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+Multi-Genre Music Transformer - Composing Full Length Musical Piece
+Abhinav Kaushal Keshari (Purdue University)
+Abstract
+In the task of generating music, the art factor plays
+a big role and is a great challenge for AI. Previ-
+ous work involving adversarial training (Dong
+et al., 2018) to produce new music pieces and
+modeling the compatibility (Huang et al., 2021)
+of variety in music (beats, tempo, musical stems)
+demonstrated great examples of learning this task.
+Though this was limited to generating mashups
+or learning features from tempo and key distri-
+butions to produce similar patterns. Compound
+Word Transformer (Hsiao et al., 2021) was able
+to represent music generation task as a sequence
+generation challenge involving musical events de-
+fined by compound words. These musical events
+give a more accurate description of notes progres-
+sion, chord change, harmony and the art factor.
+The objective of the project is to implement a
+Multi-Genre Transformer which learns to produce
+music pieces through more adaptive learning pro-
+cess involving more challenging task where gen-
+res or form of the composition is also considered.
+We built a multi-genre compound word dataset,
+implemented a linear transformer (Katharopoulos
+et al., 2020) which was trained on this dataset.
+We call this Multi-Genre Transformer, which was
+able to generate full length new musical pieces
+which is diverse and comparable to original tracks.
+The model trains 2-5 times faster than other mod-
+els discussed.
+1. Related Work
+Despite achieving great success in generation challenges
+using Artificial Intelligence in Natural Language Genera-
+tion (NLG) there is a factor of art that still makes them
+different from human like performance. In terms of NLG
+we can relate it to something like the difference between
+computer generated article and a piece of art like novels,
+biography, etc. For music art factor always come into ac-
+count and despite able to produce musical compositions
+through Adversarial networks or mixing stems using super-
+vised learning the solution still is very different from an
+original piece of music which we discuss below.
+1.1. Music Generation using GANs
+Generative adversarial networks (GANs) have provided sig-
+nificant progress in producing text, videos and images. Sim-
+ilar efforts have been made to bring neural networks to
+artistic domain of music. MuseGAN(Dong et al., 2018)
+brought a novel model for generating multi-track music.
+Until 2018, the progress in using AI to compose music had
+been able to produce
+• Single-track (monophonic) music
+• Multi-track (polyphonic) music by combining several
+monophonic melodies in chronological order
+Music usually being an art involving multiple instruments
+played together requires music to be multi-track and because
+music notes are made up of chords, arpeggios or melodies
+the idea of using a chronological order setting prevents it
+from being generalized.
+The paper(Dong et al., 2018) address this challenge in gen-
+eralising real music by discussing current technical lacks in
+neural network models and how it relates to the real world
+music.
+1. Music is an art of time and has characteristics of coher-
+ence, rhythm, tension and emotion flow. This requires
+it to have a Temporal Model.
+2. Music compositions usually involves different instru-
+ments interacting with one another making the compo-
+sitions to be harmonic. To solve this issue a Composer
+Model is required.
+3. Musical notes are built of chords, arpeggios or
+melodies and how they unfold over time; thus introduc-
+ing chronological generation of notes is not suitable.
+To address this the paper introduces using bars (seg-
+ment of time) instead of notes as the basic unit for
+composition. And then generate music bar by bar us-
+ing transposed convolutional neural networks to learn
+translation-invariant patterns.
+The paper(Dong et al., 2018) makes contributions in terms
+of both ability to artificially compose realistic music and
+use of generative adversarial framework with temporal and
+composition models. In short the contributions are:
+arXiv:2301.02385v1  [cs.SD]  6 Jan 2023
+
+Multi-Genre Music Transformer - Composing Full Length Musical Piece
+• First GAN based model for generating multi-track se-
+quence.
+• First model which can generate multi-track polyphonic
+music.
+• Same model can be used as a music accompaniment.
+• Creates a new Lakh Pianoroll Dataset (LPD) for multi-
+track piano-rolls
+• For future work metrics in the domain of artificial mu-
+sic a new set of objective metrics are proposed.
+MuseGAN model proposed considers two sub-network gen-
+erator Gtemp (temporal structure generator) and Gbar (bar
+generator) making the overall generator:
+G(z) =
+�
+Gbar(Gtemp(z)(t))
+�T
+t=1
+where z is the input noise vector. The strength of the model
+is the ability to generate samples having chord like inter-
+vals (learning features from temporal model) and melodies
+involving pitch overlap among guitar, piano and strings
+(learning features from composer model).
+The model introduces multi-track by modeling interdepen-
+dency of tracks by proposing 3 different generator model
+(Jamming, Composer and Hybrid), but the author brings up
+these based on the understanding of pop music composition.
+This possibly restricts the generator to explore on a broad
+spectrum of music and prevents it from being generalised.
+Also worth mentioning is that the work relies on multi-track
+interdependency, but misses to study about the compatibility
+of these tracks which can significantly increase the quality
+of music being generated. We will see this issue being
+addressed in the next paper.
+1.2. Modeling the Compatibility of Stem Tracks to
+Generate Music Mashups(Huang et al., 2021)
+Source separation(Jansson et al., 2017; D´efossez et al.,
+2019) makes it possible to generate a music mashup with iso-
+lated stems like vocals, drums, piano, etc. The challenge lies
+in producing music which has compatibility between these
+stems. This paper creates a mashup generation pipeline and
+trains a model to predict the compatibility by automatically
+learning to adjust key and tempo (characteristics of quality
+mashups in real world).
+General models trained for harmonic compatibility
+(Bernardes et al., 2017; Macas et al., 2018) fails to con-
+sider subtle features or surprise mixes of disparate samples
+which is quite common in this art domain. Other issue that
+arises is audio compatibility models like Neural Loop Com-
+biner (Chen et al., 2020) having lack of vocal source and
+variety of genres.
+The authors designed a self supervised learning model
+by recombining the original combination of stems before
+source separation to serve as examples of ground truth. To
+avoid highly polarized model, semi-supervised learning
+was introduced which included producing several random
+mashups by mixing different stems and treated them as
+unlabeled instances. Label smoothing regularization for
+outliers (Zheng et al., 2017) was used to assign uniform
+distribution to the unlabeled data for loss computation. This
+helps in regularization.
+The final architecture consists of 3 modules:
+1. Music Source Separation:
+Uses MSS algorithm
+(Jansson et al., 2017) to get different stems vocals,
+drums, bass and other.
+2. Mashup Database (MashupDB): Using Madmom
+(B¨ock et al., 2016) different features from the music
+clips are extracted like key, tempo and downbeat in-
+formation. Using these features and separate stem
+combinations a mashup database is created which will
+act as either harmonic or percussion stem candidates
+for mashup generation process.
+3. Mashup Generation: It uses candidate stems from
+MashupDB and adjusts key and tempo to produce
+mashups within 3 conditions - original, matched and
+unmatched.
+The model (Huang et al., 2021) is defined by p(y|V, H, P)
+where V , H, and P are input signals for respective stems
+vocal, harmonic, and percussion. The output probability p
+is used as the mashup compatibility and y ∈ {0, 1} stating
+good or bad.
+The model (Huang et al., 2021) implementation tries to
+mimic learning compatibility for producing new mashups
+and provides objective and subjective evaluation by cross
+validation among multiple different datasets. This technique
+becomes easier because of the ability of the model to ex-
+tract different stems and features and build its own mashup
+candidates. This also makes the model training process not
+dependent on human labeled data. The model is also ro-
+bust as negative data is added along with positive data for
+supervised learning. The range of music coverage is also
+extensive and the source separation step makes it easier for
+the model to be extended to different genres for training.
+But the current model design lacks the effective embedding
+of different stems while producing a mashup and makes
+it dependent on tuning of key and tempo. Currently the
+implementation comes up with fixed range of key and tempo
+difference for compatibility and does not explain in detail
+how they came up with these numbers. Although defining
+a range prevents large pitch shifting and time stretching.
+Additionally the results of the model ranks positive labeled
+data (original) over unlabeled data which might lead to
+
+Multi-Genre Music Transformer - Composing Full Length Musical Piece
+concerns of flexibility. Another major challenge of the
+model is the large training time which is around 3 days using
+an NVIDIA Tesla-V100 GPU whereas using transformer
+model significantly reduces the training time.
+1.3. Music Transformers
+With state-of-the art neural network we managed to learn
+features in music by defining certain rules on matching
+tempo, beats or compatibility. In the previous paper we
+also tried to learn compatibility with the help of supervised
+learning. The model though suffered with bias as compati-
+bility was favoured for matched key or tempo and also lacks
+generalization. Compound Word Transformer (Hsiao et al.,
+2021) considers music as sequence of events and uses a
+Transformer (neural sequence model) (Vaswani et al., 2017)
+to generate a new musical sequence.
+A musical note can be described by note’s pitch, chord, bar,
+duration, velocity (dynamics), placement (onset time). If
+we consider these as tokens we can then define music as
+sequence of tokens and these tokens are a part of pre-defined
+vocabulary. As music is multi-faceted a particular type of
+token can capture only a certain feature like melody, rhythm,
+harmony. All the neural networks until now treated these
+tokens as equal and thus lacked heterogeneity.
+Compound Word Transformer (Hsiao et al., 2021) generates
+music in a conceptually different way as it allows tokens
+to be of specific types and let them have their own proper-
+ties. Tokens can be of note type (pitch, duration) or metric
+type (beginning of new beat, bar). We then defines a mu-
+sical event by combination of such tokens which allows to
+capture co-occurrence relationship among the tokens. This
+combination of tokens are termed as compound words. So,
+now we can represent a music piece (X) as a sequence (S)
+of compound words (cp) or S = g(X) = {cpt}T
+t=1 where
+g(.) is the conversion function to convert music into time-
+ordered sequence of musical events and T is the length of
+the music sequence.
+Theoretically, the model learns over discrete-time dynamic
+directed hypergraphs. Consider a graph G = (V, E) (Figure
+1) the vertices (V ) are tokens and edges (E) are sequence of
+token. Collection of vertices can be defined as a compound
+word and hyperedge in this graph represents sequence of
+compound words. In figure 1 v1, v2, v5 are the tokens and
+the edge E1 defines a sequence of tokens whereas e1, e2
+defines a hyperedge (connecting more than 2 nodes). And
+transitioning from one hyperedge to another defines the
+sequence of composition words which we are trying to learn.
+Using a transformer we are trying to learn the next musi-
+cal event or compound word (combination of tokens). The
+self attention part of the transformer learns the dependency
+among the elements in musical sequence and different feed-
+Figure 1. Graphical Representation of Music Space
+forward head is used for tokens of different type. In short
+the implementation groups tokens to form compound words
+and then perform sequence modeling in this sequence of
+compound words, the major contributions are:
+• Compose pop-piano music of full song length.
+• Compound word sequencing with linear transformer
+providing state-of-the-art results in terms of quality
+with 5-10x faster training and inference time.
+• Music defined as Dynamic Directed Hypergraph.
+Generating a new musical event or a group of tokens to
+be combined as a compound word at each time step is the
+backbone of this model, but it relies on assuming that no
+two musical events can occur together. The new hyperedge
+generated by the Transformer decoder marks other tokens as
+[ignore] once an event of a particular token type is detected.
+Can this limit the music generation task? Additionally the
+model is trained using only pop music which limits the
+expressing power of the transformer.
+2. Implementation
+Compound Word Transformer (Hsiao et al., 2021) was able
+to represent music generation task as a sequence generation
+challenge involving musical events defined by compound
+words. Leveraging this representation we implement a neu-
+ral model which learns to produce music pieces through
+more adaptive learning process involving more challenging
+task where genres or form of the composition is also con-
+sidered. This adds the richness of music art in the learning
+process of attention driven sequential learning. We will call
+this model Multi-Genre Music Transformer and following
+are the steps involved for implementing this:
+• Building Dataset: This involves generating compound
+word dictionary for songs of different genres.
+
+Pitch
+Duration
+v1
+v2
+Velocity
+e1
+EA
+Chord
+Beat
+e2
+v5Multi-Genre Music Transformer - Composing Full Length Musical Piece
+• Implementing Transformer Model: We implement
+our Transformer class, the training steps and the gener-
+ation logic for inference.
+• Adaptive Learning: We allow our tuned model to
+be adaptable by training on a smaller and multi-genre
+dataset.
+2.1. Building Dataset
+To be able to provide a more generalised learning process
+for our transformer it needs to be trained with a piano roll
+dataset involving musical pieces of variety of genres/style.
+The dataset should be based on compound words (Hsiao
+et al., 2021) to represent different musical tokens as a com-
+bined unit for sequence modeling which is different from
+traditional musical dataset (MIDI, REMI).
+Figure 2. Dataset Building Pipeline
+This required us to build a dataset by selecting music clip-
+pings and converting them to piano roll using Onsets and
+Frames (Hawthorne et al., 2017). Extracting downbeat and
+beat information from these songs using madmom, a mu-
+sic signal processing library (B¨ock et al., 2016). Finally
+representing these metadata into a compound word repre-
+sentation using the dataset generation scripts provided in the
+compound word transformer repository1. This also adds on
+to the AILabs.tw Pop1K7 dataset (Hsiao et al., 2021) which
+currently only includes pop music. Figure 2 demonstrates
+the pipeline for creating a new dataset.
+Following the pipeline above we managed to create a Com-
+pound Word (Hsiao et al., 2021) dataset which involved
+1https://github.com/YatingMusic/compound-word-
+transformer/blob/main/dataset/Dataset.md
+piano roll for 150 musical pieces from 3 different genres
+including Electronic Dance Music (EDM), Indie and Hip-
+Hop.
+2.2. Implementing Transformer Model
+We implement a linear transformer(Katharopoulos et al.,
+2020) to address long sequence dependency which is a very
+relevant factor in music generation due to the presence of
+a context or a rhythm in the entire musical piece. Hav-
+ing an independent feed-forward head in the Transformer
+Decoder allows to improve the loss of independent tokens.
+This allows the model to scale for additional perspective
+(like genre, form or involving a particular chord progres-
+sion) in the music by adding an additional token type. We
+implement our transformer model in a generic way which
+allows user to define its own token sampling model, token
+embedding model and these can be scalable for any number
+of token types. The loss observed at each feed-forward head
+is shown in Figure 6. This shows adding a new token (for
+genre/style/form) for model to learn can be simply achieved
+by adding an independent feed-forward head for the same.
+2.2.1. TOKEN EMBEDDING
+Figure 3. Demonstrates how each token undergoes independent
+embedding before combining with Positional Encoding. Here
+T1, T2...Tk are K different tokens for our Transformer each having
+its own embedding function and dimension. We are assuming the
+Transformer supports K type of tokens.
+The input to a transformer requires positional encoding
+added to the embedding vector of our input sequence el-
+ements. As each element in our sequence is a compound
+word (Hsiao et al., 2021) which is combined of different
+tokens, we embed each token separately (allowing to have
+adaptive size) and then concatenate them. Having an adap-
+
+Youtube
+WAV Audio
+MP3 Audio
+Files
+Files
+Onsets and Frames
+Madmom
+Piano Transcription
+Beat Tracking
+Compound Word Transformer
+Scripts
+Training DataPositional
+Transformer Input
+Emedding
+Feed-Forward Layer
+Concatenate
+T1
+T2
+Embedding
+Embedding
+Embedding
+TK
+Compound WordMulti-Genre Music Transformer - Composing Full Length Musical Piece
+tive token size allows to use smaller embedding dimension
+for a token type with smaller vocabulary and when we con-
+catenate all of these we get an embedding dimension of 512
+for our model. Refer to Figure 3 for detailed steps of token
+embedding.
+2.2.2. TOKEN SAMPLING
+For inference, sampling plays a crucial role to avoid degen-
+eration and improve diversity. To avoid degeneration we
+follow Nucleus Sampling (Holtzman et al., 2019), which is
+a stochastic temperature controlled process. This method
+samples from the smallest subset of tokens whose cumu-
+lative probability mass exceeds a threshold. We also had
+each token to have a separate sampling policy by defining
+different threshold p and different temperature parameter
+τ (Ackley et al., 1985) for reshaping the probability be-
+fore sampling. We reused the inference implementation
+from Compound Word Transformer (Hsiao et al., 2021) and
+tweaked τ to have higher values for chord to allow more
+diverse chord progressions. Figure 4 shows the sampling
+process and individual feed-forward layer for each token in
+the transformer.
+Figure 4. Transformer with N self-attention layers and independent
+feed-forward head for each token. We first predict the Token Type
+for the particular time-step and then perform a nucleus sampling
+before predicting the remaining tokens.
+2.3. Adaptive Learning
+After defining the model, the next important step is to imple-
+ment the training steps. To support scalable token definition
+in our generalised transformer we make the training steps
+modular and general to variable number of token types. This
+allows easy addition of a new token and independently mon-
+itor gradient descent optimization for the respective loss.
+We trained our model in parallel for 2 different conditions.
+The first set of training was performed on the original AIL-
+abs.tw Pop1K7 dataset (Hsiao et al., 2021). The second set
+of training took into consideration to provide multi-genre
+learning environment for the transformer as it involved train-
+ing on a dictionary that was generated from 3 different
+genres (EDM, Indie, Hip-Hop).
+3. Evaluation and Results
+To train a multi-genre transformer the primary objective
+was to provide it with a dataset which is richer in variety
+than the original pop only dataset. With the help of dataset
+building pipeline we managed to create a token set which
+has a higher variance allowing the model to have a broader
+expressive power. Figure 5 shows the comparison of tokens
+between the 2 datasets used.
+Figure 5. Left image shows token distributions for the songs in the
+generated multi-genre dataset and the right image shows similar
+distribution for AILabs.tw Pop1K7 dataset (Hsiao et al., 2021).
+After training the model for both the datasets we also ob-
+serve (refer to Figure 6) the individual token loss and total
+average loss is similar and indicates the model converging.
+Additionally, the gradient descent is more gradual using the
+multi-genre dataset displaying a more settled progression.
+We trained the model with 12 self-attentions layers, 8 feed-
+forward heads with model dimension of 512 and batch size
+of 4 for 180 epochs which took around 17hrs. Then using
+the trained model we generated 20 new full length musical
+pieces with an average inference time of 12.56sec/song
+which is faster than the compound-word transformer though
+having slightly less number of average tokens per song.
+Table 1 shows a more detailed comparison.
+
+T(k-1)
+Feed-Forward Layer
+Feed-Forward Layer
+Feed-Forward Layer
+Nucleus Sampling
+Type
+Token
+Feed-Forward Layer
+T
+h
+Layer 1
+Layer 2
+Layer N
+Self-Attention Layersmean:2342.926std:1194.481
+mean:2138.370_std:775.472
+250
+10
+200
+8
+Number of
+150
+songs 6
+Number of
+songs
+100
+4
+2
+50 
++0
+0
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+2000
+4000
+6000
+8000
+Number of Tokens
+Number of TokensMulti-Genre Music Transformer - Composing Full Length Musical Piece
+Figure 6. Loss vs Epoch for different token types. The last plot corresponds to the average loss for all different token types.
+For a qualitative evaluation of the musical pieces that were
+produced we compare (Figure 7) the piano rolls of these
+with the piano rolls of original tracks that were used to train
+the model.
+Original Songs
+Generated Songs
+Figure 7. Piano roll of original and generated songs. We can see a
+rich and complete content for the generated songs similar to some
+original tracks.
+4. Conclusion
+In this project we produce music as a sequence of musical
+events produced by a trained Transformer. We leverage the
+definition of Compound Word (Hsiao et al., 2021) to define
+musical event by grouping multiple tokens. This grouping
+greatly reduces the size of our sequence and boosts long-
+range learning. This also reduces the training and inference
+time for our model remarkably. We also exploit the feature
+of each token having its independent feed-forward head for
+prediction to make the model scalable for new token types
+that can be introduced in our dictionary. This allows to add
+any new token for this transformer very easily which can be
+used for musical form, chord progression, etc. Additionally,
+we created an entire new dataset consisting of multi-genre
+compound word dictionary and trained our model with this
+to provide it a more adaptive learning environment. The
+compositions that were generated were highly rich in musi-
+cal events and were of good quality.
+Table 1. Quantitative evaluation results for Multi-Genre Transformer and Compound Word Transformer. Results for Compound Word
+Transformer comes from the implementation in the paper (Hsiao et al., 2021).
