences. Then, the trained Mobile SSD is optimized to be deployed to smartphones. The design of the 
+proposed system is shown in Fig. 2 and the components of the system are discussed below in detail. 
+ 
+ 
+Fig. 2: Decision Support System for Monitoring Ridesharing Vehicles. 
+4.3.1 
+Optimized Deep Learning (DL) Model 
+Usually, Deep learning models (DLMs) consist of many connected layers resulting in millions of parameters and 
+weight. So, running DLMs tends to be a resource-intensive process in terms of energy consumption and memory 
+accesses and hardware accelerators. To apply a DLM to IoTs and small devices such as smartphones, DLMs must be 
+compressed and quantized. Also, the DLM running on IoTs are read-only means that these compressed models cannot 
+be retrained or edited. We used a Mobile Single Shot Detector (Mobile SSD) which is based on a feed-forward 
+convolutional network that produces a fixed-size collection of bounding boxes and scores for the presence of object 
+class instances in those boxes. SSD is designed to be independent of the base network, and so it can run on top of any 
+base network such as VGG, YOLO, MobileNet (W. Liu et al., 2016). 
+ 
+The DLM is quantized where parameters are represented in 8 bits instead of 32 bits to accelerate the inference process. 
+The DLM will have a post-training quantization which is the most used form of quantization. In this approach, 
+
+CellularServersunits
+Quantization
+Compression
+OptimizedDLmodel
+Report
+101010
+01001
+Encrypt
+EncryptingViolation
+Incidence
+DecisionSupport
+Original Prediction
+Deployed
+System
+1
+Warn
+Layer
+Monitor
+conv4_x
+conv5_x
+maconv3x
+Ride-sharingVehicle
+MobileNetSingleShotDetectorNetwork
+DriverSmartPhonequantization takes place only after the model has finished training. Moreover, the network is compressed to be 10x 
+less in size of the trained network following the method presented by (Singh, Manikandan, & Namboodiri, 2019).  
+4.3.2 
+The Ridesharing Smart Phone 
+Recently smartphones' specifications are enhancing rapidly at a very reasonable cost. The average processing power 
+of most smartphones currently could host an optimized DLM and run inference smoothly. For the system to run 
+smoothly, the smartphone should have a minimum RAM (not less than 4GB) and an acceptable resolution camera 
+according to TensorFlow Guide1. 
+4.3.3 
+Decision Support System (DSS) 
+The DSS is built inside the smartphone. It is responsible for acting in case of detected violation. If a violation is 
+detected by the Mobile SSD, a warning is voiced to passengers/drivers. If violations continue to be detected, the 
+DSS will record the incidence, encrypt it, and save the incidence in a hidden folder in the ridesharing's smartphone. 
+Then, the recorded incidence is sent to the ridesharing agent and attached with encrypted media. 
+4.3.4 
+Encrypting and Transferring Violation Incidence 
+For security, images or videos that document the violation incidences are encrypted and saved in the ridesharing's 
+smartphone storage to ensure passenger/driver privacy. In addition, encryption ensures secure transfer to the 
+ridesharing agent. Otherwise, violation evidence could be deleted before it is reported to the ridesharing agent. On 
+the other hand, encryption will ensure security if the media is shared with nearby road units. So, passengers' 
+information will be kept confidential without driver intervention. Moreover, passenger/driver data is secured in case 
+the phone is stolen or hijacked.   
+4.3.5 
+Transferring Violation Evidence using The Cellular Server Units (CSU)  
+After the violation incidence is securely recorded and encrypted, it will be uploaded to the nearest cellular server 
+units. In case of no cellular connection, Violation incidences could be uploaded to the nearest on-road units in smart 
+cities infrastructure or sent in text messages. CSUs could be the driver cellular network or the networks that exist on 
+the roadsides in developed countries. So, mostly CSUs are the nearest access point where the DSS could upload 
+incident videos even if no cellular internet is turned off on the phone. 
+4.3.6 
+The Ridesharing Vehicle 
+Information about the ridesharing vehicle is saved in the ridesharing agent database. If the violation requires legal 
+authorities' interventions, the ridesharing vehicle could be found easily. In addition, the ridesharing agent requires 
+registered vehicles to have a valid title, proper condition, and valid insurance of the driver. 
+5 
+Implementation 
+The implementation process is divided into two steps. The first step is the training and testing of the Mobile SSD 
+network used in detecting violations. In this step, we collect, annotate, and pre-process relevant ridesharing violation 
+samples. Then, after the network is fully trained, it is optimized (compressed and quantized) to be easily embedded in 
+IoT(smartphone). The second section is implementing the DSS system that runs locally on the driver's smartphone. 
+The DSS act when the Mobile SSD detects violations. The DSS actions include sending warnings, encrypting violation 
+incidences, and reporting them to the ridesharing agent. In the following sections, the implementation of each section 
+is discussed in detail. 
+5.1 
+Implementing the Trained Mobile SSD Network 
+5.1.1 
+The Training Dataset 
+There is no ready to use dataset that contains images of ridesharing violations incidents.  The training and testing 
+samples used to train the Mobile SSD model are retrieved from web searches, ridesharing agents' articles, and public 
+ridesharing violations videos. The violation images were a collection of the five-standard code of conduct rules such 
+as interacting physically with the driver as specified earlier. The images collection was taken during the day and at 
+night. We also included normal images where there is no violation. 
+ 
+1 https://www.tensorflow.org/lite/guide 
+ 
+
+5.1.2 
+Annotation and Preprocessing: 
+Images were annotated in three classes: driver, passenger, and violation. Annotation was performed using Roboflow2. 
+Dataset was divided into 90% training, 6 % validation, and 4 % testing. Images were annotated and saved in 
+TFRECORD format. Every class in the image was surrounded by a bounding box representing a driver (pink box), 
+passenger (green box), and violation (purple box), as shown in fig. 3. Noted that all passengers are annotated if the 
+image has more than a passenger. Image preprocessing included resizing pictures to 416 x 416, Auto-Orientation, and 
+Auto-Adjust of Contrast through contrast stretching (Munteanu & Lazarescu, 1999). 
+ 
+ Fig. 3: Annotated Training, Validation Images Where Each Class is surrounded with A Bounding Box.  
+5.1.3 
+Hardware and Software 
+The Training was performed on the cloud Google Collaboraty using Jupiter notebooks. To accelerate the training 
+process, we used one GPU. The network was trained using TensorFlow 1 and a Keras version less than 2.0 as 
+recommend for custom models training settings3. The ridesharing DSS was created using Android studio 4.1.2 and 
+SDK 28. The device used for running and debugging is a Mac-book pro with intel core i5 2.5 GHz processor, 8 GB 
+Ram. The build and debug were performed on a Google PIXEL AVD before the system was deployed to a smartphone. 
+5.1.4 
+Optimized MobileNet Single Shot Detector Network 
+We trained a MobileNet Single Shot Detector (v2) (ssd_mobilenet_v2) that is popular for performing object detection 
+tasks on IoT devices. ssd_mobilenet_v2 is optimized for lightweight inference, enabling it to perform well natively on 
+compute-constrained mobile and embedded devices. The Mobile SSD is quantized to parameters size of 8 bit. 
+Quantization consumes less memory bandwidth. Fetching numbers in the 8-bit format from RAM requires only 25% 
+of the bandwidth of the standard 32-bit format. On the other hand, quantizing neural networks results in 2x to 4x 
+speedup during inference. We didn't train the network from scratch; rather, we transferred learning from a pre-trained 
+model with a COCO dataset.  COCO dataset training weights were uploaded to the training network in the form of 
+.h5 format.  Also, we used the same configuration for MSCOCO Dataset as shown below in table 2. 
+Table 2: Trained Network Configuration. 
+Parameter 
+Value 
+The number of layers 
+6 
+Minimum network scale 
+0.2 
+Max Network scale 
+0.95 
+Number of steps 
+100000 
+Activation Function 
+RELU_6 
+ 
+2 https://app.roboflow.com/ 
+3 Model Customization 
+
+221%After the network was trained, we ran an inference test, and then we compressed the network and quantized it to 8 bit. 
+Then, the trained network was converted into TFLITE format. The 8 bit quantization format is used to run inference 
+on IoTs such as smartphone devices. The average time spent per training step is global_step/sec:4.96517. 
+So, the entire training session duration lasted 13.7921 hours. 
+5.2 
+Implementing the SAFEMYRIDES Decision Support System  
+5.2.1 
+ Detection Response 
+The DSS was developed in Java for Android using Android Studio IDE.  The DSS primary interface runs a scene view 
+stream of the smartphone camera. The stream is not recorded. Otherwise, the detection runs on the fly using the 
+smartphone camera running scene. The system has access to both the front and rear cameras. When a violation is 
+detected with confidence level greater than 80 %, the system speaks out "Violation Detected" using the Text to sound 
+android library. If another violation is detected, the system saves the current scene into an encrypted image file and 
+stores it in a hidden folder in the ridesharing's phone. The average inference time of the system is 28 ms for detecting 
+the three classes of (passenger, driver, and violation). 
+5.2.2 
+ Encrypting Violation Incidences  
+For encryption, each violation image is converted into a byte array; then, an XOR operation is applied on each value 
+of the byte array using a predefined encryption key (2-bit key). After performing the XOR operation on each byte 
+array value, the encrypted image is written to a new image file in a jpeg format. However, the driver won't be able to 
+read the encrypted image or decrypt it. Only the ridesharing agent has the predefined key and can decrypt the image. 
+The encryption key acts as a password to encrypt and decrypt the violation image. 
+5.2.3 
+The Ridesharing Smart Phone 
+We ran experiments on two different smartphones with different specifications to monitor the performance of the 
+proposed system on different hardware specifications.  The first smartphone used during experiments was a Samsung 
+s10 + phone running on Android.  And the second phone is an LG V30+ also running on Android.  
+The system was built upon the starter code offered by android for object detection using TensorFlowLite4. The system 
+added the custom trained Mobile SSD network. The code was adjusted to detect the three classes of driver, passenger, 
+and violation. We added a text-to-speech module to send violation warnings. In addition, we added the encryption-
+decryption module for securing violation images. 
+5.2.4 
+ Results  
+These are some screenshots from the ridesharing violation application running on a Samsung s10 + phone. The 
+application can detect three classes. A detecting bounding box surrounds each detected class. The number on the right 
+top of the bounding box is the network confidence score in making this prediction. Examples of detected violations 
+by the SAFEMYRIDES are shown in Fig. 4. a, 2. b. The confidence score of the DL network measures how confident 
+is the network assigning an object a specific class.  
+6 
+Evaluation 
+The evaluation will be performed on different levels, as shown in table 3. The evaluation of the trained Mobile SSD 
+network will be functional. The network is evaluated before and after it is compressed and quantized. Functional 
+evaluation metrics include testing and training accuracy. We also traced the network testing and validation loss. The 
+system itself is performed through controlled experiments where evaluation relativeness is absolute (e.g., test if the 
+artifact achieves its goal?). Then, another evaluation is performed by conducting a comparison of the artifact 
+performance on a different device of different hardware/software specifications (Cleven, Gubler, & Hüner, 2009). The 
+stages of evaluation performed are summarized in table 3. 
+6.1 
+Evaluating the Trained Mobile SSD Network 
+6.1.1 
+Training loss/ Validation Loss 
+There are two critical measures to evaluate the Mobile SSD: the training loss and the validation loss. The loss is the 
+cost function used while training that needs to be minimized. At the end of every 100 training steps, the network is 
+ 
+4 GitHub Code 
+ 
+
+evaluated for the loss in training and validation. As long as the loss decreases, that means that the network is trained 
+correctly, and it can generalize well (Jancsary, & Rother, 2012). It is expected for the validation loss to be slightly less 
+than the training loss. If the gap between training loss and validation loss increases, this indicates network overfitting.  
+Across the 100K steps of training iterations, the training loss vs. validation loss is plotted as shown in fig. 4. The 
+training and validation loss start high and start decreasing gradually. The gap between the training and validation loss 
+is tight. However, at some steps, the validation loss is less than the training loss, which indicates that the validation 
+samples are easier to detect than training samples. The iteration with the least loss is selected in step 96800 with a 
+validation loss of 1.0199343. Then validation loss starts increasing afterward, which means that the network overfits. 
+The MAP (Mean Average Precision) is another metric used in measuring the accuracy of object detecting networks.  
+The average precision computes the average precision value for the recall values over 0 to 1 using IoU (Intersection 
+over the union) (Henderson & Ferrari, 2016). the MAP was calculated at the end of each 10000 training steps at 
+IoU=0.5 
+the 
+maximum 
+MAP 
+achieved 
+was 
+0.3816 
+at 
+step 
+55867, 
+and 
+DetectionBoxes_Recall/AR@10 = 0.4333. 
+ 
+ 
+ 
+Fig. 4: a,b  a Violation of Sitting Next to The Driver, Interacting Physically with the Driver. 
+Table 3: Evaluation Stages with Involved Components and Criteria. 
+Artifact 
+Evaluation 
+Method 
+Assessed 
+Criteria 
+Level of 
+Evaluation 
+Form of 
+Evaluation 
+Relativeness 
+of Evaluation 
+Trained 
+Mobile 
+SSD 
+Reporting 
+the 
+functional 
+performance 
+of 
+the 
+trained network 
+Training 
+accuracy 
+Loss/validation 
+accuracy Loss, 
+Instantiation 
+ 
+Quantitative 
+/ measured 
+Absolute 
+Proposed 
+System 
+Demonstration of 
+the performance of the 
+artifact with 
+several real examples 
+Goal, 
+Accuracy, 
+Latency 
+Instantiation 
+/ Real 
+examples  
+ 
+Quantitative 
+/ measured 
+Absolute 
+Proposed 
+System 
+Observing 
+the 
+differences 
+between performance 
+on different devices 
+Comparison between 
+accuracy, 
+latency 
+efficiency 
+on 
+different devices 
+Instantiation 
+ 
+Quantitative 
+/ measured 
+Relative 
+
+Rideshare Detector
+violation 92.43%10:35网M·
+driver82.39%
+violation96.20%R
+m6.1.2 
+Training and Validation Accuracy 
+We ran an inference test on the trained network before we performed optimization: compression and quantization to 
+study the effect of the optimization on accuracy and latency. The full-sized mobile SSD model sized 245 MB with a 
+32-bit inference parameter. The highest training accuracy achieved is 0.94546, and the highest validation accuracy is 
+0.93123. The average time spent by Google Collaboratory for inference testing is 3.56 seconds. 
+ 
+ 
+Fig. 5: Training loss Vs. Validation Loss of the Trained Mobile SSD. 
+6.2 
+Evaluating the SAFEMYRIDES Decision Support System 
+ 
+Firstly, the entire SAFEMYRIDES system was evaluated using a controlled simulated environment where the system 
+was applied in a real ridesharing setting. The first experiments were running on the Samsung S10+ phone. The driver/ 
+passenger performed different violation scenarios such as: interacting physically with the driver, violence, and sitting 
+next to the driver. Examples of detected violations are shown in Fig. 6 a, b.        
+6.2.1 
+Latency 
+The average system Latency is also measuring the system responsiveness that affects service quality and customer 
+satisfaction. Latency would be represented by the average system response time to detect a violation. However, the 
+response time is affected by the surrounding conditions. For example: the response time to detect a violation in daylight 
+is different than the response time at night. After setting the minimum required confidence level to 0.80 (the confidence 
+score of the Mobile SSD that the detected incidence is a violation), the average response time to violations detection 
+during daylight is 450 ms while at night is 790 ms. However, this average response time is higher to other systems 
+used in ridesharing media processing, as shown in table 4 and fig. 6.  
+Table 4: A Comparison Between SAFEMYRIDES and Other Proposed Systems. 
+Research 
+Average Latency 
+Media type 
+L. Liu et al. (2018) 
+1273ms 
+Sound 
+L. Wang et al. (2019)  
+8345 ms 
+A 60-second video data 
+Long et al. (2017) 
+2234 ms. 
+Compressed Videos. Latency (model processing time + 
+network transmission time). 
+Ran et al. (2018) 
+32100 ms. 
+video chunk is set as S= 1MB, 15 edge nodes 
+SAFEMYRIDES 
+620 ms. 
+Scene view stream 
+ 
+ 
+
+4.5 
+Training loss
+4.D 
+Validation loss
+3.5
+3.D
+25
+2D
+15
+LD 
+0
+DOEZ
+G1000
+81000
+14000
+Steps 
+ 
+ 
+ 
+Fig. 6: Detected Violations using Different Violations: Violence and Physical Interaction. 
+ 
+Fig. 7 : Average Latency of the Proposed System Versus Previously Developed Systems. 
+ 
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Liu et al. (2018)
+(Wang et al. 2019)
+(Long et al. 2017)
+(Ran et al. 2018)
+SAFEMYRIDES
+Average Latency
+
+passenger79.30%
+Rideshare Detector
+violation91.60%passenger86.72%
+violation85.76%passenger95.34%
+ideshareDetector
+violation89.28%Ridet
+violation9Z.69%6.2.2 
+Efficiency 
+In terms of the resources used by the system, the system used 29.09 MB of the phone's internal storage. The average 
+size of an encrypted media image is 120kB. Energy consumed of running the system for 15 minutes at a rate of 17 m 
+(milliamp), or 79.8 mAH (milliamp hour). Memory used by the system is 104 MB Which means it is consuming less 
+than 1% of the smartphone RAM of 8 GB.  
+6.2.3 
+ The System Accuracy and Confusion Matrix  
+In this experiment, we are evaluating the optimized trained network running on the Android S10 plus. The optimized 
+network size is 27 MB, almost 10x less than the actual model size, and runs at 8-bit inference parameters. To get an 
+in-depth analysis of how the optimized network performed, a confusion matrix was created to analyze the accuracy, 
+precision, and recall of the front and rear cameras as in table 5 and table 6. The confusion matrix was built during a 
+simulated session during the day, trying 51 different normal and violations poses. The front camera and the rear camera 
+were used to detect a violation. The rear cameras of the Samsung S10 plus are 12MP, 12 MP, and 16 MP ultra-wide 
+module cameras. The front-facing cameras are 8MP, 10 MP f/1.9 selfie cameras.  
+Precision: It tells what fraction of predictions as a positive class were genuinely positive. To calculate precision, use 
+the following formula: 𝑻𝑷/(𝑻𝑷 + 𝑭𝑷)	 or 0.9111 for the rear camera and 0.906 for the front camera according to the 
+confusion matrix.  
+Recall: also known as Sensitivity, Probability of Detection. It tells what fraction of all positive samples were correctly 
+predicted as positive by the classifier. To calculate Recall, use the following formula: 𝑻𝑷/(𝑻𝑷 + 𝑭����) or 0.976 for 
+the rear camera and 0.928 for the front camera. 
+Accuracy: is the measure of what fraction of violation incidences were correctly detected by the system; the accuracy 
+formula is Accuracy = (𝑇𝑃 + 𝑇𝑁)/(𝑃 + 𝑁). Based on the formula, the system's accuracy in detecting correct 
+violation incidences is 46/51=0.9019 for the front camera and 0.862. 
+Table 5: Confusion Matrix of The Proposed System (Rear Camera). 
+True/Detected 
+True  
+False 
+True 
+41 (TP) 
+1 (FN) 
+False 
+4 (FP) 
+5 (TN) 
+Table 6: Confusion Matrix of the proposed system (Front camera). 
+True/Detected 
+True  
+False 
+True 
+39 (TP) 
+3 (FN) 
+False 
+4 (FP) 
+5 (TN) 
+6.2.4 
+Privacy 
+Surveillance systems have raised apprehension about their threat to individuals' privacy rights (Cheung, Venkatesh, 
+Paruchuri, Zhao, & Nguyen, 2009). Privacy protection is a core objective in the proposed system through ensuring 
+transparency, authorization, and encryption.  Transparency means that both the driver and the passenger are aware 
+that the ridesharing session is monitored. A warning of "Violation Detected" is declared when the system detects a 
+violation. Media surveillance is privacy-intrusive because it allows the observation of certain information that is 
+considered privacy intrusive such as a person's identity or characteristics (age, race, gender) (Senior, 2009) . Therefore, 
+authorization of users is required to monitor ridesharing sessions. Moreover, negotiation of privacy preference 
+(Consortium, 2002) or flexible authorization is offered when users prefer monitoring at night sessions or in remote 
+destinations. 
+Privacy protection technologies have focused mainly on different visual obfuscation techniques. Encryption of 
+violation incidences is applied to protect sensitive personal information from being saved and transferred securely 
+(Diffie & Landau, 2010) . In the proposed system, an XOR operation with a (2 bit) encryption key is applied for every 
+detected violation incidence and saved in a secured folder in the ridesharing phone. The detected violation incidences 
+are only shared with ridesharing agents or official legal authorities. 
+
+6.3 
+Evaluating SAFEMYRIDES on Different Devices 
+In this section, the SAFEMYRIDES is applied to a different device to notice the discrepancy in the system 
+performance with different hardware specifications. In the previous section, the system was applied to a Samsung S10 
+plus device. In this section, the system will be applied to an LG V30 plus phone to compare and contrast the system's 
+performance on different hardware settings.  
+LG V30 plus has only two rear cameras (16 MP and 12 MP) and one 5MP front camera compared to three rear cameras 
+and two front cameras in the Samsung S 10 plus. The LG V30 Adreno 540 GPU is compared to an Adreno 640 GPU 
+in S10+. Also, the S10 + has a Li-Ion 4100 mAh Battery, and 8 GB RAM is compared to Li-Po 3300 mAh and 4 GB 
+in LG V30.  Table 7 and Fig. 7 compares the performance of SAFEMYRIDES on the two devices in terms of average 
+latency, average accuracy, and efficiency (the resources used) for running the system for 15 minutes. The comparison 
+is performed using the rear cameras because of higher resolution and night shots adjustment features. 
+While the Samsung S10 + achieved higher performance and less latency than The LG V30, the resources consumed 
+(battery and RAM were higher). Samsung S10 + has three cameras which consumed more battery power and RAM in 
+return. There is almost a 4 % difference in the accuracy between the LG V30 and Samsung S10 + and more than 350 
+ms difference in Latency. 
+Table 7:  Performance Comparison for Applying SAFEMYRIDES on Different Devices. 
+Metric/ Device 
+LG V30 (Rear Camera) 
+Samsung S 10 plus (Rear Camera) 
+Accuracy 
+0.8845 
+0.9019 
+Latency 
+1297 ms  
+920 ms. 
+RAM 
+89 MB 
+104 MB 
+Battery Consumption 
+48.7 mAH (milliamp hour) 
+79.8 mAH (milliamp hour). 