+MODEL
+TRAINING TIME
+GPU
+INFERENCE TIME (/SONG)
+AVG TOKENS (/SONG)
+MULTI-GENRE TRANSFORMER
+17 HRS
+9.8GB
+12.56 SEC
+9190
+COMPOUND TRANSFORMER
+1.3 DAYS
+9.5GB
+19.8 SEC
+9546
+
+tempo loss vs epoch
+chord loss vs epoch
+bar-beat loss vs epoch
+type loss vs epoch
+Pop Dataset
+14 
+Pop Dataset
+14
+Pop Dataset
+Pop Dataset
+14 
+ Multi-Genre Dataset
+ Multi-Genre Dataset
+ Multi-Genre Dataset
+1.2
+0.6 
+Multi-Genre Dataset
+1.2 
+1.2 
+1.0 
+1.0 
+0.5 
+10 
+0.8 
+0.8 
+0.4 
+0.8
+0.6 
+0.6
+0.6 
+0.4
+E'O
+0.4 
+0.4 
+0.2 
+0.2 
+0.2 
+0.2 
+0.0 
+75100125150175
+75100125150175
+0.1
+0
+25
+50
+75100125
+150175
+25
+50
+0
+25
+50
+0
+25
+50
+75100125150175
+pitch loss vs epoch
+duration loss vs epoch
+velocity loss vs epoch
+average loss vs epoch
+OE
+Pop Dataset
+18
+ Pop Dataset
+1.8 
+ Pop Dataset
+16 
+PopDataset
+Multi-Genre Dataset
+ Multi-Genre Dataset
+Multi-Genre Dataset
+Multi-Genre Dataset
+2.5 
+16
+16
+14 
+2.0 
+14 
+1.2 
+14 
+1.2 
+1.0 
+1.5
+1.2
+10 
+0.8 
+1.0 
+0.8
+10 
+0.6 
+0.5 
+0.8
+0.4 
+0.6 
+0
+25
+50
+100
+150
+175
+25
+50
+125
+150
+175
+0
+50
+75
+100125 150
+175
+0
+0
+50
+125150175.:
+84
+(ow)
+".
+(ow)
+.
+72
+72
+.
+60
+60
+48
+36 
+LLLL
+36 
+:
+24
+0
+20
+40
+60
+80
+100
+120
+140
+160
+0
+50
+100
+150
+200
+time (sec)
+time (sec)96
+(aw)
+72
+72
+itch
+60
+60 
+48
+8
+96
+36
+0
+50
+100
+150
+0
+50
+100
+150
+200
+time (sec)
+time (sec)Multi-Genre Music Transformer - Composing Full Length Musical Piece
+References
+Ackley, D. H., Hinton, G. E., and Sejnowski, T. J.
+A
+learning algorithm for boltzmann machines.
+Cog-
+nitive science, 9(1):147–169, 1985.
+URL https:
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+article/abs/pii/S0364021385800124.
+Bernardes, G., Davies, M. E., and Guedes, C.
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+hierarchical harmonic mixing method.
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+Hawthorne, C., Elsen, E., Song, J., Roberts, A., Simon,
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+

3dE0T4oBgHgl3EQfeACo/content/tmp_files/load_file.txt

@@ -0,0 +1,507 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf,len=506
+page_content='Multi-Genre Music Transformer - Composing Full Length Musical Piece  Abhinav Kaushal Keshari (Purdue University)  Abstract  In the task of generating music, the art factor plays  a big role and is a great challenge for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Previ-  ous work involving adversarial training (Dong  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2018) to produce new music pieces and  modeling the compatibility (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021)  of variety in music (beats, tempo, musical stems)  demonstrated great examples of learning this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Though this was limited to generating mashups  or learning features from tempo and key distri-  butions to produce similar patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Compound  Word Transformer (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) was able  to represent music generation task as a sequence  generation challenge involving musical events de-  fined by compound words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' These musical events  give a more accurate description of notes progres-  sion, chord change, harmony and the art factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The objective of the project is to implement a  Multi-Genre Transformer which learns to produce  music pieces through more adaptive learning pro-  cess involving more challenging task where gen-  res or form of the composition is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  We built a multi-genre compound word dataset,  implemented a linear transformer (Katharopoulos  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2020) which was trained on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  We call this Multi-Genre Transformer, which was  able to generate full length new musical pieces  which is diverse and comparable to original tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The model trains 2-5 times faster than other mod-  els discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Related Work  Despite achieving great success in generation challenges  using Artificial Intelligence in Natural Language Genera-  tion (NLG) there is a factor of art that still makes them  different from human like performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' In terms of NLG  we can relate it to something like the difference between  computer generated article and a piece of art like novels,  biography, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' For music art factor always come into ac-  count and despite able to produce musical compositions  through Adversarial networks or mixing stems using super-  vised learning the solution still is very different from an  original piece of music which we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Music Generation using GANs  Generative adversarial networks (GANs) have provided sig-  nificant progress in producing text, videos and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Sim-  ilar efforts have been made to bring neural networks to  artistic domain of music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' MuseGAN(Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2018)  brought a novel model for generating multi-track music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Until 2018, the progress in using AI to compose music had  been able to produce  Single-track (monophonic) music  Multi-track (polyphonic) music by combining several  monophonic melodies in chronological order  Music usually being an art involving multiple instruments  played together requires music to be multi-track and because  music notes are made up of chords, arpeggios or melodies  the idea of using a chronological order setting prevents it  from being generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The paper(Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2018) address this challenge in gen-  eralising real music by discussing current technical lacks in  neural network models and how it relates to the real world  music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Music is an art of time and has characteristics of coher-  ence, rhythm, tension and emotion flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This requires  it to have a Temporal Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Music compositions usually involves different instru-  ments interacting with one another making the compo-  sitions to be harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' To solve this issue a Composer  Model is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Musical notes are built of chords, arpeggios or  melodies and how they unfold over time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' thus introduc-  ing chronological generation of notes is not suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  To address this the paper introduces using bars (seg-  ment of time) instead of notes as the basic unit for  composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' And then generate music bar by bar us-  ing transposed convolutional neural networks to learn  translation-invariant patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The paper(Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2018) makes contributions in terms  of both ability to artificially compose realistic music and  use of generative adversarial framework with temporal and  composition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' In short the contributions are:  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='02385v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='SD]  6 Jan 2023  Multi-Genre Music Transformer - Composing Full Length Musical Piece  First GAN based model for generating multi-track se-  quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  First model which can generate multi-track polyphonic  music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Same model can be used as a music accompaniment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Creates a new Lakh Pianoroll Dataset (LPD) for multi-  track piano-rolls  For future work metrics in the domain of artificial mu-  sic a new set of objective metrics are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  MuseGAN model proposed considers two sub-network gen-  erator Gtemp (temporal structure generator) and Gbar (bar  generator) making the overall generator:  G(z) =  �  Gbar(Gtemp(z)(t))  �T  t=1  where z is the input noise vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The strength of the model  is the ability to generate samples having chord like inter-  vals (learning features from temporal model) and melodies  involving pitch overlap among guitar, piano and strings  (learning features from composer model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The model introduces multi-track by modeling interdepen-  dency of tracks by proposing 3 different generator model  (Jamming, Composer and Hybrid), but the author brings up  these based on the understanding of pop music composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  This possibly restricts the generator to explore on a broad  spectrum of music and prevents it from being generalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Also worth mentioning is that the work relies on multi-track  interdependency, but misses to study about the compatibility  of these tracks which can significantly increase the quality  of music being generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We will see this issue being  addressed in the next paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Modeling the Compatibility of Stem Tracks to  Generate Music Mashups(Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021)  Source separation(Jansson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' D´efossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=',  2019) makes it possible to generate a music mashup with iso-  lated stems like vocals, drums, piano, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The challenge lies  in producing music which has compatibility between these  stems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This paper creates a mashup generation pipeline and  trains a model to predict the compatibility by automatically  learning to adjust key and tempo (characteristics of quality  mashups in real world).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  General models trained for harmonic compatibility  (Bernardes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Macas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2018) fails to con-  sider subtle features or surprise mixes of disparate samples  which is quite common in this art domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Other issue that  arises is audio compatibility models like Neural Loop Com-  biner (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2020) having lack of vocal source and  variety of genres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The authors designed a self supervised learning model  by recombining the original combination of stems before  source separation to serve as examples of ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' To  avoid highly polarized model, semi-supervised learning  was introduced which included producing several random  mashups by mixing different stems and treated them as  unlabeled instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Label smoothing regularization for  outliers (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2017) was used to assign uniform  distribution to the unlabeled data for loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This  helps in regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The final architecture consists of 3 modules:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Music Source Separation:  Uses MSS algorithm  (Jansson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2017) to get different stems vocals,  drums, bass and other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Mashup Database (MashupDB): Using Madmom  (B¨ock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2016) different features from the music  clips are extracted like key, tempo and downbeat in-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Using these features and separate stem  combinations a mashup database is created which will  act as either harmonic or percussion stem candidates  for mashup generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Mashup Generation: It uses candidate stems from  MashupDB and adjusts key and tempo to produce  mashups within 3 conditions - original, matched and  unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The model (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) is defined by p(y|V, H, P)  where V , H, and P are input signals for respective stems  vocal, harmonic, and percussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The output probability p  is used as the mashup compatibility and y ∈ {0, 1} stating  good or bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The model (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) implementation tries to  mimic learning compatibility for producing new mashups  and provides objective and subjective evaluation by cross  validation among multiple different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This technique  becomes easier because of the ability of the model to ex-  tract different stems and features and build its own mashup  candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This also makes the model training process not  dependent on human labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The model is also ro-  bust as negative data is added along with positive data for  supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The range of music coverage is also  extensive and the source separation step makes it easier for  the model to be extended to different genres for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  But the current model design lacks the effective embedding  of different stems while producing a mashup and makes  it dependent on tuning of key and tempo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Currently the  implementation comes up with fixed range of key and tempo  difference for compatibility and does not explain in detail  how they came up with these numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Although defining  a range prevents large pitch shifting and time stretching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Additionally the results of the model ranks positive labeled  data (original) over unlabeled data which might lead to  Multi-Genre Music Transformer - Composing Full Length Musical Piece  concerns of flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Another major challenge of the  model is the large training time which is around 3 days using  an NVIDIA Tesla-V100 GPU whereas using transformer  model significantly reduces the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Music Transformers  With state-of-the art neural network we managed to learn  features in music by defining certain rules on matching  tempo, beats or compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' In the previous paper we  also tried to learn compatibility with the help of supervised  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The model though suffered with bias as compati-  bility was favoured for matched key or tempo and also lacks  generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Compound Word Transformer (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=',  2021) considers music as sequence of events and uses a  Transformer (neural sequence model) (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2017)  to generate a new musical sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  A musical note can be described by note’s pitch, chord, bar,  duration, velocity (dynamics), placement (onset time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' If  we consider these as tokens we can then define music as  sequence of tokens and these tokens are a part of pre-defined  vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' As music is multi-faceted a particular type of  token can capture only a certain feature like melody, rhythm,  harmony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' All the neural networks until now treated these  tokens as equal and thus lacked heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Compound Word Transformer (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) generates  music in a conceptually different way as it allows tokens  to be of specific types and let them have their own proper-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Tokens can be of note type (pitch, duration) or metric  type (beginning of new beat, bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We then defines a mu-  sical event by combination of such tokens which allows to  capture co-occurrence relationship among the tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This  combination of tokens are termed as compound words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' So,  now we can represent a music piece (X) as a sequence (S)  of compound words (cp) or S = g(X) = {cpt}T  t=1 where  g(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=') is the conversion function to convert music into time-  ordered sequence of musical events and T is the length of  the music sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Theoretically, the model learns over discrete-time dynamic  directed hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Consider a graph G = (V, E) (Figure  1) the vertices (V ) are tokens and edges (E) are sequence of  token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Collection of vertices can be defined as a compound  word and hyperedge in this graph represents sequence of  compound words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' In figure 1 v1, v2, v5 are the tokens and  the edge E1 defines a sequence of tokens whereas e1, e2  defines a hyperedge (connecting more than 2 nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' And  transitioning from one hyperedge to another defines the  sequence of composition words which we are trying to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Using a transformer we are trying to learn the next musi-  cal event or compound word (combination of tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The  self attention part of the transformer learns the dependency  among the elements in musical sequence and different feed-  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Graphical Representation of Music Space  forward head is used for tokens of different type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' In short  the implementation groups tokens to form compound words  and then perform sequence modeling in this sequence of  compound words, the major contributions are:  Compose pop-piano music of full song length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Compound word sequencing with linear transformer  providing state-of-the-art results in terms of quality  with 5-10x faster training and inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Music defined as Dynamic Directed Hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Generating a new musical event or a group of tokens to  be combined as a compound word at each time step is the  backbone of this model, but it relies on assuming that no  two musical events can occur together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The new hyperedge  generated by the Transformer decoder marks other tokens as  [ignore] once an event of a particular token type is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Can this limit the music generation task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Additionally the  model is trained using only pop music which limits the  expressing power of the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Implementation  Compound Word Transformer (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) was able  to represent music generation task as a sequence generation  challenge involving musical events defined by compound  words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Leveraging this representation we implement a neu-  ral model which learns to produce music pieces through  more adaptive learning process involving more challenging  task where genres or form of the composition is also con-  sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This adds the richness of music art in the learning  process of attention driven sequential learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We will call  this model Multi-Genre Music Transformer and following  are the steps involved for implementing this:  Building Dataset: This involves generating compound  word dictionary for songs of different genres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Pitch  Duration  v1  v2  Velocity  e1  EA  Chord  Beat  e2  v5Multi-Genre Music Transformer - Composing Full Length Musical Piece  Implementing Transformer Model: We implement  our Transformer class, the training steps and the gener-  ation logic for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Adaptive Learning: We allow our tuned model to  be adaptable by training on a smaller and multi-genre  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Building Dataset  To be able to provide a more generalised learning process  for our transformer it needs to be trained with a piano roll  dataset involving musical pieces of variety of genres/style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The dataset should be based on compound words (Hsiao  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) to represent different musical tokens as a com-  bined unit for sequence modeling which is different from  traditional musical dataset (MIDI, REMI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Dataset Building Pipeline  This required us to build a dataset by selecting music clip-  pings and converting them to piano roll using Onsets and  Frames (Hawthorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Extracting downbeat and  beat information from these songs using madmom, a mu-  sic signal processing library (B¨ock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Finally  representing these metadata into a compound word repre-  sentation using the dataset generation scripts provided in the  compound word transformer repository1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This also adds on  to the AILabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='tw Pop1K7 dataset (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) which  currently only includes pop music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Figure 2 demonstrates  the pipeline for creating a new dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Following the pipeline above we managed to create a Com-  pound Word (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) dataset which involved  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='com/YatingMusic/compound-word-  transformer/blob/main/dataset/Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='md  piano roll for 150 musical pieces from 3 different genres  including Electronic Dance Music (EDM), Indie and Hip-  Hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Implementing Transformer Model  We implement a linear transformer(Katharopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=',  2020) to address long sequence dependency which is a very  relevant factor in music generation due to the presence of  a context or a rhythm in the entire musical piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Hav-  ing an independent feed-forward head in the Transformer  Decoder allows to improve the loss of independent tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  This allows the model to scale for additional perspective  (like genre, form or involving a particular chord progres-  sion) in the music by adding an additional token type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We  implement our transformer model in a generic way which  allows user to define its own token sampling model, token  embedding model and these can be scalable for any number  of token types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The loss observed at each feed-forward head  is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This shows adding a new token (for  genre/style/form) for model to learn can be simply achieved  by adding an independent feed-forward head for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' TOKEN EMBEDDING  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Demonstrates how each token undergoes independent  embedding before combining with Positional Encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Here  T1, T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Tk are K different tokens for our Transformer each having  its own embedding function and dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We are assuming the  Transformer supports K type of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The input to a transformer requires positional encoding  added to the embedding vector of our input sequence el-  ements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' As each element in our sequence is a compound  word (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) which is combined of different  tokens, we embed each token separately (allowing to have  adaptive size) and then concatenate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Having an adap-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Youtube  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='WAV Audio  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='MP3 Audio  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Files  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Files  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Onsets and Frames  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Madmom  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Piano Transcription  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Beat Tracking  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Compound Word Transformer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Scripts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Training DataPositional  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Transformer Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Emedding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Feed-Forward Layer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Concatenate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='T1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='T2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Embedding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Embedding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Embedding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='TK  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='Compound WordMulti-Genre Music Transformer - Composing Full Length Musical Piece  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='tive token size allows to use smaller embedding dimension  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='for a token type with smaller vocabulary and when we con-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='catenate all of these we get an embedding dimension of 512  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Refer to Figure 3 for detailed steps of token  embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' TOKEN SAMPLING  For inference, sampling plays a crucial role to avoid degen-  eration and improve diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' To avoid degeneration we  follow Nucleus Sampling (Holtzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2019), which is  a stochastic temperature controlled process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This method  samples from the smallest subset of tokens whose cumu-  lative probability mass exceeds a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We also had  each token to have a separate sampling policy by defining  different threshold p and different temperature parameter  τ (Ackley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 1985) for reshaping the probability be-  fore sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We reused the inference implementation  from Compound Word Transformer (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) and  tweaked τ to have higher values for chord to allow more  diverse chord progressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Figure 4 shows the sampling  process and individual feed-forward layer for each token in  the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Transformer with N self-attention layers and independent  feed-forward head for each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We first predict the Token Type  for the particular time-step and then perform a nucleus sampling  before predicting the remaining tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Adaptive Learning  After defining the model, the next important step is to imple-  ment the training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' To support scalable token definition  in our generalised transformer we make the training steps  modular and general to variable number of token types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This  allows easy addition of a new token and independently mon-  itor gradient descent optimization for the respective loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  We trained our model in parallel for 2 different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  The first set of training was performed on the original AIL-  abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='tw Pop1K7 dataset (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The second set  of training took into consideration to provide multi-genre  learning environment for the transformer as it involved train-  ing on a dictionary that was generated from 3 different  genres (EDM, Indie, Hip-Hop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Evaluation and Results  To train a multi-genre transformer the primary objective  was to provide it with a dataset which is richer in variety  than the original pop only dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' With the help of dataset  building pipeline we managed to create a token set which  has a higher variance allowing the model to have a broader  expressive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Figure 5 shows the comparison of tokens  between the 2 datasets used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Left image shows token distributions for the songs in the  generated multi-genre dataset and the right image shows similar  distribution for AILabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='tw Pop1K7 dataset (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  After training the model for both the datasets we also ob-  serve (refer to Figure 6) the individual token loss and total  average loss is similar and indicates the model converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Additionally, the gradient descent is more gradual using the  multi-genre dataset displaying a more settled progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  We trained the model with 12 self-attentions layers, 8 feed-  forward heads with model dimension of 512 and batch size  of 4 for 180 epochs which took around 17hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Then using  the trained model we generated 20 new full length musical  pieces with an average inference time of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='56sec/song  which is faster than the compound-word transformer though  having slightly less number of average tokens per song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Table 1 shows a more detailed comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  T(k-1)  Feed-Forward Layer  Feed-Forward Layer  Feed-Forward Layer  Nucleus Sampling  Type  Token  Feed-Forward Layer  T  h  Layer 1  Layer 2  Layer N  Self-Attention Layersmean:2342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='926std:1194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='481  mean:2138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='370_std:775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='472  250  10  200  8  Number of  150  songs 6  Number of  songs  100  4  2  50  +0  0  0  1000  2000  3000  4000  5000  6000  7000  2000  4000  6000  8000  Number of Tokens  Number of TokensMulti-Genre Music Transformer - Composing Full Length Musical Piece  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Loss vs Epoch for different token types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The last plot corresponds to the average loss for all different token types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  For a qualitative evaluation of the musical pieces that were  produced we compare (Figure 7) the piano rolls of these  with the piano rolls of original tracks that were used to train  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Original Songs  Generated Songs  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Piano roll of original and generated songs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We can see a  rich and complete content for the generated songs similar to some  original tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Conclusion  In this project we produce music as a sequence of musical  events produced by a trained Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We leverage the  definition of Compound Word (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', 2021) to define  musical event by grouping multiple tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This grouping  greatly reduces the size of our sequence and boosts long-  range learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This also reduces the training and inference  time for our model remarkably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' We also exploit the feature  of each token having its independent feed-forward head for  prediction to make the model scalable for new token types  that can be introduced in our dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' This allows to add  any new token for this transformer very easily which can be  used for musical form, chord progression, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Additionally,  we created an entire new dataset consisting of multi-genre  compound word dictionary and trained our model with this  to provide it a more adaptive learning environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' The  compositions that were generated were highly rich in musi-  cal events and were of good quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Quantitative evaluation results for Multi-Genre Transformer and Compound Word Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' Results for Compound Word  Transformer comes from the implementation in the paper (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
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+page_content='  MODEL  TRAINING TIME  GPU  INFERENCE TIME (/SONG)  AVG TOKENS (/SONG)  MULTI-GENRE TRANSFORMER  17 HRS  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  A  learning algorithm for boltzmann machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Cog-  nitive science, 9(1):147–169, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
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+page_content='  Bernardes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=', Davies, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
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+page_content='  A  hierarchical harmonic mixing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  In Inter-  national Symposium on Computer Music Multidis-  ciplinary Research,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
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+page_content=' Springer,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
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+page_content=' In International Conference  on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' 5156–5165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  URL http://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='mlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='press/v119/  katharopoulos20a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
+page_content='  Macas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
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+page_content='computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
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+page_content='thecvf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'}
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+Federated Learning under Heterogeneous and
+Correlated Client Availability
+Angelo Rodio∗, Francescomaria Faticanti∗, Othmane Marfoq∗†, Giovanni Neglia∗, Emilio Leonardi‡
+∗Inria, Universit´e Cˆote d’Azur, France. Email: {firstname.lastname}@inria.fr,
+†Accenture Labs, Sophia-Antipolis, France. Email: {firstname.lastname}@accenture.com,
+‡Politecnico di Torino, Turin, Italy. Email: {firstname.lastname}@polito.it
+Abstract—The enormous amount of data produced by mobile and
+IoT devices has motivated the development of federated learning
+(FL), a framework allowing such devices (or clients) to collabora-
+tively train machine learning models without sharing their local
+data. FL algorithms (like FedAvg) iteratively aggregate model
+updates computed by clients on their own datasets. Clients may
+exhibit different levels of participation, often correlated over time
+and with other clients. This paper presents the first convergence
+analysis for a FedAvg-like FL algorithm under heterogeneous
+and correlated client availability. Our analysis highlights how
+correlation adversely affects the algorithm’s convergence rate
+and how the aggregation strategy can alleviate this effect at
+the cost of steering training toward a biased model. Guided
+by the theoretical analysis, we propose CA-Fed, a new FL
+algorithm that tries to balance the conflicting goals of maximizing
+convergence speed and minimizing model bias. To this purpose,
+CA-Fed dynamically adapts the weight given to each client and
+may ignore clients with low availability and large correlation. Our
+experimental results show that CA-Fed achieves higher time-
+average accuracy and a lower standard deviation than state-of-
+the-art AdaFed and F3AST, both on synthetic and real datasets.
+Index Terms—Federated Learning, Distributed Optimization.
+I. INTRODUCTION
+The enormous amount of data generated by mobile and IoT de-
+vices motivated the emergence of distributed machine learning
+training paradigms [1], [2]. Federated Learning (FL) [3]–[6]
+is an emerging framework where geographically distributed
+devices (or clients) participate in the training of a shared
+Machine Learning (ML) model without sharing their local
+data. FL was proposed to reduce the overall cost of collecting
+a large amount of data as well as to protect potentially
+sensitive users’ private information. In the original Federated
+Averaging algorithm (FedAvg) [4], a central server selects
+a random subset of clients from the set of available clients
+and broadcasts them the shared model. The sampled clients
+perform a number of independent Stochastic Gradient Descent
+(SGD) steps over their local datasets and send their local
+model updates back to the server. Then, the server aggregates
+the received client updates to produce a new global model, and
+a new training round begins. At each iteration of FedAvg, the
+server typically samples randomly a few hundred devices to
+participate [7], [8].
+This research was supported by the French government through the 3IA
+Cˆote d’Azur Investments in the Future project by the National Research
+Agency (ANR) with reference ANR-19-P3IA-0002, and by Groupe La Poste,
+sponsor of Inria Foundation, in the framework of FedMalin Inria Challenge.
+A first version of this work has been accepted at IEEE INFOCOM 2023.
+In real-world scenarios, the availability/activity of clients is
+dictated by exogenous factors that are beyond the control of
+the orchestrating server and hard to predict. For instance, only
+smartphones that are idle, under charge, and connected to
+broadband networks are commonly allowed to participate in
+the training process [4], [9]. These eligibility requirements can
+make the availability of devices correlated over time and space
+[7], [10]–[12]. For example, temporal correlation may origin
+from a smartphone being under charge for a few consecutive
+hours and then ineligible for the rest of the day. Similarly,
+the activity of a sensor powered by renewable energy may
+depend on natural phenomena intrinsically correlated over
+time (e.g., solar light). Spatial correlation refers instead to
+correlation across different clients, which often emerges as
+consequence of users’ different geographical distribution. For
+instance, clients in the same time zone often exhibit similar
+availability patterns, e.g., due to time-of-day effects.
+Temporal correlation in the data sampling procedure is known
+to negatively affect the performance of ML training even in
+the centralized setting [13], [14] and can potentially lead to
+catastrophic forgetting: the data used during the final training
+phases can have a disproportionate effect on the final model,
+“erasing” the memory of previously learned information [15],
+[16]. Catastrophic forgetting has also been observed in FL,
+where clients in the same geographical area have more similar
+local data distributions and clients’ participation follows a
+cyclic daily pattern (leading to spatial correlation) [7], [10],
+[11], [17]. Despite this evidence, a theoretical study of the
+convergence of FL algorithms under both temporally and
+spatially correlated client participation is still missing.
+This
+paper
+provides
+the
+first
+convergence
+analysis
+of
+FedAvg [4] under heterogeneous and correlated client avail-
+ability. We assume that clients’ temporal and spatial availabil-
+ity follows an arbitrary finite-state Markov chain: this assump-
+tion models a realistic scenario in which the activity of clients
+is correlated and, at the same time, still allows the analytical
+tractability of the system. Our theoretical analysis (i) quantifies
+the negative effect of correlation on the algorithm’s conver-
+gence rate through an additional term, which depends on the
+spectral properties of the Markov chain; (ii) points out a trade-
+off between two conflicting objectives: slow convergence to
+the optimal model, or fast convergence to a biased model, i.e.,
+a model that minimizes an objective function different from the
+initial target. Guided by insights from the theoretical analysis,
+1
+arXiv:2301.04632v1  [cs.LG]  11 Jan 2023
+
+we propose CA-Fed, an algorithm which dynamically assigns
+weights to clients and achieves a good trade-off between
+maximizing convergence speed and minimizing model bias.
+Interestingly, CA-Fed can decide to ignore clients with low
+availability and high temporal correlation. Our experimental
+results demonstrate that excluding such clients is a simple, but
+effective approach to handle the heterogeneous and correlated
+client availability in FL. Indeed, while CA-Fed achieves a
+comparable maximum accuracy as the state-of-the-art methods
+F3AST [18] and AdaFed [19], its test accuracy exhibits
+higher time-average and smaller variability over time.
+The remainder of this paper is organized as follows. Section II
+describes the problem of correlated client availability in FL
+and discusses the main related works. Section III provides
+a convergence analysis of FedAvg under heterogeneous and
+correlated client participation. CA-Fed, our correlation-aware
+FL algorithm, is presented in Section IV. We evaluate CA-Fed
+in Section V, comparing it with state-of-the-art methods on
+synthetic and real-world data. Section VII concludes the paper.
+II. BACKGROUND AND RELATED WORKS
+We consider a finite set K of N clients. Each client k ∈ K
+holds a local dataset Dk. Clients aim to jointly learn the
+parameters w ∈ W ⊆ Rd of a global ML model (e.g., the
+weights of a neural network architecture). During training, the
+quality of the model with parameters w on a data sample
+ξ ∈ Dk is measured by a loss function f(w; ξ). The clients
+solve, under the orchestration of a central server, the following
+optimization problem:
+min
+w∈W ⊆Rd
+�
+F(w) :=
+�
+k∈K
+αkFk(w)
+�
+,
+(1)
+where Fk(w) :=
+1
+|Dk|
+�
+ξ∈Dk f(w; ξ) is the average loss
+computed on client k’s local dataset, and α = (αk)k∈K are
+positive coefficients such that �
+k αk = 1. They represent
+the target importance assigned by the central server to each
+client k. Typically (αk)k∈K are set proportional to the clients’
+dataset size |Dk|, such that the objective function F in (1)
+coincides with the average loss computed on the union of the
+clients’ local datasets D = ∪k∈KDk.
+Under proper assumptions, precised in Section III, Problem (1)
+admits a unique solution. We use w∗ (resp. F ∗) to denote
+the minimizer (resp. the minimum value) of F. Moreover, for
+k∈K, Fk admits a unique minimizer on W. We use w∗
+k (resp.
+F ∗
+k ) to denote the minimizer (resp. the minimum value) of Fk.
+Problem (1) is commonly solved through iterative algo-
+rithms [4], [8] requiring multiple communication rounds be-
+tween the server and the clients. At round t > 0, the server
+broadcasts the latest estimate of the global model wt,0 to
+the set of available clients (At). Client k ∈ At updates the
+global model with its local data through E ≥ 1 steps of local
+Stochastic Gradient Descent (SGD):
+wk
+t,j+1 = wk
+t,j − ηt∇Fk(wk
+t,j, Bk
+t,j)
+j = 0, . . . , E − 1, (2)
+where ηt
+> 0 is an appropriately chosen learning rate,
+referred to as local learning rate; Bk
+t,j is a random batch
+sampled from client k’ local dataset at round t and step j;
+∇Fk(·, B) :=
+1
+|B|
+�
+ξ∈B ∇f(·, ξ) is an unbiased estimator of
+the local gradient ∇Fk. Then, each client sends its local model
+update ∆k
+t := wk
+t,E − wk
+t,0 to the server. The server computes
+∆t := �
+k∈At qk ·∆k
+t , a weighted average of the clients’ local
+updates with non-negative aggregation weights q = (qk)k∈K.
+The choice of the aggregation weights defines an aggregation
+strategy (we will discuss different aggregation strategies later).
+The aggregated update ∆t can be interpreted as a proxy for
+−∇F(wt,0); the server applies it to the global model:
+wt+1,0 = ProjW (wt,0 + ηs · ∆t)
+(3)
+where ProjW (·) denotes the projection over the set W, and
+ηs > 0 is an appropriately chosen learning rate, referred to as
+the server learning rate.1
+The aggregate update ∆t is, in general, a biased estimator
+of −∇F(wt,0), where each client k is taken into account
+proportionally to its frequency of appearance in the set At and
+to its aggregation weight qk. Indeed, under proper assumptions
+specified in Section III, one can show (see Theorem 2) that the
+update rule described by (2) and (3) converges to the unique
+minimizer of a biased global objective FB, which depends
+both on the clients’ availability (i.e., on the sequence (At)t>0)
+and on the aggregation strategy (i.e., on q = (qk)k∈K):
+FB(w) := �N
+k=1 pkFk(w), with pk :=
+πkqk
+�N
+h=1 πhqh ,
+(4)
+where πk := limt→∞ P(k ∈ At) is the asymptotic availability
+of client k. The coefficients p = (pk)k∈K can be interpreted
+as the biased importance the server is giving to each client k
+during training, in general different from the target importance
+α. In what follows, w∗
+B (resp. F ∗
+B) denotes the minimizer
+(resp. the minimum value) of FB.
+In some large-scale FL applications, like training Google
+keyboard next-word prediction models, each client participates
+in training at most for one round. The orchestrator usually
+selects a few hundred clients at each round for a few thousand
+rounds (e.g., see [5, Table 2]), but the available set of clients
+may include hundreds of millions of Android devices. In this
+scenario, it is difficult to address the potential bias unless there
+is some a-priori information about each client’s availability.
+Anyway, FL can be used by service providers with access
+to a much smaller set of clients (e.g., smartphone users that
+have installed a specific app). In this case, a client participates
+multiple times in training: the orchestrating server may keep
+track of each client’s availability and try to compensate for
+the potentially dangerous heterogeneity in their participation.
+Much previous effort on federated learning [4], [17]–[19],
+[22]–[25] considered this problem and, under different as-
+1The aggregation rule (3) has been considered also in other works, e.g., [8],
+[20], [21]. In other FL algorithms, the server computes an average of clients’
+local models. This aggregation rule can be obtained with minor changes to (3).
+2
+
+sumptions on the clients’ availability (i.e., on (At)t>0), de-
+signed aggregation strategies that unbias ∆t through an appro-
+priate choice of q. Reference [22] provides the first analysis of
+FedAvg on non-iid data under clients’ partial participation.
+Their analysis covers both the case when active clients are
+sampled uniformly at random without replacement from K and
+assigned aggregation weights equal to their target importance
+(as assumed in [4]), and the case when active clients are
+sampled iid with replacement from K with probabilities α
+and assigned equal weights (as assumed in [23]). However,
+references [4], [22], [23] ignore the variance induced by the
+clients stochastic availability. The authors of [24] reduce such
+variance by considering only the clients with important up-
+dates, as measured by the value of their norm. References [17]
+and [25] reduce the aggregation variance through clustered and
+soft-clustered sampling, respectively.
+Some recent works [18], [19], [26] do not actively pursue the
+optimization of the unbiased objective. Instead, they derive
+bounds for the convergence error and propose heuristics to
+minimize those bounds, potentially introducing some bias.
+Our work follows a similar development: we compare our
+algorithm with F3AST from [18] and AdaFed from [19].
+The novelty of our study is in considering the spatial and
+temporal correlation in clients’ availability dynamics. As dis-
+cussed in the introduction, such correlations are also intro-
+duced by clients’ eligibility criteria, e.g., smartphones being
+under charge and connected to broadband networks. The effect
+of correlation has been ignored until now, probably due to the
+additional complexity in studying FL algorithms’ convergence.
+To the best of our knowledge, the only exception is [18], which
+scratches the issue of spatial correlation by proposing two
+different algorithms for the case when clients’ availabilities
+are uncorrelated and for the case when they are positively
+correlated (there is no smooth transition from one algorithm
+to the other as a function of the degree of correlation).
+The effect of temporal correlation on centralized stochastic
+gradient methods has been addressed in [12]–[14], [27]: these
+works study a variant of stochastic gradient descent where
+samples are drawn according to a Markov chain. Refer-
+ence [12] extends its analysis to a FL setting where each client
+draws samples according to a Markov chain. In contrast, our
+work does not assume a correlation in the data sampling but
+rather in the client’s availability. Nevertheless, some of our
+proof techniques are similar to those used in this line of work
+and, in particular, we rely on some results in [14].
+III. ANALYSIS
+A. Main assumptions
+We consider a time-slotted system where a slot corresponds
+to one FL communication round. We assume that clients’
+availability over the timeslots t ∈ N follows a discrete-time
+Markov chain (At)t≥0.2
+2In Section III-D we will focus on the case where this chain is the
+superposition of N independent Markov chains, one for each client.
+Assumption 1. The Markov chain (At)t≥0 on the finite state
+space [M] is time-homogeneous, irreducible, and aperiodic. It
+has transition matrix P and stationary distribution π.
+Markov chains have already been used in the literature to
+model the dynamics of stochastic networks where some nodes
+or edges in the graph can switch between active and inactive
+states [28], [29]. The previous Markovian assumption, while
+allowing a great degree of flexibility, still guarantees the
+analytical tractability of the system. The distance dynamics
+between current and stationary distribution of the Markov
+process can be characterized by the spectral properties of its
+transition matrix P [30]. Let λ2(P ) denote the the second
+largest eigenvalue of P in absolute value. Previous works [14]
+have shown that:
+max
+i,j∈[M] |[P t]i,j − πj| ≤ CP · λ(P )t,
+for t ≥ TP ,
+(5)
+where the parameter λ(P ) := (λ2(P ) + 1)/2, and CP , TP
+are positive constants whose values are reported in [14,
+Lemma 1].3 Note that λ(P ) quantifies the correlation of the
+Markov process (At)t≥0: the closer λ(P ) is to one, the slower
+the Markov chain converges to its stationary distribution.
+In our analysis, we make the following additional assumptions.
+Let w∗, w∗
+B denote the minimizers of F and FB on W,
+respectively.
+Assumption 2. The hypothesis class W is convex, compact,
+and contains in its interior the minimizers w∗, w∗
+B, w∗
+k.
+The following assumptions concern clients’ local objective
+functions {Fk}k∈K. Assumptions 3 and 4 are standard in
+the literature on convex optimization [31, Sections 4.1, 4.2].
+Assumption 5 is a standard hypothesis in the analysis of
+federated optimization algorithms [8, Section 6.1].
+Assumption 3 (L-smoothness). The local functions {Fk}N
+k=1
+have L-Lipschitz continuous gradients: Fk(v) ≤ Fk(w) +
+⟨∇Fk(w), v − w⟩ + L
+2 ∥v − w∥2
+2, ∀v, w ∈ W.
+Assumption
+4
+(Strong convexity). The local functions
+{Fk}N
+k=1
+are
+µ-strongly
+convex:
+Fk(v)
+≥
+Fk(w) +
+⟨∇Fk(w), v − w⟩ + µ
+2 ∥v − w∥2
+2 , ∀v, w ∈ W.
+Assumption 5 (Bounded variance). The variance of stochastic
+gradients in each device is bounded: E ∥∇Fk(wk
+t,j, ξk
+t,j) −
+∇Fk(wk
+t,j)∥2 ≤ σ2
+k, k = 1, . . . , N.
+Assumptions 2–5 imply the following properties for the local
+functions, described by Lemma 1 (proof in Appendix B).
+Lemma 1. Under Assumptions 2–5, there exist constants D,
+G, and H > 0, such that, for w ∈ W and k ∈ K, we have:
+∥∇Fk(w)∥ ≤ D,
+(6)
+E ∥∇Fk(w, ξ)∥2 ≤ G2,
+(7)
+|Fk(w) − Fk(w∗
+B)| ≤ H.
+(8)
+3Note that (5) holds for different definitions of λ(P ) as far as λ(P ) ∈
+(λ2(P ), 1). The specific choice for λ(P ) changes the constants CP and TP .
+3
+
+Similarly to other works [8], [22], [23], [32], we introduce a
+metric to quantify the heterogeneity of clients’ local datasets:
+Γ := max
+k∈K{Fk(w∗) − F ∗
+k }.
+(9)
+If the local datasets are identical, the local functions {Fk}k∈K
+coincide among them and with F, w∗ is a minimizer of each
+local function, and Γ = 0. In general, Γ is smaller the closer
+the distributions the local datasets are drawn from.
+B. Main theorems
+Theorem 1 (proof in Appendix A) decomposes the error of
+the target global objective as the sum of an optimization error
+for the biased global objective and a bias error.
+Theorem 1 (Decomposing the total error). Under Assump-
+tions 2–4, the optimization error of the target global objective
+ϵ = F(w) − F ∗ can be bounded as follows:
+ϵ ≤ 2κ2(FB(w) − F ∗
+B)
+�
+��
+�
+:=ϵopt
++ 2κ4χ2
+α∥pΓ
+�
+��
+�
+:=ϵbias
+,
+(10)
+where κ := L/µ, and χ2
+α∥p := �N
+k=1 (αk − pk)2/pk.
+Theorem 2 below proves that the optimization error ϵopt asso-
+ciated to the biased objective FB, evaluated on the trajectory
+determined by scheme (3), asymptotically vanishes. The non-
+vanishing bias error ϵbias captures the discrepancy between
+F(w) and FB(w). This latter term depends on the chi-square
+divergence χ2
+α∥p between the target and biased probability
+distributions α = (αk)k∈K and p = (pk)k∈K, and on
+Γ, that quantifies the degree of heterogeneity of the local
+functions. When all local functions are identical (Γ = 0),
+the bias term ϵbias also vanishes. For Γ > 0, the bias error
+can still be controlled by the aggregation weights assigned
+to the devices. In particular, the bias term vanishes when
+qk ∝ αk/πk, ∀k ∈ K. Since it asymptotically cancels the bias
+error, we refer to this choice as unbiased aggregation strategy.
+However, in practice, FL training is limited to a finite number
+of iterations T (typically a few hundreds [5], [7]), and the
+previous asymptotic considerations may not apply. In this
+regime, the unbiased aggregation strategy can be suboptimal,
+since the minimization of ϵbias not necessarily leads to the
+minimization of the total error ϵ ≤ ϵopt + ϵbias. This motivates
+the analysis of the optimization error ϵopt.
+Theorem 2 (Convergence of the optimization error ϵopt). Let
+Assumptions 1–5 hold and the constants M, L, D, G, H, Γ,
+σk, CP , TP , λ(P ) be defined as above. Let Q = �
+k∈K qk.
+Let the stepsizes satisfy:
+�
+t ηt = +∞,
+�
+t ln(t) · η2
+t < +∞.
+(11)
+Let T denote the total communication rounds. For T ≥ TP ,
+the expected optimization error can be bounded as follows:
+E[FB( ¯wT,0) − F ∗
+B] ≤
+1
+2 q⊺Σq+υ
+π⊺q
++ ψ +
+φ
+ln(1/λ(P ))
+(�T
+t=1 ηt)
+,
+(12)
+where ¯wT,0 :=
+�T
+t=1 ηtwt,0
+�T
+t=1 ηt
+, and
+Σ = diag(σ2
+kπk
+�
+t η2
+t ),
+υ = 2
+E ∥w0,0 − w∗∥2 + 1
+4MQ �
+t(η2
+t + 1
+t2 ),
+ψ = 4L(EQ + 2)Γ �
+t η2
+t + 2
+3(E − 1)(2E − 1)G2 �
+t η2
+t ,
+Jt =min {max {⌈ln (2CP Ht)/ln (1/λ(P ))⌉ , TP } , t},
+φ = 2EDGQ �
+t ln(2CP Ht)η2
+t−Jt.
+Theorem 2 (proof in Appendix B) proves convergence of
+the expected biased objective FB to its minimum F ∗
+B under
+correlated client participation. Our bound (12) captures the
+effect of correlation through the factor ln (1/λ(P )): a high
+correlation worsens the convergence rate. In particular, we
+found that the numerator of (12) has a quadratic-over-linear
+fractional dependence on q. Minimizing ϵopt leads, in general,
+to a different choice of q than minimizing ϵbias.