+Internal Storage 
+29.8 MB 
+28.09 MB 
+Average CPU 
+0.72 
+0.64 
+ 
+ 
+Fig. 8: Performance Comparison between LG and Samsung using Rear Camera. 
+ 
+
+Accuracy
+Latency
+FAM
+Baltery Consumption
+Intemal Storage 
+7 
+Discussion and Implications 
+While most of current edge computing approaches focus on optimizing processing and data transfer on edge units, 
+little research focus on empowering IoT to decrease dependency on task offloading. So, we developed a decentralized-
+control edge computing model where computation and decision making are moved to the IoT level. Moving the 
+computation to IoT (onsite) will enhance efficiency, responsiveness, accuracy, and privacy issues of many IoT 
+applications. The decentralized-control model is fueled by technologies such as TF lite and IoT optimization 
+techniques which empower IoTs to run cognitive tasks at humble consumption of resources. As a proof of concept, 
+we build a scene-aware ridesharing monitoring system to detect violations at runtime with an average latency of 620 
+ms. The system was evaluated technically and functionally using controlled experiments by stimulating different 
+violation poses in real ridesharing settings. 
+The system was designed to ensure safety and privacy. So, violation incidences are saved in hidden folders in an 
+encrypted format with an average file size of 120KB. The system responsiveness (latency) was comapred against 
+previously developed systems. The system achieved the least latency and reduced the cost of transferring images for 
+processing on edge nodes. Instead, the system detects violations locally on the driver’s smartphone.  Given the limited 
+number of training samples, the system achieved a 90.1% accuracy rate for detecting 51 different violations poses 
+with a 94% sensitivity rate. On the other hand, we can notice the significant difference in average inference time 
+(delay) between the full network of 3.56 seconds and the optimized network of 920 ms which gives optimized DL 
+networks an additional advantage to be applied for real time response.  
+When the system was applied to different devices, a variance in performance was detected which affected the detection 
+accuracy and latency. The higher is the resolution of the phone's camera, the better is the accuracy. Moreover, the 
+higher is the GPU/CPU, the less is the system latency. In addition, detection using rear cameras is better than front 
+cameras. Moreover, we noticed there is not much difference in detection accuracy during the day, but there is a 
+difference in detection accuracy at night. In general, the lower detection accuracy at night calls for increasing the 
+number of night training samples. Moreover, since most smartphone cameras don’t have night vision, the camera 
+ability to balance colors of night images affects detection accuracy. Also, the false-positive rate calls for training the 
+network with more regular ridesharing poses. The system is efficient to use in terms of storage (29 MB) and memory 
+consumption (less than 1% of active memory (RAM) consumption).  
+8 
+Limitations and Future Work 
+The implementation of the ridesharing monitoring system was performed as a proof of concept for the decentralized 
+control edge computing model. The size of the dataset used was limited. Increasing the training and validation samples 
+will positively affect the accuracy and the latency of predictions. In addition, the encryption of violation incidences 
+was performed using a simple XOR function. More sophisticated encryption methods using a 128-Bit key are 
+preferred. Also, additional privacy preservation mechanisms such as video inpainting and data-hiding schemes will 
+add extra confirmation on user privacy (Cheung et al., 2009). Since the system achieved lower accuracy and decreased 
+sensitivity in detecting violations during the night, the deep learning network needs to be trained with more night 
+samples, where poor light obstructs definite detection of violations. 
+Future research on the optimization of camera usage is required in terms of battery consumption as the energy 
+consumption of cameras is considered above average, assuming that a ridesharing driver will be using the system for 
+hours. Moreover, the system should be deployed in a real setting to address other intangibles quality measures such 
+as reliability, understanding, and courtesy. 
+Other future work includes implementing the ridesharing monitoring system on other platforms such as IOS and 
+Raspberry PI. The system should be tested on different platforms for benchmarking. Benchmarking will help 
+determine the minimum required for a smartphone to run the system. While the system used compression and 
+quantization techniques such as optimization methods for the Mobile SSD, applying other optimization techniques 
+such as pruning might enhance the optimized network accuracy.  
+The proposed system is trying to solve the safety and privacy issues related to ridesharing, and it has several 
+implications to practice. Firstly, the system allows local ridesharing monitoring on the driver's smartphone, saving the 
+cost of an additional device for monitoring. Having the monitoring system running on the same device used in the 
+ridesharing will ensure efficiency and privacy. Secondly, the system uses no network communication while detecting 
+violations and limits the network communication to reporting the violation incidences to the ridesharing agent. 
+
+Minimizing network communication ensures privacy, security, and efficiency. So, the system could be used where 
+poor network connections exist. Thirdly, encryption of ridesharing incidences ensures privacy and security. So, the 
+passenger/driver data are only shared directly with the ridesharing agent in case of detected violation. 
+This research contributes to the Information System research by applying the concept of decentralized control where 
+an IoT (smartphone) can independently detect violations and make decisions. The research is one of the early attempts 
+to apply different optimization methods to deep learning networks to run efficiently on an IoT. The research sheds 
+light on the importance of limiting cloud/edge computation and limiting network communications while enhancing 
+the decisions making process and analytics of the IoTs. 
+9 
+Conclusion 
+This research is introducing a decentralized-control edge computing model where computation and decision making 
+is moved to the IoT level. The model addresses the efficiency, responsiveness, and privacy issues that current edge 
+computing models still contemplate. Although battery powered, IoTs could run cognitive tasks at reasonable 
+consumption of resources using different optimization techniques. We applied the model to build a scene-aware 
+ridesharing monitoring system to detect violations at runtime. The system uses an optimized deep learning model on 
+IoTs that can locally detect violations in ridesharing sessions on the driver's smartphone. The system records violations 
+incidents only and saves them in an encrypted format. So, the system addresses the safety issue without violating 
+privacy. On the technical level, the decentralized control edge model is reducing communication cost and latency, 
+ensure privacy and less prone to data transfer security risks. 
+10 
+Declaration of interests 
+The authors declare the following financial interests/personal relationships which may be considered as potential 
+competing interests. This research was supported in part by 2019 Small Grants Award from Virginia Commonwealth 
+School of Business Information Systems Department. 
+ 
+11 
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+

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+CONTCOMMRTD: A DISTRIBUTED CONTENT-BASED
+MISINFORMATION-AWARE COMMUNITY DETECTION SYSTEM
+FOR REAL-TIME DISASTER REPORTING
+Elena-Simona Apostol1, Ciprian-Octavian Truic˘a1, Adrian Paschke2
+1University Politehnica of Bucharest, Bucharest, Romania
+2Fraunhofer Institute for Open Communication Systems, Berlin, Germany
+[email protected], [email protected], [email protected]
+[email protected]
+ABSTRACT
+Real-time social media data can provide useful information on evolving hazards. Alongside traditional
+methods of disaster detection, the integration of social media data can considerably enhance disaster
+management. In this paper, we investigate the problem of detecting geolocation-content communities
+on Twitter and propose a novel distributed system that provides in near real-time information on
+hazard-related events and their evolution. We show that content-based community analysis leads
+to better and faster dissemination of reports on hazards. Our distributed disaster reporting system
+analyzes the social relationship among worldwide geolocated tweets, and applies topic modeling
+to group tweets by topics. Considering for each tweet the following information: user, timestamp,
+geolocation, retweets, and replies, we create a publisher-subscriber distribution model for topics. We
+use content similarity and the proximity of nodes to create a new model for geolocation-content based
+communities. Users can subscribe to different topics in specific geographical areas or worldwide and
+receive real-time reports regarding these topics. As misinformation can lead to increase damage if
+propagated in hazards related tweets, we propose a new deep learning model to detect fake news.
+The misinformed tweets are then removed from display. We also show empirically the scalability
+capabilities of the proposed system.
+Keywords Distributed System · Spatial-Temporal System · Community Detection · Topic Modeling · Misinformation
+Detection
+1
+Introduction
+Worldwide, hazards frequently have a dramatic impact on rural and urban societies, or the environment. There are
+different classes of hazards, e.g., natural (geophysical, hydrological, etc.), anthropogenic (chemical, major accident,
+technological, biological, etc.). In recent years many studies have recognized the value of community activity during
+disasters due to hazards [58, 51]. Following these analyses, it was observed that people who are physically nearby a
+location where a hazard occurred tend to produce more disaster-related information on social media [6, 31].
+Alongside traditional methods of hazard detection, e.g., satellite or airborne sensing platforms, topographic data sources,
+Internet of Things, the integration of social media data would considerably enhance disaster management, especially
+in areas with little to no infrastructure, and, therefore, will be of greater importance in the future [49, 45]. Twitter
+is one of the largest social networking platforms, and it has a remarkable capability of continuous retrieval of data
+and content-sharing services [19]. It is also a suitable place where citizens can present their concerns in a real-time
+manner [16].
+Social media has a significant impact on public perception of different hazards and their contingency plans. This is
+why is extremely important not to consider tweets that contain misinformation. Although several papers examined the
+arXiv:2301.12984v1  [cs.SI]  30 Jan 2023
+
+negative impact of misinformation spread on Twitter during disasters that can cause mass panic or financial loss [59], in
+the current literature there is no work that mitigates their harm in real time.
+In this study, we attempt to answer a fundamental question: Can the social and geographical disparities of Twitter be
+used to assist citizens and government organizations in informing or disaster management? To answer this question,
+we consider classifying tweets in different disaster-related topics and construct a graph model based on content and
+geolocation. We also recognise the ethical issues caused by the fact that tweets collected from unreliable sources and
+showed to other users may increase the spread of misinformation [56]. Thus, we also apply fake news detection before
+displaying the data to the end user.
+In this paper, we 1) investigate the problem of detecting content-based communities in order to provide valuable
+information on hazard-related events and their evolution, and 2) propose a distributed content-based misinformation-
+aware community detection system for real-time disaster reporting (ContCommRTD). Besides the proposed system, the
+main contributions are:
+1) We propose a new distributed solution to analyze the social relationship among worldwide geolocated tweets.
+2) We define a new social network model for detecting geolocation-content based communities and named
+entities.
+3) We create a meta-knowledge dictionary to collect topic-specific tweets containing hazard-related keywords
+and hashtags.
+4) We apply topic modeling to discover topics and to classify tweets into different topics. A user can then
+subscribe to one of these topics and receive in real-time all the tweets that discuss the event of interest.
+5) We propose an efficient distributed publish-subscribe based model using Apache Kafka [18, 23] and Mon-
+goDB [8] that allows real-time collection, storage, and distribution of Twitter posts relevant to the chosen type
+of hazard.
+6) We propose a new Deep Learning architecture, FN-BERT-TFIDF, to determine if the tweets of interest are
+fake news.
+The rest of this paper is structured as follows: In Section 2, we present the state-of-the-art. In Section 4, we describe the
+proposed system. In Section 5, we present the datasets and analyze our results. Section 6 concludes the paper and hints
+at future research.
+2
+Related Work
+Real-time disaster detection using data from dynamic social media environments, e.g., Twitter, and considering different
+locations is a challenging task that recently received much attention [21, 22, 34]. [15] propose detecting disaster-related
+topics on Twitter data using an LDA (Latent Dirichlet Allocation) topic model enhanced with a set of seed words from
+older Tweets of the same geographic area. This solution is applicable when there is a single topic of interest, e.g.,
+earthquakes or hurricanes, and similar data is available. Several solutions use in the detection task, not only the textual
+data from tweets but also incorporate images from the posts or satellites [47, 24, 29]. [3] present an unsupervised
+learning solution that detects large-scale hazard-related sub-events in Tweets. The authors use an ontology containing
+crisis management vocabulary to rank the candidate sub-events and then cluster the most important sub-events using
+spectral clustering.
+Geolocation is also important when considering natural hazards analysis using social media data. To the best of our
+knowledge, the current real-time unsupervised disaster detection solutions consider only geotagged tweets [63, 64].
+However, tweets do not necessarily come with geoinformation. Several offline classifier-based disaster detection
+solutions also try to determine the location by searching the tweets’ content for names of cities and countries [33, 52].
+These solutions use corpora of labeled tweets. [33] propose using GeoText to search for names of cities and countries.
+Another solution proposes location mention recognition from labeled crisis-related tweets using BERT (Bidirectional
+Encoder Representations from Transformers [12]) based classification [52].
+In the current literature, different graph-based methods are applied to citizen science-based disaster detection solutions.
+Dou et al. [13] use a semantic graph-based topic detection method to identify fine-grained topics during natural disasters
+in social media. A community detection algorithm, i.e., the algorithm of modularity optimization [9], is also used to
+extract topics that denote the same class of event information. A similar pipeline is also proposed in article [36]: 1) a
+graph generation algorithm is used to transform the text data into a graph of the keywords, 2) a community detection
+algorithm is applied to discover hazard-related topics. Nguyen et al. [38] present a framework for 1) filtering and
+classifying tweets and 2) identifying and summarizing important disaster-related topics. The authors use a graph-based
+ranking algorithm to select and summarize important tweets.
+2
+
+Another issue to consider regarding this research topic is the impact of misinformation. Misinformation (false or
+inaccurate information) and disinformation (intentionally spreading misinformation) in the form of fake news are tools
+used to manipulate public opinion on particular topics, distort public perceptions, and generate social unrest while
+lacking the rigor of traditional journalism [26, 56]. Singh et al. [48] propose content and context aware RNN-based
+solution for fake news detection during natural disasters. Their solution uses the user profile information and the
+temporal and textual features of the analysed events. Pelrine et al. [40] analyze the performance for misinformation
+detection of several transformer language models (e.g., BERT [12], RoBERTa [32], ALBERT [30]) on different datasets.
+Based on their results, these transformers models have very good performance metrics for large-enough datasets.
+However, if we consider smaller datasets, such as the case of disaster-related Twitter datasets (e.g., COAID [10]), the
+performance decreases. Although several studies have analysed the impact of misinformation on different types of
+hazards, e.g., Hurricane [25], COVID-19 [17], none offer a real-time hazard-related event detection solution while
+removing misinformation.
+3
+Problem definition
+We define an undirected graph G = (V, E, C, L), where V is the set of vertices, E is the set of edges, C is the set that
+stores the textual content of each vertex, and L the set of geolocations for each node. Thus, for our problem, we define
+the social network as n undirected graphs Γ = {Gi = (Vi, Ei, Ci, Li)|u, v ∈ Vi ∧ (u, v) ∈ Ei ∧ cu, cv ∈ Ci ∧ lu, lu ∈
+Li ∧ i = 1, N}, where each two vertices u, v ∈ Vi are linked by through an edge (u, v) ∈ Ei that represents the social
+relationship between them. Thus, Γ is a graph with n disconnected components Gi. The content cu, cv of vertices u, v
+is stored in Ci, while their geolocation lu, lv are stored in Li. We define a social relationship as the linked that occurs
+between these nodes in either of the cases:
+1) u retweets the content of v,
+2) u replies to v.
+For each hazard related event, we have a Meta-Knowledge dictionary that contains a list H with the most relevant
+keywords and hashtags. The dictionary is constructed separately. Using H, we collect the data from the social network.
+For each record u, we verify if the location lu is given. If lu is missing, we use Name Entity Recognition (NER) to
+extract any mention of locations and then map the name with its coordinates (geolocation). Using these information
+together with the social relationships, we construct the list of n undirected graphs Γ, i.e., the social graph.
+Algorithm 1 presents the collection and creation of the social graph Γ, which receives as input the list of relevant
+keywords and hashtags for an event H. Lines 1 to 2 initializes the social graph Γ and the disconnected graph Ω used for
+extracting the connected components Gi. Lines 3 to 6 initializes the components of Ω The records are collected into R
+using a Social Network API (Line 7). For each record u ∈ R, we add elements to the V, C, L components of Ω (Lines 8
+to 15). Line 10 applies some preprocessing techniques on the textual content. Lines 12 to 14 determines the coordinates
+based on locations if the geolocation is not provided in the record. The edges are added separately after we have all the
+nodes (Lines 16 to 19). We construct Ω (Line 20), extract the connected components to build Γ (Line 21), and return
+the social graph (Line 22).
+Using the graph structures in Γ, we want to determine communities where there is a high social media activity for
+users’ topics of interest. These communities can contain multiple graphs Gi ∈ Γ that are not directly interconnected.
+To build the communities, first we filter the content of a graph Gi and remove any nodes and their edges that spread
+misinformation. At the end of the misinformation detection process we obtaining Γ′ that contain only the clean
+undirected graphs G′
+i.
+Algorithm 2 presents the misinformation detection algorithm, which receives as input the social graphs Γ and outputs
+the clean graph Γ′. After the clean graph is initialized Γ′ (Line 1), the veracity of all the nodes’ content from a graph
+G ∈ Γ is verified (Lines 2 to 12). If the the content is deemed as Fake, the record is removed form the graph (Lines 4
+to 10. The corresponding graph G′
+i is constructed using the updated V ′, E′, C′, L′ and it is added to Γ′ (Lines 11 to 12).
+When the verification are finished for all the graphs, the algorithm returns the clean graph Γ′.
+With the clean content Γ′, we build the content-based communities. In order to achieve this, we utilize:
+1) the content similarity between the nodes given the membership level m of the content of a node cu to a topic
+tk, and
+2) the proximity of nodes to a core point p that defines a geographic area A.
+3
+
+Algorithm 1: SocialGraph - Construct the social graph
+Input
+:the keywords and hashtags list H
+Output :the undirected social graph Γ
+1 Γ ← ∅
+2 Ω ← ∅
+3 V ← ∅
+4 E ← ∅
+5 C ← ∅
+6 L ← ∅
+7 R ← SocialNetworkAPI(H)
+8 foreach u ∈ R do
+9
+V ← V ∪ {u}
+10
+c′
+u ← Preprocess(cu)
+11
+C ← C ∪ {c′
+u}
+12
+if lu = Nil then
+13
+λ ← NER(cu)
+14
+lu ← GeoLocation(λ)
+15
+L ← L ∪ {lu}
+16 foreach u ∈ V do
+17
+foreach v ∈ V do
+18
+if Retweet(u, v) ∨ Reply(u, v) then
+19
+E ← E ∪ {(u, v)}
+20 Ω = (V, E, C, L)
+21 Γ ← ConnectedComponents(Ω)
+22 return Γ
+Algorithm 2: MisinformationDetection - Content veracity detection
+Input
+:the social graphs Γ
+Output :the clean undirected graph Γ′
+1 Γ′ ← ∅
+2 foreach G = (V, E, C, L) ∈ Γ do
+3
+C′ ← C
+4
+foreach cu ∈ C do
+5
+veracity ← DetectV eracity(cu)
+6
+if veracity = Fake then
+7
+V ← V \ {u}
+8
+E ← E \ {(u, v)|(u, v) ∈ E}
+9
+C′ ← C′ \ {cu}
+10
+L ← L \ {lu}
+11
+G′ = (V, E, C′, L)
+12
+Γ′ = Γ′ ∪ {G}
+13 return Γ′
+To compute the content similarity, we first use a topic modeling algorithm to extract k topics T within our graphs. Then
+we determine the membership of a node’s content cu to belong to a topic tj ∈ T (j = 1, k) using the cosine similarity
+sim(cu, tj). Given the similarity sim(cu, tj) between a node’s content and a topic, if the similarity is over a given
+threshold εc (sim(cu, tj) ≤ εc) then cu belongs to topic tj. Based on the threshold, the same node’s content can belong
+to multiple topics. Using the topics T and the graph Γ′, we construct the topic graphs Θ.
+Algorithm 3 presents the topic graphs extraction. The algorithm receives as input the the undirected clean graphs Γ′,
+the number of topics k, and the similarity threshold εc. The output is the topic graphs Θ. Line 1 initializes Θ. Lines 2
+to 5 concatenates the contenct of all the graphs G and extract k topics stored in T. We iterate trough each graph and
+each topic to determine the membership of the content of a node to a topic using the similarity and create a topic graph
+G′ which is added to the topic graphs Θ (Lines 6 to 19). At the end of the iteration, the topic graphs Θ are returned
+(Line 20).
+4
+
+Algorithm 3: TopicGraphs - Topic graphs extraction
+Input
+:the undirected clean graphs Γ′
+the number of topics k
+the similarity threshold εc
+Output :the topic graphs Θ
+1 Θ ← ∅
+2 D ← ∅
+3 foreach G = (V, E, C, L) ∈ Γ′ do
+4
+D ← D ∪ C
+5 T ← Topics(D, k)
+6 foreach t ∈ T do
+7
+V ′ ← ∅
+8
+E′ ← ∅
+9
+C′ ← ∅
+10
+L′ ← ∅
+11
+foreach G = (V, E, C, L) ∈ Γ′ do
+12
+foreach cu ∈ C do
+13
+if sim(cu, t) ≥ εc then
+14
+V ′ ← V ′ ∪ {u}
+15
+E′ ← E′ ∪ {(u, v)|(u, v) ∈ E}
+16
+C′ ← C′ ∪ {cu}
+17
+L′ ← L′ ∪ {lu}
+18
+G′ = (V ′, E′, C′, L′)
+19
+Θ ← Θ ∪ {G′}
+20 return Θ
+The proximity a node u to a core point p is computed using its geolocation coordinates lu. The intuition behind this
+assumption is based on the fact that nodes that are near each other will fall within the same geographic area A. Each
+geographic area A represents a cluster. The area A is given by a core point p and a radius εl that specifies the cluster’s
+maximum extend. Thus, the proximity of a node u is computed as distance δ(lu, p) between its geolocation lu and
+the core point p. If δ(lu, p) ≤ εl then u ∈ A. The area A and the core point p is determined using a data clustering
+algorithm, e.g., DBSCAN [14]. Using the content similarity and the proximity of nodes, we create geolocation-content
+based communities Σ = (VΣ, EΣ, CΣ, LΣ). Within the same area, there can be multiple communities.
+Algorithm 4 presents the construction of the geolocation-content based communities. The algorithm receives as input
+the topic graph θ ∈ Θ and the proximity threshold εl and outputs the communities Σ. Line 1 initializes Σ. Lines 2 to 5
+extract all the geolocations from θ and determines the areas A for which the radius is equal to εl. We iterate trough each
+graph and each area to construct the communities Σ using the condition δ(lu, p) ≤ εl (Lines 6 to 19). At the end of the
+iteration, the geolocation-content based communities Σ are returned (Line 20).
+Algorithm 5 presents ContCommRTD, the solution to our problem. The algorithm receives as input the the keywords
+and hashtags list H, the number of topics k, the similarity threshold εc, and the proximity threshold εl. Firstly, the
+social graph Γ is build using H and Algorithm 1 (Line 2). Secondly, the graph Γ′ is constructed by removing from
+Γ all the nodes that contain misinformation (Line 3). Thirdly, the topic graphs are extracted (Line 4). Lastly, the
+geolocation-content based communities Σ are determined for each topic graph and returned (Lines 5 to 7).
+4
+System Description
+The architecture of the proposed system ContCommRTD, which implements Algorithm 5, is presented in Figure 1.
+4.1
+Twitter Data Collector
+This module is used to fetch Twitter data in real-time, using the Twitter Developer API and it implements Line 7 from
+Algorithm 1. It collects tweets based on the chosen type of hazard, e.g., extreme hydrological hazards associated with
+water-related events. The initial selection is done by searching tweets that contain specific words. For this purpose, we
+created a meta-knowledge dictionary as follows. For each new type of hazard, we add a new entry containing a list of
+5
+
+Algorithm 4: CommunityGraphs - Geolocation-content based communities extraction
+Input
+:the undirected topic graph θ
+the proximity threshold εl
+Output :the communities Σ′ = (VΣ, EΣ, CΣ, LΣ)
+1 Σ′ ← ∅
+2 P ← ∅
+3 foreach (V, E, C, L) ∈ θ do
+4
+P ← P ∪ L
+5 A ← Areas(P, εl)
+6 foreach p ∈ A do
+7
+V ′ ← ∅
+8
+E′ ← ∅
+9
+C′ ← ∅
+10
+L′ ← ∅
+11
+foreach (V, E, C, L) ∈ θ do
+12
+foreach lu ∈ L do
+13
+if δ(lu, p) ≤ εl then
+14
+V ′ ← V ′ ∪ {u}
+15
+E′ ← E′ ∪ {(u, v)|(u, v) ∈ E}
+16
+C′ ← C′ ∪ {cu}
+17
+L′ ← L′ ∪ {lu}
+18
+G′ = (V ′, E′, C′, L′)
+19
+Σ′ ← Σ′ ∪ {G′}
+20 return Σ′
+Algorithm 5: ContCommRTD - Geolocation community graphs extraction
+Input
+: the keywords and hashtags list H
+the number of topics k
+the similarity threshold εc
+the proximity threshold εl
+Output :the communities Σ = (VΣ, EΣ, CΣ, LΣ)
+1 Σ ← ∅
+2 Γ ← SocialGraph(H)
+3 Γ′ ← MisinformationDetection(Γ)
+4 Θ ← TopicGraphes(Γ′, k, εc)
+5 foreach G = (V, E, C, L) ∈ Θ do
+6
+Σ ← Σ ∪ CommunityGraphs(G, εl)
+7 return Σ
+Figure 1: Architecture of ContCommRTD (database colors denote the collections used by each module)
+top-k keywords and hashtags for the chosen hazardous event. To extract the keywords we employ TLATR [55] using
+the following pipeline:
+6
+
+Twitter Data
+Even Aggregator and User Interface
+Collector
+Kafka Cluster
+User Interface
+Zookeper
+Data
+Kafka Broker
+Preprocessing
+Kafka
+/Community Detection &
+Topic 1
+Consumer 1
+Community Graphs
+Misinformation
+Detection
+Kafka
+Topic 2
+Text Analysis &
+Consumer 2
+Topic Graphs
+Kafka
+t2
+tk
+tweets t1
+Kafka
+Producer
+Topic K
+Consumer K1) extract the topics and keywords including hashtags
+2) label the topics.
+When users searcher for new events, they will select from a list of topics’ labels and the system will automatically load
+all the topic’s keywords and hashtags, which in turn are used to filter the stream of tweets. The collected tweets are
+stored in tweets collection in a NoSQL Document-Oriented Distributed Database Management system, i.e., MongoDB.
+We choose MongoDB, because benchmarks show that this system is fast, reliable, and offers good performances when
+dealing with textual data [53, 54].
+Using these filters, we collect the tweet and its retweets and replies. For a tweet, we store the following information: id,
+language, retweet flag, creation date, user information, geolocation information, and text. For a retweet, we also keep
+the retweeter’s coordinates and location. Each retweet information is also updated in real-time with the original tweet’s
+data. The same information is also kept for replies. We use geolocation as a sharding key to distribute the data between
+different sites and improve querying.
+4.2
+Data Preprocessing and Geolocation enhancement
+This module implements constructs the social graph Γ implemented by Algorithm 1.
+4.2.1
+Data Preprocessing.
+We extract the textual content of all the Tweets from the database and apply the following preprocessing steps to extract
+a clean text:
+1) remove user mentions (terms that start with @), http links, punctuation, double spacing, and numbers from the
+text;
+2) Transform the text to lowercase;
+3) Extract terms through tokenization;
+4) Remove stop words and tokens with a length smaller than 3 characters;
+5) Extract the stem of the remaining terms using the Porter Stemmer [41] to reduce the number of terms by
+removing inflected and derived words, thus minimizing the vocabulary.
+Using the clean text and the normalized Term Frequency-Inverse Document Frequency TFIDFn, we built the
+document-term matrix W = [wij] needed by the analysis algorithm, where wij = TFIDFn(tij, di, D), tij is a
+term appearing in document di, and D is the entire collection of tweets. We use the scikit-learn [39] TFIDF
+implementation.
+4.2.2
+Analyzing the Geolocation of Tweets
+We utilize several techniques to obtain the geolocation information of a tweet, retweet, or reply. The most basic
+approach was to get the coordinates of a tweet. For this to work, the user must have the location turned on. There are
+various studies (e.g., [50]) that show that only a relatively small percentage of users use this feature. However, we
+observe that a large majority of the collected hazard-related tweets are either geotagged or contain location information
+in the text, e.g., "Heavy rain in Masjid Al Haram". A tweeter that contains a Place has 4 pairs of coordinates that
+define the area for that Place. If the tweet doesn’t have a geolocation tag either, we apply Named Entity Recognition to
+identify possible locations from the text. Once identified, we use GeoNames [61] to get the coordinates. This is an
+improvement over many existing real-time systems that solely rely on geolocation information embedded within the
+tweet to determine events [63, 64].
+4.3
+Misinformation Detection
+This module is used to filter and mark any tweets that spread misinformation, i.e., Fake News (FN). This module
+implements Algorithm 2. As the Twitter datasets that contain disaster-related events have usually a limited size, we
+use Transfer Learning techniques to transfer knowledge gained from these larger but more generic Twitter datasets
+to misinformation detection on disaster-related datasets. For the task of misinformation detection, we propose a new
+Deep Learning architecture, FN-BERT-TFIDF (Figure 2a), that receives as input the TF-IDF vector for a tweet as
+well as tweet embedding extracted with BERT. For each input, the model contains one BiLSTM layer, a CNN layer,
+and a MaxPooling layer. We choose a BiLSTM layer because it enables the network to use both previous and future
+elements by looking forward and backward in the words sequence. We use a CNN layer to create new features using
+7
+
+the convolution operation between the text window and every distinct filter. We use a MaxPooling layer to decrease the
+size of the feature channels by grouping elements into fix lengths sequences and choosing only the feature with the
+maximum value. The output of the MaxPooling layers are concatenated and sent as input to the classification layer. To
+test the efficiency of our model, we perform ablation testing and compare the results of the FN-TFIDF-BERT model
+with the results obtained when only TF-IDF, i.e., FN-TFIDF (Figure 2b), or BERT, i.e., FN-BERT (Figure 2c), is used
+as input. After a tweet passes throughout the detection model, its record in MongoDB is updated with a new field that
+encodes its veracity. If a tweet is detected as fake, then it will not be used in the next modules. For implementing the
+Deep Learning models, we use the Keras interface of the TensorFlow [1]. For building the BERT embeddings, we use
+simpletransformers with the HuggingFace [62] BERT model.
+(a) FN-BERT-TFIDF
+(b) FN-TFIDF
+(c) FN-BERT
+Figure 2: Deep Learning Architectures
+4.4
+Text Analysis and Topic Graphs
+This module implements Algorithm 3 to extract the topic graphs. To Analyze the tweets and determine the context
+regarding the hazard discussed in the text, we use Online Latent Dirichlet Allocation (OLDA) [2]. OLDA is a generative
+statistical model used for topic modeling that groups together terms that are syntactically different but have similar
+meaning and represent the same concepts. The algorithm determines for each tweet a specific topic by calculating the
+similarity between the tweet and all of the topics. Thus, OLDA assigns a tweet to a mixture of topics, i.e., each tweet is
+a combination of one or more topics. The model is built in real-time, as more tweets are added to the database, the
+model is retrained with the new information. We use a 50% threshold (i.e., εc) to determine if a tweet is relevant to a
+topic as a tweet can belong to multiple topics. We use the gensim [43] OLDA implementation.
+4.5
+Kafka Modules
+The Kafka Producer module takes <tweet, topic> pairs and sends them to the Kafka Consumers. Kafka Cluster is used
+to create an environment that assures the distributed exchange of messages between the Kafka Producer and the Kafka
+Consumers. It contains the Kafka Broker and the Zookeeper service. The job orchestration and cluster topology is done
+by the Zookeeper service, a distributed coordination service for distributed applications [4]. The Kafka Broker’s main
+role is to act as a transition channel. It takes messages in the form of <tweet, topic> pairs and delivers them to the right
+Kafka Consumer. Thus, the receiving of data by the Kafka Consumers is done in parallel in real-time in a non-blocking
+way. Each Kafka Consumer stores messages in their own topic-dependent collections within MongoDB and stores
+<topic, [(tweet_ids, geolocation)]> pairs, i.e., a topic and a list of all the related tweets ids with their geolocation. The
+geolocation is used for sharding.
+4.6
+Community Graphs and User interface
+This module is used to present in real-time to users one or more topics of interest from one or more hazard related fields
+on Google maps (Figure 3a). Users can choose and subscribe to desired topics (Figure 3b) and can monitor a specific
+area on the map just by searching for it. For each subscribed topic, the user sees on the map a colored pins updated
+automatically in almost real-time. A user can also view the formed communities based on the chosen topics (Figure 3c).
+Each pin represent a community for a specific topic within the social graph. To form these communities, we use
+Algorithm 4 with DBSCAN (Density Based Spatial Clustering of Applications with Noise) [14], a density-based
+8
+
+TF-IDF
+BiLSTM
+CNN
+MaxPooling
+Concatenate
+BERT
+BiLSTM
+CNN
+MaxPooling
+DenseTF-IDF
+BiLSTM
+CNN
+MaxPooling
+DenseBERT
+BiLSTM
+CNN
+MaxPooling
+Dense(a) Google map view
+(b) Subscriptions
+(c) Communities per topic
+Figure 3: User interface
+clustering algorithm. DBSCAN computes the density of points (core points) p around which we form the area A. Thus,
+the clustering algorithm measures density as the number of points within the radius εl of a point from the analyzed
+dataset. It can estimate the connected components of the λ-density level set x : f(x) ≥ λ given n samples from an
+unknown density f [27].
+The user has access to the <topic, tweets> pairs. The topic is extracted from the MongoDB based on the user’s
+subscribed topics. By matching the tweet_ids presented in the tweets and topic collections, we extract all the geographical
+coordinates and group them together using DBSCAN. Thus, the same topic can appear multiple times on the map
+in different geographic locations. For these geographic locations, we present to the user the latest information as the
+tweets are updated in real-time by the Twitter Data Collector module. This component employs strong synchronization,
+in order to retrieve information from the database in near real-time.
+If a user unsubscribes from a topic, then the corresponding pins are removed from the map. We use timestamped
+matching queries to retrieve the relevant information. The pins for the topics that have not beet updated in more than
+24h are removed automatically from the map. Their topics are kept in tweets collection for retraining and improving
+performance of the OLDA model.
+4.7
+Design Challenges
+With real-time data stream collection systems, it is important to take into account that the system may be disconnected
+from the service. To mitigate this challenge, we implemented a background process checker that verifies every 30
+seconds that the Twitter Data Collection is working. Also, Twitter is limiting the number of tweets that can be collect in
+an hour. We address this challenge by using filters to select only relevant tweets.
+To build the topic graph, we use OLDA. The reason behind choosing OLDA, instead of LSI or NMF, is that OLDA is
+a probabilistic model while NMF and LSI are a matrix factorization and multivariate analysis techniques. With the
+increase of the volume of collected data, the runtime performance of OLDA can decrease. Thus, we systematically
+start in background a new OLDA training process each hour. When a training process is finished, the new model is
+uploaded in the application and the old one is removed from production and archived. Furthermore, we also use the
+same approach to fine-tune the misinformation model to improve performance. We use the same strategy for building
+the community graph with DBSCAN as we do for constructing the topic graph with OLDA. We use DBSCAN as we
+do not need to know apriori the number of clusters. We use sharding for our database to increase localization, load
+balancing, and querying through geodistribution. For all the collections, we use the geolocation coordinates as the shard
+key. We also use replication to create three member replica sets for each shard. Through this mechanism, we increase
+data availability, add fault and partition tolerance, and eliminate the single point of failure. These design choices are
+also used to seamlessly scale the datasets horizontally.
+9
+
+La Verendry
+Wildlife
+Reserve
+Algonquin
+Manitoulin
+Provincial
+Island
+Park
+SCONSIN
+MICHIGAN
+NEWYOR
+PENNSYLVA
+LLINOIS
+INDIANA
+OHIO
+MARYLAND NE
+WEST
+DELA
+VIRGINIA
+KENTUCKY
+VIRGINIATopic 1
+Topic 2
+Topic 3
+Topic 4
+Topic 55
+Evaluation
+In this section, we present the experimental results of our proposed system, ContCommRTD. We first focus on the
+ablation testing for the misinformation detection models. Secondly, we present two use cases of ContCommRTD for
+the disaster reporting:
+1) Hydrological Hazards and
+2) COVID-19 Infection Hazards.
+We conclude this section with ContCommRTD scalability tests.
+5.1
+Data Collection
+We collect two datasets for the two use cases. First, for the Hydrological Hazards use case we collect 356 483 Tweets
+using the hydrological data dictionary and store them in the distributed MongoDB database. This dataset is also used to
+train the first OLDA model and to evaluate the algorithm’s runtime performance and topic quality for the first. Second,
+for the COVID-19 Infection Hazards we collect 50 230 and present the community detection results. Table 1 presents a
+subset of the used keywords and hashtags for both hydrological and COVID-19 use cases.
+Table 1: Example of filter keywords (subset)
+Keywords
+flood
+flowage, rain, precipitation, floodplain, groundwater, overflow, deluge
+water level, water flow, rainfall, inundation, torrent, groundwater flood
+tsunami, torential, costal flooding, costal storm, river flooding, hurricane, ...
+Hashtags
+flood
+#flood, #precipitation, #rainfall, #deluge, #torrent, #inundation, #rain
+#floods, #waterlevel, #hurricane, #tornado, #torential, #storm, #flowage, ...
+Keywords
+Covid-19
+coronavirus, corona, COVID, COVID-19, pandemic, quarantine, lockdown,
+corona virus, hand sanitizer, infection, wash your hands, mask,
+personal protective equipment, covid quarentined, intensive care ...
+Hashtags
+Covid-19
+#covid, #sarscov2, #corona, #vaccine, #stayhealthy, #stayathome,
+#SARSCoV2 #viruscorona, #quarentined, #quarentinelife, ...
+5.2
+Evaluation Metrics
+5.2.1
+Misinformation Detection
+To evaluate the quality of the Deep Learning architectures employed for Misinformation Detection, we use Accuracy,
+Precision, and Recall.
+5.2.2
+Topic Modeling
+To evaluate the quality of the topic model, we employ perplexity and topic coherence [37]. Perplexity is a measure that
+determines how well a probability model predicts a sample [20]. A low perplexity denotes that the distribution predicts
+correctly the sample. For topic modeling, an algorithm that achieves a low perplexity indicates that it fit the data better.
+Topic Coherence measures the human-interpretability of a topic [35]. We use the CV [44] to measure the coherence of
+our topics. For perplexity, we use gensim’s log_perplexity() implementation, while for CV we use palmetto [44].
+5.2.3
+Community Detection
+To evaluate the quality of the communities, we employ Davies-Bouldin [11], Calinski-Harabasz [5], and Silhouette [46].
+The Davies-Bouldin score measures the separation between clusters by computing the ratio between within-cluster
+distances and between-cluster distances which determines the average similarity of each cluster with its most similar
+cluster. The Davies-Bouldin score bounded in the range [0, 1], whit a score closer to 0 showing better separation. The
+Calinski-Harabasz score measures how well are the clusters defined by computing the ratio between the within-cluster
+dispersion and the between-cluster dispersion. A higher Calinski-Harabasz shows core shows clusters that are dense
+and well separated. Unfortunately, the Calinski-Harabasz score is not bounded. The Silhouette score is another
+measurement for determining how well are the clusters defined by computing the mean intra-cluster distance and the
+mean nearest-cluster distance. The Silhouette is defined in [−1, 1] range, with scores closer −1 for incorrect clusters,
+scores closer to +1 for highly dense clusters, and scores around 0 for overlapping clusters.
+10
+
+5.3
+Misinformation Detection
+To test the proposed models, we use two publicly available dataset: Covid-19 dataset [7] and LIAR [60] using 2 labels,
+i.e., fake or real. For the experiments, we use a 70%-30% train-test split with random shuffle, while maintaining the
+the class ratio for the two sets. We use 256 units for the BiLSTM with a dropout of 0.2. For the CNN layer, we use
+64 filters and a kernel size of 128. We used a 5 000-dimensions TF-IDF vector and the pretrained 1 024-dimension
+BERT transformer from HuggingFace [62], i.e., bert-large-uncased. We observed that FN-BERT-TFIDF outperforms
+the other two models on both dataset (Table 2).
+Table 2: Misinformation ablation results
+Dataset
+Model
+Accuracy
+Precision
+Recall
+F1 Score
+LIAR
+FN-TFIDF
+56.01 ± 0.33
+55.81 ± 0.52
+55.85 ± 0.54
+55.83 ± 0.53
+FN-BERT
+58.75 ± 0.44
+57.97 ± 0.50
+57.09 ± 0.46
+57.53 ± 0.48
+FN-BERT-TFIDF
+60.92 ± 0.59
+60.65 ± 0.90
+60.66 ± 0.97
+60.65 ± 0.93
+LSTM [42]
+∼58
+N/A
+N/A
+N/A
+Covid-19
+FN-TFIDF
+81.93 ± 0.37
+78.48 ± 0.64
+76.90 ± 0.11
+77.68 ± 0.19
+FN-BERT
+86.86 ± 0.07
+86.86 ± 0.07
+86.86 ± 0.07
+86.86 ± 0.07
+FN-BERT-TFIDF
+87.92 ± 0.04
+87.91 ± 0.02
+87.92 ± 0.03
+87.82 ± 0.02
+VAE + MLP [7]
+N/A
+N/A
+N/A
+85.98 ± 0.10
+5.4
+Topic and Communities detection
+When training OLDA, we need to initialize two parameters:
+1) Alpha the document-topic density (a larger value means that a document contain a lager number of topics), and
+2) Beta the topic-word density (a larger value means that more words are considered to belong to the same topic).
+We initialize these values to "auto" in order to learn them automatically from the corpus. After training the OLDA
+model 10 times on the initial corpus, we obtain an average runtime of 17min.
+5.4.1
+Use case 1: Hydrological Hazards.
+We extract 3 topics (Table 3) and evaluate the quality of the model using perplexity, obtaining a score of 7.83 for the
+entire dataset. This score shows that OLDA manages to predict well the data sample from our dataset. Finally, we
+compute the CV score to determine topic coherence and human-interpretability. For the OLDA model train on the initial
+corpus we obtain a CV = 0.48, meaning that the topics are readable by users. The evaluation of the community graphs
+shows a good separation and well-bounded clusters. The scores we obtain are the following: 0.0286 Davies-Bouldin,
+∼ 3.60e9 Calinski-Harabasz, and 0.9758 Silhouette.
+Table 3: Example of OLDA topics for hydrological hazards
+Topic
+Keywords and importance
+1
+rain
+wind
+temp
+disaster
+humidity
+weather
+0.133
+0.039
+0.025
+0.022
+0.015
+0.011
+2
+hurricane
+water
+overflow
+help
+time
+authority
+0.044
+0.034
+0.021
+0.019
+0.018
+0.018
+3
+thunderstorm
+severe
+warning
+storm
+tornado
+county
+0.053
+0.046
+0.044
+0.032
+0.019
+0.018
+Table 4 presents some examples of tweet-topic distribution.
+5.4.2
+Use case 2: COVID-19 Infection Hazards.
+As in the first use case, we train an OLDA model to create geolocation-content based communities on COVID-19
+related topics. Figure 4a depicts the communities for three major Coronavirus related topics, while Figure 4b shows the
+word cloud for the "blue" topic from Figure 4a (#1 in Table 5). We extract 3 topics (Table 5) for which we obtain a
+perplexity score of 9.01 and a CV score of 4.8 for the entire dataset. We obtain the following score for the community
+graphs: 0.0197 Davies-Bouldin, ∼ 2.62e9 Calinski-Harabasz, and 0.9853 Silhouette. By evaluating these scores, we
+observe that we obtain well separated and bounded clusters.
+11
+
+Table 4: Example of matching tweets to topics
+Topic
+Tweet
+1
+2
+3
+4
+5
+6
+After a hurricane, a guy found this pittie on
+the roof of a submerged car waiting for help
+0.02
+0.02
+0.02
+0.02
+0.90
+0.021
+Happening now: A 33,000 litres of truck laden
+with diesel was prevented from causing another
+national disaster by Officers and men of Federal
+Fire Service at Ojuelegba bridge. Situation under
+control and normalcy restored.
+0.0
+0.0
+0.0
+0.96
+0.0
+0.0
+(a) Communities
+(b) Word cloud for "blue" community
+Figure 4: COVID-19 Infection Hazards
+Table 5: Example of OLDA topics for COVID-19 hazards
+Topic
+Keywords and importance
+1
+cases
+home
+intensive
+care
+#coronavirus
+quarantine
+0.090
+0.080
+0.080
+0.079
+0.072
+0.070
+2
+coronavirus
+covid-19
+hospital
+virus
+wuhan
+chinese
+0.116
+0.078
+0.035
+0.035
+0.035
+0.034
+3
+#coronavirus
+outbreak
+health
+vaccine
+spread
+died
+0.198
+0.119
+0.090
+0.080
+0.072
+0.069
+5.5
+Scalability
+To test the scalability of the proposed system, we simulate multiple scenarios that simultaneously use two types of
+processes in a pseudo-distributed environment. The first type of process connects to the MongoDB database, inserts a
+new tweet in the tweets collections, and updates its topic-dependent collection. The second type of process simulates
+multiple clients that count all the tweets present at a given moment for a topic of interest. This set of experiments are
+run on an IBM System x3550 M4 with 64GB of RAM and an Intel(R) Xeon(R) CPU E5-2670 v2 @ 2.50GHz with 40
+cores.