+C. Minimizing the total error ϵ ≤ ϵopt + ϵbias
+Our analysis points out a trade-off between minimizing ϵopt
+or ϵbias. Our goal is to find the optimal aggregation weights q∗
+that minimize the upper bound on total error ϵ(q) in (10):
+minimize
+q
+ϵopt(q) + ϵbias(q);
+subject to
+q ≥ 0,
+∥q∥1 = Q.
+(13)
+In Appendix E we prove that (13) is a convex optimization
+problem, which can be solved with the method of Lagrange
+multipliers. However, the solution is not of practical utility
+because the constants in (10) and (12) (e.g., L, µ, Γ, CP ) are
+in general problem-dependent and difficult to estimate during
+training. In particular, Γ poses particular difficulties as it is
+defined in terms of the minimizer of the target objective F, but
+the FL algorithm generally minimizes the biased function FB.
+Moreover, the bound in (10), similarly to the bound in [32],
+diverges when setting some qk equal to 0, but this is simply
+an artifact of the proof technique. A result of more practical
+interest is the following (proof in Appendix C):
+Theorem 3 (An alternative decomposition of the total er-
+ror ϵ). Under the same assumptions of Theorem 1, let Γ′ :=
+maxk{Fk(w∗
+B) − F ∗
+k }. The following result holds:
+ϵ ≤ 2κ2(FB(w) − F ∗
+B)
+�
+��
+�
+:=ϵopt
++ 8κ4d2
+T V (α, p)Γ′
+�
+��
+�
+:=ϵ′
+bias
+,
+(14)
+where dT V (α, p) := 1
+2
+�N
+k=1|αk − pk| is the total variation
+distance between the probability distributions α and p.
+The new constant Γ′ is defined in terms of w∗
+B, and then
+it is easier to evaluate during training. However, Γ′ depends
+on q, because it is evaluated at the point of minimum of FB.
+This dependence makes the minimization of the right-hand
+side of (14) more challenging (for example, the corresponding
+problem is not convex). We study the minimization of the two
+terms ϵopt and ϵ′
+bias separately and learn some insights, which
+we use to design the new FL algorithm CA-Fed.
+4
+
+D. Minimizing ϵopt
+The minimization of ϵopt is still a convex optimization problem
+(Appendix D). In particular, at the optimum non-negative
+weights are set accordingly to q∗
+k = a(λ∗πk − θ∗) with
+a, λ∗, and θ∗ positive constants (see (29)). It follows that
+clients with smaller availability get smaller weights in the
+aggregation. In particular, this suggests that clients with the
+smallest availability can be excluded from the aggregation,
+leading to the following guideline:
+Guideline A: to speed up the convergence, we can exclude,
+i.e., set q∗
+k = 0, the clients with lowest availability πk.
+This guideline can be justified intuitively: updates from clients
+with low participation may be too sporadic to allow the FL
+algorithm to keep track of their local objectives. They act as
+a noise slowing down the algorithm’s convergence. It may be
+advantageous to exclude these clients from participating.
+We observe that the choice of the aggregation weights q does
+not affect the clients’ availability process and, in particular,
+λ(P ). However, if the algorithm excludes some clients, it
+is possible to consider the state space of the Markov chain
+that only specifies the availability state of the remaining
+clients, and this Markov chain may have different spectral
+properties. For the sake of concreteness, we consider here
+(and in the rest of the paper) the particular case when the
+availability of each client k evolves according to a two-
+states Markov chain (Ak
+t )t≥0 with transition probability ma-
+trix Pk and these Markov chains are all independent. In
+this case, the aggregate process is described by the product
+Markov chain (At)t≥0 with transition matrix P = �
+k∈K Pk
+and λ(P ) = maxk∈K λ(Pk), where Pi
+� Pj denotes the
+Kronecker product between matrices Pi and Pj [30, Exer-
+cise 12.6]. In this setting, it is possible to redefine the Markov
+chain (At)t≥0 by taking into account the reduced state space
+defined by the clients with a non-null aggregation weight, i.e.,
+P ′ = �
+k′∈K|qk′>0 Pk′ and λ(P ′) = maxk′∈K|qk′>0 λ(Pk′),
+which is potentially smaller than the case when all clients
+participate to the aggregation. These considerations lead to
+the following guideline:
+Guideline B: to speed up the convergence, we can exclude,
+i.e., set q∗
+k = 0, the clients with largest λ(Pk).
+Intuition also supports this guideline. Clients with large λ(Pk)
+tend to be available or unavailable for long periods of time.
+Due to the well-known catastrophic forgetting problem affect-
+ing gradient methods [33], [34], these clients may unfairly
+steer the algorithm toward their local objective when they
+appear at the final stages of the training period. Moreover,
+their participation in the early stages may be useless, as their
+contribution will be forgotten during their long absence. The
+FL algorithm may benefit from directly neglecting such clients.
+We observe that guideline B strictly applies to this specific
+setting where clients’ dynamics are independent (and there
+is no spatial correlation). We do not provide a corresponding
+Algorithm 1: CA-Fed (Correlation-Aware FL)
+Input : w0,0, α, q(0), {ηt}T
+t=1, ηs, E, β, τ
+1 Initialize ˆF (0), ˆF ∗, ˆΓ
+′(0), ˆπ(0), and ˆλ(0);
+2 for t = 1, . . . , T do
+3
+Receive set of active client At, loss vector F (t);
+4
+Update ˆF (t), ˆΓ
+′(t), ˆπ(t), and ˆλ(t);
+5
+Initialize q(t) =
+α
+ˆπ(t) ;
+6
+q(t) ← get(q(t), α, ˆF (t), ˆF ∗, ˆΓ
+′(t), ˆπ(t), ˆλ(t));
+7
+q(t) ← get(q(t), α, ˆF (t), ˆF ∗, ˆΓ
+′(t), ˆπ(t), �ˆπ(t));
+8
+for client {k ∈ At; q(t)
+k
+> 0}, in parallel do
+9
+for j = 0, . . . , E − 1 do
+10
+wk
+t,j+1 = wk
+t,j − ηt∇Fk(wk
+t,j, Bk
+t,j) ;
+11
+∆k
+t ← wt,E − wt,0;
+12
+wt+1,0 ← ProjW (wt,0 + ηs
+�
+k∈At q
+(t)
+k · ∆k
+t );
+13 Function get(q, α, F , F ∗, Γ, π, ρ):
+14
+K ← sort by descending order in ρ;
+15
+ˆϵ ← ⟨F −F ∗, π ˜⊙q⟩ + d2
+T V (α, π ˜⊙q) · Γ;
+16
+for k ∈ K do
+17
+q+
+k ← 0;
+18
+ˆϵ+ ← ⟨F −F ∗, π ˜⊙q+⟩ + d2
+T V (α, π ˜⊙q+) · Γ;
+19
+if ˆϵ − ˆϵ+ ≥ τ then
+20
+ˆϵ ← ˆϵ+;
+21
+q ← q+;
+22
+return q
+guideline for the case when clients are spatially correlated (we
+leave this task for future research). However, in this more gen-
+eral setting, it is possible to ignore guideline B but still draw
+on guidelines A and C, or still consider guideline B if clients
+are spatially correlated (see discussion in Section VI-B).
+E. Minimizing ϵ′
+bias
+The bias error ϵ′
+bias in (14) vanishes when the total variation
+distance between the target importance α and the biased
+importance p is zero, i.e., when qk ∝ αk/πk, ∀k ∈ K. Then,
+after excluding the clients that contribute the most to the
+optimization error and particularly slow down the convergence
+(guidelines A and B), we can assign to the remaining clients an
+aggregation weight inversely proportional to their availability,
+such that the bias error ϵ′
+bias is minimized.
+Guideline C: to reduce the bias error, we set q∗
+k ∝ αk/πk for
+the clients that are not excluded by the previous guidelines.
+IV. PROPOSED ALGORITHM
+Guidelines A and B in Section III suggest that the minimiza-
+tion of ϵopt can lead to the exclusion of some available clients
+from the aggregation step (3), in particular those with low
+availability and/or high correlation. For the remaining clients,
+guideline C proposes to set their aggregation weight inversely
+proportional to their availability to reduce the bias error ϵ′
+bias.
+Motivated by these insights, we propose CA-Fed, a client
+sampling and aggregation strategy that takes into account the
+problem of correlated client availability in FL, described in
+5
+
+Algorithm 1. CA-Fed learns during training which are the
+clients to exclude and how to set the aggregation weights of the
+other clients to achieve a good trade-off between ϵopt and ϵ′
+bias.
+While guidelines A and B indicate which clients to remove,
+the exact number of clients to remove at round t is identified
+by minimizing ϵ(t) as a proxy for the bound in (14):4
+ϵ(t) := FB(wt,0)−F ∗
+B + d2
+T V (α, p)Γ′.
+(15)
+A. CA-Fed’s core steps
+At each communication round t, the server sends the current
+model wt,0 to all active clients and each client k sends back
+a noisy estimate F
+(t)
+k
+of the current loss computed on a batch
+of samples Bk
+t,0, i.e., F
+(t)
+k
+=
+1
+|Bk
+t,0|
+�
+ξ∈Bk
+t,0 f(wt,0, ξ) (line 3).
+The server uses these values and the information about the
+current set of available clients At to refine its own estimates
+of each client’s loss ( ˆF (t) = ( ˆF
+(t)
+k )k∈K), and each client’s
+loss minimum value ( ˆF ∗ = ( ˆF ∗
+k )k∈K), as well as of Γ′, πk,
+λk, and ϵ(t), denoted as ˆΓ
+′(t), ˆπ
+(t)
+k , ˆλ
+(t)
+k , and ˆϵ(t), respectively
+(possible estimators are described below) (line 4).
+The server decides whether excluding clients whose avail-
+ability pattern exhibits high correlation (high ˆλ
+(t)
+k ) (line 6).
+First, the server considers all clients in descending order of
+ˆλ(t) (line 14), and evaluates if, by excluding them (line 17),
+ˆϵ(t) appears to be decreasing by more than a threshold τ ≥ 0
+(line 19). Then, the server considers clients in ascending order
+of ˆπ(t), and repeats the same procedure to possibly exclude
+some of the clients with low availability (low ˆπ
+(t)
+k ) (lines 7).
+Once the participating clients (those with qk > 0) have
+been selected, the server notifies them to proceed updating
+the current models (lines 9–10) according to (2), while the
+other available clients stay idle. Finally, model’s updates are
+aggregated according to (3) (line 12).
+B. Estimators
+We now briefly discuss possible implementation of the esti-
+mators ˆF
+(t)
+k , ˆF ∗
+k , ˆΓ
+′(t), ˆπ
+(t)
+k , and ˆλ
+(t)
+k . Server’s estimates for the
+clients’ local losses ( ˆF (t) = ( ˆF
+(t)
+k )k∈K) can be obtained from
+the received active clients’ losses (F (t) = (F
+(t)
+k )k∈At) through
+an auto-regressive filter with parameter β ∈ (0, 1]:
+ˆF
+(t) = (1 − β1At) ⊙ ˆF (t−1) + β1At ⊙ F
+(t),
+(16)
+where ⊙ denotes the component-wise multiplication between
+vectors, and 1At is a N-dimensions binary vector whose k-th
+component equals 1 if and only if k is active at round t, i.e.,
+k ∈ At. The server can keep track of the clients’ loss minimum
+values and estimate F ∗
+k as ˆF ∗
+k = mins∈[0,t] ˆF
+(s)
+k . The values of
+FB(wt,0), F ∗
+B, Γ′, and ϵ(t) can be estimated as follows:
+ˆF
+(t)
+B − ˆF ∗
+B = ⟨ ˆF (t) − ˆF ∗, ˆπ(t) ˜⊙q(t)⟩,
+(17)
+ˆΓ
+′(t) = maxk∈K( ˆF
+(t)
+k
+− ˆF ∗
+k ),
+(18)
+ˆϵ(t) = ˆF
+(t)
+B − ˆF ∗
+B + d2
+T V (α, ˆπ(t) ˜⊙q(t)) · ˆΓ
+′(t).
+(19)
+4Following (14), one could reasonably introduce a hyper-parameter to
+weigh the relative importance of the optimization and bias terms in the sum.
+We discuss this additional optimization of CA-Fed in Section VI-A.
+where π ˜⊙q ∈ RN, such that
+�
+π ˜⊙q
+�
+k =
+πkqk
+�N
+h=1 πhqh , k ∈ K.
+For ˆπ
+(t)
+k , the server can simply keep track of the total number
+of times client k was available up to time t and compute
+ˆπ
+(t)
+k
+using a Bayesian estimator with beta prior, i.e., ˆπ
+(t)
+k
+=
+(�
+s≤t 1k∈As +nk)/(t+nk +mk), where nk and mk are the
+initial parameters of the beta prior.
+For ˆλ
+(t)
+k , the server can assume the client’s availability evolves
+according to a Markov chain with two states (available and
+unavailable), track the corresponding number of state tran-
+sitions, and estimate the transition matrix
+ˆP
+(t)
+k
+through a
+Bayesian estimator similarly to what done for ˆπ
+(t)
+k . Finally,
+ˆλ
+(t)
+k is obtained computing the eigenvalues of ˆP
+(t)
+k .
+C. CA-Fed’s computation/communication cost
+CA-Fed aims to improve training convergence and not to
+reduce its computation and communication overhead. Never-
+theless, excluding some available clients reduces the overall
+training cost, as we will discuss in this section referring, for
+the sake of concreteness, to neural networks’ training.
+The available clients not selected for training are only re-
+quested to evaluate their local loss on the current model once
+on a single batch instead than performing E gradient updates,
+which would require roughly 2 × E − 1 more calculations
+(because of the forward and backward pass). For the selected
+clients, there is no extra computation cost as computing the
+loss corresponds to the forward pass they should, in any case,
+perform during the first local gradient update.
+In terms of communication, the excluded clients only transmit
+the loss, a single scalar, much smaller than the model update.
+Conversely, participating clients transmit the local loss and the
+model update. Still, this additional overhead is negligible and
+likely fully compensated by the communication savings for
+the excluded clients.
+V. EXPERIMENTAL EVALUATION
+A. Experimental Setup
+a) Federated system simulator: In our experiments, we sim-
+ulate the clients’ availability dynamics featuring different
+levels of temporal correlations. We model the activity of each
+client as a two-state homogeneous Markov process with state
+space S = {“active”, “inactive”}. We use pk,s to denote the
+probability that client k ∈ K remains in state s ∈ S.
+In order to simulate the statistical heterogeneity present in the
+federated learning system, we consider an experimental setting
+with two disjoint groups of clients Gi, i = 1, 2, to which
+we associate two different data distributions Pi, i = 1, 2,
+to be precised later. Let ri = |Gi|/N, i = 1, 2 denote the
+fraction of clients in group i = 1, 2. In order to simulate
+the heterogeneity of clients’ availability patterns in realistic
+federated systems, we split the clients of each group in two
+classes uniformly at random: “more available” clients whose
+steady-state probability to be active is πk,active = 1/2 + g and
+“less available” clients with πk,active = 1/2 − g, where g ∈
+6
+
+Inactive,
+excluded
+Inactive,
+included
+Active,
+excluded
+Active,
+included
+More Available
+Less Available, Weakly Correlated
+0
+20
+40
+60
+80
+100
+120
+140
+Communication round
+Less Available, Correlated
+Clients
+Fig. 1: Clients’ activities and CA-Fed’s clients selection on the synthetic dataset.
+More Available
+Less Available
+Correlated
+Less Available
+Weakly Correlated
+Clients
+Cumulative weight
+Unbiased
+CA-Fed
+AdaFed
+F3AST
+Target
+Fig. 2: Importance given to the clients by the different algorithms
+throughout a whole training process on the synthetic dataset.
+(0, 1/2) is a parameter controlling the heterogeneity of clients
+availability. We furthermore split each class of clients in two
+sub-classes uniformly at random: “correlated” clients that tend
+to persist in the same state (λk = ν with values of ν close to
+1), and “weakly correlated” clients that are almost as likely
+to keep as to change their state (λk ∼ N(0, ε2), with ε close
+to 0). In our experiments, we suppose that r1 = r2 = 1/2,
+g = 0.4, ν = 0.9, and ε = 10−2.
+b) Datasets and models: All experiments are performed on
+a binary classification synthetic dataset (described in Ap-
+pendix F) and on the real-world MNIST dataset [35], using
+N = 24 clients. For MNIST dataset, we introduce statistical
+heterogeneity across the two groups of clients (i.e., we make
+the two distributions P1 and P2 different), following the same
+approach in [36]: 1) every client is assigned a random subset
+of the total training data; 2) the data of clients from the second
+group is modified by randomly swapping two pairs of labels.
+We maintain the original training/test data split of MNIST and
+use 20% of the training dataset as validation dataset. We use a
+linear classifier with a ridge penalization of parameter 10−2,
+which is a strongly convex objective function, for both the
+synthetic and the real-world MNIST datasets.
+c) Benchmarks:
+We compare CA-Fed, defined in Algo-
+rithm 1, with the Unbiased aggregation strategy, where all
+the active clients participate and receive a weight inversely
+proportional to their availability, and with the state-of-the-
+art FL algorithms discussed in Section II: F3AST [18] and
+AdaFed [19]. We tuned the learning rates η, ηs via grid
+search, on the grid η : {10−3, 10−2.5, 10−2, 10−1.5, 10−1},
+ηs : {10−2, 10−1.5, 10−1, 10−0.5, 100}. For CA-Fed, we used
+τ = 0, β = 0.2. We assume all algorithms can access an oracle
+providing the true availability parameters for each client. In
+0
+20
+40
+60
+80
+100
+120
+140
+Communication round
+35
+40
+45
+50
+55
+60
+65
+70
+75
+Time-average test accuracy
+Unbiased
+F3AST
+AdaFed
+CA-Fed (Ours)
+(a) Synthetic
+0
+20
+40
+60
+80
+100
+120
+140
+Communication round
+10
+20
+30
+40
+50
+60
+Time-average test accuracy
+Unbiased
+F3AST
+AdaFed
+CA-Fed (Ours)
+(b) MNIST
+Fig. 3: Test accuracy vs number of communication rounds.
+practice, Unbiased, AdaFed, and F3AST rely on the exact
+knowledge of πk,active, and CA-Fed on πk,active and λk. 5
+B. Experimental Results
+Figure 1 shows the availability of each client during a training
+run on the synthetic dataset. Clients selected (resp. excluded)
+by CA-Fed are highlighted in black (resp. red). We observe
+that excluded clients tend to be those with low average
+availability or high correlation.
+Figure 2 shows the importance pk (averaged over time) given
+by different algorithms to each client k during a full training
+run. We observe that all the algorithms, except Unbiased,
+depart from the target importance α. As suggested by guide-
+lines A and B, CA-Fed tends to favor the group of “more
+available” clients, at the expense of the “less available” clients.
+Figure 3 shows the time-average accuracy up to round t of
+the learned model averaged over three different runs. On both
+datasets, CA-Fed achieves the highest accuracy, which is
+about a percentage point higher than the second best algorithm
+(F3AST). Table I shows for each algorithm: the average over
+three runs of the maximum test accuracy achieved during train-
+ing, the time-average test accuracy achieved during training,
+together with its standard deviation within the second half of
+the training period. Results show that while CA-Fed achieves
+a maximum accuracy which is comparable to the Unbiased
+baseline and state-of-the-art AdaFed and F3AST, it gets a
+higher time-average accuracy (1.24 percentage points) in com-
+parison to the second best (F3AST), and a smaller standard
+deviation (1.5×) in comparison to the second best (F3AST).
+5The authors have provided public access to their code and data at:
+https://github.com/arodio/CA-Fed.
+7
+
+TABLE I: Maximum and time-average test accuracy, together with
+their standard deviations, on the Synthetic / MNIST datasets.
+TEST ACCURACY
+MAXIMUM
+TIME-AVERAGE
+STANDARD DEVIATION
+UNB I AS ED
+78.94 / 64.87
+75.32 / 61.39
+0.48 / 1.09
+F3AST
+78.97 / 64.91
+75.33 / 61.52
+0.40 / 0.94
+ADAFED
+78.69 / 63.77
+74.81 / 60.48
+0.59 / 1.37
+CA-FE D
+79.03 / 64.94
+76.22 / 62.76
+0.28 / 0.61
+VI. DISCUSSION
+In this section, we discuss some general concerns and remarks
+on our algorithm.
+A. Controlling the number of excluded clients
+Theorems 1 and 3 suggest that the condition number κ2 can
+play a meaningful role in the minimization of the total error ϵ.
+Our algorithm uses a proxy (ϵ(t)) of the total error. To take into
+account the effect of κ2, we can introduce a hyper-parameter
+that weights the relative importance of the optimization and
+bias error in (15):
+ϵ′(t) := FB(wt,0) − F ∗
+B + ¯κ2 · d2
+T V (α, p)Γ′.