+Table 6 presents the results for 10 executions. For a tiny number of write and read operations, i.e., the number of tweets
+writes ≤ 1 000 and the number of clients reads ≤ 1 000, the system is stable and handles the burst of requests in under a
+second. In this case, the maximum response time of 0.55 seconds is registered for 1 000 write and 1 000 read operations.
+As the number of operations increases, the system handles the request in under 1.5 minutes. We note that the probability
+of receiving a high number of tweets and having a large number of client requests, even during a disaster, is very low.
+12
+
+gas realising europelikely fargate opening
+said
+lookcase milanoyrdon
+n old fast
+like china
+store.
+del
+ speaklspy countryTable 6: Runtime in seconds: write vs. read operations
+No. Clients
+100
+1 000
+10 000
+100 000
+No. Tweets
+100
+0.13 ± 0.01
+0.54 ± 0.01
+4.49 ± 0.16
+44.10 ± 0.30
+1 000
+0.36 ± 0.01
+0.55 ± 0.01
+4.43 ± 0.05
+43.67 ± 1.21
+10 000
+3.45 ± 0.11
+3.59 ± 0.03
+4.80 ± 0.02
+48.05 ± 0.05
+100 000
+33.37 ± 0.48
+34.60 ± 0.07
+34.83 ± 0.77
+87.67 ± 1.04
+6
+Conclusions
+In this paper, we present ContCommRTD, a new distributed system that determines geolocation-content based com-
+munities depending on the topics of interest and user geolocation, and takes into account misinformation on social
+networks. Moreover, we have also shown how our system can be applied to track the evolution of different hazards.
+This is of significant importance, especially for government organizations that can subscribe to topics of interest in a
+specific geographical area or can subscribe to receive information on a topic regardless of the location. Our system also
+provides an interactive graphical interface where a user can select what are the topics of interest. On the user map, the
+notifications within the relevant topics will appear colored. All the notifications that came from the detection of other
+information apart from the chosen topics are displayed in gray color. Furthermore, our approach can be generalized
+for other types of hazards or social events if new terms are added to the dictionary used to collect specific tweets. As
+tweets collected from unreliable or unchecked sources may spread misinformation, we consider detecting these tweets
+and removing them from our active communities. For this task, we propose FN-BERT-TFIDF, a new Deep Learning
+BERT-based model. Moreover, the proposed system scales well as the number of read and write operations increases.
+As future work, we plan to add more information in order to analyze human behaviors during disasters, as follows.
+Between the filtering and preprocessing process, the tweet could undergo a more detailed analysis, such as domain-
+specific rule-based filtering and disambiguation, and text-based sentiment analysis [34, 28, 57].
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+ACM Transactions on Intelligent Systems and Technology, 9(3):1–24, feb 2018. doi: 10.1145/3066166.
+16
+

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+On Excess Mass Behavior in Gaussian Mixture
+Models with Orlicz-Wasserstein Distances
+Aritra Guha⋆
+Nhat Ho†
+XuanLong Nguyen⋆
+Department of Statistics, University of Michigan⋆
+Department of Statistics and Data Sciences, The University of Texas at Austin†
+January 30, 2023
+Abstract
+Dirichlet Process mixture models (DPMM) in combination with Gaussian kernels
+have been an important modeling tool for numerous data domains arising from
+biological, physical, and social sciences. However, this versatility in applications does
+not extend to strong theoretical guarantees for the underlying parameter estimates,
+for which only a logarithmic rate is achieved. In this work, we (re)introduce and
+investigate a metric, named Orlicz-Wasserstein distance, in the study of the Bayesian
+contraction behavior for the parameters. We show that despite the overall slow
+convergence guarantees for all the parameters, posterior contraction for parameters
+happens at almost polynomial rates in outlier regions of the parameter space. Our
+theoretical results provide new insight in understanding the convergence behavior
+of parameters arising from various settings of hierarchical Bayesian nonparametric
+models. In addition, we provide an algorithm to compute the metric by leveraging
+Sinkhorn divergences and validate our findings through a simulation study.
+1
+Introduction
+From their origin in the work of Pearson Pearson (1894), mixture models have been widely
+used by statisticians McLachlan & Basford (1988); Lindsay (1995); Mengersen et al. (2011)
+in variety of modern interdisciplinary domains such as medical science Schlattmann (2009),
+bioinformatics Ji et al. (2005), survival analysis Tsodikov et al. (2003), psychometry Gu et al.
+(2018) and image classification Permuter et al. (2006), to name just a few. The heterogeneity
+in data populations and associated quantities of interest has inspired the use of a variety of
+kernels, each with its own advantages and characteristics. Gaussian kernels are particularly
+popular in various inferential problems, especially those related to density estimation and
+clustering analysis Kotz et al. (2001); Bailey et al. (1994.); Roeder & Wasserman (1997); Robert
+(1996); Banfield & Raftery (1993). In addition to the choice of kernels, the Bayesian mixture
+modelers are also guided by the selection of prior distributions for the quantities of interest. In
+particular, Bayesian nonparametric priors (BNP) for mixture models are increasingly embraced,
+thanks to computational ease and the modeling flexibility that these rich priors entail Escobar
+& West (1995); MacEachern (1999).
+On the theoretical front, convergence rates for (Gaussian) mixture models received extensive
+treatments in the Bayesian paradigm Ghosal et al. (2000); Barron et al. (1999); Ghosal &
+van der Vaart (2007). There have been enormous recent progress on both density estimation
+and parameter estimation problems. The density estimation problem under Gaussian mixture
+models with BNP priors was extensively studied by Ghosal & van der Vaart (2001) who
+obtained attractive polynomial rates of contraction relative to the Hellinger distance metric.
+In the parameter estimation problem, the metric of choice is Wasserstein distance, which
+1
+arXiv:2301.11496v1  [math.ST]  27 Jan 2023
+
+proved to be a natural tool to analyze the convergence of mixture parameters Nguyen (2013).
+Moreover, Nguyen (2013) showed that the fast rates for density estimation with BNP Gaussian
+mixtures do not extend themselves to parameter estimation scenarios. Meanwhile, practitioners
+have employed successfully BNP mixture models, which yield useful estimates for model
+parameters that provide meaningful information about the data population’s heterogeneity.
+This state of affairs leaves a gap in the theoretical understanding and the practical usage
+of Bayesian mixture models. In this paper, we aim to bridge this gap by capturing more
+accurately the heterogenenous behavior in the rates of parameter estimation. We proceed to
+describe this in further detail.
+1.1
+Gaussian Mixture Models
+Consider discrete mixing (probability) measure G = �k
+i=1 piδθi. Here, p = (p1, . . . , pk) is a
+vector of mixing weights, while atoms {θi}k
+i=1 are elements in a given space Θ ⊂ Rd. Here
+k is used to denote the number of components, which can potentially be infinite. Mixing
+measure G is combined with a Gaussian kernel with known covariance matrix Σ, denoted by
+f(·|θ) ∼ N(θ, Σ) (to avoid notational cluttering, we remove Σ from notation in the remainder
+of the paper), with respect to the Lebesgue measure µ to yield a mixture density:
+pG(.) :=
+�
+f(·|θ)dG(θ) =
+k
+�
+i=1
+pif(·|θi).
+(1)
+The atoms θi’s are representatives of the underlying subpopulations. Let X1, . . . , Xn be i.i.d.
+samples from a mixture density pG0(x) =
+�
+f(x|θ)dG0(θ), where G0 = �k0
+i=1 p0
+i δθ0
+i is a true but
+unknown discrete mixing measure with unknown number of support points k0 ∈ N ∪ {∞}. We
+assume in this work that all the masses {p0
+i }k0
+i=1 are strictly positive and the atoms {θ0
+i : i ≤ k0}
+are distinct.
+A Bayesian mixture modeler places a prior distribution Πn on a suitable space (specifically,
+G(Θ) of discrete measures on Θ). The posterior distribution corresponding to Πn, both of
+which may vary with sample size, can be computed as:
+Πn(G ∈ B
+��X1:n) =
+�
+B
+�n
+i=1 pG(Xi)dΠn(G)
+�
+G(Θ)
+�n
+i=1 pG(Xi)dΠn(G).
+(2)
+Dirichlet process Gaussian mixture models: In the absense of the knowledge of the
+number of mixture components k0, the learning of mixture models is carried out by the use of
+Bayesian non-parametric (BNP) priors, leading to the infinite mixture setting. One of the
+most popular such priors is the Dirichlet process prior Antoniak (1974), which uses sample
+draws from a base measure H to define the random components and weights of the mixture
+model, leading to the popular Dirichlet Process Gaussian Mixture Models (DPGMM) Lo
+(1984); Escobar & West (1995). In essence, the Dirichlet process prior places zero probability
+on mixing measures with a finite number of supporting atoms and enables the addition of
+more atoms in the supporting set as the number of data points increase. The DPGMM is
+formulated as follows:
+G
+∼
+DP(α, H),
+θ1, . . . , θn
+i.i.d.
+∼
+G,
+Xi|θi
+∼
+f(Xi|θi),
+∀i = 1, . . . , n,
+(3)
+2
+
+where DP stands for Dirichlet process, the base measure H is a distribution on Θ, and α > 0
+is a concentration parameter which controls the rate at which new atoms may be considered,
+by varying the tail-behavior of mixture weights. A parametric counterpart of DPGMM is the
+mixture of finite Gaussian mixtures prior (MFM) Miller & Harrison (2018), which places all
+its mass on mixing measures with finite number of supporting atoms. BNP priors other than
+DPGMM may have the effects of pushing the atoms away from each other Xie & Xu (2017) or
+encouraging the weights of mixture to have a polynomial tail behavior De Blasi et al. (2015).
+The popularity of BNP priors may partially have been promoted due to a misconception
+that it ”automatically” determines the number of components in the posterior inference process.
+This issue was highlighted by Miller & Harrison (2014), who demonstrated that Dirichlet
+Process priors overestimate the true number of components, k0, almost surely. Subsequent
+work Guha et al. (2021) has provided post-processing techniques to determine k0 consistently
+with Dirichlet Process priors. Their method depends on the knowledge of the parameter
+contraction rate, with respect to the Euclidean Wasserstein metric, i.e., Wasserstein metric
+with underlying distance metric ℓ2, a rate that is extremely slow for the Gaussian kernels
+Nguyen (2013).
+The inconsistency of estimating k0 arises primarily because Dirichlet priors typically tend
+to create a large number of extraneous components. While some of these components may
+be in the neighborhood of the true supports, others may be outliers and in practice, can be
+easily eliminated from consideration by careful truncation techniques. However, the Euclidean
+Wasserstein distance treats both the scenarios similarly and in turn yields slow convergence
+rates for both sets of extraneous atoms. This calls for alternative metrics for investigating
+parameter estimation rates. In a recent work, Manole & Ho (2022) argued that Wasserstein
+metrics capture only the worst-case uniform rates of parameter estimation and therefore can
+yield extremely slow rates in comparison to the local rates observed in practice, which may
+vary drastically based on the likelihood curvature in the parameter neighborhood. Employing
+alternate distance metrics via the use of Voronoi tessallations, they showed that in the finite
+Gaussian mixture setting with overfitted components (where ∞ > k > k0), even though the
+uniform convergence rates may be slow as k increases, there may still be some atoms which
+enjoy much faster rates of convergence.
+The infinite Gaussian mixture setting is generally more challenging to address, (a) since the
+”true” atoms are not guaranteed to be well-separated, (b) each true atom may be surrounded
+by potentially infinitely many atoms a posteriori and (c) a posteriori samples can potentially
+have a significant portion of atomic masses attributed to outlier regions of the parameter space.
+We argue in this work that in the infinite Gaussian mixture setting, the rates captured by
+Wasserstein distances for outlier masses are inadequately slow and will demonstrate that with
+the help of a new suitably defined choice of metric this difficulty can be alleviated.
+1.2
+Contribution
+As a primary contribution of this work we study a generalized class of metrics called Orlicz-
+Wasserstein metrics, in the context of parameter estimation arising in infinite mixture models.
+We show that an in-depth analysis using this metric helps alleviate a number of the concerns
+attributable to the use of Wasserstein distances for quantifying the rates of parameter con-
+vergence arising in infinite Gaussian mixtures. This class of distance metrics generalizes the
+Wasserstein metric relative to the Orlicz norm using a variety of choices of convex functions.
+They encompass a very wide range of distances on the space of probability measures, including
+the Euclidean Wasserstein metrics as a special case. By making appropriate choices of convex
+3
+
+functions we can obtain a fast, almost polynomial contraction rates for atomic masses in
+outlier regions of the parameter space. This is very different from the slow local contraction
+behavior around the true atoms under the standard Wasserstein metric. This helps us establish
+informative and useful finer details about the convergence behavior of parameter estimates
+underlying the usage of Gaussian mixture models in clustering. We believe the usage of
+Orlicz-Wasserstein metrics for parameter estimation in Dirichlet process Gaussian mixture
+models opens a new range of directions for future research that aim for developing statistically
+sound and computationally efficient strategies for posterior sampling with mixture models.
+Organization. The remainder of the paper is organized as follows. Section 2 provides
+necessary backgrounds about posterior contraction of parameters in Gaussian mixture models
+under Wasserstein distances. Section 3.1 introduces Orlicz-Wasserstein distances and some of
+its key properties. Section 3.4 provides computational approximations to calculating Orlicz-
+Wasserstein metrics for two mixing measures. Section 3.2 presents exact lower bounds for the
+Hellinger metric with respect to Orlicz-Wasserstein distances for Gaussian kernels. Section 3.3
+uses the results in Section 3.2 to provide the key results in the paper with regards to contraction
+behavior using Orlicz-Wasserstein metrics. Proofs of results are deferred to the Appendices.
+Notation. For any function g : X → R, we denote �g(ω) as the Fourier transformation of
+function g. Given two densities p, q (with respect to the Lebesgue measure µ), the squared
+Hellinger distance is given by h2(p, q) = (1/2)
+�
+(
+�
+p(x) −
+�
+q(x))2dµ(x). For any metric d
+on Θ, we define the open ball of d-radius ϵ around θ0 ∈ Θ as Bd(ϵ, θ0). Additionally, the
+expression an ≳ bn will be used to denote the inequality up to a constant multiple where
+the value of the constant is independent of n. We also denote an ≍ bn if both an ≳ bn and
+an ≲ bn hold. Furthermore, we denote Ac as the complement of set A for any set A while
+B(x, r) denotes the ball, with respect to the l2 norm, of radius r > 0 centered at x ∈ Rd.
+The expression D(ϵ, P, d) used in the paper denotes the ϵ-packing number of the space P
+relative to the metric d. d is replaced by h to denote the hellinger norm. Finally, we use
+Diam(Θ) = sup{∥θ1 − θ2∥ : θ1, θ2 ∈ Θ} to denote the diameter of a given parameter space Θ
+relative to the l2 norm, ∥ · ∥, for elements in Rd. Regarding the space of mixing measures,
+let Ek := Ek(Θ) and Ok := Ok(Θ) respectively denote the space of all mixing measures with
+exactly and at most k support points, all in Θ. Additionally, denote G := G(Θ) =
+∪
+k∈N+ Ek the
+set of all discrete measures with finite supports on Θ. G(Θ) denotes the space of all discrete
+measures (including those with countably infinite supports) on Θ. Finally, M(Θ) stands for
+the space of all probability measures on Θ.
+2
+Posterior contraction under Wasserstein distance
+Following the work of Nguyen (2013), Wasserstein distances have been used to explore
+parameter estimation rates of mixture models, embodied through their mixing measures.
+In this section, we outline the basic concepts as follows. Let Θ ⊂ Rd. Moreover, define
+M(Θ) = {P : P is a probability measure on Θ}.
+Definition 2.1. Given µ, ν ∈ M(Θ) and the l2 metric ∥ · ∥ on Rd, the Wasserstein dis-
+tance Villani (2009) of order r seeks a joint measure π ∈ Π minimizing
+Wr(µ, ν) :=
+�
+inf
+π∈Π
+�
+Θ×Θ
+∥θ1 − θ2∥rdπ(θ1, θ2)
+�1/r
+.
+(4)
+4
+
+Here, Π is the set of couplings of µ and ν denoted by Π = {π : γ1
+#π = µ, γ2
+#π = ν}, where γ1,
+γ2 are functions that project onto the first and second coordinates of Θ × Θ respectively.
+In particular, as shown by Nguyen (2013), given two discrete measures G = �k
+i=1 piδθi and
+G′ = �k′
+i=1 p′
+iδθ′
+i, a coupling between p and p′ is a joint distribution q on [1 . . . , k] × [1, . . . , k′],
+which is expressed as a matrix q = (qij)1≤i≤k,1 ≤j≤k′ ∈ [0, 1]k×k′ with marginal probabilities
+�k
+i=1 qij = p′
+j and � j = 1k′qij = pi for any i = 1, 2, . . . , k and j = 1, 2, . . . , k′. We use
+Q(p, p′) to denote the space of all such couplings of p and p′. For any r ≥ 1, the r-th order
+Wasserstein distance between G and G′ is given by
+Wr(G, G′)
+=
+inf
+q∈Q(p,p′)
+��
+i,j
+qij∥θi − θ′
+j∥r
+�1/r
+.
+(5)
+Heinrich & Kahn (2018) show that with Gaussian kernels, the minimax rate for estimation
+is dependent on the number of extra components and goes down as the number of potential
+components increases, meaning it gets harder to accurately cluster the observations as we
+have more and more extra components. The Gaussian kernel being smooth fits in as many
+components as possible without changing the mixture density and therefore achieves a very
+slow parameter contraction rate. With potentially infinitely many extra components (while
+using Dirichlet Process priors), rates are even slower. In fact, Nguyen (2013) shows that for
+DPGMM with posterior distribution Πn(·|X1:n), the following holds true.
+Πn
+�
+G ∈ G(Θ) : W2(G, G0) ≲ (log n)−1/2
+����X1:n
+�
+→ 1
+(6)
+in pG0-probability. On the other hand, it has been shown that ordinary-smooth kernels need
+only a power of − log(ϵ) components to approximate an infinite component mixing density upto
+ϵ- approximation in Lq distance Nguyen (2013); Gao & van der Vaart (2016). Correspondingly,
+Laplace kernels need a polynomial power of (1/ϵ) many components for the same degree
+of approximation. This combined with (6) suggests that BNP priors use a lot more extra
+components to fit the true mixture distribution than is necessary, especially with Gaussian
+kernel. The extra components can potentially arise from two different sources, (i) multiple
+supporting atoms in the posterior trying to approximate each true atom, (ii) or excessively
+many outlier atoms in the posterior sample. If condition (ii) is true, this may potentially have
+negative consequences for using Gaussian kernels for clustering purposes. From Eq. (6), we
+are only able to conclude that
+Πn
+�
+G =
+�
+piδθi :
+�
+j
+pj1{∥θj−θ0
+i ∥>η ∀i} ≳
+log(n)/η2
+����X1:n
+�
+→ 1
+(7)
+which states that masses attributed to outlier atoms (those > η distance from any ”true” atom)
+vanish at only a slow logarithmic rate. Clearly, while standard Wasserstein distances are the
+popular choices of metrics, they do not help differentiate between the sources of extra atoms,
+and thereby are not useful while discarding outlier atoms. To facilitate this distinction of
+the source of excess atoms, in this paper we consider a generalisation of standard (Euclidean)
+Wasserstein metrics called Orlicz-Wasserstein distances which allow placement of higher weight
+penalties on outliers and thereby help to identify outlier atoms better. We proceed in the
+following sections to describe this in further detail.
+5
+
+3
+A generalized metric for contraction of mixing measures
+In existing literature thus far, the rates of parameter estimation have been extensively studied
+with respect to Euclidean Wasserstein distances, in the works of Nguyen (2013); Ho & Nguyen
+(2016b,a); Gao & van der Vaart (2016); Guha et al. (2021). As part of this work, we extend
+such results to the regime of Orlicz-Wassertein metrics which take a more careful consideration
+of the geometry of the parameter space. In that regard, for the sake of completeness, we first
+introduce the reader to the notion of Orlicz norms and spaces as follows.
+3.1
+Orlicz-Wasserstein distance
+The Orlicz norm is defined as follows Wellner (2017).
+Definition 3.1. Let µ be a σ−finite measure on a space X with metric ∥ · ∥. Assume that
+Φ : [0, ∞) → [0, ∞) be a convex function satisfying:
+(i) Φ(x)
+x
+→ ∞,
+as x → ∞,
+(ii) Φ(x)
+x
+→ 0,
+as x → 0.
+Then, the Orlicz space is defined as follows:
+LΦ :=
+�
+f : X → R| ∃ λ ∈ R+ s.t.
+�
+X
+Φ(∥f(x)∥/λ) dµ(x) ≤ 1
+�
+.
+(8)
+Moreover, the Orlicz norm corresponding to f ∈ LΦ is given by:
+∥f∥Φ := inf{λ ∈ R+ :
+�
+X
+Φ(∥f(x)∥/λ) dµ(x) ≤ 1}.
+(9)
+Without loss of generalisation, we will assume X = Rd, with ∥ · ∥ denoting the standard
+Euclidean metric. Notice that when Φ(x) = xp with p ≥ 1, the Orlicz norm, ∥f∥Φ is the
+same as the Lp-norm. In this sense, the Orlicz norm generalizes the concept of Lp-norm for
+p ≥ 1. Recall that, a coupling between two probability measures ν1 and ν2 on Rd is a joint
+distribution on Rd × Rd with corresponding marginal distributions ν1 and ν2. Corresponding
+to the Orlicz norms, we define the Orlicz-Wasserstein metric which generalizes the Wr-metric
+as follows.
+Definition 3.2. Let ν1, ν2 be probability measures on (Rd, ∥ · ∥). Assume that Φ : [0, ∞) →
+[0, ∞) is a convex function satisfying conditions (i) and (ii) in Definition 3.1. We define the
+Orlicz-Wasserstein distance between ν1 and ν2 as follows:
+WΦ(ν1, ν2) :=
+inf
+ν∈Q(ν1,ν2) inf{λ ∈ R+ :
+�
+Rd×Rd Φ(∥x − y∥/λ) dν(x, y) ≤ 1},
+(10)
+where Q(ν1, ν2) is the set of all possible couplings of ν1 and ν2.