+A small value of ¯κ2 penalizes the bias term in favor of the
+optimization error, resulting in a larger number of clients
+excluded by CA-Fed. On the other hand, CA-Fed tends to
+include more clients for a large value of ¯κ2. Asymptotically,
+for ¯κ2 → +∞, CA-Fed reduces to the Unbiased baseline.
+To further improve the performance of CA-Fed, a finer tuning
+of the values of ¯κ2 can be performed.
+B. CA-Fed in presence of spatial correlation
+Although CA-Fed is mainly designed to handle temporal
+correlation, it does not necessarily perform poorly in presence
+of spatial correlation, as well.
+Consider the following spatially-correlated scenario: clients
+are grouped in clusters, each cluster c ∈ C is characterized
+by an underlying Markov chain, which determines when all
+clients in the cluster are available/unavailable, the Markov
+chains of different clusters are independent. Let λc denote
+the second largest eigenvalue in module of cluster-c’s Markov
+chain. In this case, one needs to exclude all clients in the
+cluster ¯c = arg maxc∈C λc to reduce the eigenvalue of the
+aggregate Markov chain.
+In this setting, CA-Fed would associate similar eigenvalue
+estimates to all clients in the same cluster, then it would
+correctly start considering for exclusion the clients in cluster
+¯c and potentially remove sequentially all clients in the same
+cluster. These considerations suggest that CA-Fed may still
+operate correctly even in presence of spatial correlation.
+C. About CA-Fed’s fairness
+A strategy that excludes clients from the training phase,
+such as CA-Fed, may naturally raise fairness concerns. The
+concept of fairness in FL does not have a unified definition in
+the literature [37, Chapter 8]: fairness goals can be captured by
+a suitable choice of the target weights in (1). For example, per-
+client fairness can be achieved by setting αk equal for every
+client, while per-sample fairness by setting αk proportional
+to the local dataset size |Dk|. If we assume that the global
+objective in (1) indeed reflects also fairness concerns, then
+CA-Fed is intrinsically fair, in the sense that it guarantees
+that the performance objective of the learned model is as close
+as possible to its minimum value.
+VII. CONCLUSION
+This paper presented the first convergence analysis for a
+FedAvg-like FL algorithm under heterogeneous and corre-
+lated client availability. The analysis quantifies how correla-
+tion adversely affects the algorithm’s convergence rate and
+highlights a general bias-versus-convergence-speed trade-off.
+Guided by the theoretical analysis, we proposed CA-Fed, a
+new FL algorithm that tries to balance the conflicting goals
+of maximizing convergence speed and minimizing model bias.
+Our experimental results demonstrate that adaptively excluding
+clients with high temporal correlation and low availability is an
+effective approach to handle the heterogeneous and correlated
+client availability in FL.
+APPENDIX
+A. Proof of Theorem 1
+We bound the optimization error of the target objective as the
+optimization error of the biased objective plus a bias term:
+F(w) − F ∗
+(a)
+≤
+1
+2µ ∥∇F(w)∥2
+(b)
+≤ L2
+2µ ∥w − w∗∥2
+(c)
+≤ L2
+µ (∥w − w∗
+B∥2 + ∥w∗
+B − w∗∥2)
+(d)
+≤ 2L2
+µ2 (FB(w) − F ∗
+B)
+�
+��
+�
+:=ϵopt
++ 2L2
+µ2 (F(w∗
+B) − F ∗)
+�
+��
+�
+:=ϵbias
+,
+where (a), (b), and (d) follow from the Assumptions 3, 4,
+and the inequality (c) follows from (a + b)2 ≤ 2a2 + 2b2.
+In particular, (b) requires ∇Fk(w∗
+k) = 0. Theorem 2 further
+develops the optimization error ϵopt. We now expand ϵbias:
+∥∇F(w∗
+B)∥
+(e)=
+����N
+k=1(αk − pk)∇Fk(w∗
+B)
+���
+(f)
+≤ L �N
+k=1|αk − pk| ∥w∗
+B − w∗
+k∥
+(20)
+(g)
+≤ L
+�
+2
+µ
+�N
+k=1
+|αk−pk|
+√pk
+�
+pk(Fk(w∗
+B) − F ∗
+k ),
+where (e) uses ∇FB(w∗
+B) = 0; (f) applies first the triangle
+inequality, then the L-smoothness, and (g) follows from the
+µ-strong convexity. In addition, (f) requires ∇Fk(w∗
+k) = 0.
+Similarly to [32], in (g) we multiply numerator and denomi-
+nator by √pk. By direct calculations, it follows that:
+∥∇F(w∗
+B)∥2
+(h)
+≤ 2L2
+µ
+� �N
+k=1
+|αk−pk|
+√pk
+�
+pk(Fk(w∗
+B) − F ∗
+k )
+�2
+(i)
+≤ 2L2
+µ
+�
+N�
+k=1
+(αk−pk)2
+pk
+��
+N�
+k=1
+pk(Fk(w∗
+B) − F ∗
+k )
+�
+(j)
+≤ 2L2
+µ χ2
+α∥pΓ,
+8
+
+where (i) uses the Cauchy–Schwarz inequality, and (j) used:
+�N
+k=1 pk(Fk(w∗
+B) − F ∗
+k ) ≤ �N
+k=1 pk(Fk(w∗) − F ∗
+k ) ≤ Γ.
+Finally, by strong convexity of F, we conclude that:
+F(w∗
+B) − F ∗ ≤
+1
+2µ ∥∇F(w∗
+B)∥2 ≤ L2
+µ2 χ2
+α∥pΓ.
+B. Proof of Theorem 2
+1) Additional notation: let wk
+t,j be the model parameter vector
+computed by device k at the global round t, local iteration j.
+We define:
+gt(At) = �
+k∈At qk
+�E−1
+j=0 ∇Fk(wk
+t,j, ξk
+t,j),
+and ¯gt(At) = Eξ|At[gt(At)].
+Following (2) and (3), the update rule of CA-Fed is:
+wt+1,0 = ProjW (wt,0 − ηtgt(At)).
+(21)
+2) Key lemmas and results: we provide useful lemmas and
+results to support the proof of the main theorem.
+Proof of Lemma 1. The boundedness of W gives a bound on
+(wt,0)t≥0 based on the update rules in (2) and (3). From the
+convexity of {Fk}k∈K, it follows that:
+D :=
+sup
+w∈W,k∈K
+∥∇Fk(w)∥ < +∞.
+Items (6), (8) are directly derived from the previous observa-
+tion. Item (7) follows combining (6) and Assumption 5:
+E ∥∇Fk(w, ξ)∥2 ≤ D2 + max
+k∈K {σ2
+k} := G2.
+Lemma 2 (Convergence under heterogeneous client availabil-
+ity). Let the local functions {Fk}k∈K be convex, Assump-
+tions 3, 5 hold. If ηt ≤
+1
+2L(EQ+1), we have:
+�
+t ηt E[�
+k∈At qk (Fk(wt,0) − Fk(w∗
+B))] ≤
++ 2
+E ∥w0,0 − w∗
+B∥2 + 2 �N
+k=1 πkq2
+kσ2
+k
+�
+t η2
+t
++ 2
+3
+�N
+k=1 πkqk(E − 1)(2E − 1)G2 �
+t η2
+t
++ 2L(EQ + 2) �N
+k=1 πkqkΓ �
+t η2
+t := C1 < +∞.
+Proof of Lemma 2.
+∥wt+1,0 − w∗
+B∥2 = ∥ProjW (wt,0 − ηtgt) − ProjW (w∗
+B)∥2
+≤ ∥wt,0 − ηtgt − w∗
+B + ηt¯gt − ηt¯gt∥2 = A1 + A2 + A3,
+where:
+A1 = ∥wt,0 − w∗
+B − ηt¯gt∥2 ,
+A2 = 2ηt⟨wt,0 − w∗
+B − ηt¯gt, ¯gt − gt⟩,
+A3 = η2
+t ∥gt − ¯gt∥2 .
+Note E[A2] = 0. We bound A1, A3 using the key steps in [22]:
+(1) the variance of gt(At) is bounded if the variance of the
+stochastic gradients at each device is bounded:
+A3 = EB|At ∥gt − ¯gt∥2 =
+= �
+k∈At q2
+k
+�E−1
+j=0 EB|At
+��∇Fk(wk
+t,j, ξk
+t,j)−∇Fk(wk
+t,j)
+��2
+≤ E �
+k∈At q2
+kσ2
+k;
+(2) the distance of the local model wk
+t,E from the global
+model wt,0 is bounded since the expected squared norm of
+the stochastic gradients is bounded:
+EB|At
+�
+k∈At qk
+�E−1
+j=0
+��wk
+t,j − wt,0
+��2 =
+= EB|At
+�
+k∈At qk
+�E−1
+j=1 η2
+t
+���
+�j−1
+j′=0 ∇Fk(wk
+t,j′, ξk
+t,j′)
+���
+2
+≤ η2
+t
+�
+k∈At qk
+�E−1
+j=1 j �j−1
+j′=0 EB|At
+��∇Fk(wk
+t,j′, ξk
+t,j′)
+��2
+≤ η2
+t
+�
+k∈At qkG2 �E−1
+j=1 j2
+= 1
+6η2
+t
+�
+k∈At qkE(E − 1)(2E − 1)G2.
+Lemma 3 (Optimization error after Jt steps). Let Assump-
+tions 1, 2 hold, the local functions {Fk}k∈K be convex, D, H
+be defined as in (6), (8), and Jt defined as in Theorem 2.
+Then:
+�
+t ηt E[�
+k∈At qk(Fk(wt−Jt,0) − Fk(wt,0))]
+≤ EDGQ �
+t Jtη2
+t−Jt
+�N
+k=1 πkqk :=
+C3
+ln(1/λ(P )) < +∞.
+For the proof of Lemma 3, we introduce the following results:
+|Fk(v) − Fk(w)| ≤ D · ∥v − w∥ , ∀v, w ∈ W,
+(22)
+EBk
+t,0,...,Bk
+t,E−1 ∥wt+1,0 − wt,0∥ ≤ ηtGE(�
+k∈At qk). (23)
+Equation (22) is due to convexity of {Fk}k∈K, which gives:
+⟨∇Fk(v), v − w⟩ ≤ ∥Fk(v) − Fk(w)∥ ≤ ⟨∇Fk(w), v − w⟩;
+the Cauchy–Schwarz inequality concludes:
+|Fk(v) − Fk(w)| ≤ max{∥∇Fk(v)∥ , ∥∇Fk(w)∥} ∥v − w∥
+≤ D · ∥v − w∥ .
+Equation (23) follows combining equations (7) and (21):
+EB|At ∥wt+1,0 − wt,0∥ ≤
+≤ ηt EB|At
+����
+k∈At qk
+�E−1
+j=0 ∇Fk(wk
+t,j, ξk
+t,j)
+���
+≤ ηt
+�
+k∈At qk
+�E−1
+j=0 EB|At
+��∇Fk(wk
+t,j, ξk
+t,j)
+��
+≤ ηtGE(�
+k∈At qk).
+Proof of Lemma 3. The evolution of the local objectives after
+Jt communication rounds is bounded:
+�
+tηt E[�
+k∈At qk(Fk(wt−Jt,0) − Fk(wt,0))]
+(a)
+≤ D �
+t ηt E[�
+k∈At qk EB ∥wt−Jt,0 − wt,0∥]
+(b)
+≤ D �
+t ηt
+�t−1
+d=t−Jt E[�
+k∈At qk EB ∥wd,0 − wd+1,0∥]
+(c)
+≤ EDG �
+t
+�t−1
+d=t−Jt ηtηd E[�
+k∈At qk
+�
+k′∈Ad qk′]
+(d)
+≤ EDG
+2
+�
+t
+�t−1
+d=t−Jt(η2
+t + η2
+d) E[�
+k∈At qk
+�
+k′∈Ad qk′]
+(e)
+≤ EDGQ �
+t Jtη2
+t−Jt
+�N
+k=1 πkqk :=
+C3
+ln(1/λ(P )),
+9
+
+where (a) follows from (22); (b) applies the triangle inequal-
+ity; (c) uses (23); (d) applies the Cauchy–Schwarz inequality;
+(e) uses ηt < ηd ≤ ηt−Jt and �N
+k=1 qk = Q.
+3) Core of the proof: The proof consists in two main steps:
+1. �
+t ηt
+�N
+k=1 πkqk E[FB(wt−Jt,0) − F ∗
+B)]≤C2+
+C3
+ln(1/λ(P ));
+2. �
+t ηt
+�N
+k=1 πkqk E[FB(wt,0)−FB(wt−Jt,0)]≤
+C3
+ln(1/λ(P )).
+Step 1. Combining Lemma 2 and 3, we get:
+�
+t ηt E[ �
+k∈At
+qk(Fk(wt−Jt,0) − Fk(w∗
+B))] ≤ C1 +
+C3
+ln(1/λ(P )).
+The constant Jt, introduced in [14], is an important parameter
+for the analysis and frequently used. Combining its definition
+in Theorem 2 and equation (5), it follows:
+��[P Jt]i,j − πj
+�� ≤ CP λ(P )Jt ≤
+1
+2Ht,
+∀i, j ∈ [M].
+(24)
+Assume t ≥ TP . We derive an important lower bound:
+EAt|At−Jt [�
+k∈At qk(Fk(wt−Jt,0) − Fk(w∗
+B))]
+(a)= �M
+I=1 P(At=I|At−Jt) �
+k∈I qk(Fk(wt−Jt,0)−Fk(w∗
+B))
+(b)= �M
+I=1 [P Jt]At−Jt,I
+�
+k∈I qk (Fk(wt−Jt,0) − Fk(w∗
+B))
+(c)
+≥ �M
+I=1
+�
+π(I) −
+1
+2Ht
+� �
+k∈I qk(Fk(wt−Jt,0) − Fk(w∗
+B))
+(d)
+≥ (�N
+k=1 πkqk) · (FB(wt−Jt,0) − F ∗
+B) − 1
+2tMQ,
+(25)
+where (a) is the definition of the conditional expectation, (b)
+uses the Markov property, (c) follows from (24), and (d) is
+due to (8). Taking total expectations:
+( �N
+k=1 πkqk) �
+t ηt E[FB(wt−Jt,0) − F ∗
+B]
+≤ �
+t ηt E[�
+k∈At qk(Fk(wt−Jt,0) − Fk(w∗
+B))]
++ 1
+4MQ �
+t(η2
+t + 1
+t2 ) = C2 +
+C3
+ln(1/λ(P )),
+(26)
+where C2 = C1 + 1
+4MQ �
+t(η2
+t + 1
+t2 ).
+Step 2. By direct calculation (similar to Lemma 3):
+(�N
+k=1 πkqk) �
+t ηt E[FB(wt,0) − FB(wt−Jt,0)]≤
+C3
+ln(1/λ(P )).
+Summing Step 1 and 2, and applying Jensen’s inequality:
+(�T
+t=1 ηt)(�N
+k=1 πkqk) E[FB( ¯wT,0) − F ∗
+B] ≤
+(�N
+k=1 πkqk) �T
+t=1 ηt E[FB(wt,0) − F ∗
+B] ≤ C2 +
+2C3
+ln(1/λ(P )),
+where ¯wT,0 :=
+�T
+t=1 ηtwt,0
+�T
+t=1 ηt
+, and the constants are in (12).
+C. Proof of Theorem 3
+It follows the same lines of Theorem 1, developing (20) as:
+∥∇F(w∗
+B)∥ ≤ L
+�
+2
+µ
+�N
+k=1|αk − pk|
+�
+(Fk(w∗
+B) − F ∗
+k )
+≤ 2L
+�
+2
+µdT V (α, p)
+√
+Γ′,
+where dT V (α, p) := 1
+2
+�N
+k=1|αk − pk| is the total variation
+distance between the probability measures α and p.
+D. Minimizing ϵopt
+Equation 12 defines the following optimization problem:
+minimize
+q
+f(q) =
+1
+2 q⊺Aq+B
+π⊺q
++ C;
+subject to
+q ≥ 0,
+π⊺q > 0,
+∥q∥1 = Q.
+Let us rewrite the problem by adding a variable s := 1/π⊺q
+and then replacing y := sq. Note that the objective function is
+the perspective of a convex function, and is therefore convex:
+min
+y,s
+f(y, s) =
+1
+2sy⊺Ay + Bs + C
+(27a)
+s.t.
+y ≥ 0, s > 0, π⊺y = 1, ∥y∥1 = Qs.
+(27b)
+The Lagrangian function L is as follows:
+L(y, s, λ, θ, µ) =
+1
+2sy⊺Ay + Bs + C+
++λ(1 − π⊺y) + θ(∥y∥1 − Qs) − µ⊺y.
+(28)
+Since the constraint s > 0 defines an open set, the set defined
+by the constraints in (27b) is not closed. However, the solution
+is never on the boundary s = 0 because L∗ → +∞ as s → 0+,
+and we can consider s ≥ 0. The KKT conditions for y∗
+k read:
+if y∗
+k > 0: y∗
+k =
+s∗
+A[kk](λ∗πk − θ∗); y∗
+k = 0 otherwise. (29)
+Since λ∗ ≥ 0, the clients with smaller πk may have q∗
+k = 0.
+E. Convexity of ϵopt + ϵbias
+In Appendix D, we proved that ϵopt(q) is convex. To prove
+that ϵbias(q) is also convex, we need to study the convexity of
+χ2
+α∥p = �N
+k=1(fk ◦ gk)(q), where fk(pk) = (pk − αk)2/pk,
+and gk(q) = (πkqk)/ �N
+h=1 πhqh. We observe that fk(pk)
+is convex, and gk(q) is a particular case of linear-fractional
+function [38]. By direct inspection, it can be proved that
+(fk◦gk)(q) is convex in dom(fk◦gk) = {q : ∥q∥1 = Q > 0}.
+F. Synthetic dataset
+Our synthetic datasets has been generated as follows:
+1) For client k ∈ K, sample group identity ik from a
+Bernoulli distribution of parameter 1/2;
+2) Sample model parameters w∗ ∼ N(0, Id) from the d-
+dimensional normal distribution;
+3) For client k ∈ K and sample index j ∈ {1, . . . , 150},
+sample clients input data x(j)
+k
+∼ N(0, Id) from the d-
+dimensional normal distribution;
+4) For client k ∈ K such that ik = 0 and sample index j ∈
+{1, . . . , 150}, sample the true labels y(j)
+k
+from a Bernoulli
+distribution with parameter equal to sigmoid(⟨w∗, x(j)
+k ⟩);
+5) For client k
+∈
+K such that ik
+=
+1 and sample
+index j ∈ {1, . . . , 150}, sample the true labels y(j)
+k
+from a Bernoulli distribution with parameter equal to
+0.8·sigmoid(⟨w∗, x(j)
+k ⟩)+0.2·(1−sigmoid(⟨w∗, x(j)
+k ⟩)).
+10
+
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+FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS OF SURFACES
+C. CASAGRANDE
+Dedicated to Lorenzo, Sabrina, and Fabrizio
+Abstract. Let X be a smooth, complex Fano 4-fold, and ρX its Picard num-
+ber. We show that if ρX > 12, then X is a product of del Pezzo surfaces. The
+proof relies on a careful study of divisorial elementary contractions f : X → Y
+such that dim f(Exc(f)) = 2, together with the author’s previous work on Fano
+4-folds. In particular, given f : X → Y as above, under suitable assumptions we
+show that S := f(Exc(f)) is a smooth del Pezzo surface with −KS = (−KY )|S.
+1. Introduction
+Smooth, complex Fano varieties have been classically intensively studied, and
+have attracted a lot of attention also in the last decades, due to their role in
+the framework of the Minimal Model Program. The Fano condition is a natural
+positivity condition of the tangent bundle, and it ensures a rich geometry, from
+both the points of view of birational geometry and of families of rational curves.
+It has been known since the 90’s that Fano varieties form a bounded family in
+each dimension. Del Pezzo surfaces are known classically, and the classification
+of Fano 3-folds have been in achieved in the 80’s, there are 105 families.
+Starting from dimension 4, there are probably too many families to get a com-
+plete classification; still we aim to better understand and describe the behavior
+and properties of these varieties. In this paper we focus on Fano 4-folds X with
+“large” Picard number ρX; let us recall that since X is Fano, ρX is equal to the
+second Betti number b2(X). We show the following result.
+Theorem 1.1. Let X be a smooth Fano 4-fold with ρX > 12. Then X ∼= S1 ×S2,
+where Si are del Pezzo surfaces.
+To the author’s knowledge, all known examples of Fano 4-folds which are not
+products of surfaces have ρ ≤ 9, so that we do not know whether the condition
+ρ > 12 in Th. 1.1 is sharp. We refer the reader to [Cas22b, §6] for an overview
+of known Fano 4-folds with ρ ≥ 6; there are few examples and it is an interesting
+problem to construct new ones.
+As ρS1×S2 = ρS1 + ρS2, and del Pezzo surfaces have ρ ≤ 9, Th. 1.1 implies the
+following.