+Orlicz Wasserstein distances have been briefly introduced in the works of Kell (2017);
+Sturm (2011), however, the utility of the metrics for contraction properties of parameter
+estimation has remained hitherto unexplored. Also, following Lemma 3.1 of Sturm (2011),
+we see under some minor regularity conditions, for every Φ, ν1, ��2, there exists λmin and νopt
+6
+
+such that λmin = WΦ(ν1, ν2) and
+�
+Rd×Rd Φ(∥x − y∥/λmin) dνopt(x, y) = 1. This combined with
+Fubini’s theorem establishes the equivalence of the definitions in this work and those of Sturm
+(2011); Kell (2017).
+Note that when Φ(x) = xr for r ≥ 1, then WΦ(ν1, ν2) = Wr(ν1, ν2), the usual Wasserstein
+distance of order r between ν1 and ν2.
+The following lemma demonstrates that Orlicz-
+Wasserstein defines a proper metric on (Rd, ∥ · ∥).
+Lemma 3.3. The Orlicz-Wasserstein WΦ is a distance metric on the set of probability
+measures on (Rd, ∥·∥), namely, it is symmetric and satisfies the identity and triangle inequality
+properties.
+The proof of Lemma 3.3 is in Appendix B.1. The notion of Orlicz-Wasserstein distance may
+encompass a stronger notion of metrics than that of the usual Wasserstein distance to compare
+probability measures as evidenced by the following lemma.
+Lemma 3.4. Let ν1, ν2 be probability measures on (Rd, ∥ · ∥). Also assume Φ, Ψ are convex
+functions satisfying conditions (i) and (ii) in Definition 3.1. Suppose that for all x > 0,
+Φ(x) ≤ Ψ(x). Then, we have
+WΦ(ν1, ν2) ≤ WΨ(ν1, ν2).
+The proof of Lemma 3.4 is in Appendix B.2. Note that the supremum of convex functions
+is also a convex function. Therefore, as a corollary to the above lemma we obtain the following
+inequality.
+Corollary 3.1. Let Φ1(·) be a polynomial convex function and Φ2(·) an exponential convex
+function. Ψ is the supremum of Φ1(·) and Φ2(·). Then the following holds, for any G, G′,
+1 > α > 0.
+WΨ(G, G′)
+≥
+WαΦ1+(1−α)Φ2(G, G′)
+(11)
+≥
+αWΦ1(G, G′) + (1 − α)WΦ2(G, G′)
+An important property of the Wasserstein distances is that if one mixing measure is close
+to another in Wasserstein distance, it provides a way to control the corresponding contraction
+rates of the atoms and the masses associated with them. The following lemma provides a
+similar result for Orlicz-Wasserstein norms.
+Lemma 3.5. Let G0 = �k0
+i=1 p0
+i δθ0
+i , G = �k
+i=1 piδθi be mixing measures such that θj, θ0
+i ∈ Rd
+for all i, j. Assume that Φ : [0, ∞) → [0, ∞) is a convex function satisfying conditions (i) and
+(ii) in Definition 3.1. Then
+�
+j
+pj1{∥θj−θ0
+i ∥>η for all i} ≤
+�
+Φ
+�
+η
+WΦ(G, G0)
+��−1
+.
+(12)
+Here, k0, k can also take the value ∞.
+The proof of Lemma 3.5 is in Appendix B.5. Lemma 3.5 allows us to identify the amount of
+mass transferred over large distances, when the mass transfer occurs between two measures G
+and G0. Note that the constraint on Φ is very minimal, thereby lending flexibility to the result.
+Since operations like supremums of convex functions or compositions of a convex function
+with a non-decreasing convex function (this is the outer function), also yield convex functions,
+Lemma 3.5 is a standalone result of interest as a generalisation of Bernstein/Hoeffding type
+inequalities for mixing measures.
+7
+
+3.2
+Lower bound of Hellinger distance based on Orlicz-Wasserstein metric
+In the previous section, we state results to control the cost of mass transfer attributable
+to large transportation distances using Orlicz-Wasserstein distances. This is an important
+result in understanding the contraction behaviors of support points in the outlier regions of
+parameter space. Traditionally, contraction behavior has been extensively studied Ghosal &
+van der Vaart (2001) in the regime of mixture densities pG. The following results help us
+connect our understanding of posterior contraction on space of mixture densities to that of
+mixing measure, relative to that of Orlicz-Wasserstein distances. This is stated as follows in
+the next theorem.
+Theorem 3.2. Let Φ be a convex function satisfying conditions (i) and (ii) in Definition 3.1
+such that Φ(x) ≤ exp(xβ) − 1 for some 16/15 > β > 1. Then, as Θ = [−¯θ, ¯θ]d, for any mixing
+measures G, G′, we have
+WΦ(G, G′) ≲ C
+�
+¯θ5/4
+(log(1/h(pG, pG′)))1/8 +
+�
+1
+log(1/h(pG, pG′))
+�11/8
+(13)
++
+�
+1
+log(c/h(pG, pG′)(log(1/h(pG, pG′)))d/4)
+�1/2�
+(14)
+for some constant C dependent on the dimension and known covariance matrix.
+The proof of Theorem 3.2 is in Appendix A.1. The key technical novelty of the proof lies in
+the idea of convolving the mixing measures with a mollifier which is exponentially integrable
+while its Fourier transform is smoother than the Gaussian location kernel. This helps to
+smoothly transition the problem of bounding distances on mixing measures to the Fourier
+transform domain of corresponding mixture densities. We make a few comments about the
+above theorem.
+(i) The upper bound on the RHS of equation (13) depends on a power of log-Hellinger
+distance between the corresponding mixture densities. This strengthens the result in Theorem
+2 of Nguyen (2013), who obtained a (log(1/h))−1/2 upper bound for W2(G, G′). The result in
+Theorem 3.2 is obtained in terms of Orlicz-Wasserstein distances relative to an exponential
+convex function, thus lending it more flexibility.
+(ii) The key object to obtaining this result is to find a suitable mollifier Zδ, which we
+choose as c1
+δ (
+�
+exp(−itx/δ) exp(−t4)dt)2 with c being the constant of proportionality for the
+proof of Theorem 3.2. However, we believe a more refined choice of mollifier can yield sharper
+estimates on the RHS of equation (13).
+(iii) The result is obtained with exact computation of the involvement of ¯θ. Therefore, it
+can also be used for posterior contraction rates with sieve priors, although for this work we
+study only compactly supported priors.
+Outline of proof of Theorem 3.2:
+Here, we provide a proof strategy for Theorem 3.2,
+which relies on the following triangle inequality with Orlicz-Wasserstein distance between G
+and G′:
+WΦ(G, G′) ≤ WΦ(G, G ∗ Zδ,d) + WΦ(G′, G′ ∗ Zδ,d) + WΦ(G ∗ Zδ,d, G′ ∗ Zδ,d),
+(15)
+where Zδ,d(x1, . . . , xd) := �d
+i=1 ζδ(xi) and ζδ(x) := c1
+δ (
+�
+exp(−itx/δ) exp(−t4)dt)2, with c
+being the constant of proportionality. To control both WΦ(G, G ∗ Zδ,d) and WΦ(G′, G′ ∗ Zδ,d),
+we use the following lemma:
+8
+
+Lemma 3.6. Assume that ν2 = ν1 ∗ Zδ,d where ν1 is a given probability measure on (Rd, ∥ · ∥).
+Furthermore, suppose that Φ(x) ≤ exp(xα) − 1 for some 1 < α < 4/3. Then, there exists
+universal constant Cα depending only on α such that
+WΦ(ν1, ν2) ≤ Cαδ.
+The proof of Lemma 3.6 is in Appendix B.3. For the final term WΦ(G ∗ Zδ,d, G′ ∗ Zδ,d), we can
+upper bound it using the following result:
+Lemma 3.7. Let ν1, ν2 be probability measures on (Rd, ∥ · ∥) and let Φ be a convex function
+satisfying conditions (i) and (ii) in Definition 3.1. Then, we obtain that
+WΦ(ν1, ν2)
+≤ 2 inf{λ ∈ R+ :
+(16)
+�
+Rd Φ(∥x∥/λ) d|ν1(x) − ν2(x)| ≤ 1}.
+The proof of Lemma 3.7 is in Appendix B.4. Using triangle inequality and Lemmas 3.6 and 3.7,
+we obtain
+WΦ(G, G′) ≾ δ + inf{λ ∈ R+ :
+�
+Rd Φ(∥x∥/λ) · |(G − G′) ∗ Zδ,d(x)|dx ≤ 1}.
+We then decompose the integral with respect to Rd into two integrals: one with respect to
+∥x∥ ≤ M and one with respect to ∥x∥ > M, and after some algebraic manipulations, we have
+inf
+�
+λ ∈ R+ :
+�
+Rd Φ(∥x∥/λ) · |(G − G′) ∗ Zδ,d(x)|dx ≤ 1
+�
+≾
+M
+log(C/(h(pG, pG0) exp(α2dδ−4)Md/2)) +
+(d¯θ)5/4
+log(3/2)M1/4 + δ5/4
+M1/4 ,
+for any M > 0 where C is some universal constant. Collecting these results leads to
+WΦ(G, G′) ≾ inf
+δ,M
+�
+δ +
+M
+log(C/(h(pG, pG0) exp(α2dδ−4)Md/2)) +
+(d¯θ)5/4
+log(3/2)M1/4 + δ5/4
+M1/4
+�
+.
+Solving the minimization problem, we obtain the conclusion of Theorem 3.2.
+In the next section, we use Theorem 3.2 to establish posterior contraction bounds of
+parameter estimating in Dirichet Process Gaussian mixtures.
+3.3
+Posterior contraction with Orlicz Wasserstein distances
+On the parametric estimation front, Nguyen (2013); Guha et al. (2021); Ohn & Lin (2020)
+establish logarithmic rates for estimating mixing measures in Dirichlet Process Gaussian
+mixtures. While Nguyen (2013) establishes an approximately log(n)−1/2 rate of contraction
+relative to the W2 metric, more recently, Ohn & Lin (2020) establish minimax type ≈ log(n)
+rates relative to the W1 metric. Putting the results in context with Lemma 3.5, both those
+results imply, �
+j pj1∥θj−θ0
+i ∥>η for all i ≈ log(n), meaning the mass of posterior sample atoms
+in the region of parameter space not populated by atoms of the true (data-generating) mixing
+measure decays logarithmically. This puts the use of DPGMMs for clustering in a negative
+light.
+9
+
+In this section, we show that a much stronger almost polynomial rate can be established
+for this objective, facilitated by the use of Orlicz-Wasserstein metrics.
+To facilitate our
+presentation, we consider the following notation.
+EX η(Θ, r) :=
+�
+G =
+�
+piδθi ∈ G(Θn1) :
+�
+j
+pj1{∥θj−θ0
+i ∥>η for all i} ≥ r
+�
+.
+(17)
+EX η(Θ, r) here denotes the set of mixing measures which devote at least r probability mass
+to atoms which are away from the atoms of G0 by distance η. To study the contraction of
+mixing measure of DPGMMs, we impose the following assumption on the base distribution H.
+(P.1) The base distribution H is supported on Θ = [−¯θ, ¯θ]d, and absolutely continuous
+with respect to the Lebesgue measure µ on Θ and admits a density function g(·). Also, H is
+approximately uniform, i.e., minθ∈Θ g(θ) >
+c0
+µ(Θ) > 0.
+Let f1(n, d) := (log(n)/(d + 2) − log(log n))−1/8.
+Theorem 3.3. Given the Dirichlet Process Gaussian mixture models (3), if Φ satisfies the
+assumptions in Theorem 3.2, then for any η > 0 the following holds:
+Πn
+�
+G ∈ G(Θ)
+: WΦ(G, G0) ≥ f1(n, d)
+���� X1:n
+� P n
+G0
+→ 0.
+The proof of Theorem 3.3 is in Appendix A.2. The following result is a simple corollary of
+Theorem 3.3.
+Corollary 3.4. Given all the assumptions in Theorem 3.3,
+Πn
+�
+G ∈ EX η
+�
+Θ, 2 exp
+�
+−η log(n)1/8
+(d + 2)
+�� ���� X1:n
+�
+PG0
+→ 0.
+(18)
+The proof of Corollary 3.4 is in Appendix A.3.
+Remarks: (i) Corollary 3.4 suggests that if η can be chosen sufficiently small so that each
+η-neighborhood contains at most one true atom, Gaussian mixture models can be useful choices
+in clustering as well since outlier atoms vanish at almost polynomial rates.
+(ii) We believe the rate of contraction can be optimized further with a more refined choice
+of Φ(·), however, we make no such attempts in this work. Corollary 3.4 reveals the power of
+Orlicz-Wasserstein distances for Gaussian mixture models. On the other hand, this exponential
+choice of Φ does not improve on the bound for heavy tailed kernels such as Laplace location
+mixtures.
+We show in this section that Orlicz-Wasserstein metrics provide strong theoretical guaran-
+tees for mixing measures. This raises the natural question as to how such a metric can be
+computed for arbitrary choices of Φ. We provide some guidance in that regard in the following
+section.
+3.4
+Computation of the Orlicz-Wasserstein
+In practice, the Euclidean Wasserstein distance is computed for samples of the respective
+distributions. The exact computation turns out to be a linear programming problem which
+scales to the order of O(n3 log(n)), where n is the combined sample size of the two sampling
+distributions for which the distance is being calculated. Cuturi (2013) shows that using entropic
+10
+
+(a) Standard Wasserstein W1
+(b) Orlicz-Wasserstein with exponential Φ
+Figure 1. Transportation plans. (a) Entropic OT produces more global plans and is unable
+to capture local structure of mass transfers. (b) Entropic Orlicz-Wasserstein penalizes mass
+transfers over large distances
+regularization this can be drastically improved to O(n2) Altschuler et al. (2017); Lin et al.
+(2019, 2022). Further speed-ups and easiness of computation via the use of dual formulation
+of the entropic regularization has been explored by the works of Seguy et al. (2017); Genevay
+et al. (2016); Genevay (2019).
+Here we consider the entropic regularized version of the
+Orlicz-Wasserstein metrics.
+Computational procedure: In that respect, we consider solving the following problem as a
+surrogate to equation (10).
+W λ
+Φ(ν1, ν2)
+:= infν∈Q(ν1,ν2) AΦ(ν1, ν2),
+(19)
+P λ
+Φ(ν1, ν2)
+:= arg infν∈Q(ν1,ν2) AΦ(ν1, ν2),
+(20)
+where AΦ(ν1, ν2) := inf{η ∈ R+ :
+�
+Rd×Rd Φ(∥x − y∥/η) dν(x, y) − (1/λ)(H(ν)) ≤ 1} with H(µ)
+used to denote the Shannon entropy of distribution µ. To obtain solutions for equation (19),
+we resort to using outputs from Sinkhorn divergence computations.
+Consider two discrete probability measures, r (with m atoms, {xi}m
+i=1) and c (with n
+atoms, {yi}m
+i=1). Let Mn×m be a distance matrix such that Mij = c(xi, yj) for some cost
+function c(·, ·). Let S(M, λ, r, c) be used to denote the Sinkhorn divergence optimized objective
+function for cost matrix M, regularization parameter λ and d(M, λ, r, c) = ⟨S(M, λ, r, c), M⟩
+be used to denote the transport cost. Algorithm 1 defines a procedure to obtain a regularised
+Orlicz-Wasserstein distance between ν1 = �
+i riδxi and ν2 = �
+i ciδyi in such a scenario by
+iteratively updating the value of Orlicz-Wasserstein distance until convergence. The crucial
+intuition behind Algorithm 1 is that infν∈Q(ν1,ν2)
+�
+Rd×Rd Φ(∥x−y∥/η) dν(x, y)−(1/λ)(H(ν)) is
+a monotonically non-increasing function of η. Therefore the solution to the Orlicz Wasserstein
+distance can be obtained by a binary search once upper and lower limits are known. This is
+rigorously explained in Proposition C.1 in Appendix C.
+Simulations settings: We provide a demonstration of the utility of using Orlicz-Wassestein
+distances in Figure 1. We consider two mixing densities, ν1 on the y-axis is a 3-mixture of
+univariate normal distributions with means at [3, 4, 5], common σ = 0.3 and mixture weights
+[0.37, 0.3, 0.33]. On the other hand ν2 represented in the x-axis is a 4-mixture of univariate
+Laplace kernels with means at [7, 8, 9, 6], scale parameters [0.3, 0.3, 0.3, 0.1]] and mixture
+weights [0.30, 0.32, 0.32, 0.06]. The left plot of Figure 1 shows the transportation plan for
+output of Sinkhorn mechanism with regularisation parameter 0.01, while the right plot shows
+11
+
+07
+10
+15
+2507
+10
+15
+25Algorithm 1: Computing Orlicz Wasserstein distances between two discrete proba-
+bility measures
+1: Input M, λ, r, c, ϵ.
+2: Output W λ
+Φ(ν1, ν2).
+3: I = (r > 0); r = r(I); M = M(I, :);
+4: xupp = max(M)/Φ−1(1),
+xlow = [S(M, λ, r, c) + 1
+2λ(H(r) + H(c))]/Φ−1(1 + 1
+λ(H(r) + H(c))
+5: fxupp = S(Φ(M/xupp), λ, r, c),fxlow = S(Φ(M/xlow), λ, r, c).
+6: while |xlow − xupp| < ϵ not converged do
+7: xnew = (xlow ∗ fxupp − xupp ∗ fxlow)/(fxupp − fxlow)).
+8: if xnew < xupp and xnew > xlow do
+9: fxnew = S(Φ(M/xupp), λ, r, c)
+10: if fxnew < 1, xupp = xnew, fxupp = fxnew.
+11: else: xlow = xnew, fxlow = fxnew
+12: end if
+13: else xnew = (xlow + xupp)/2. repeat Step 9-12.
+14: end if
+15: end while
+16: return W λ
+Φ(ν1, ν2) := xupp.
+the same for transportation plan obtained via Algorithm 1 with λ = 0.01 (Φ(·) = exp(·/β) − 1,
+β = 1.1). We have the following remarks.
+Remark: The entropic Orlicz-Wasserstein procedure produces sharper transport plans. This
+indicates that it performs a shrinkage procedure on the space of transportation plans. This can
+have potential benefits towards obtaining robust plans and provide a promising direction of
+future research. Additionally, while entropic Euclidean Wasserstein transport plans distribute
+the mass of the outlier atom of ν2 (mean=6, weight= 0.06), its Orlicz-Wasserstein counterpart
+manages to avoid it entirely.
+By penalizing mass transfers over large distances, Orlicz-
+Wasserstein distances are able to restrict attention to localised transportation plans. This in
+turn helps capture the small outlier mass associated with aposteriori DPGMM samples, as
+seen in Section 3.3.
+4
+Conclusion
+In this work, we discuss the shortcomings of traditional Wasserstein metrics to perform
+clustering with Gaussian mixture models. We re-introduce a novel metric, called Orlicz-
+Wasserstein distances, in the context of estimating parameter convergence rates of hierarchical
+and mixture models and provide sound theoretical justifications of its ability to address the
+concerns associated with traditional Wasserstein distances. We also provide a theoretically
+sound approximate algorithm to compute the distance metric, and also show that convergence
+rates of Orlicz-Wasserstein distances carry over to the approximate distance. Lastly, we
+provide a preliminary simulation study to initiate a discussion on future research with Orlicz-
+Wasserstein distances. Since they allow low/high penalty on mass transfers over large distances,
+depending on the choice of function Φ, this lends flexibility to extending mass transfers over
+local/global regions and consequentially may be used as a device for smoothing/sharpening
+12
+
+standard OT plans. Combined with dimension reduction techniques this can lend usage to a
+number of application domains such as anomaly detection and robust optimal transport.
+Acknowledgements
+We thank Professor Fedor Nazarov and Professor Mark Rudelson for discussions on the paper.
+This research is supported in part by grants NSF CAREER DMS-1351362, NSF CNS-1409303,
+a research gift from Adobe Research and a Margaret and Herman Sokol Faculty Award.
+Nhat Ho acknowledges support from the NSF IFML 2019844 and the NSF AI Institute for
+Foundations of Machine Learning.
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+
+Supplement to “On Excess Mass Behavior in Gaussian
+Mixture Models with Orlicz-Wasserstein Distances”
+In this supplementary material, we present proofs of key results in Appendix A and proofs of
+lemmas in Appendix B. We then provide theoretical guarantee for the algorithm to compute
+the entropic regularized Orlicz-Wasserstein in Appendix C.
+A
+Proofs of key results
+Notation revisited
+For any function g : X → R, we denote �g(ω) as the Fourier transfor-
+mation of function g. Given two densities p, q (with respect to the Lebesgue measure µ), the
+squared Hellinger distance is given by h2(p, q) = (1/2)
+�
+(
+�
+p(x) −
+�
+q(x))2dµ(x). For any
+metric d on Θ, we define the open ball of d-radius ϵ around θ0 ∈ Θ as Bd(ϵ, θ0). Additionally,
+the expression an ≳ bn will be used to denote the inequality up to a constant multiple where
+the value of the constant is independent of n. We also denote an ≍ bn if both an ≳ bn and
+an ≲ bn hold. Furthermore, we denote Ac as the complement of set A for any set A while
+B(x, r) denotes the ball, with respect to the l2 norm, of radius r > 0 centered at x ∈ Rd.
+The expression D(ϵ, P, d) used in the paper denotes the ϵ-packing number of the space P
+relative to the metric d. d is replaced by h to denote the hellinger norm. Finally, we use
+Diam(Θ) = sup{∥θ1 − θ2∥ : θ1, θ2 ∈ Θ} to denote the diameter of a given parameter space Θ
+relative to the l2 norm, ∥ · ∥, for elements in Rd. Regarding the space of mixing measures,
+let Ek := Ek(Θ) and Ok := Ok(Θ) respectively denote the space of all mixing measures with
+exactly and at most k support points, all in Θ. Additionally, denote G := G(Θ) =
+∪
+k∈N+ Ek the
+set of all discrete measures with finite supports on Θ. Moreover, G(Θ) denotes the space of
+all discrete measures (including those with countably infinite supports) on Θ. Finally, M(Θ)
+stands for the space of all probability measures on Θ.