+Corollary 1.2. Let X be a smooth Fano 4-fold. Then ρX ≤ 18.
+2020 Mathematics Subject Classification. 14J45,14J35,14E30.
+1
+arXiv:2301.11953v1  [math.AG]  27 Jan 2023
+
+2
+C. CASAGRANDE
+Let us note that Th. 1.1 and Cor. 1.2 generalize to dimension 4 the analogous
+result for Fano 3-folds, established by Mori and Mukai in the 80’s:
+Theorem 1.3 ([MM86], Th. 1.2). Let X be a smooth Fano 3-fold with ρX > 5.
+Then X ∼= S × P1 where S is a del Pezzo surface. In particular ρX ≤ 10.
+The proof of Th. 1.1 relies on a careful study of elementary contractions of X of
+type (3, 2), together with the author’s previous work on Fano 4-folds. To explain
+this, let us introduce some notation.
+Let X be a Fano 4-fold. A contraction is a surjective morphism f : X → Y , with
+connected fibers, where Y is normal and projective; f is elementary if ρX−ρY = 1.
+As usual, an elementary contraction can be of fiber type, divisorial, or small.
+We say that an elementary contraction f : X → Y is of type (3, 2) if it is
+divisorial with dim S = 2, where E := Exc(f) and S := f(E) ⊂ Y . Such f
+can have at most finitely many 2-dimensional fibers; outside the images of these
+fibers, Y and S are smooth, and f is just the blow-up of the surface S. If y0 ∈ S is
+the image of a two-dimensional fiber, then either Y or S are singular at y0; these
+singularities have been described by Andreatta and Wi´sniewski, see Th. 2.1. In
+any case, Y has at most isolated locally factorial and terminal singularities, while
+S can be not normal.
+We denote by N1(X) the real vector space of one-cycles with real coefficients,
+modulo numerical equivalence; we have dim N1(X) = ρX. For any closed subset
+Z ⊂ X, we set
+N1(Z, X) := ι∗(N1(Z)) ⊂ N1(X)
+where ι: Z �→ X is the inclusion, so that N1(Z, X) is the subspace of N1(X)
+spanned by classes of curves in Z, and dim N1(Z, X) ≤ ρZ.
+We study an elementary contraction f : X → Y of type (3, 2) under the hy-
+pothesis that:
+dim N1(E, X) ≥ 4.
+In particular this implies that Y is Fano too (Lemma 2.2).
+We would like to compare (−KY )|S to −KS, but since S may be singular,
+we consider the minimal resolution of singularities µ: S′ → S and set L :=
+µ∗((−KY )|S), a nef and big divisor class on S′. We show that KS′+L is semiample
+(Lemma 3.1). Then our strategy is to look for curves in S′ on which KS′ + L is
+trivial, using other elementary contractions of X of type (3, 2) whose exceptional
+divisor intersects E in a suitable way.
+Hence let us assume that X has another elementary contraction g1 of type (3, 2)
+whose exceptional divisor E1 intersects E, and such that E ·Γ1 = 0 for a curve Γ1
+contracted by g1. Set D := f(E1) ⊂ Y . We show that an irreducible component
+C1 of D ∩ S is a (−1)-curve contained in the smooth locus Sreg, and such that
+−KY · C1 = 1 (Lemma 3.2, see Fig. 3.1 on p. 7). If C′
+1 ⊂ S′ is the transform of
+C1, we have (KS′ + L) · C′
+1 = 0.
+Finally let us assume that X has three elementary contractions g1, g2, g3, all of
+type (3, 2), satisfying the same assumptions as g1 above. We also assume that
+
+FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS
+3
+E1 · Γ2 > 0 and E1 · Γ3 > 0, where E1 = Exc(g1) and Γ2, Γ3 are curves contracted
+by g2, g3 respectively. Then we show that S is a smooth del Pezzo surface with
+−KS = (−KY )|S (Th. 3.6 and Prop. 3.10); let us give an overview of the proof.
+The previous construction yields three distinct (−1)-curves C′
+1, C′
+2, C′
+3 ⊂ S′ such
+that (KS′ + L) · C′
+i = 0 and C′
+1 intersects both C′
+2 and C′
+3. This shows that the
+contraction of S′ given by KS′ +L cannot be birational, namely KS′ +L is not big.
+We also rule out the possibility of a contraction onto a curve, and conclude that
+KS′ + L ≡ 0. Finally we show that ωS ∼= OY (KY )|S, where ωS is the dualizing
+sheaf of S, and conclude that S is smooth and del Pezzo.
+We believe that these results can be useful in the study of Fano 4-folds besides
+their use in the present work. It would be interesting to generalize this technique
+to higher dimensions.
+Let us now explain how we use these results to prove Th. 1.1. We define the
+Lefschetz defect of X as:
+δX := max
+�
+codim N1(D, X) | D ⊂ X a prime divisor
+�
+.
+This invariant, introduced in [Cas12], measures the difference between the Picard
+number of X and that of its prime divisors; we refer the reader to [Cas22b] for a
+survey on δX.
+Fano 4-folds with δX ≥ 3 are classified, as follows.
+Theorem 1.4 ([Cas12], Th. 3.3). Let X be a smooth Fano 4-fold. If δX ≥ 4,
+then X ∼= S1 × S2 where Si are del Pezzo surfaces, and δX = maxi ρSi − 1.
+Theorem 1.5 ([CRS22], Prop. 1.5). Smooth Fano 4-folds with δX = 3 are clas-
+sified. They have 5 ≤ ρX ≤ 8, and if ρX ∈ {7, 8} then X is a product of surfaces.
+Therefore in our study of Fano 4-folds we can assume that δX ≤ 2, that is,
+codim N1(D, X) ≤ 2 for every prime divisor D ⊂ X. To prove that ρX ≤ 12, we
+look for a prime divisor D ⊂ X with dim N1(D, X) ≤ 10.
+To produce such a divisor, we look at contractions of X. If X has an elementary
+contraction of fiber type, or a divisorial elementary contraction f : X → Y with
+dim f(Exc(f)) ≤ 1, it is not difficult to find a prime divisor D ⊂ X such that
+dim N1(D, X) ≤ 3, hence ρX ≤ 5 (Lemmas 2.5 and 2.6).
+The case where X has a small elementary contraction is much harder and is
+treated in [Cas22a], where the following result is proven.
+Theorem 1.6 ([Cas22a], Th. 1.1). Let X be a smooth Fano 4-fold. If X has a
+small elementary contraction, then ρX ≤ 12.
+We are left with the case where every elementary contraction f : X → Y is of
+type (3, 2). In this case we show (Th. 4.1) that, if ρX ≥ 8, we can apply our
+previous study of elementary contractions of type (3, 2), so that if E := Exc(f)
+and S := f(E) ⊂ Y , then S is a smooth del Pezzo surface. This implies that
+dim N1(S, Y ) ≤ ρS ≤ 9, dim N1(E, X) = dim N1(S, Y ) + 1 ≤ 10, and finally that
+ρX ≤ 12, proving Th. 1.1.
+
+4
+C. CASAGRANDE
+The structure of the paper is as follows. In §2 we gather some preliminary
+results.
+Then in §3 we develop our study of elementary contractions of type
+(3, 2), while in §4 we prove Th. 1.1.
+1.1. Notation
+We work over the field of complex numbers. Let X be a projective variety.
+We denote by N1(X) (respectively, N 1(X)) the real vector space of one-cycles
+(respectively, Cartier divisors) with real coefficients, modulo numerical equiva-
+lence; dim N1(X) = dim N 1(X) = ρX is the Picard number of X.
+Let C be a one-cycle of X, and D a Cartier divisor. We denote by [C] (respec-
+tively, [D]) the numerical equivalence class in N1(X) (respectively, N 1(X)). We
+also denote by D⊥ ⊂ N1(X) the orthogonal hyperplane to the class [D].
+The symbol ≡ stands for numerical equivalence (for both one-cycles and divi-
+sors), and ∼ stands for linear equivalence of divisors.
+NE(X) ⊂ N1(X) is the convex cone generated by classes of effective curves,
+and NE(X) is its closure. An extremal ray R is a one-dimensional face of NE(X).
+If D is a Cartier divisor in X, we write D·R > 0, D·R = 0, and so on, if D·γ > 0,
+D · γ = 0, and so on, for a non-zero class γ ∈ R. We say that R is K-negative if
+KX · R < 0.
+Suppose that X has terminal and locally factorial singularities, and is Fano.
+Then NE(X) is a convex polyhedral cone. Given a contraction f : X → Y , we
+denote by NE(f) the convex subcone of NE(X) generated by classes of curves
+contracted by f; we recall that there is a bijection between contractions of X
+and faces of NE(X), given by f �→ NE(f). Moreover dim NE(f) = ρX − ρY , in
+particular f is elementary if and only if NE(f) is an extremal ray.
+When dim X = 4, we say that an extremal ray R is of type (3, 2) if the as-
+sociated elementary contraction f is of type (3, 2), namely if f is divisorial with
+dim f(Exc(f)) = 2. We also set ER := Exc(f) and denote by CR ⊂ ER a general
+fiber of f|ER; note that ER · CR = −1.
+We will also consider the cones Eff(X) ⊂ N 1(X) of classes of effective divisors,
+and mov(X) ⊂ N1(X) of classes of curves moving in a family covering X. Since
+X is Fano, both cones are polyhedral; we have the duality relation Eff(X) =
+mov(X)∨.
+2. Preliminaries
+In this section we gather some preliminary results that will be used in the sequel.
+Andreatta and Wi´sniewski have classified the possible 2-dimensional fibers of
+an elementary contraction of type (3, 2) of a smooth Fano 4-fold. In doing this,
+they also describe precisely the singularities both of the target, and of the image
+of the exceptional divisor, as follows.
+Theorem 2.1 ([AW98], Theorem on p. 256). Let X be a smooth Fano 4-fold and
+f : X → Y an elementary contraction of type (3, 2). Set S := f(Exc(f)).
+
+FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS
+5
+Then f can have at most finitely many 2-dimensional fibers. Outside the images
+of these fibers, Y and S are smooth, and f is the blow-up of S.
+Let y0 ∈ S ⊂ Y be the image of a 2-dimensional fiber; then one of the following
+holds:
+(i) S is smooth at y0, while Y has an ordinary double point at y0, locally factorial
+and terminal;
+(ii) Y is smooth at y0, while S is singular at y0. More precisely either S is not
+normal at y0, or it has a singularity of type 1
+3(1, 1) at y0 (as the cone over
+a twisted cubic).
+In particular the singularities of Y are at most isolated, locally factorial, and
+terminal.
+Now we give some simple preliminary results on extremal rays of type (3, 2).
+Lemma 2.2. Let X be a smooth Fano 4-fold and f : X → Y an elementary
+contraction of type (3, 2); set E := Exc(f). If dim N1(E, X) ≥ 4, then E · R ≥ 0
+for every extremal ray R of X different from NE(f), and Y is Fano.
+Proof. It follows from [Cas17, Lemma 2.16 and Rem. 2.17] that NE(f) is the
+unique extremal ray of X having negative intersection with E, −KX + E =
+f ∗(−KY ) is nef, and (−KX + E)⊥ ∩ NE(X) = NE(f), so that −KY is ample.
+■
+Lemma 2.3. Let X be a smooth Fano 4-fold and R1, R2 extremal rays of X of
+type (3, 2) such that dim N1(ER1, X) ≥ 4 and ER1 · R2 = 0.
+Then ER2 ·R1 = 0 and R1+R2 is a face of NE(X) whose associated contraction
+is birational, with exceptional locus ER1 ∪ ER2.
+Proof. Let H be a nef divisor on X such that H⊥ ∩ NE(X) = R2, and set H′ :=
+H + (H · CR1)ER1. Then H′ · CR1 = H′ · CR2 = 0, and if R3 is an extremal ray
+of NE(X) different from R1 and R2, we have ER1 · R3 ≥ 0 by Lemma 2.2, hence
+H′·R3 > 0. Therefore H′ is nef and (H′)⊥∩NE(X) = R1+R2 is a face of NE(X).
+If Γ ⊂ X is an irreducible curve with [Γ] ∈ R1 + R2, then H′ · Γ = 0, so that
+either ER1 · Γ < 0 and Γ ⊂ ER1, or H · Γ = 0, [Γ] ∈ R2 and Γ ⊂ ER2. This shows
+that the contraction of R1 + R2 is birational with exceptional locus ER1 ∪ ER2.
+Finally we have ER2 · R1 = 0 by [Cas13b, Lemma 2.2(b) and its proof].
+■
+Lemma 2.4. Let X be a smooth Fano 4-fold and R1, R2 distinct extremal rays of
+X of type (3, 2) with dim N1(ERi, X) ≥ 4 for i = 1, 2. If there exists a birational
+contraction g: X → Z with R1, R2 ⊂ NE(g), then ER1 · R2 = ER2 · R1 = 0.
+Proof. We note first of all that ERi · Rj ≥ 0 for i ̸= j by Lemma 2.2. Suppose
+that ER1 · R2 > 0. Then ER1 · (CR1 + CR2) = ER1 · CR2 − 1 ≥ 0. Moreover
+ER2 · R1 > 0 by Lemma 2.3, so that ER2 · (CR1 + CR2) ≥ 0. On the other hand
+for every prime divisor D different from ER1, ER2 we have D · (CR1 + CR2) ≥ 0,
+therefore [CR1 + CR2] ∈ Eff(X)∨ = mov(X). Since [CR1 + CR2] ∈ NE(g), g should
+be of fiber type, a contradiction.
+■
+
+6
+C. CASAGRANDE
+Lemma 2.5. Let X be a smooth Fano 4-fold with δX ≤ 2, and g: X → Z a
+contraction of fiber type. Then ρZ ≤ 4.
+Proof. This follows from [Cas12]; for the reader’s convenience we report the proof.
+If dim Z ≤ 1, then ρZ ≤ 1. If Z is a surface, take any prime divisor D ⊂ X such
+that g(D) ⊊ Z, so that N1(g(D), Z) = {0} if g(D) = {pt}, and N1(g(D), Z) =
+R[g(D)] if g(D) is a curve. Consider the pushforward of one-cycles g∗ : N1(X) →
+N1(Z), and note that dim ker g∗ = ρX−ρZ. We have g∗(N1(D, X)) = N1(g(D), Z)
+and dim N1(g(D), Z) ≤ 1, thus codim N1(D, X) ≥ ρZ − 1, and δX ≤ 2 yields
+ρZ ≤ 3.
+If dim Z = 3, then as in [Cas12, proof of Cor. 1.6] one shows that there exists
+a prime divisor D ⊂ X such that dim N1(g(D), Z) ≤ 2, and reasoning as before
+we get ρZ ≤ 4.
+■
+Lemma 2.6 ([Cas17], Rem. 2.17(1)). Let X be a smooth Fano 4-fold. If X has
+a divisorial elementary contraction not of type (3, 2), then ρX ≤ 5.
+3. Showing that S is a del Pezzo surface
+In this section we study elementary contractions of type (3, 2) of a Fano 4-fold. We
+focus on the surface S which is the image of the exceptional divisor; as explained
+in the Introduction, our goal is to show that under suitable assumptions, S is a
+smooth del Pezzo surface.
+Recall that S has isolated singularities by Th. 2.1.
+Lemma 3.1. Let X be a smooth Fano 4-fold and f : X → Y an elementary
+contraction of type (3, 2). Set E := Exc(f) and S := f(E), and assume that
+dim N1(E, X) ≥ 4.
+Let µ: S′ → S be the minimal resolution of singularities, and set L := µ∗((−KY )|S).
+Then KS′ + L is semiample.
+Proof. Note that −KY is Cartier by Th. 2.1, and ample by Lemma 2.2, so that
+L is nef and big on S′, and for every irreducible curve Γ ⊂ S′, we have L · Γ = 0
+if and only if Γ is µ-exceptional.
+Consider the pushforward of one-cycles f∗ : N1(X) → N1(Y ). Then f∗(N1(E, X)) =
+N1(S, Y ), therefore ρS′ ≥ ρS ≥ dim N1(S, Y ) ≥ 3.
+Let R be a KS′-negative extremal ray of NE(S′). The contraction associated
+to R can be onto a point (if S′ ∼= P2), onto a curve (so that ρS′ = 2), or the
+blow-up of a smooth point (see for instance [Mat02, Th. 1-4-8]). Since ρS′ > 2, R
+is generated by the class of a (−1)-curve Γ, that cannot be µ-exceptional, because
+µ is minimal. Then L · Γ > 0 and (KS′ + L) · Γ = L · Γ − 1 ≥ 0.
+Moreover, if γ ∈ NE(S′)KS′≥0, then (KS′ + L) · γ = KS′ · γ + L · γ ≥ 0.
+By the Cone Theorem, we conclude that KS′+L is nef on S′, and also semiample
+by the Base-Point-Free Theorem.
+■
+Lemma 3.2. Let X be a smooth Fano 4-fold and f : X → Y an elementary
+contraction of type (3, 2). Set E := Exc(f) and S := f(E), and assume that
+
+FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS
+7
+Figure 3.1. The varieties in Lemma 3.2.
+g
+E
+X
+ER1
+T1
+Y
+S
+C1
+D = f(ER1)
+f
+Z
+h(E)
+h(ER1)
+h
+dim N1(E, X) ≥ 4. Let µ: S′ → S be the minimal resolution of singularities, and
+set L := µ∗((−KY )|S).
+Suppose that X has an extremal ray R1 of type (3, 2) such that:
+E · R1 = 0
+and
+E ∩ ER1 ̸= ∅.
+Set D := f(ER1) ⊂ Y .
+Then D|S = C1 + · · · + Cr where Ci are pairwise disjoint (−1)-curves contained
+in Sreg, ER1 = f ∗(D), and f∗(CR1) ≡Y Ci. Moreover if C′
+i ⊂ S′ is the transform
+of Ci, we have (KS′ + L) · C′
+i = 0 for every i = 1, . . . , r.
+Proof. By Lemma 2.3 we have ER1·NE(f) = 0 and NE(f)+R1 is a face of NE(X),
+whose associated contraction h: X → Z is birational with Exc(h) = E ∪ER1. We
+have a diagram (see Fig. 3.1):
+(3.4)
+X
+f
+�
+h
+�
+Y
+g
+� Z
+where g is an elementary, K-negative, divisorial contraction, with Exc(g) = D
+(recall that Y is is locally factorial by Th. 2.1, and Fano by Lemma 2.2).
+Since ER1·NE(f) = E·R1 = 0, both h(E) and h(ER1) are surfaces in Z, and the
+general fiber of h over these surfaces is one-dimensional. Moreover h(E) ∩ h(ER1)
+is finite, and the connected components of E ∩ ER1 are 2-dimensional fibers of h
+over these points.
+Using the classification of the possible 2-dimensional fibers of h in [AW98], as
+in [Cas22a, Lemma 4.15] we see that every connected component Ti of E ∩ ER1
+
+8
+C. CASAGRANDE
+(which is non-empty by assumption) is isomorphic to P1 ×P1 with normal bundle
+O(−1, 0) ⊕ O(0, −1), for i = 1, . . . , r. Set Ci := f(Ti), so that D ∩ S = f(E ∩
+ER1) = f(∪iTi) = ∪iCi. Then Ci ∼= P1, Ci ∩ Cj = ∅ if i ̸= j, and f has fibers of
+dimension one over Ci, therefore Ci ⊂ Sreg and Ci ⊂ Yreg by Th. 2.1.
+Moreover g(D) = h(ER1) is a surface, namely g is of type (3, 2), and Ci is a
+one-dimensional fiber of g contained in Yreg, hence KY · Ci = D · Ci = −1. We
+also have ER1 = f ∗(D) and f∗(CR1) ≡Y Ci.
+Since Ci ⊂ Sreg, it is a Cartier divisor in S, and we can write D|S = m1C1 +
+· · · + mrCr with mi ∈ Z>0 for every i = 1, . . . , r. In S we have Ci · Cj = 0 for
+i ̸= j, hence for i ∈ {1, . . . , r} we get
+−1 = D · Ci = (m1C1 + · · · + mrCr) · Ci = miC2
+i
+and we conclude that mi = 1 and C2
+i = −1, so that Ci is a (−1)-curve in S.
+Finally −KS · Ci = −KY · Ci = 1, hence if C′
+i ⊂ S′ is the transform of Ci, we
+have (KS′ + L) · C′
+i = 0.
+■
+Corollary 3.5. Let X be a smooth Fano 4-fold and f : X → Y an elementary
+contraction of type (3, 2). Set E := Exc(f), and assume that dim N1(E, X) ≥ 4.
+Suppose that X has an extremal ray R1 of type (3, 2) such that E · R1 = 0.
+Then R′
+1 := f∗(R1) is an extremal ray of Y of type (3, 2), and ER1 = f ∗(ER′
+1).
+Proof. If E∩ER1 ̸= ∅, we are in the setting of Lemma 3.2; consider the elementary
+contraction g: Y → Z as in (3.4). Then NE(g) = f∗(R1) = R′
+1 is an extremal ray
+of Y of type (3, 2), and f ∗(ER′
+1) = ER1.
+If E ∩ ER1 = ∅, then we still have a diagram as (3.4), where g is locally
+isomorphic to the contraction of R1 in X, and the statement is clear.
+■
+Theorem 3.6. Let X be a smooth Fano 4-fold and f : X → Y an elementary
+contraction of type (3, 2). Set E := Exc(f) and S := f(E), and assume that
+dim N1(E, X) ≥ 4.
+Suppose that X has two extremal rays R1, R2 of type (3, 2) such that:
+ER1 · R2 > 0 and E · Ri = 0, E ∩ ERi ̸= ∅ for i = 1, 2.
+Then one of the following holds:
+(i) S is a smooth del Pezzo surface and −KS = (−KY )|S;
+(ii) ER1 · CR2 = ER2 · CR1 = 1.
+Proof. We apply Lemma 3.2 to f, R1 and to f, R2. Write f(ER1)|S = C1+· · ·+Cr,
+and let Γ2 be an irreducible component of f(ER2)|S, so that C1, . . . , Cr, Γ2 are
+(−1)-curves contained in Sreg, and Γ2 ≡ f∗(CR2). Then
+(3.7)
+0 < ER1 · CR2 = f ∗(f(ER1)) · CR2 = f(ER1) · Γ2 = (C1 + · · · + Cr) · Γ2,
+hence Ci · Γ2 > 0 for some i, say i = 1.
+Let µ: S′ → S be the minimal resolution of singularities, and set L := µ∗((−KY )|S).
+Moreover let Γ′
+2 and C′
+1 in S′ be the transforms of Γ2 and C1 respectively;
+
+FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS
+9
+then Γ′
+2 and C′
+1 are disjoint from the µ-exceptional locus, are (−1)-curves in
+S′, (KS′ + L) · C′
+1 = (KS′ + L) · Γ′
+2 = 0, and C′
+1 · Γ′
+2 > 0.
+Recall that KS′ + L is semiample by Lemma 3.1. In particular, the face (KS′ +
+L)⊥ ∩ NE(S′) contains the classes of two distinct (−1)-curves which meet. This
+means that the associated contraction cannot be birational, and we have two
+possibilities: either KS′ + L ≡ 0, or the contraction associated to KS′ + L is onto
+a curve. We show that these two cases yield respectively (i) and (ii).
+Suppose first that KS′ + L ≡ 0; in particular −KS′ is nef and big, namely S′ is
+a weak del Pezzo surface.
+Set for simplicity F := OY (KY )|S, invertible sheaf on S, and let ωS be the
+dualizing sheaf of S. We have KS′ ≡ µ∗(F), and since S′ is rational, we also have
+OS′(KS′) ∼= µ∗(F). By restricting to the open subset µ−1(Sreg), we conclude that
+(ωS)|Sreg ∼= F|Sreg. Now we use the following.
+Lemma 3.8. Let S be a reduced and irreducible projective surface with isolated
+singularities, and ωS its dualizing sheaf. If there exists an invertible sheaf F on
+S such that (ωS)|Sreg ∼= F|Sreg, then S is normal and ωS ∼= F.
+This should be well-known to experts, we include a proof for lack of references.
+We postpone the proof of Lemma 3.8 and carry on with the proof of Th. 3.6.
+By Lemma 3.8 we have that S is normal and ωS ∼= F, in particular ωS is
+locally free. If y0 is a singular point of S, then by Th. 2.1 y0 is a singularity of
+type 1
+3(1, 1), but this contradicts the fact that ωS is locally free. We conclude
+that S is smooth, and finally that −KS = (−KY )|S is ample, so that S is a del
+Pezzo surface, and we have (i).
+Assume now that KS′ +L yields a contraction g: S′ → B onto a smooth curve.
+Let F ⊂ S′ be a general fiber F of g, so that −KS′ · F = L · F. Since F is
+not µ-exceptional, we have L · F > 0 and hence −KS′ · F > 0. Thus there is
+a non-empty open subset B0 ⊆ B such that (−KS′)|g−1(B0) is g-ample, therefore
+g|g−1(B0) : g−1(B0) → B0 is a conic bundle, F ∼= P1, and −KS′ · F = 2.
+The curves C′
+1 and Γ′
+2 are components of the same fiber F0 of g, and −KS′ ·F0 =
+2 = −KS′ · (C′
+1 + Γ′
+2). For any irreducible curve C0 contained in F0 we have
+−KS′ · C0 = L · C0 ≥ 0, so that if C0 is different from C′
+1 and Γ′
+2, we must have
+−KS′ · C0 = L · C0 = 0 and C0 is µ-exceptional. Thus C0 ∩ (C′
+1 ∪ Γ′
+2) = ∅, and
+since F0 is connected, we conclude that F0 = C′
+1 + Γ′
+2 and F0 ⊂ g−1(B0), hence
+F0 is isomorphic to a reducible conic.
+This also shows that C′
+i for i > 1 are contained in different fibers of g, so that
+C1 · Γ2 = Γ2 · C1 = 1
+and
+Ci · Γ2 = 0
+for every i = 2, . . . , r,
+and finally using (3.7)
+ER1 · CR2 = (C1 + · · · + Cr) · Γ2 = 1.
+Similarly we conclude that ER2 · CR1 = 1.
+■
+
+10
+C. CASAGRANDE
+Remark 3.9. In the setting of Th. 3.6(i), we cannot conclude that Y is smooth.
+A priori Y could have isolated singularities at some y0 ∈ S; by [AW98] in this
+case f −1(y0) ∼= P2.
+Proof of Lemma 3.8. Recall that S has isolated singularities. The surface S is
+reduced, thus it satisfies condition (S1), namely
+depth OS,y ≥ 1
+for every y ∈ S.
+Then by [Har07, Lemma 1.3] the dualizing sheaf ωS satisfies condition (S2):
+depth ωS,y ≥ 2
+for every y ∈ S,
+where depth ωS,y is the depth of the stalk ωS,y as an OS,y-module.
+Then, for every open subset U ⊂ S such that S ∖ U is finite, we have ωS =
+j∗((ωS)|U), where j : U �→ S is the inclusion, see [Har07, Rem. 1.8].
+This is
+analogous to the properties of reflexive sheaves on normal varieties, see [Har80,
+Propositions 1.3 and 1.6], and can be proved using local cohomology [Gro67].
+Hence we have ωS = j∗((ωS)|Sreg), where j : Sreg �→ S is the inclusion. Since F
+is locally free, we get
+ωS = j∗((ωS)|Sreg) ∼= j∗(F|Sreg) = F,
+in particular ωS is an invertible sheaf and for every y ∈ Y we have ωS,y ∼= OS,y
+as an OS,y-module, thus depth OS,y = 2. Therefore S has property (S2), and it is
+normal by Serre’s criterion.
+■
+Proposition 3.10. Let X be a smooth Fano 4-fold and f : X → Y an elementary
+contraction of type (3, 2). Set E := Exc(f) and S := f(E), and assume that
+dim N1(E, X) �� 4.
+Suppose that X has three distinct extremal rays R1, R2, R3 of type (3, 2) such
+that:
+E · Ri = 0, E ∩ ERi ̸= ∅ for i = 1, 2, 3, and ER1 · Rj > 0 for j = 2, 3.
+Then S is a smooth del Pezzo surface and −KS = (−KY )|S.
+Proof. We apply Th. 3.6 to f, R1, R2 and to f, R1, R3.
+Let us keep the same
+notation as in the proof of Th. 3.6; moreover we denote by Γ3 an irreducible
+component of f(ER3)|S and Γ′
+3 ⊂ S′ its transform. We show that KS′ + L ≡ 0,
+which yields the statement by the proof of Th. 3.6.
+Otherwise, KS′ + L yields a contraction g: S′ → B onto a curve, and F0 =
+C′
+1 + Γ′
+2 is a fiber of g. On the other hand also Γ′
+3 is contained in a fiber of g, it
+is different from C′
+1 and Γ′
+2, and C′
+1 · Γ′
+3 > 0, which is impossible.
+■
+Corollary 3.11. Let X be a smooth Fano 4-fold with δX ≤ 2. Suppose that X
+has four distinct extremal rays R0, R1, R2, R3 of type (3, 2) such that:
+ER0 · Ri = 0 for i = 1, 2, 3, and ER1 · Rj > 0 for j = 2, 3.
+Then one of the following holds:
+(i) dim N1(ERi, X) ≤ 3 for some i ∈ {0, 1, 2, 3}, in particular ρX ≤ 5;
+
+FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS
+11
+(ii) dim N1(ER0, X) ≤ 10, in particular ρX ≤ 12.
+Moreover if f : X → Y is the contraction of R0 and S := f(ER0), then S is
+a smooth del Pezzo surface and −KS = (−KY )|S.
+Proof. We assume that dim N1(ERi, X) ≥ 4 for every i = 0, 1, 2, 3, and prove (ii).
+We show that ER0 ∩ ERi ̸= ∅ for every i = 1, 2, 3.
+If ER0 ∩ ERi = ∅ for
+some i ∈ {1, 2, 3}, then for every curve C ⊂ ER0 we have ERi · C = 0, so that
+[C] ∈ (ERi)⊥, and N1(ER0, X) ⊂ (ERi)⊥.
+Since the classes [ER1], [ER2], [ER3] ∈ N 1(X) generate distinct one dimensional
+faces of Eff(X) (see [Cas13a, Rem. 2.19]), they are linearly independent, hence in
+N1(X) we have
+codim
+�
+(ER1)⊥ ∩ (ER2)⊥ ∩ (ER3)⊥�
+= 3.
+On the other hand codim N1(ER0, X) ≤ δX ≤ 2, thus N1(ER0, X) cannot be
+contained in the above intersection. Then N1(ER0, X) ̸⊂ (ERh)⊥ for some h ∈
+{1, 2, 3}, hence ER0 ∩ ERh ̸= ∅. In particular, since ER0 · Rh = 0, there exists an
+irreducible curve C ⊂ ER0 with [C] ∈ Rh.
+For j = 2, 3 we have ER1 · Rj > 0, and by Lemma 2.3 also ERj · R1 > 0. This
+implies that ER0 ∩ ERi ̸= ∅ for every i = 1, 2, 3. For instance say h = 3: then
+ER1 · R3 > 0 yields ER1 ∩ C ̸= ∅, hence ER0 ∩ ER1 ̸= ∅. Then there exists an
+irreducible curve C′ ⊂ ER0 with [C′] ∈ R1, and ER2 ·R1 > 0 yields ER0 ∩ER2 ̸= ∅.
+Finally we apply Prop. 3.10 to get that S is a smooth del Pezzo surface and
+−KS = (−KY )|S.
+Therefore dim N1(S, Y ) ≤ ρS ≤ 9 and dim N1(ER0, X) =
+dim N1(S, X) + 1 ≤ 10, so we get (ii).
+■
+4. Proof of Th. 1.1
+In this section we show how to apply the results of §3 to bound ρX; the following
+is our main result.
+Theorem 4.1. Let X be a smooth Fano 4-fold with δX ≤ 2 and ρX ≥ 8, and with
+no small elementary contraction.
+Then ρX ≤ δX + 10 ≤ 12. Moreover every elementary contraction f : X → Y
+is of type (3, 2), and S := f(Exc(f)) ⊂ Y is a smooth del Pezzo surface with
+−KS = (−KY )|S.
+In the proof we will use the following terminology: if R1, R2 are distinct one-
+dimensional faces of a convex polyhedral cone C, we say that R1 and R2 are
+adjacent if R1 + R2 is a face of C. A facet of C is a face of codimension one, and
+RC is the linear span of C. We will also need the following elementary fact.
+Lemma 4.2 ([Ewa96], Lemma II.2.6). Let C be a convex polyhedral cone not
+containing non-zero linear subspaces, and R0 a one-dimensional face of C. Let
+R1, . . . , Rm be the one-dimensional faces of C that are adjacent to R0. Then the
+linear span of R0, R1, . . . , Rm is RC.
+
+12
+C. CASAGRANDE
+Proof of Th. 4.1. Let f : X → Y be an elementary contraction; note that ρY =
+ρX − 1 ≥ 7.
+Then f is not of fiber type by Lemma 2.5, and not small by
+assumption, so that f is divisorial. Moreover f is of type (3, 2) by Lemma 2.6.
+Set E := Exc(f) and S := f(E) ⊂ Y ; we have dim N1(E, X) ≥ ρX − δX ≥ 6,
+and if R′ ̸= NE(f) is another extremal ray of X, we have E · R′ ≥ 0 by Lemma
+2.2. Moreover, if R′ is adjacent to NE(f), then E ·R′ = 0. Indeed the contraction
+g: X → Z of the face R′ + NE(f) cannot be of fiber type by Lemma 2.5, thus it
+is birational and we apply Lemma 2.4.
+We are going to show that there exists three extremal rays R′
+1, R′
+2, R′
+3 adjacent
+to NE(f) such that ER′
+1 · R′
+j > 0 for j = 2, 3, and then apply Cor. 3.11.
+Let us consider the cone NE(Y ). It is a convex polyhedral cone whose extremal
+rays R are in bijection with the extremal rays R′ of X adjacent to NE(f), via
+R = f∗(R′), see [Cas08, §2.5].
+By Cor. 3.5, R is still of type (3, 2), and f ∗(ER) = ER′. Thus for every pair
+R1, R2 of distinct extremal rays of Y , with Ri = f∗(R′
+i) for i = 1, 2, we have
+ER1 · R2 = ER′
+1 · R′
+2 ≥ 0.
+If R1 and R2 are adjacent, we show that ER1·R2 = ER2·R1 = 0. Indeed consider
+the contraction Y → Z of the face R1 + R2 and the composition g: X → Z,
+which contracts R′
+1 and R′
+2. Again g cannot be of fiber type by Lemma 2.5, thus
+it is birational and we apply Lemma 2.4 to get ER′
+1 · R′
+2 = ER′
+2 · R′
+1 = 0, thus
+ER1 · R2 = ER2 · R1 = 0.
+Fix an extremal ray R1 of Y . We show that there exist two distinct extremal
+rays R2, R3 of Y with ER1 · Rj > 0 for j = 2, 3.
+Indeed since ER1 is an effective divisor, there exists some curve C ⊂ Y with
+ER1 · C > 0, hence there exists some extremal ray R2 with ER1 · R2 > 0.
+By contradiction, let us assume that ER1 · R = 0 for every extremal ray R of Y
+different from R1, R2. This means that the cone NE(Y ) has the extremal ray R1
+in the halfspace N1(Y )ER1<0, the extremal ray R2 in the halfspace N1(Y )ER1>0,
+and all other extremal rays in the hyperplane (ER1)⊥.
+Fix R ̸= R1, R2, and let τ be a facet of NE(Y ) containing R and not R1. Note
+that Rτ ̸= (ER1)⊥, as ER1 and −ER1 are not nef. By Lemma 4.2 the rays adjacent
+to R in τ cannot be all contained in (ER1)⊥. We conclude that R2 is adjacent to
+R, therefore ER2 · R = 0, namely R ⊂ (ER2)⊥.
+Summing up, we have shown that every extremal ray R ̸= R1, R2 of Y is
+contained in both (ER1)⊥ and (ER2)⊥. On the other hand these rays include all
+the rays adjacent to R1, so by Lemma 4.2 their linear span must be at least a
+hyperplane. Therefore (ER1)⊥ = (ER2)⊥ and the classes [ER1], [ER2] ∈ N 1(Y ) are
+proportional, which is impossible, because they generate distinct one dimensional
+faces of the cone Eff(Y ) (see [Cas13a, Rem. 2.19]).
+We conclude that there exist two distinct extremal rays R2, R3 of Y with ER1 ·
+Rj > 0 for j = 2, 3.
+
+FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS
+13
+For i = 1, 2, 3 we have Ri = f∗(R′
+i) where R′
+i is an extremal ray of X adjacent to
+NE(f), so that E · R′
+i = 0. Moreover for j = 2, 3 we have ER′
+1 · R′
+j = ER1 · Rj > 0.
+We apply Cor. 3.11 to NE(f), R′
+1, R′
+2, R′
+3. We have already excluded (i), and
+(ii) yields the statement.
+■
+We can finally prove the following more detailed version of Th. 1.1.
+Theorem 4.3. Let X be a smooth Fano 4-fold which is not a product of surfaces.
+Then ρX ≤ 12, and if ρX = 12, then there exist X
+ϕ
+��� X′
+g→ Z where ϕ is a
+finite sequence of flips, X′ is smooth, g is a contraction, and dim Z = 3.
+Proof. Since X is not a product of surfaces, we have δX ≤ 3 by Th. 1.4. Moreover
+δX = 3 yields ρX ≤ 6 by Th. 1.5, while δX ≤ 2 yields ρX ≤ 12 by Theorems 1.6
+and 4.1.
+If ρX = 12, the statement follows from [Cas22a, Theorems 2.7 and 9.1].
+■
+References
+[AW98]
+M. Andreatta and J.A. Wi´sniewski, On contractions of smooth varieties, J. Algebraic
+Geom. 7 (1998), 253–312.
+[Cas08]
+C. Casagrande, Quasi-elementary contractions of Fano manifolds, Compos. Math.
+144 (2008), 1429–1460.
+[Cas12]
+,
+On
+the
+Picard
+number
+of
+divisors
+in
+Fano
+manifolds,
+Ann. Sci. ´Ec. Norm. Sup´er. 45 (2012), 363–403.
+[Cas13a]
+, On the birational geometry of Fano 4-folds, Math. Ann. 355 (2013), 585–628.
+[Cas13b]
+, Numerical invariants of Fano 4-folds, Math. Nachr. 286 (2013), 1107–1113.
+[Cas17]
+, Fano 4-folds, flips, and blow-ups of points, J. Algebra 483 (2017), 362–414.
+[Cas22a]
+, Fano 4-folds with a small contraction, Adv. Math. 405 (2022), 1–55, paper
+no. 108492.
+[Cas22b]
+, The Lefschetz defect of Fano varieties, Rend. Circ. Mat. Palermo (2), pub-
+lished online 19 December, 2022.
+[CRS22] C. Casagrande, E.A. Romano, and S.A.Secci, Fano manifolds with Lefschetz defect 3,
+J. Math. Pures Appl. 163 (2022), 625–653, Corrigendum: 168 (2022), 108–109.
+[Ewa96] G. Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Math-
+ematics, vol. 168, Springer-Verlag, 1996.
+[Gro67]
+A. Grothendieck, Local cohomology, Lecture Notes in Math., vol. 41, Springer-Verlag,
+1967.
+[Har80]
+R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176.
+[Har07]
+, Generalized divisors and biliaison, Illinois J. Math. 51 (2007), 83–98.
+[Mat02]
+K. Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, 2002.
+[MM86]
+S. Mori and S. Mukai, Classification of Fano 3-folds with b2 ≥ 2, I, Algebraic and
+Topological Theories – to the memory of Dr. Takehiko Miyata (Kinosaki, 1984), Ki-
+nokuniya, Tokyo, 1986, pp. 496–545.
+Universit`a di Torino, Dipartimento di Matematica, via Carlo Alberto 10, 10123
+Torino - Italy
+Email address: [email protected]
+

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+arXiv:2301.00989v1  [cs.CV]  3 Jan 2023
+STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES
+1
+A New Perspective to Boost Vision
+Transformer for Medical Image Classification
+Yuexiang Li
+[email protected]
+Yawen Huang
+[email protected]
+Nanjun He
+[email protected]
+Kai Ma
+[email protected]
+Yefeng Zheng
+[email protected]
+Tencent Jarvis Lab
+Shenzhen
+China
+Abstract
+Transformer has achieved impressive successes for various computer vision tasks.
+However, most of existing studies require to pretrain the Transformer backbone on a
+large-scale labeled dataset (e.g., ImageNet) for achieving satisfactory performance, which
+is usually unavailable for medical images. Additionally, due to the gap between medical
+and natural images, the improvement generated by the ImageNet pretrained weights sig-
+nificantly degrades while transferring the weights to medical image processing tasks. In
+this paper, we propose Bootstrap Own Latent of Transformer (BOLT), a self-supervised
+learning approach specifically for medical image classification with the Transformer
+backbone. Our BOLT consists of two networks, namely online and target branches,
+for self-supervised representation learning. Concretely, the online network is trained to
+predict the target network representation of the same patch embedding tokens with a dif-
+ferent perturbation. To maximally excavate the impact of Transformer from limited med-
+ical data, we propose an auxiliary difficulty ranking task. The Transformer is enforced
+to identify which branch (i.e., online/target) is processing the more difficult perturbed
+tokens. Overall, the Transformer endeavours itself to distill the transformation-invariant
+features from the perturbed tokens to simultaneously achieve difficulty measurement and
+maintain the consistency of self-supervised representations. The proposed BOLT is eval-
+uated on three medical image processing tasks, i.e., skin lesion classification, knee fatigue
+fracture grading and diabetic retinopathy grading. The experimental results validate the
+superiority of our BOLT for medical image classification, compared to ImageNet pre-
+trained weights and state-of-the-art self-supervised learning approaches.