+A.1
+Proof of Theorem 3.2
+We present the proof of Theorem 3.2 for the lower bound of Hellinger distance between mixing
+density functions based on Orlicz-Wasserstein metric between their corresponding mixing
+measures.
+In this proof, we denote a ≲ b to imply that a ≤ C · b for a universal constant C dependent
+on α, d, and ¯θ. Also, f ∗ g will denote the outcome of convolution operation on functions f
+and g. Now, we consider the following density function in R:
+K(x) := c
+�� ∞
+−∞
+exp(−itx) exp(−t4)dt
+�2
+,
+(21)
+where c is a proportionality constant so that
+� ∞
+−∞ K(x)dx = 1. Lemma B.1 shows that K(·) is
+integrable.
+Moreover, Lemma B.2 shows that the characteristic function ˆK(·), corresponding to K(·)
+satisfies,
+| ˆK(x)| ≲ exp(−(x/2)4).
+The strategy to obtain upper bounds for WΦ(G, G′) is to convolve G with mollifiers, Zδ,d(·),
+of the form Zδ,d(x) = �d
+i=1
+1
+δ K(xi/δ) for δ > 0, where x = (x1, . . . , xd). In particular, by
+17
+
+triangle inequality and following Lemma 3.3 we can write:
+WΦ(G, G′) ≤ W1(G, G ∗ Zδ,d) + WΦ(G′, G′ ∗ Zδ,d) + WΦ(G ∗ Zδ,d, G′ ∗ Zδ,d).
+For Φ(x) = exp((7/32)x) − 1, following Lemma 3.6 we find that
+WΦ(G, G ∗ Zδ,d) ≤ Cαδ.
+Therefore, we can write
+WΦ(G, G′) ≤ 2Cαδ + WΦ(G ∗ Zδ,d, G′ ∗ Zδ,d).
+For every M > 0, we have
+WΦ(G ∗ Zδ,d, G′ ∗ Zδ,d) ≤ 2 inf{λ ∈ R+ :
+�
+Rd Φ(∥x∥/λ) · |(G − G′) ∗ Zδ,d(x)|dx ≤ 1}
+≤ 2 inf{λ ∈ R+ : s1 ≤ 1/2 and s2 ≤ 1/2},
+≤ 2 max{inf{λ ∈ R+ : s1 ≤ 1/2}, inf{λ ∈ R+ : s2 ≤ 1/2}},
+(22)
+with the first inequality following from Lemma 3.7 and the third inequality comes from the
+monotonicity of function Φ. Here, we denote
+s1 =
+�
+∥x∥2≤M
+Φ(∥x∥/λ) · |(G − G′) ∗ Zδ,d(x)|dx,
+s2 =
+�
+∥x∥2>M
+Φ(∥x∥/λ) · |(G − G′) ∗ Zδ,d(x)|dx.
+We now proceed to bound T1 = inf{λ ∈ R+ : s1 ≤ 1/2} and T2 = inf{λ ∈ R+ : s2 ≤ 1/2}.
+Bounding for T1:
+Using Holder’s inequality, we obtain
+inf{λ ∈ R+ :
+�
+∥x∥2≤M
+Φ(∥x∥/λ) · |(G − G′) ∗ Zδ,d(x)|dx ≤ 1/2}
+≤
+inf{λ > 0 :
+�
+∥x∥≤M
+exp((∥x∥/λ)β) · |(G − G′) ∗ Zδ,d(x)|dx ≤ 3/2}
+≤
+inf
+�
+λ > 0 :
+�
+�
+∥x∥≤M
+exp((M/λ)β)dx
+�1/2�
+�
+∥x∥≤M
+|(G − G′) ∗ Zδ,d(x)|2dx
+�1/2
+≤ 3/2
+�
+≤
+inf
+�
+λ > 0 :
+πd/4
+�
+( d
+2 + 1)Γ(d/2)
+Md/2 exp((M/λ)β)∥(G − G′) ∗ Zδ,d(x)∥2 ≤ 3/2
+�
+=
+M
+(log(cd/(∥(G − G′) ∗ ζδ,d∥2Md/2)))1/β .
+(23)
+18
+
+Since f is Gaussian distribution, we have ˜f(ω) ≥ cf exp(−α �d
+i=1 ω2
+i ) for some cf, α > 0.
+Given that inequality, we find that
+∥(G − G′) ∗ Zδ,d∥2
+2
+=
+�
+| �G − �
+G′|2(ω)| �Kδ,d(ω)|2dω =
+�
+| �f( �G − �
+G′)|2(ω)| �Kδ,d(ω)|2
+| �f(ω)|2 dω
+≤
+∥pG − pG′∥2
+2 sup
+ω∈Rd
+| �Kδ,d(ω)|2
+| �f(ω)|2
+≤
+4∥f∥∞h2(pG, pG0) sup
+ω∈Rd
+� 1
+c2
+f
+·
+d
+�
+i=1
+exp(−δ4|ωi|4) exp(2α|ωi|2)
+�
+.
+By taking derivatives, we obtain the maximum as
+sup
+ωi∈R
+�
+exp(−δ4|ωi|4) exp(2α|ωi|2)
+�
+= exp(α2/δ4).
+Plugging these results into equation (23) leads to
+inf{λ ∈ R+ :
+�
+∥x∥2≤M
+Φ(∥x∥/λ) · |(G − G′) ∗ Zδ,d(x)|dx ≤ 1/2}
+≤
+M
+(log(c/(h(pG, pG0) exp(α2dδ−4)Md/2)))1/β
+(24)
+for some universal constant c.
+Bounding for T2:
+For any M > 0, we denote
+k′
+=
+inf{λ ∈ R+ : EX∼(G−G′)(Φ(∥X∥5/4/λM1/4) ≤ 1/2},
+k′′
+=
+inf{λ ∈ R+ : EY ∼Zδ,d(Φ(∥Y ∥5/4/λM1/4) ≤ 1/2}.
+(25)
+Then, by the convexity of Φ we have
+inf{λ ∈ R+ : EX∼G−G′,Y ∼Zδ,d(Φ(∥X + Y ∥5/4/λM(1/4)) ≤ 1/2} ≤ 21/4(k′ + k”).
+The above inequality is because of the following inequalities:
+EX∼G−G′,Y ∼Zδ,d(Φ(∥X + Y ∥5/4/21/4(k′ + k”)M(1/4))
+≤
+EX∼G−G′,Y ∼Zδ,d(Φ(21/4(∥X∥5/4 + ∥Y ∥5/4)/21/4(k′ + k”)M(1/4)))
+=
+EX∼G−G′,Y ∼Zδ,d
+�
+Φ
+��
+k′∥X∥5/4 + ∥Y ∥5/4�
+/(k′ + k”)M(1/4)��
+≤
+EX∼G−G′,Y ∼Zδ,d
+�
+Φ
+�
+k′
+k′ + k”
+�
+∥X∥5/4
+k′M(1/4)
+�
++
+k”
+k′ + k”
+�
+∥Y ∥5/4
+k”M(1/4)
+���
+≤
+EX∼G−G′,Y ∼Zδ,d
+k′
+k′ + k”Φ
+�
+∥X∥5/4
+k′M(1/4)
+�
++
+k”
+k′ + k”Φ
+�
+∥Y ∥5/4
+k”M(1/4)
+�
+≤ 1
+2.
+(26)
+The first inequality follows from ∥a + b∥P ≤ 2p−1(∥a∥p + ∥b∥p). The second last inequality
+follows from convexity of Φ and the final inequality follows from equation (25). Therefore, we
+19
+
+obtain that
+inf{λ ∈ R+ :
+�
+∥x∥2>M
+Φ(∥x∥/λ) · |(G − G′) ∗ Zδ,d(x)|dx ≤ 1/2}
+≤
+inf{λ ∈ R+ :
+�
+∥x∥2>M
+Φ(∥x∥5/4/λM(1/4)) · |(G − G′) ∗ Zδ,d(x)|dx ≤ 1/2}
+≤
+inf{λ ∈ R+ : EX∼G−G′,Y ∼Zδ,d(Φ(∥X + Y ∥5/4/λM(1/4)) ≤ 1/2}
+≲
+inf{λ > 0 :
+�
+Rd
+exp((∥x∥5/4/λM1/4)β) · |(G − G′)(x)|dx ≤ 3/2}
++
+inf{λ > 0 :
+�
+Rd
+exp((∥x∥5/4/λM1/4)β) · |Zδ,d(x)|dx ≤ 3/2}
+≲
+(d¯θ)5/4
+M1/4 + Cδ5/4/M 1/4,
+(27)
+where C = inf{λ > 0 :
+�
+Rd
+exp((∥x∥5/4/λ)β) · |K1,d(x)|dx < ∞ as K1,d(x) ∼ O(exp(−|x|4/3))
+for large |x|, by Lemma B.1. Hence, using these results we get
+WΦ(G, G′)
+≲
+δ + max
+�(d¯θ)5/4
+M1/4 + Cδ5/4/M 1/4,
+M
+(log(c/(h(pG, pG0) exp(α2dδ−4)Md/2)))1/β
+�
+≤
+δ + (d¯θ)5/4
+M1/4 + Cδ5/4/M 1/4 +
+M
+(log(c/(h(pG, pG0) exp(α2dδ−4)Md/2)))1/β . (28)
+Choosing M = (log(1/h(pG, pG0)))1/2 and δ =
+2α2
+log(1/h(pG, pG0)) in equation (28) we obtain,
+WΦ(G, G′) ≲ (log(1/h(pG, pG0)))−1
++
+(d¯θ)5/4
+(log(1/h(pG, pG0)))1/8 +
+�
+1
+log(1/h(pG, pG0))
+�11/8
++
+�
+1
+log(c/h(pG, pG0)(log(1/h(pG, pG0)))d/4)
+�(1/β)−(1/2)
+(29)
+As a consequence, we obtain the conclusion of the theorem.
+A.2
+Proof of Theorem 3.3
+The proof of this result follows by an application of Lemma B.3, B.4 and B.5 in combination
+with Theorem 2.1 in Ghosal et al. (2000). To facilitate the presentation, we break the proof
+into several steps.
+Step 1:
+First we compute the contraction rate relative to the Hellinger metric, i.e., assume
+that
+¯θd
+ϵd+2
+n
+log
+� ¯θ
+ϵn
+�
+= o(n) and nϵ2
+n → ∞.
+Then we show that
+Πn(G ∈ G(Θ) : h(pG, pG0) ≥ Lϵn|X1, . . . , Xn)
+PG0
+→ 0.
+(30)
+20
+
+We apply Theorem 7.1 in Ghosal et al. (2000), with ϵ = Lϵn and D(ϵ) = exp
+�
+c1
+�
+¯θ
+√λminϵn
+�d
+log
+�
+e + 32e¯θ2
+λminϵ2n
+��
+,
+where L ≥ 2 is a large constant to be chosen later and c1 is the constant in equation (49).
+Lemma B.4 shows the validity of this choice of D(ϵ). Then there exists a test function φn that
+satisfies
+P n
+G0φn
+≤
+exp
+�
+c1
+�
+¯θ
+√λminϵn
+�d
+log
+�
+e + 32e¯θ2
+λminϵ2n
+��
+× exp(−KnL2ϵ2
+n)
+1
+1 − exp(−KnL2ϵ2n),
+sup
+G∈G(Θ):h(pG,pG0)≥Lϵn
+P n
+G(1 − φn)
+≤
+exp(−KnL2ϵ2
+n).
+(31)
+Now, we have
+EPG0Πn(G ∈ G(Θ) : h(pG, pG0) ≥ Lϵn|X1, . . . , Xn)φn
+≤ P n
+G0φn ≤ 2 exp
+�
+c1
+�
+¯θ
+√λminϵn
+�d
+log
+�
+e + 32e¯θ2
+λminϵ2n
+�
+− KnL2ϵ2
+n
+�
+.(32)
+Based on computation with the posterior,
+Πn(G : h(pG, pG0 ≥ ϵn)|X1, . . . , Xn)(1 − φn)
+=
+�
+G∈G(Θ):h(pG,pG0)≥ϵn
+�n
+i=1
+pG(Xi)
+pG0(Xi)dΠn(G)(1 − φn)
+�
+G∈G(Θ)
+�n
+i=1
+pG(Xi)
+pG0(Xi)dΠn(G)
+≤
+�
+G∈G(Θ):h(pG,pG0)≥ϵn
+�n
+i=1
+pG(Xi)
+pG0(Xi)dΠn(G)(1 − φn)
+�
+G∈G(Θ):K(pG0,pG)≲ϵ2
+n,K2(pG0,pG)≲ϵ2
+n(log(M/ϵn))2
+�n
+i=1
+pG(Xi)
+pG0(Xi)dΠn(G)
+,
+where M = exp(dλ−1
+min(5¯θ2
+0 + 4¯θ2)), with λmin being the minimum eigenvalue of Σ.
+Step 1.1:
+In this step we show that
+�
+G∈G(Θ):K(pG0,pG)≲ϵ2
+n,K2(pG0,pG)≲ϵ2
+n(log(M/ϵn))2
+n
+�
+i=1
+pG(Xi)
+pG0(Xi)dΠn(G)
+≳ exp(−(1 + C)nλminϵ2
+n)
+Γ(γ)(c0γπd/2)D
+(2Γ(d/2 + 1))D(2D)D−1
+�√λminϵn
+2¯θ
+�2(D−1)+dD
+(33)
+with pn
+G0 probability → 1,
+for all C > 0 and ϵn > 0 is sufficiently small, where D = D(√λminϵn, Θ, ∥.∥) ≈
+� ¯θ
+ϵn
+�d
+stands
+for the maximal √λminϵn-packing number for Θ under ∥.∥ norm, and Γ(·) is the gamma
+function. First we show that
+{G ∈ G(Θ) : W2(G, G0) ≲
+�
+λminϵn}
+⊂ {G ∈ G(Θ) : K(pG0, pG) ≲ ϵ2
+n, K2(pG0, pG) ≲ ϵ2
+n(log(M/ϵn))2},
+(34)
+21
+
+for ϵn sufficiently small.
+Since
+� (pG0(x))2
+pG(x)
+µ(dx) ≤ M by Lemma B.5, it follows by an application of Theorem 5
+in Wong & Shen (1995) that for ϵn < 1/2(1 − e−1)2,
+h(pG, pG0) ≲ ϵ2
+n =⇒ K2(pG0, pG) ≲ ϵ2
+n(log(M/ϵn))2.
+Following Example 1 in Nguyen (2013), h2(pG, pG0) ≤ W 2
+2 (G, G0)
+8λmin
+for Gaussian location
+mixtures.
+Similarly, from Nguyen (2013) it also follows that K(pG, pG0) ≤ W 2
+2 (G, G0)
+2λmin
+. Combining
+the above displays, equation (34) follows.
+Following Lemma 8.1 in Ghosal et al. (2000), for every C, ϵ, M > 0 and any measure Π on
+the set {G ∈ G(Θ) : K(pG0, pG) ≲ ϵ2
+n, K2(pG0, pG) ≲ ϵ2
+n(log(M/ϵn))2}, we have,
+P n
+G0
+��
+n
+�
+i=1
+pG(Xi)
+pG0(Xi)dΠn(G) ≤ exp(−(1 + C)nϵ2)
+�
+≤
+1
+C2nϵ2(log(M/ϵ))2 .
+(35)
+The result in equation (33) now follows by an application of Lemma B.3 in combination with
+equations (34) and (35) using the fact that nϵ2
+n → ∞.
+Step 1.2:
+Let the event in (33) be denoted as Tn. Then
+EPG0 [Πn(G : h(pG, pG0) ≥ Lϵn)|X1, . . . , Xn)(1 − φn)] ≤ PG0(T C
+n )
++
+PG0(Tn)
+exp((1 + C)nλminϵ2
+n)
+Γ(γ)(c0γπd/2)D
+(2Γ(d/2+1))D(2D)D−1
+� √λminϵn
+2¯θ
+�2(D−1)+dD
+sup
+G∈G(Θ):h(pG,pG0)≥Lϵn
+P n
+G(1 − φn)
+≲
+exp((1 + C)nλminϵ2
+n)
+Γ(γ)(c0γπd/2)D
+(2Γ(d/2+1))D(2D)D−1
+� √λminϵn
+2¯θ
+�2(D−1)+dD exp(−KnL2ϵ2
+n) + o(1).
+(36)
+The final step follows from simple computation similar to that of the Proof of Theorem 2.1
+in Ghosal et al. (2000) and using the fact that
+¯θd
+ϵd+2
+n
+log
+� ¯θ
+ϵn
+�
+= o(n). Combining equations (32)
+and (36) and using the condition
+¯θd
+ϵd+2
+n
+log
+� ¯θ
+ϵn
+�
+= o(n), it follows that for L large enough
+Πn(G ∈ G(Θ) : h(pG, pG0) ≥ Lϵn|X1, . . . , Xn)
+PG0
+→ 0.
+(37)
+Step 2:
+For some sufficiently large L with ϵn = L(log n)n−1/(d+2) satisfies
+¯θd
+ϵd+2
+n
+log
+� ¯θ
+ϵn
+�
+=
+o(n). Therefore we get, from the result in Step 1of this proof
+Πn
+�
+G ∈ G(Θ) : h(pG, pG0) ≥ L(log n)
+n1/(d+2)
+���� X1:n
+� P n
+G0
+→ 0.
+Now, from Theorem 3.2, we have
+Πn
+�
+G ∈ G(Θ) : WΦ(G, G0) ≥ f1(n, d)
+���� X1:n
+� P n
+G0
+→ 0,
+where f1(n, d) := (log(n)/(d + 2) − log(log n))−1/8.
+22
+
+A.3
+Proof of Corollary 3.4
+Let G0 = �k0
+i=1 p0
+i δθ0
+i , G = �k
+j=1 piδθi. Suppose q = (qij)1≤i≤k0,1 ≤j≤k ∈ [0, 1]k0×k is a coupling
+between p0 = (p0
+1, . . . , p0
+k0) and p = (p1, . . . , pk), with Q(p, p′) represents the space of all such
+couplings of p0 and p. Using the proof technique similar to Lemma 3.5, we get
+�
+qij exp((∥θ0
+i − θj∥/k)β)
+≥
+�
+qij1{∥θ0
+i −θj∥≥η} exp((η/k)β)
+≥
+�
+pj1{∥θ0
+i −θj∥≥η for all i} exp((η/k)β),
+for all 1 < β < 16/15.
+We denote K = inf{λ ≥ 0 : � pj1{∥θ0
+i −θj∥≥η for all i} exp((η/λ)β) ≤ 2}. Then, we find that
+K ≥ η
+�
+log
+�
+1
+� pj1{∥θ0
+i −θj∥≥η for all i}
+��−1/β
+,
+and
+�
+j
+pj1{∥θj−θ0
+i ∥>η for all i} ≤ 2 exp
+�
+−η
+WΦ(G, G0)
+�
+.
+Putting these results together with Theorem 3.3 leads to
+Πn
+�
+G ∈ EX η
+�
+�Θ, 2 exp
+�
+�−
+�
+η log(n)1/8
+(d + 2)
+�β�
+�
+�
+�
+���� X1:n
+�
+PG0
+→ 0
+in P n
+G0 probability. Since this result holds for all 1 < β < 16/15, we obtain the conclusion.
+B
+Proofs for Lemmas
+We now present the proofs for all lemmas in Section 3.
+B.1
+Proof of Lemma 3.3
+We need to show the following properties of Orlicz-Wasserstein:
+(i) WΦ(ν1, ν2) = WΦ(ν2, ν1) for any probability measures ν1, ν2 on (Rd, ∥ · ∥).
+(ii) WΦ(µ, µ) = 0 for any probability measure µ on (Rd, ∥ · ∥).
+(iii) WΦ(ν1, ν2) ≤ WΦ(ν1, ν3)+WΦ(ν3, ν2) for any probability measures ν1, ν2, ν3 on (Rd, ∥·∥).
+(i) follows easily from the fact ∥x − y∥ = is symmetric with respect to x, y ∈ Rd .
+For (ii) consider the coupling, ν(x, y) = µ(x)1x=y, then it is clear to see that for any k > 0,
+�
+Rd×Rd Φ(∥x − y∥/k) dν(x, y) = 1 and therefore WΦ(µ, µ) = 0.
+For part (iii), assume that WΦ(ν1, ν3) = k1, WΦ(ν3, ν2) = k2. Then, it is enough to show
+that there exists a coupling ν of ν1 and ν2 such that
+�
+Rd×Rd Φ(∥x − y∥/(k1 + k2)) dν(x, y) ≤ 1.
+By results from Villani (2003, 2009), there exists a coupling µ1 of ν1 and ν3 and a coupling
+µ2 of ν2 and ν3 such that,
+�
+Rd×Rd Φ(∥x − z∥/k1) dµ1(x, z)
+≤
+1
+�
+Rd×Rd Φ(∥z − y∥/k2) dµ2(y, z)
+≤
+1.
+(38)
+23
+
+Then, by a result in probability theory there exists a probability measure µ on Rd × Rd × Rd
+such that
+�
+x∈Rd µ(dx, y, z)
+=
+µ2(y, z)
+�
+x∈Rd µ(x, dy, z)
+=
+µ1(x, z)
+(39)
+Define ν(x, y) :=
+�
+z∈Rd µ(x, y, dz). Then, we obtain that
+�
+Rd×Rd Φ(∥x − y∥/(k1 + k2)) dν(x, y)
+=
+�
+Rd×Rd×Rd Φ(∥x − y∥/(k1 + k2)) dµ(x, y, z)
+≤
+�
+Rd×Rd×Rd Φ((∥x − z∥ + ∥y − z∥)/(k1 + k2)) dµ(x, y, z)
+≤
+�
+Rd×Rd×Rd Φ
+�
+k1
+k1 + k2
+∥x − z∥
+k1
++
+k2
+k1 + k2
+∥y − z∥
+k2
+�
+dµ(x, y, z)
+≤
+k1
+k1 + k2
+�
+Rd×Rd Φ
+�∥x − z∥
+k1
+�
+dµ1(x, z)
++
+k2
+k1 + k2
+�
+Rd×Rd Φ
+�∥y − z∥
+k2
+�
+dµ2(y, z) ≤ 1.
+The first inequality follows from the triangle inequality property of ∥·∥, while the last inequality
+follows from the convexity of Φ.
+B.2
+Proof of Lemma 3.4
+Fix a coupling ν of ν1 and ν2. Consider λ satisfying
+�
+Rd×Rd Φ(∥x − y∥/λ) dν(x, y) < ∞,
+�
+Rd×Rd Ψ(∥x − y∥/λ) dν(x, y) < ∞,
+�
+Rd×Rd Φ(∥x − y∥/λ) dν(x, y) ≤
+�
+Rd×Rd Ψ(∥x − y∥/λ) dν(x, y),
+and thus, we find that
+�
+λ :
+�
+Rd×Rd Ψ(∥x − y∥/λ) dν(x, y) ≤ 1
+�
+(40)
+⊂
+�
+λ :
+�
+Rd×Rd Φ(∥x − y∥/λ) dν(x, y) ≤ 1
+�
+.
+As a consequence, we obtain the conclusion of Lemma 3.4 since infimum of a set is smaller
+than the infimum of its subset.
+24
+
+B.3
+Proof of Lemma 3.6
+Consider X ∼ ν1 and Y ∼ Zδ,d. Let K be such that
+�
+R
+exp((7/32)|yi/K|α − (7/16)|yi/δ|4/3)dyi < ∞.
+Then, we find that
+inf
+µ
+� �
+Rd×Rd Φ(∥x − y∥/λ) dµ(x, y) : µ ∈ Q(ν1, ν2)
+�
+≤
+�1
+δ
+�d �
+Rd exp((7/32)∥y∥α/λα)
+d
+�
+i=1
+K1(yi/δ)
+d
+�
+i=1
+dyi − 1
+≤
+d
+�
+i=1
+�1
+δ
+� �
+R
+exp((7/32)|yi|α/λα)K1(yi/δ)dyi − 1
+=
+d
+�
+i=1
+�1
+δ
+� �
+R
+φ(yi)2 exp((7/32)|yi/λ|α − (7/16)|yi/δ|4/3)dyi − 1,
+where φ(·) is the function in Lemma B.1. The second inequality follows from the fact that
+∥x∥p ≤ ∥x∥q when p ≥ q, where ∥ · ∥p is the Lp norm.
+The final equality follows from
+Lemma B.1. Now, as |φ(x)| ≤ Cφ for some constant Cφ < ∞, we have following the result in
+Lemma 3.4,
+WΦ(ν1, ν2) ≤ Cαδ
+where
+Cα = inf
+�
+k > 0 :
+�
+R
+exp(|y/k|α − |y|4/3)dy − 1 ≤ 1
+C2
+φ
+�
+.
+Note that, Cα as defined above exists because α ≤ 4/3. As a consequence, we obtain the
+conclusion of the lemma.
+B.4
+Proof of Lemma 3.7
+Consider a coupling, ν between ν1 and ν2 that keeps fixed all the mass shared between ν1 and
+ν2, and redistributes the remaining mass independently, i.e.,
+ν(x, y) = (ν1(x)
+�
+ν2(y))1x=y +
+1
+(ν1 − ν2)+(Rd)(ν1(x) − ν2(x))+(ν2(y) − ν1(y))+
+(41)
+Let k0 be defined as
+k0 := inf{k ∈ R+ :
+�
+Rd Φ(∥x∥/k) d|ν1(x) − ν2(x)| ≤ 1}.
+(42)
+Then, using ν as defined in the above display we get
+�
+Rd×Rd Φ(∥x − y∥/2k0) dν(x, y)
+(43)
+=
+�
+Rd×Rd Φ(∥x − y∥/2k0) ·
+1
+(ν1 − ν2)+(Rd)(ν1(x) − ν2(x))+(ν2(y) − ν1(y))+
+≤
+�
+Rd×Rd Φ(∥x∥/k0)(ν1(x) − ν2(x))+ ≤ 1
+25
+
+Therefore,
+WΦ(ν1, ν2) ≤ 2 inf{k ∈ R+ :
+�
+Rd Φ(∥x∥/k) d|ν1(x) − ν2(x)| ≤ 1}.
+As a consequence, we reach the conclusion of the lemma.
+Lemma B.1. Let f(x) = exp(−x4), and ˜f(t) = (1/2π)
+� ∞
+−∞ exp(−itx)f(x)dx. Then,
+| ˜f(t)| ≤ φ(t) exp(−7/32|t|4/3),
+(44)
+where φ(t) is an absolutely bounded real-valued function.
+Proof. Consider a rectangle on the complex plane, with vertices at R, −R, R + iζ, −R + iζ
+respectively. Following Goursat’s Theorem Stein & Shakarchi (2010) for integration along
+rectangular contours on the complex plane, the contour integral along a closed rectangle is 0.
+Therefore,
+� R
+−R
+exp(−itx)f(x)dx +
+� R+iζ
+R
+exp(−itx)f(x)dx +
+� −R
+−R+iζ
+exp(−itx)f(x)dx
++
+� −R+iζ
+R+iζ
+exp(−itx)f(x)dx = 0.
+Now,
+|
+� R+iζ
+R
+exp(−itx)f(x)dx| = |
+� ζ
+0
+exp(itR − tx)f(R + ix)idx| ≤ C exp(−R4) → 0,
+as R → ∞. Similarly,
+|
+� −R
+−R+iζ
+exp(−itx)f(x)dx| → 0,
+as R → ∞.
+Therefore,
+lim
+R→∞
+� R+iζ
+−R+iζ
+exp(−itx)f(x)dx = lim
+R→∞
+� R
+−R
+exp(−itx)f(x)dx = 2π ˜f(t).
+Now,
+lim
+R→∞
+� R
+−R
+exp(−itx)f(x)dx = 2π ˜f(t)
+=
+lim
+R→∞
+� R+iζ
+−R+iζ
+exp(−itx)f(x)dx
+=
+lim
+R→∞
+� R
+−R
+exp(it(x + iζ))f(x + iζ)dx.
+=
+lim
+R→∞
+� R
+−R
+exp(−itx − tζ)) exp(−(x + iζ)4)dx.
+Expanding the above expression,
+˜f(t) = (1/2π) limR→∞
+� R
+−R exp(−itx − 4ix3ζ + 4ixζ3 − tζ − (x2 − 3ζ2)2 + 8ζ4)dx.
+26
+
+Substituting ζ = 1
+4 sign(t)|t|1/3 in the above equationa,
+| ˜f(t)| ≤ (1/2π) exp(−(7/32)|t|4/3) ·
+� ∞
+−∞
+exp(−(x2 − (1/3)|t|1/2)2)dx.
+(45)
+The proof is complete when we note that φ(t) := (1/2π)
+� ∞
+−∞ exp(−(x2 − (1/3)|t|1/2)2)dx is an
+absolutely bounded function.
+Lemma B.2. Let k(t) = c ˜f(t)2, where ˜f(t) = (1/2π)
+� ∞
+−∞ exp(−itx) exp(−x4)dx and c is a
+constant of proportionality so that
+� ∞
+−∞ k(t)dt = 1. Then,
+|
+� ∞
+−∞
+exp(itx)k(t)dt| ≲ exp(−(x/2)4)
+(46)
+Proof. Define f(x) = exp(−x4). Then, by a version of the Fourier inversion theorem,
+� ∞
+−∞
+exp(itx)k(t)dt = f ∗ f(x),
+where ∗ is the convolution operator. Since convolution of even functions is even, it is enough
+to show the result for x > 0. Then,
+f ∗ f(x)
+=
+� ∞
+−∞ exp(−y4) exp(−(y − x)4)dy
+=
+� ∞
+x/2 exp(−y4) exp(−(y − x)4)dy +
+� x/2
+−∞ exp(−y4) exp(−(y − x)4)dy
+≤ exp(−(x/2)4)
+� ∞
+x/2 exp(−(y − x)4)dy + exp(−(x/2)4)
+� x/2
+−∞ exp(−y4)dy
+≤ 2 exp(−(x/2)4)
+� ∞
+−∞ exp(−y4)dy.
+(47)
+The result holds with C = 2
+� ∞
+−∞ exp(−y4)dy since
+� ∞
+−∞ exp(−y4)dy < ∞.
+B.5
+Proof of Lemma 3.5
+Suppose q = (qij)1≤i≤k0,1 ≤j≤k ∈ [0, 1]k0×k is a coupling between p0 = (p0
+1, . . . , p0
+k0) and
+p = (p1, . . . , pk), with Q(p, p′) representing the space of all such couplings of p and p′. Then,
+for fixed k we have
+�
+qijΦ(∥θ0
+i − θj∥/k) ≥
+�
+qij1{∥θ0
+i −θj∥≥η}Φ(η/k)
+≥
+�
+pj1{∥θ0
+i −θj∥≥η for all i}Φ(η/k).
+Let K = inf{k ≥ 0 : � pj1{∥θ0
+i −θj∥≥η for all i}Φ(η/k) ≤ 1}. Then,
+K ≥ η
+�
+Φ−1
+�
+1
+� pj1{∥θ0
+i −θj∥≥η for all i}
+��−1
+,
+(48)
+where Φ−1 is the inverse function of the function Φ. Note that, this function exists and is
+concave as Φ is monotonic increasing and convex. Moreover, by Lemma 3.4(i), we would have
+that WΦ(G, G0) ≥ K, where,
+WΦ(G, G0) := infq∈Q(p,p′){inf{k ≥ 0 : � qijΦ(∥θ0
+i − θj∥/k) ≤ 1}}
+Combining the results from equations (48) and (49) we obtain the conclusion of the lemma.
+27
+
+B.6
+Prior mass on Wasserstein ball
+Lemma B.3. Let G ∼ DP(γ, Hn). Fix r ≥ 1. Assume G0 ∈ M(Θ), where Θ = [−¯θ, ¯θ]d. If
+Hn admits condition (P.1), then the following holds
+Π (W r
+r (G, G0) ≤ (2r + 1)ϵr) ≥
+Γ(γ)(c0γπd/2)D
+(2Γ(d/2 + 1))D(2D)D−1
+� ϵ
+2¯θ
+�r(D−1)+dD
+for all ϵ sufficiently small so that D(ϵ, Θ, ∥.∥) > γ.
+Here, D = D(ϵ, Θ, ∥.∥) stands for the maximal ϵ-packing number for Θ under ∥.∥ norm,
+and Γ(·) is the gamma function.
+Proof. From Lemma 5 in Nguyen (2013),
+Π (W r
+r (G, G0) ≤ (2r + 1)ϵr) ≥ Γ(γ)γD
+(2D)D−1
+�
+ϵ
+Diam(Θ)
+�r(D−1)
+sup
+S
+D
+�
+i=1
+Hn(Si),
+where, S := (S1, ..., SD) denotes the D disjoint ϵ/2-balls that form a maximal ϵ-packing of Θ.
+The supremum is taken over all such packings.
+Now, Hn(A) ≥
+� c0
+µ(Θ)
+�
+µ(A). Moreover, �D
+i=1 µ(Si) ≥
+�
+(√πϵ)d
+2Γ(d/2 + 1)
+�D
+. Using this, we
+arrive at the result.
+B.7
+Metric entropy with Hellinger distance
+Lemma B.4. Let G0 be a discrete mixing measure with all its atoms in Θ = [−˜θ, ˜θ]d ⊂ Rd.
+Let PG(Θ) := {pG : G ∈ G(Θ)}. Then, if the kernel f is multivariate Gaussian with covariance
+matrix Σ,
+log D(ϵ/2, {pG ∈ PG(Θ) : ϵ < h(pG, pG0) ≤ 2ϵ}, h) ≤ c1
+�
+˜θ
+√λminϵ
+�d
+log
+�
+e + 32e˜θ2
+λminϵ2
+�
+(49)
+for some universal constant c1.
+Proof. Let N(ϵ, P, d) denote the ϵ-covering number of the space P relative to the metric d.
+It is related to the packing number by the following identity:
+N(ϵ, P, h) ≤ D(ϵ, P, d) ≤ N(ϵ/2, P, h).
+(50)
+Using the result in Example 1 of Nguyen (2013), when fΣ(·|θ) ∼ Nd(θ, Σ),
+h2(fΣ(·|θi), fΣ(·|θ′
+j)) = 1 − exp
+�
+−1
+8∥θi − θ′
+j∥2
+Σ−1
+�
+≤
+∥θi − θ′
+j∥2
+8λmin
+,
+(51)
+where ∥z∥Σ−1 :=
+√
+z′Σ−1z.
+Let G0 = �k0
+i=1 p0
+i δθ0
+i and G = �k′
+j=1 p′
+jδθ′
+j be mixing measures in G(Θ), with k0, k′ ∈ [1, ∞].
+Let q = (qij)1≤i≤k0,1 ≤j≤k′ ∈ [0, 1]k0×k′ denote a coupling of p0 and p′.
+28
+
+Using Lemma 2 of Nguyen (2013) with φ(x) = 1
+2(√x − 1)2, gives us:
+h2(pG, pG0) ≤
+inf
+q∈Q(p0,p′)
+�
+i,j
+qij
+∥θi − θ′
+j∥2
+8λmin
+= W2(G, G0)2
+8λmin
+,
+(52)
+where Q(p0, p′) is the set of all couplings of p0 and p′. Therefore, it immediately follows that:
+log D(ϵ/2, {pG ∈ PG(Θ) : ϵ < h(pG, pG0) ≤ 2ϵ}, h)
+≤
+log D(
+�
+2λminϵ, {G : G ∈ G(Θ)}, W2) ≤ N
+��
+λmin
+8
+ϵ, Θ, ∥ · ∥
+�
+log
+�
+e + 32e˜θ2
+λminϵ2
+�
+.
+The last inequality follows by applying Eq. (50) followed by Lemma 4 part (b) of Nguyen
+(2013). The result then follows immediately.
+B.8
+Computation of M corresponding to KL ball
+Lemma B.5. Let G be a discrete mixing measure with all its atoms in
+�
+−˜θ, ˜θ
+�d
+for some
+˜θ > 0. Furthermore, assume the atoms of G0 lie in
+�
+−¯θ, ¯θ
+�d where ¯θ > 0 is given. Then, the
+following holds if the kernel f is multivariate Gaussian,
+�
+(pG0(x))2
+pG(x)
+µ(dx) ≤ exp(dλ−1
+min(5¯θ2 + 4˜θ2)).
+(53)
+Here µ is the Lebesgue measure on Rd.
+Proof. For the multivariate Gaussian kernel with covariance matrix Σ, similar to the multi-
+variate Laplace case, using lemma 2 of Nguyen (2013) with φ(x) = 1
+x, gives us:
+�
+(pG0(x))2
+pG(x)
+µ(dx) ≤
+inf
+q∈Q(p0,p′)
+�
+i,j
+qij
+�
+(fΣ(x|θ0
+i ))2
+fΣ(x|θ′
+j) µ(dx),
+(54)
+where Q(p0, p′) is the set of all couplings of p0 and p′, and fΣ(·|θ) is the multivariate Gaussian
+kernel with covariance parameter Σ and mean parameter θ.
+�
+(fΣ(x|θ0
+i ))2
+fΣ(x|θ′
+j) µ(dx)
+=
+�
+fΣ(x|θ0
+i ) exp
+�
+−∥x − θ0
+i ∥2
+Σ−1 + ∥x − θ′
+j∥2
+Σ−1
+2
+�
+µ(dx)
+(55)
+=
+�
+fΣ(x|θ0
+i ) exp
+�
+−∥θ′
+j − θ0
+i ∥2
+Σ−1
+2
++ ⟨x − θ′
+j, Σ−1θ′
+j − θ0
+i ⟩
+�
+µ(dx),
+where the second equality follows by simple calculation using x − θ0
+i = (x − θ′
+j) + (θ′
+j − θ0
+i ).
+If MΣ(t|θ) is the moment generating function of the Gaussian distribution with mean θ
+and covariance Σ, then
+MΣ(t|θ) = exp(⟨θ, t⟩ + 1
+2⟨t, Σt⟩).
+29
+
+Using this result , we can rewrite Eq. (55) as
+�
+(fΣ(x|θ0
+i ))2
+fΣ(x|θ′
+j) µ(dx)
+=
+exp(⟨θ′
+j − θ0
+i , Σ−1θ0
+i + θ′
+j⟩) ≤ exp(2dλ−1
+min(˜θ + ¯θ)2 + dλ−1
+min¯θ2),
+The bound on
+�
+(pG0(x))2/pG(x)µ(dx) then follows immediately.
+C
+Theoretical guarantee of Algorithm 1
+We show in this section that the output of Algorithm 1 converges to the Entropy regularised
+version of the Orlicz-Wasserstein distance in equation (19).
+Proposition C.1. Let ˆW λ
+Φ(ν1, ν2) be the output of Algorithm 1 and W λ
+Φ(ν1, ν2) be as in
+equation (19). Then
+| ˆW λ
+Φ(ν1, ν2) − W λ
+Φ(ν1, ν2)| < ϵ.
+(56)
+Proof. Here M is the cost matrix such that Mij = ∥xi − yj∥.
+Note that S(Φ(M/W λ
+Φ(ν1, ν2)), λ, r, c) < 1 and if S(Φ(M/η), λ, r, c) < 1, then η <
+W λ
+Φ(ν1, ν2)).
+If ˆxupp = max(M)/Φ−1(1), ˆxlow = d(M, λ, r, c)/Φ−1(1 + d(M, λ, r, c) − S(M, λ, r, c)) it is
+enough to show that
+fxupp = S(Φ(M/ˆxupp), λ, r, c) < 1,fxlow = S(Φ(M/ˆxlow), λ, r, c) > 1, since it would imply
+xupp := ˆW λ
+Φ(ν1, ν2) < W λ
+Φ(ν1, ν2) < xlow and therefore if |xupp − xlow| < ϵ, the result holds
+directly.
+We need to show
+(i) S(Φ(M/ˆxupp), λ, r, c) < 1.
+(ii) S(Φ(M/ˆxlow), λ, r, c) > 1.
+For (i), observe that
+S(Φ(M/ˆxupp), λ, r, c) =
+inf
+ν∈Q(ν1,ν2)
+�
+Rd×Rd Φ(∥x − y∥/ˆxupp) dν(x, y) − (1/λ)(H(ν))
+(57)
+=
+inf
+ν∈Q(ν1,ν2)
+�
+Rd×Rd Φ(Φ−1(1)∥x − y∥/max(M)) dν(x, y) − (1/λ)(H(ν)) ≤ 1
+(58)
+The last inequality holds by monotonicity of Φ combined with ∥x − y∥/max(M) < 1 with
+ν-probability 1, and the fact that H(ν) > 0.
+For (ii),note that for any ν ∈ Q(ν1, ν2), it holds that
+�
+Rd×Rd Φ(∥x − y∥/η) dν(x, y) − H(ν)/λ
+≥ Φ
+��
+Rd×Rd(∥x − y∥/η) dν(x, y)
+�
+− (H(r) + H(c))/λ
+(59)
+≥ Φ((S(M, λ, r, c) + (H(r) + H(c))/2λ)/η)) − (H(r) + H(c))/λ.
+(60)
+Both the inequalities hold by monotonicty and convexity of Φ combined with the fact that
+∀ν ∈ Q(ν1, ν2), it holds that H(r) + H(c) ≥ H(ν) ≥ (H(r) + H(c))/2.
+Now Φ((S(M, λ, r, c) + (H(r) + H(c))/2λ)/η)) − (H(r) + H(c))/λ ≥ 1, for any η ≤ ˆxupp,
+this completes the proof.
+30
+
+D
+Estimation of number of components for mixing measures
+In this section, we consider how Orlicz-Wasserstein distances could be used to improved
+estimation of the number of components with Gaussian mixture models. Gaussian Mixture
+models have been used for the purpose of clustering both historically Goldberger & Roweis
+(2004) as well as in modern applications Athey & Vogelstein (2019); Chakravarti et al. (2019);
+Jiao et al. (2022). From the Bayesian perspective, often used BNP priors for mixture models
+tend to overestimate the number of components drastically by producing multiple extraneous
+components around the ”true” components Miller & Harrison (2014). This makes it difficult
+to estimate the number of components, where it may of interest MacEachern & Muller (1998);
+Green & Richardson (2001).
+Several recent works have explored the consistent estimation of the number of compo-
+nents with mixture models, both with in-processing Manole & Khalili (2021) and post-
+processing Guha et al. (2021) techniques. However, while Manole & Khalili (2021) restricts
+attention to the overfit setting only, Guha et al. (2021) requires the knowledge of explicit
+contraction rates of respective parameters in Wasserstein distances. As parameter convergence
+rates of Dirichlet Process Gaussian Mixture models are extremely slow Nguyen (2013), this
+would also affect the estimation of the number of components negatively. The procedures in
+both the works Guha et al. (2021); Manole & Khalili (2021) consist of two smaller steps, trun-
+cation of extraneous outlier atoms and merging of atoms which are close to the ”true” atoms.
+The results in this work specifically, Theorem 3.3 provide a low threshold for truncating outlier
+atoms thereby eliminating outlier atoms more efficiently. Combined with an understanding of
+convergence behavior around the ”true” atoms would allow efficient estimation of the number
+of components with Dirichlet Process Gaussian Mixture models.
+31
+

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+Genuinely noncyclic geometric gates in two-pulse schemes
+Nils Eivarsson1 and Erik Sj¨oqvist1, ∗
+1Department of Physics and Astronomy, Uppsala University, Box 516, Se-751 20 Uppsala, Sweden
+(Dated: January 16, 2023)
+While most approaches to geometric quantum computation is based on geometric phase in cyclic evolution,
+noncyclic geometric gates have been proposed to further increase flexibility. While these gates remove the dy-
+namical phase of the computational basis, they do not in general remove it from the eigenstates of the time
+evolution operator, which makes the geometric nature of the gates ambiguous. Here, we resolve this ambiguity
+by proposing a scheme for genuinely noncyclic geometric gates. These gates are obtained by evolving the com-
+putational basis along open paths consisting geodesic segments, and simultaneously assuring that no dynamical
+phase is acquired by the eigenstates of the time evolution operator. While we illustrate the scheme for the sim-
+plest nontrivial case of two geodesic segments starting at each computational basis state of a single qubit, the
+scheme can be straightforwardly extended to more elaborate paths, more qubits, or even qudits.
+I.
+INTRODUCTION
+Quantum computation is a form of information processing
+that uses quantum mechanical properties, such as superpo-
+sition and entanglement, to perform calculations. Although
+quantum computers of today do not have the number of qubits
+required for realising programmable large-scale computation,
+progress towards this goal has been achieved [1] on different
+experimental platforms [2–4].
+It is not, however, enough to scale up quantum computers;
+they also need to be resilient to noise and decoherence un-
+der which qubits would lose their desired quantum mechan-
+ical properties. The state within a quantum computer can-
+not be completely isolated and decoherence is therefore in-
+evitable. It is possible to overcome this problem according
+to the threshold theorem [5, 6], which entails that an arbitrar-
+ily long quantum computation can be sustained by error cor-
+rection techniques provided the error rate per gate is below a
+certain threshold value.
+One approach to reach below this threshold is to use ge-
+ometric quantum gates [7–12].
+These are based on the
+(Abelian) geometric phase (GP) of quantum systems [13, 14].
+When a quantum state undergoes a cyclic evolution it can gain
+a phase factor. This phase factor can be split into a dynam-
+ical part and a geometric part, where the geometric part is
+only dependent on the path a quantum state takes through its
+state space. By choosing an evolution for which the dynami-
+cal phase effect is trivial, it is possible to implement quantum
+logic gates that are purely dependent on the geometry of the
+path. This dependency on only geometry can be shown to be
+resilient against certain types of noise [16, 17].
+While geometric quantum gates show promise in the im-
+plementation of robust quantum computers, there are still im-
+provements to be made before they are fully operational. One
+possible improvement is to reduce the number of pulses and
+the run time needed to implement the gates, as this would re-
+duce the qubits’ exposure to the environment, thereby limit-
+ing the effect of decoherence, as well as simplifying the pulse
+schemes. To reduce the run time, noncyclic evolution in the
+∗ [email protected]
+implementation of geometric quantum gates have been pro-
+posed [18–25] and experimentally implemented [26]. These
+gate are based on computational states evolving along open
+paths while still only acquiring a GP in noncyclic evolution
+[27, 28]. However, in cyclic evolution of the computational
+states, the GPs are the eigenvalues of the time evolution oper-
+ator but this is not the case in the noncyclic case. This makes
+the geometric meaning of these noncyclic gates ambiguous.
+Here, we propose precise conditions for geometric gates.
+Under these conditions the meaning of noncyclic geomet-
+ric gates become unambiguous.
+This provides a notion of
+genuinely noncyclic geometric gates. Furthermore, geomet-
+ric gates under cyclic evolution of the computational states,
+which are gates that trivially satisfy these conditions, are
+Abelian (they diagonalize in the computational basis) and
+therefore fail universality. In order to complete the univer-
+sal set, we shall thus see that at least one genuinely noncyclic
+geometric gate is needed.
+II.
+CONDITIONS FOR GEOMETRIC GATES
+We consider Schr¨odinger evolution iℏ| ˙ψ(t)⟩ = H(t)|ψ(t)⟩,
+where H(t) is the Hamiltonian of the system with Hilbert
+space H. Let U(t, 0) = Te− i
+ℏ
+� t
+0 H(t′)dt′ be the corresponding
+time evolution operator. We assume that the computational
+system consists of n qubits with state space M ⊆ H spanned
+by 2n ≤ dim H predetermined computational state vectors
+{|⃗q⟩ = |q1, . . . , qn⟩}q1,...,qn=0,1, fixed by the final read out of the
+computation.
+Let U(τ, 0) be the desired gate realized during the time in-
+terval t ∈ [0, τ]. Consider the eigenvalue equation
+U(τ, 0)|ψk⟩ = eiϕk|ψk⟩, k = 1, . . . , 2n.
+(1)
+Here, {|ψk⟩} is an orthonormal set of vectors since U(τ, 0) is
+a normal operator [5]. We assume that U(τ, 0) preserves M,
+i.e., |ψk⟩ = �
+⃗q c(k)
+⃗q |⃗q⟩, ∀k, even in the case where 2n < dim H.
+Thus, U(0, τ)U†(τ, 0) is a projection operator on M.
+The action of U(τ, 0) on each computational state |⃗q⟩ can be
+arXiv:2301.05627v1  [quant-ph]  13 Jan 2023
+
+2
+understood in terms of its GP
+Γ⃗q = arg⟨⃗q|U(τ, 0)|⃗q⟩
++i
+� τ
+0
+⟨⃗q|U†(t, 0) ˙U(t, 0)|⃗q⟩dt,
+(2)
+provided ⟨⃗q|U(τ, 0)|⃗q⟩ is nonzero. Γ⃗q is real-valued and a prop-
+erty of the path U(t, 0)|⃗q⟩⟨⃗q|U†(t, 0) in state space, as it is
+invariant under monotonic reparametrizations t �→ s(t) and
+time-local phase changes U(t, 0) �→ eif(t)U(t, 0) [28]. These
+Γ⃗q:s are generically noncyclic GPs as the paths in state space
+are typically open. On the other hand, the eigenvectors |ψk⟩ of
+U(τ, 0) are the cyclic states of the evolution U(t, 0), t ∈ [0, τ],
+and may thus be analyzed by using the Aharonov-Anandan
+GP [14]. In this framework, each cyclic phase ϕk contains a
+geometric (γk) and a dynamical (δk) part, given by
+γk = arg⟨ψk|U(τ, 0)|ψk⟩
++i
+� τ
+0
+⟨ψk|U†(t, 0) ˙U(t, 0)|ψk⟩dt.
+(3)
+and
+δk = −1
+ℏ
+� τ
+0
+⟨ψk|U†(t, 0)H(t)U(t, 0)|ψk⟩dt,
+(4)
+such that ϕk = γk + δk.
+Based on the above, we can now define a genuinely non-
+cyclic geometric gate (GNGG) as a unitary U(τ, 0) that satis-
+fies the following two conditions:
+(i) Γ⃗q − Γ⃗0 = arg⟨⃗q|U(τ, 0)|⃗q⟩ − ⟨⃗0|U(τ, 0)|⃗0⟩, mod 2π,
+(ii) δk − δ1 = 0, mod 2π,
+for binary vectors ⃗q and k. In essence, such a gate depends
+only on GPs for both its eigenstates and the predetermined
+computational states.
+Before addressing the physical realization of GNGGs in the
+next section, we consider some qubit gates assuming them to
+be genuinely noncyclic, in order to gain some further con-
+ceptual insights. First, let us consider a genuinely noncyclic
+geometric Hadamard gate H. Such a gate takes the form
+H =
+1√
+2
+�
+|0⟩ + |1⟩
+�
+⟨0| + eiπ 1√
+2
+�
+|1⟩ − |0⟩
+�
+⟨1|
+(5)
+in the computational basis [29, 30], and
+H =
+1
+4 − 2
+√
+2
+�
+|0⟩ −
+�
+1 −
+√
+2
+�
+|1⟩
+� �
+⟨0| −
+�
+1 −
+√
+2
+�
+⟨1|
+�
++eiπ
+1
+4 + 2
+√
+2
+�
+|0⟩ −
+�
+1 +
+√
+2
+�
+|1⟩
+� �
+⟨0| −
+�
+1 +
+√
+2
+�
+⟨1|
+�
+,
+(6)
+in the eigenbasis, with the phase factor eiπ = −1 assumed to be
+geometric in both cases. We thus see that the geometric phase
+difference for both forms is π, i.e., Γ1 − Γ0 = γ2 − γ1 = π.
+As a second example, we consider rotation gates Uz(ϑz) =
+e−i ϑz
+2 σz and Uy(ϑy) = e−i
+ϑy
+2 σy that can be used to describe
+an arbitrary single-qubit rotation [31].
+First, we note that
+Uz(ϑz) is diagonal in the computational basis, i.e., the eigen-
+basis coincides with the computational basis. This implies
+that the gate is a standard cyclic geometric gate [8], for which
+Γ1 −Γ0 = γ2 −γ1 = ϑz. More interesting is the genuinely non-
+cyclic geometric implementation of Uy(ϑy). We may write
+Uy =
+�
+cos ϑy
+2 |0⟩ + sin ϑy
+2 |1⟩
+�
+⟨0|
++
+�
+− sin ϑy
+2 |0⟩ + cos ϑy
+2 |1⟩
+�
+⟨1|
+(7)
+and
+Uy = e−i
+ϑy
+2 |y+⟩⟨y+| + ei
+ϑy
+2 |y−⟩⟨y−|,
+|y±⟩ =
+1√
+2
+�|0⟩ ± i|1⟩�
+(8)
+in the computational basis and eigenbasis, respectively. Pro-
+vided ϑy � π, we find Γ1 − Γ0 = 0 and γ2 − γ1 = ϑy, i.e., while
+the noncyclic GP difference of the computational basis states
+is trivial, the cyclic GP difference is generally not. In the case
+where ϑy = π, the noncyclic phases are not defined as the com-
+putational states are each mapped on orthogonal states. It fol-
+lows, more generally, that π rotations around any axis in the xy
+plane cannot be implemented in a genuinly noncyclic manner.
+Curiously, this implies that there is no genuinely noncyclic
+implementation of CNOT, while the GNGG scheme can be
+used to realize other entangling two-qubit control gates with
+noncyclic and cyclic GPs 0, 0, Γ10, Γ11 and 0, 0, γ3, γ4, respec-
+tively.
+III.
+TWO-PULSE SINGLE-QUBIT GATES
+A key point of the GNGG technique is that it can be used
+to reduce the number of pulses to implement a universal set of
+geometric gates. To make this point explicit, we shall now ex-
+amine the physical realization of genuinely noncyclic single-
+qubit gates and demonstrate single-qubit universality, by using
+the simplest nontrivial case of two pulses. To achieve univer-
+sality with geometric gates in such schemes, adhering to the
+proposed conditions (i) and (ii) above, noncyclic geometric
+gates are required, as cyclic gates are inherently Abelian and
+thus insufficient for universality [29].
+Let the two pulses be applied during [0, t1] and [t1, τ], re-
+spectively. By choosing the pulses such that they move the
+computational basis along a pair of geodesic segments, the
+corresponding dynamical phases vanish. After constructing
+these gates, the dynamical phases of the eigenstates of the time
+evolution operator are studied, to find which gates are gen-
+uinely geometric, i.e., satisfy the condition 2δ = 0, mod 2π,
+where we have used that δ0 = −δ1 ≡ δ. As we shall see,
+this requires a careful tuning of rotation axes and precession
+angles associated with the two pulses.
+A path is geodesic on the Bloch sphere when the axis n of
+rotation is orthogonal to the initial Bloch vector. Since the
+first pulse acts on the computational basis {|q⟩}q=0,1, it should
+thus correspond to a rotation around any axis n1 in the xy-
+plane, as this would result in evolution of these states along
+
+3
+FIG. 1. The rotation of the qubit during the first pulse (upper panel).
+θ1 is the rotational angle and n1 is the axis of rotation. The rotation
+of the qubit during the second pulse (lower panel). θ2 is the rotational
+angle and φ determines the axis of rotation n2. n2 is orthogonal to
+the final state (with Bloch vector r1) of the first pulse. The paths
+gradually shifts from black (dark) at the start to yellow (bright) at the
+end.
+parts of a great circle that pass through the poles of the Bloch
+sphere. Choosing one axis over another only rotates the finale
+eigenvector of the evolution around the z-axis. We can there-
+fore limit ourselves to one specific axis of rotation and later
+generalize to other axes in the xy-plane by simply rotating the
+gate. We choose the y axis, i.e., n1 = ey, which defines the
+Hamiltonian
+H1 = 1
+2ℏω σy
+(9)
+with corresponding time evolution operator
+U1(t, 0) = e− i
+2 ωt|y+⟩⟨y+| + e
+i
+2 ωt|y−⟩⟨y−|.
+(10)
+At the final time t1 of the first pulse, the qubit has rotated an
+angle θ1 = ωt1. This rotation can be seen in Fig. 1(a).
+For the second pulse to move the qubit along a geodesic,
+the axis of rotation must be orthogonal to the final state of
+the first pulse. Starting at |0⟩, the final state of the first pulse
+can be described by the Bloch vector r1 = sin θ1ex + cos θ1ez,
+which serves as initial state for the second pulse. The axis of
+rotation can therefore be taken as
+n2 = − cos θ1 cos φ ex + sin φ ey + sin θ1 cos φ ez.
+(11)
+This lies in the plane spanned by the vector − cos θ1 ex +
+sin θ1 ez, orthogonal to r1, and the y-axis. φ is the rotational
+angle around r1 relative the xz-plane, see Fig. 1(b). With the
+axis of rotation defined, the Hamiltonian becomes
+H2 = ℏω
+2
+�
+− cos θ1 cos φ σx + sin φ σy
++ sin θ1 cos φ σz
+�
+(12)
+with eigenvalues and eigenvectors
+λ± = ±1
+2ℏω,
+|κ±⟩ =
+�
+cos2 θ1 cos2 φ + sin2 φ
+2 ∓ 2 sin θ1 cos φ
+×
+�
+|0⟩ +
+∓1 + sin θ1 cos φ
+cos θ1 cos φ + i sin φ|1⟩
+�
+.
+(13)
+The time evolution operator of the second rotation is
+U2(t, t1) = e− i
+2 ω(t−t1)|κ+⟩⟨κ+| + e
+i
+2 ω(t−t1)|κ−⟩⟨κ−|,
+(14)
+which rotates the qubit an additional angle θ2 = ω(τ−t1). This
+rotation is shown in Fig. 1(b) as the geodesic path connecting
+r1 and r2. The time evolution during the full time interval
+[0, τ] can be written as
+U(t, 0) =
+�
+U1(t, 0),
+0 ≤ t ≤ t1,
+U2(t, t1)U1(t1, 0), t1 ≤ t ≤ τ
+(15)
+with the gate U(τ, 0) = U2(τ, t1)U1(t1, 0) being characterized
+by the angles θ1, θ2, and φ, i.e., U(τ, 0) ≡ U(θ1, θ2, φ).
+By inserting Eqs. (10) and (14) into (15), we find the eigen-
+values and eigenvectors of U(θ1, θ2, φ):
+λ± = cos
+�θ1
+2
+�
+cos
+�θ2
+2
+�
+− sin
+�θ1
+2
+�
+sin
+�θ2
+2
+�
+sin φ ± i
+�
+1 −
+�
+cos
+�θ1
+2
+�
+cos
+�θ2
+2
+�
+− sin
+�θ1
+2
+�
+sin
+�θ2
+2
+�
+sin φ
+�2
+≡ e±iϕ,
+|ψ±⟩ = N±
+���������|0⟩ +
+cos
+� θ1
+2
+� �
+− cos
+� θ2
+2
+�
++ i cos φ sin
+� θ2
+2
+��
++ sin
+� θ1
+2
+�
+sin
+� θ2
+2
+�
+sin (φ) + e±iϕ
+sin
+� θ1
+2
+� �
+− cos
+� θ2
+2
+�
++ i cos φ sin
+� θ2
+2
+��
+− cos
+� θ1
+2
+�
+sin
+� θ2
+2
+�
+sin φ
+|1⟩
+���������
+(16)
+
+zez
+r1
+01
+n1
+XZ
+r1
+x
+02
+r24
+with normalization factors
+N± =
+1
+2
+√
+2
+�������������
+2 − cos θ1
+�
+1 + cos θ2 + 2 cos(2φ) sin
+� θ2
+2
+�2�
++ 2 sin θ1 sin θ2 sin(φ)
+1 + cos
+� θ1
+2
+� �
+− cos
+� θ2
+2
+�
+cos ϕ ± sin
+� θ2
+2
+�
+cos φ sin ϕ
+�
++ sin
+� θ1
+2
+�
+sin
+� θ2
+2
+�
+sin φ cos ϕ
+�������������
+1/2
+.
+(17)
+FIG. 2. The roots of 2δ = 0. θ1 (θ2) is the first (second) rotation angle,
+and φ is the angle determining the axis of rotation for the second
+rotation.
+Next, we find the dynamical phases of the evolution by us-
+ing Eq. (4), yielding
+δ± = −1
+ℏ
+� t1
+0
+⟨ψ±|H1|ψ±⟩dt
+−1
+ℏ
+� τ
+t1
+⟨ψ±|U†
+1(t1, 0)H2U1(t1, 0)|ψ±⟩dt,
+(18)
+where we have taken into account that H1 and H2 commute
+with U1 and U2, respectively. Explicitly,
+U†
+1(t1, 0)H2U1(t1, 0) = ℏω
+2
+�
+− cos φ σx + sin φ σy
+�
+. (19)
+This shows that both integrands on the right-hand side of
+FIG. 3. GP corresponding to the roots of 2δ = 0. θ1 (θ2) is the first
+(second) rotation angle, and φ is the angle determining the axis of
+rotation for the second rotation.
+Eq. (18) are time-independent, which implies
+δ± = −θ1
+2 ⟨ψ±|σy|ψ±⟩
+−θ2
+2 ⟨ψ±|
+�
+− cos φ σx + sin φ σy
+�
+|ψ±⟩ = ±δ.
+(20)
+To find geometric gates, we need to solve for which choices of
+θ1, θ2, and φ the dynamical phases become trivial, i.e., satisfy
+the condition 2δ = 0, mod 2π. We thus look for numerical
+solutions of
+θ1⟨ψ±|σy|ψ±⟩ + θ2⟨ψ±|
+�
+− cos φ σx + sin φ σy
+�
+|ψ±⟩
+= 0,
+mod 2π.
+(21)
+We restrict to 2δ = 0 in the following.
+
+Roots of 2s
+3
+2.5
+—01/元 = 0.1
+01/π = 0.3
+2
+—01/元 = 0.5
+2
+1.5
+01/π = 0.7
+01/元 = 0.9
+0.5
+0
+0.5
+1
+1.5
+2
+Φ/元Roots of 2s
+3
+2.5
+—01/元 = 1.1
+2
+01/元 = 1.3
+01/π = 1.5
+1.5
+01/π = 1.7
+1
+01/元 = 1.9
+0.5
+0
+0
+0.5
+1
+1.5
+2
+Φ/元Geometric phase for roots of 2d
+2
+1.8
+01/元 = 0.1
+1.6
+01/元 = 0.3
+1.4
+01/元 = 0.5
+1.2
+01/π = 0.7
+1
+01/π = 0.9
+0.8
+0.6
+0
+0.5
+1.5
+2Geometric phase for roots of 2
+2
+1.5
+01/π = 1.1
+-01/元 = 1.3
+/
+01/π = 1.5
+01/π = 1.7
+01/元 = 1.9
+0.5
+0
+0
+0.5
+1.5
+25
+It is possible to find multiple roots for the same set of
+{θ1, φ}. In Fig. 2, roots where θ1 and θ2 are positive and give
+rise to the smallest total precession angles θ1 + θ2, as function
+of φ, are shown. A positive θ1 (θ2) corresponds to a clock-
+wise rotation driven by the first (second) pulse. Roots can be
+found for counter-clockwise rotations to have the same shape
+but reflected or inverted. For θ1 > 0, θ2 < 0, the roots are
+inverted through the point (φ, θ2) = (π, 0); for θ1 < 0, θ2 > 0,
+on the other hand, they are reflected in the line φ = π. When
+both rotations are counter-clockwise the roots are reflected in
+the line θ1 = 0. In Fig. 3, the GPs corresponding to these
+roots are shown. For both counter-clockwise rotations where
+θ1 > 0, θ2 < 0 and θ1 < 0, θ2 > 0 the GPs are reflected in the
+line φ = π and for the case when both rotations are counter-
+clockwise the GPs are the same as in Fig. 3.
+One motivating aspect of noncyclic geometric schemes is
+that they may shortened the exposure to decoherence effects
+by shortening the run time of the gates. To test this in the two-
+pulse realization of GNGGs, we use the total precession angle
+θ1+θ2 as a natural measure of run time. A closer inspection of
+Fig. 2 entails that this angle is at least 2π for all roots. It thus
+appears that the run time cannot be shortened in the proposed
+noncyclic scheme, as compared to cyclic geometric gates with
+pulses driving the computational basis states along ‘orange
+slice’ shaped loops on the Bloch sphere [33, 34].
+FIG. 4. Continuous set of eigenvectors of gates with 2γ = π with the
+first rotation taken around the y-axis. The eigenvectors are swept so
+as to form surfaces inside the Bloch sphere of the qubit.
+To implement a specific gate, both eigenvalues and eigen-
+vectors must match the desired gate. We have shown that
+it is possible to find any eigenvalue but it is also necessary
+to find the corresponding eigenvectors. To demonstrate this,
+eigenvectors corresponding to 2γ = π are displayed in Fig. 4.
+These eigenvectors cover the entire z-axis. Keep in mind that
+only a first rotation around the y-axis has been considered so
+far.
+GNGGs with the same eigenvalue and an eigenvector
+only differing by a rotation around the z-axis can be found by
+changing the first axis of rotation to another in the xy-plane,
+while keeping the relation between the first and the second
+axis. With this finding any GNGGs with eigenvalue 2γ = π,
+such as the Hadamard gate H, can be realized with only two
+pulses and it can similarly be shown for other γ as well.
+FIG. 5. The path of the |0⟩-state (upper panel) and one of the eigen-
+states (lower panel) when acted upon by a geometric Hadamard gate.
+The paths gradually shifts from black (dark) at the start to yellow
+(bright) at the end. The computational state evolves along geodesic
+segments.
+To give an example, we show how a Hadamard gate can
+be implemented. For this, 2γ = π and eigenvectors corre-
+spond to ± ex+ez
+√
+2 on the Bloch sphere. These eigenvectors are
+not present in Fig. 4, but can be found by a suitable rotation
+around the z axis. Fig. 5 illustrates this Hadamard gate acting
+on the computational state |0⟩ and one of the eigenstates.
+IV.
+CONCLUSIONS
+We have proposed a notion of noncyclic geometric gates
+in which both the computational basis and the eigenbasis ac-
+quire purely geometric phases. We have demonstrated a phys-
+ical realization of such genuinely noncyclic geometric gates
+in the single qubit case. This is achieved by using pulse pairs
+that drive the computational states along pairs of geodesic seg-
+ments on the Bloch sphere, and simultaneously make the dy-
+namical phase difference of the eigenstates to vanish. The pro-
+posed concept removes the ambiguity of standard noncyclic
+geometric gates [18–25], in which the computational basis
+undergoes purely geometric evolution, while the eigenstates
+generally do not. Our scheme takes advantage of noncyclic
+geometric phase in order to achieve universality.
+While the analysis focuses on gates using only two pulses,
+
+yZez
+r2
+yz
+q1
+y6
+it can straightforwardly be extended to three or more pulses.
+This may open up for reduction of accumulated rotation angle
+as well as more elaborate paths of the computational states, so
+as to reduce the detrimental effect of noise and decoherence.
+The scheme may further be extended to nongeodesic evolu-
+tion of the computational basis to further improve the error
+resilience of the gates. This extension would require simulta-
+neous removal of the dynamical phase effects of the computa-
+tional basis and the eigenstates of the gates.
+ACKNOWLEDGMENT
+E.S. acknowledges financial support from the Swedish Re-
+search Council (VR) through Grant No. 2017-03832.
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