+1
+Introduction
+Recently, vision Transformer (ViT) [10] and its variants [23, 32, 36] has been introduced
+for various computer vision tasks (e.g., image classification [10, 18], object detection [9,
+© 2022. The copyright of this document resides with its authors.
+It may be distributed unchanged freely in print or electronic forms.
+
+2
+STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES
+41], semantic segmentation [34, 39] and medical image processing [11, 15, 16, 31, 38])
+and gained increasing attentions from the community. The common ViT usually requires
+pretrainig on large-scale natural image datasets, e.g., ImageNet, to achieve the satisfactory
+performance. For natural images, the labels for pretraining dataset can be efficiently obtained
+by crowdsourcing, as even ordinary people possess the ability to effectively identify and label
+objects in natural images. However, the same strategy cannot be adopted for medical images,
+as professional expertise is mandatory for high-quality medical image annotations. Hence,
+the limited amount of annotated medical data is the major obstacle for the improvement of
+diagnosis accuracy even with the powerful vision Transformer.
+Self-supervised learning (SSL) approach is a potential solution to tackle the challenge of
+insufficient annotated data. The typical self-supervised learning formulates a proxy task to
+extract representative features from unlabeled data, which can boost the accuracy of subse-
+quent target task. Existing studies have proposed various proxy tasks, including grayscale
+image colorization [19], patch re-ordering [25], and context restoration [27]. The SSL was
+firstly brought to medical image processing by Zhang et al. [37]. Concretely, the neural
+network was pretrained with a proxy task that sorted the 2D slices from the conventional
+3D medical volumes for the subsequent fine-grained body part recognition. Zhu et al. [40]
+enforced 3D networks to play a Rubik’s cube game for pretraining, which can be seen as an
+extension of 2D Jigsaw puzzles [24]. Contrastive learning [13] has been recently popular-
+ized for self-supervised representation learning. These approaches enforce neural networks
+to spontaneously exploit useful information from pairs of positive and negative samples,
+instead of permuting the contextual information of images for self-supervised signal for-
+mulation. He et al. [14] firstly introduced the idea of contrastive learning into the area
+of self-supervised learning. They proposed an approach, namely MoCo, which addressed
+the problem of large number of negative samples for contrastive learning by maintaining a
+memory bank of negative samples. Following the direction, various contrastive-learning-
+based self-supervised approaches have been proposed [4, 6, 7, 12, 26, 33, 35]. Inspired by
+the success of self-supervised learning for CNNs, researchers began to make their efforts to
+ViT. Atito et al. [1] directly utilized the existing SSL approaches, including rotation pre-
+diction, contrastive learning and image restoration, to pretrain vision Transformers. Several
+studies [2, 3] have been proposed along this direction. However, taking the architecture dif-
+ference between CNN and ViT into account, i.e., CNN takes the whole image as input, while
+the input of ViT is the embedding tokens of image tiles, the self-supervised learning approach
+specifically for ViT is worthwhile to develop.
+In the recent study, Chen et al. [7] proposed MoCo V3 as a token-based constrastive
+learning approach, specifically for ViT to extract self-supervised features from raw data.
+The network pretrained with MoCo V3 outperformed the ImageNet-pretrained one, which
+demonstrated the effectiveness of token-based self-supervised learning. In this paper, we
+follow the direction and propose a token-wise perturbation based self-supervised learning
+framework specifically for medical image classification with vision Transformer, namely
+Bootstrap Own Latent of Transformer (BOLT). Similar to the existing Bootstrap Your Own
+Latent (BYOL) [12], our BOLT consists of two networks, namely online and target branches,
+for self-supervised representation learning. Instead of image-wise transformation adopted by
+BYOL, the online network of our BOLT is trained to predict the target network representa-
+tion of the same patch embedding tokens with a different perturbation. Moreover, to encour-
+age the vision Transformer to deeply exploit useful information from limited medical data,
+we propose an auxiliary difficulty ranking task. The difference between the original patch
+embedding tokens and the perturbed ones is measured as the difficulty (i.e., the larger dif-
+
+STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES
+3
+Linear Projection of Flattened Patches
+Permutation
+Linear Projection
+Split
+Sliding Window
+Token Permutation Module
+x
+Vision Transformer ��
+Vision Transformer ��
+Patch 
+Embedding
+Token  
+Perturbation 
+Module
+Token 
+Perturbation 
+Module
+��
+��
+��
+��
+��(��)
+�� (��)
+Difficulty-awareness Loss
+Exponential Moving Average
+��
+��
+Similarity Loss
+Online
+Target
+Embedded Token ��
+Content perturbed Token ��
+Long Token ��
+Figure 1: The architecture of our BOLT framework. Compared to the original BYOL, our
+BOLT consists of two main revisions: 1) The proposed BOLT generates two views of em-
+bedding tokens for self-supervised learning; 2) A novel difficulty-awareness loss is proposed
+to encourage the ViT to deeply exploit useful information from raw data. sg(.) means stop-
+gradient.
+ference means more difficult for the vision Transformer to process), which is then adopted
+as the supervision signal. In other words, the vision Transformer is required to identify
+which branch (online/target) is processing the more difficult perturbed tokens. Under the
+co-supervision of the two tasks, the vision Transformer is encouraged to endeavour itself to
+distill the transformation-invariantfeatures from the perturbed tokens, which should be capa-
+ble for simultaneous difficulty measurement and maintain the consistency of self-supervised
+representations. In summary, the main contributions of our work can be concluded into
+four-fold:
+• A token perturbation based self-supervised learning approach, namely BOLT, specif-
+ically designed for vision Transformer is proposed. A token perturbation module is
+integrated to the existing BYOL framework for the more effective ViT pretraining.
+• An auxiliary self-supervised task, i.e., difficulty ranking, is proposed to encourage
+ViTs to deeply exploit useful information from limited medical data. The self-supervised
+signal of this auxiliary task also derives from the perturbed tokens generated by our
+perturbation module. To our best knowledge, this is the first SSL framework based on
+the difficulty-awareness paradigm.
+• The proposed BOLT is evaluated on three medical image processing tasks, i.e., skin
+lesion classification, knee fatigue fracture grading and diabetic retinopathy grading.
+The experimental results demonstrate the superiority of our BOLT, compared to the
+widely-used ImageNet pretrained weights.
+• Last but not least, we pretrain the ViT using different self-supervised learning ap-
+proaches on a large-scale private fundus image dataset captured from a collaborating
+hospital for diabetic retinopathy grading task. The dataset consists of 350,000 fundus
+images of normal cohort and patients with various diseases, which may be the largest
+fundus image dataset in the worldwide. The pretraining on our private large-scale
+dataset is verified to benefit the related downstream target task. To advance the de-
+velopment of automated fundus image processing, we will release the ViT pretrained
+models to the community.
+
+4
+STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES
+Linear Projection of Flattened Patches
+1
+2
+3
+4
+5
+6
+7
+8
+9
+6
+1
+5
+7
+3
+9
+8
+2
+4
+Permutation
+6 1 5
+7 3 9
+8 2 4
+Linear Projection
+1
+2
+3
+Split
+6
+1
+5
+7
+3
+9
+8
+2
+4
+Sliding Window
+Token Permutation Module
+Vision Transformer ��
+Vision Transformer ��
+Patch 
+Embedding
+Token  
+Permutation 
+Module
+Token  
+Permutation 
+Module
+��
+��
+��
+��
+�� ��
+�� ��
+Difficulty awareness Loss
+Exponential Moving Average
+��
+��
+Similarity Loss
+Online
+Target
+Embedded Token ��
+Content-perturbed Token ��
+Long Token ��
+Figure 2: The architecture of the proposed token perturbation module. The module consists
+of three operations (i.e., permutation, linear projection and split) to perturb the order and
+content of embedded tokens. Note that nine embedding tokens in this figure are taken as an
+example. The exact number (N) of embedding tokens is decided by HW
+P2 , where H and W are
+the height and width of the original image, respectively, and (P, P) is the size of each image
+patch.
+2
+Method
+In this section, we introduce the proposed BOLT framework in details. The pipeline of our
+Bootstrap Own Latent of Transformer (BOLT) is illustrated in Fig. 1. Similar to BYOL, the
+proposed BOLT adopts two branches to extract useful information from raw data, i.e., the
+online and target branches. The online branch consists of a set of weights θ, including a
+vision Transformer fθ, a projector gθ and a predictor qθ. The target branch is of the same
+architecture with a different set of weights ξ. The target branch generates the regression
+targets for the online branch to learn, and its parameters ξ are an exponential moving average
+of the online branch parameters θ, which can be defined as:
+ξ ← τξ + (1 − τ)θ
+(1)
+where τ ∈ [0,1] is the decay rate.
+Compared to the existing BYOL [12], the proposed BOLT has two differences: First,
+instead of image-based perturbation, we implement a token-based perturbation module for
+the constrastive learning. The underlying reason for the token-based perturbation is that the
+vision Transformer is insensitive to the order of input embedded tokens due to the mechanism
+of self-attention, which neutralizes the effectiveness of typical image-based transformation
+(e.g., Jigsaw puzzle permutation [24]) made to the self-supervised learning of ViT. Inspired
+by recent studies [8, 36], our token perturbation module involves permutation, fusion and
+split operations to simultaneously disarrange the order and content of tokens. Second, since
+the recent study [29] demonstrated the difficulty-awareness can boost the performance of
+CNNs, a difficulty-awareness auxiliary task, i.e., requiring the ViT to identify which branch
+(online/target) is processing the more difficult perturbed tokens, is integrated to the existing
+BYOL framework.
+
+STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES
+5
+2.1
+Token Perturbation Module
+Instead of permuting the image content, we propose a token perturbation module to per-
+turb the order and content of embedded tokens for the self-supervised learning of a vision
+Transformer. The architecture of our token perturbation module is presented in Fig. 2, which
+involves three operations, i.e., permutation, linear projection and split.
+Permutation. Similar to the typical vision Transformer, the input image x ∈ RH×W×C is
+cropped into a sequence of flattened 2D patches xp ∈ RN×(P2C), where H and W are the
+height and width of the original image, respectively, C is the number of channels, (P, P) is
+the size of each image patch, and N = HW
+P2 is the resulting number of patches. Therefore, the
+embedded tokens zo can be written as:
+zo = [x1
+pE;x2
+pE;··· ;xN
+p E],
+(2)
+where E ∈ R(P2C)×D is a trainable linear projection (D is the latent vector size of the vision
+Transformer). Then, the permuted tokens zp are obtained using a permutation operation
+(Perm(·)), which randomly disarranges the order of zo: zp = Perm(zo). Fig. 2 shows an
+example, the order of zo is disarranged to [z6
+o;z1
+o;z5
+o;z7
+o;z3
+o;z9
+o;z8
+o;z2
+o;z4
+o].
+Linear Projection. After the permutation, we concatenate M adjacent tokens using a sliding
+window with a stride S = W
+P , which results in K = N
+S long tokens (z′
+p) with the length of
+M × D. The obtained tokens are then fed to a linear projection layer (Efuse ∈ RMD×SD) for
+information fusion, which yields K content-perturbed long tokens (zl):
+zl = z′
+pEfuse.
+(3)
+Split. As previously mentioned, the typical vision Transformer uses the constant latent vec-
+tor size D through all of its layers; hence, the fused tokens with the length of S×D need to be
+reshaped back to the length of D to fulfill the input requirement of ViT. To achieve that, the
+proposed token perturbation module adopts a split operation to separate each long token into
+S D-length tokens. The splitted tokens (zs) is then fed to ViT for self-supervised learning.
+2.2
+Loss Function
+As shown in Fig. 1, our BOLT is jointly supervised by two loss functions, i.e., similarity
+loss and difficulty-awareness loss. The similarity loss is consistent to the existing BYOL
+framework. Concretely, for a set of embedded tokens zo, our BOLT produces two augmented
+perturbed tokens zt and z′
+t for online and target branches, respectively. The perturbed tokens
+zt are then fed to a ViT fθ, which yields a representation yθ = fθ(zt) and a projection zθ =
+gθ(yθ). For the perturbed tokens for the target branch, a representation yξ = fξ(z′
+t) and a
+projection zξ = gξ(yξ) are accordingly generated. Consistent to BYOL, a prediction network
+qθ(.) is adopted to yield the prediction of zξ and l2-norm is calculated for network training:
+Lθ =
+��qθ(zθ)− zξ
+��2
+2
+(4)
+where θ denotes the network weights of the online branch including fθ, gθ and qθ. The loss
+LBOLT
+θ
+= Lθ + ˜Lθ only optimizes the weights of online branch θ, where ˜Lθ is the symmetric
+loss of Lθ by feeding z′
+t and zt to online and target branches, respectively.
+
+6
+STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES
+Difficulty-awareness Loss. Apart from the similarity loss, inspired by the curriculum learn-
+ing [17], we propose an auxiliary task—identifying which branch is processing the tokens
+with a larger level of perturbation. Such an auxiliary task can drive ViTs to self-adaptively
+pay more attention on the hard case and accordingly better exploit the semantic information
+from the embedded tokens, since they are required to understand the content of tokens for
+the accurate difficulty ranking.
+To formulate the auxiliary task, the self-supervised signal needs to be first generated. As-
+suming the perturbed tokens feeding to online and target branches as zt and z′
+t, respectively,
+the self-supervised signal ysel f can be defined as:
+ysel f =
+�
+0,
+MSE(Perm−1
+zt (zt)−zo) < MSE(Perm−1
+z′t (z′
+t)−zo)
+1,
+MSE(Perm−1
+zt (zt)−zo) ⩾ MSE(Perm−1
+z′t (z′
+t)−zo)
+(5)
+where MSE(.) is the mean squared error function; Perm−1(.) is the inverse permutation
+operation rearranging the perturbed tokens back to the original order.
+After the self-supervision is obtained, the features extracted by the online and target
+ViTs (i.e., yθ and yξ) are concatenated (Cat(.)) and sent to a fully-connected layer (FC(.))
+for difficulty classification. Specifically, the process can be written as:
+LDif f
+fθ
+= −ysel f ∗ log(p)− (1 − ysel f)∗ log(1 − p))
+(6)
+where p = FC(Cat(yθ,yξ))) is the probability of ysel f = 1. Similar to LBOLT
+θ,ξ
+, the difficulty-
+awareness loss only optimizes the online branch (fθ).
+We notice that the recent study [29] has already proposed a difficulty-awareness loss for
+scleral spur localization. Hence, it is worthwhile to emphasize the difference between it and
+our loss function. Concretely, Tao et al. [29] explicitly enforced networks to predict the Dice
+score of input images using segmentation ground truth to achieve difficulty-awareness. Due
+to the lack of manual annotations, few study introduces the idea of difficulty-awareness for
+self-supervised learning (SSL). In this study, we obtain the difficulty-related information in
+a self-supervised manner using the token perturbation module, and implicitly formulate the
+difficulty-ranking proxy task. To our best knowledge, this is the first SSL framework based
+on the difficulty-awareness paradigm.
+Overall Objective. Combining the aforementioned loss functions (LBOLT and LDif f ), the
+full objective L for the optimization of the online branch can be written as:
+L = LBOLT
+θ
++ αLDif f
+fθ
+(7)
+where α = 0.1 is the loss weight of LDif f
+fθ
+. According to Eq. (1), the weights of target branch
+ξ are updated via exponential moving average.
+3
+Experiments
+We evaluate the proposed BOLT on three target tasks, including skin lesion classification,
+knee fatigue grading and diabetic retinopathy grading, using publicly available and private
+datasets. Conventional self-supervised learning approaches often pretrain the models on
+a large-scale unlabeled dataset (i.e., proxy set), and then finetune them on the relatively
+smaller target set. In this paper, three different medical image processing tasks are involved
+
+STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES
+7
+for performance evaluation and the corresponding proxy and target datasets (example images
+are shown in Supplementary Material) for each task are introduced in the followings:
+Skin Lesion Classification. The publicly available ISIC 2019 dataset1 is used to validate
+the effectiveness of the proposed BOLT. Specifically, the dataset [30] is provided by the
+ISIC 2019 challenge, which encourages researchers to develop the automated systems pre-
+dicting eight skin disease categories with dermoscopic images, i.e., squamous cell carci-
+noma, melanocytic nevus, benign keratosis, actinic keratosis, dermatofibroma, basal cell
+carcinoma, vascular lesion, and melanoma. The whole ISIC 2019 dataset, consisting of over
+20,000 dermoscopic images, is adopted as the proxy set. Due to the class imbalance prob-
+lem of original ISIC dataset, consistent to [21], 628 images are randomly sampled from each
+class to establish a balanced target set. It is worthwhile to mention that the images from the
+two classes consisting of fewer than 628 images are all taken into the target set. After that,
+the balanced target set with 4,260 images is randomly separated into training, validation and
+test sets based on the ratio of 70:10:20. Note that the ViT is first pretrained on the proxy set
+and finetuned on the training and validation sets, and then evaluated on the test set.
+Knee Fatigue Grading. The publicly available MURA dataset2 (musculoskeletal radio-
+graphs) [28], which is a large dataset of bone X-rays (over 40,000 images), is adopted as the
+proxy set to pretrain ViTs for the subsequent target task (i.e., knee fatigue grading). For the
+knee fatigue grading, 2,725 X-ray images are collected from a collaborating hospital as the
+target set [20]. The positions of fatigue fracture are different, i.e., navicular bone, tibia and
+fibula. Each X-ray image is labeled by three physicians, and the final grade is decided via
+majority-voting. In particular, the target set has 1,785 normal, 190 grade-1, 452 grade-2, 196
+grade-3 and 102 grade-4 cases, respectively. For the evaluation on our private knee fatigue
+grading dataset, the target set is divided to training, validation and test sets according to the
+ratio of 70:10:20. Similar to [20], due to the imbalance problem (normal vs. fatigue fracture
+and grade-2 vs. other fracture grades), an equal number (20) of test images from each cate-
+gory are randomly sampled to form an uniform-distribution set for performance evaluation,
+instead of using the whole test set.
+Diabetic Retinopathy Grading. For the diabetic retinopathy grading task, we pretrain the
+ViT on a large-scale private dataset captured from a collaborating hospital (proxy set), with
+approval obtained from the institutional review board of the hospital. The dataset consists of
+350,000 fundus images of normal cohort and patients with various diseases. Then, the pre-
+trained ViT is finetuned on the publicly available APTOS 2019 blindness detection dataset
+(target set) for performance evaluation.3 In particular, there are 3,662 fundus images con-
+tained in the target set and the severity of diabetic retinopathy (DR) can be classified to
+four grades, i.e., normal (1,805), mild DR (370), moderate DR (999), severe DR (193) and
+proliferative DR (295). Consistent to [22], a five fold cross-validation is conducted on this
+dataset.4
+Baselines & Evaluation Criterion. To demonstrate the effectiveness of our BOLT pretrain-
+ing, we finetune ViTs with ImageNet pretrained weights on the target tasks and evaluate
+1https://challenge2019.isic-archive.com/
+2https://stanfordmlgroup.github.io/competitions/mura/
+3https://www.kaggle.com/c/aptos2019-blindness-detection
+4The ViT pretrained on our private large-scale dataset may benefit the related downstream target tasks. To
+advance the development of automated fundus image processing, we will release the ViT pretrained models to the
+community soon.
+
+8
+STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES
+Table 1: The classification accuracy (ACC) presented in percentage (%) of ViTs using dif-
+ferent training strategies with different amounts of training data on the ISIC 2019 test set.
+100%
+50%
+10%
+Train-from-scratch
+39.4
+35.2
+31.3
+ImageNet Pretrained
+80.5
+76.1
+62.1
+SimSam [5]
+79.9
+75.9
+61.2
+BYOL [12]
+80.1
+75.4
+61.3
+MoCo V3 [7]
+80.3
+75.2
+61.2
+BOLT w./o. LDi f f
+80.8
+75.8
+62.1
+BOLT (ours)
+81.5
+76.6
+62.4
+ImageNet Pretrained ResNet-50
+75.7
+72.5
+61.2
+their performances on the test set. Consistent to MoCo V3 [7], the basic ViT-B/16 is adopted
+as backbone. The original BYOL [12], state-of-the-art self-supervised learning approach
+SimSam [5] and token-based self-supervised learning approach MoCo V3 [7] are assessed
+for comparison. It is worthwhile to mention that the backbones of representation networks
+of BYOL and SimSam implemented in this study are ViT-B/16. The average classification
+accuracy (ACC) is adopted as metric for the performance evaluation.
+3.1
+Performance Evaluation
+In this section, we evaluate the effectiveness of different training strategies on different
+datasets and present the experimental results. The widely-used ImageNet pretrained ResNet-
+50 is also adopted as a baseline for comparison. Some detailed discussions are presented in
+Supplementary Material.
+Skin Lesion Classification. First, the different training strategies are evaluated on the pub-
+licly available ISIC 2019 dataset. The evaluation results of models finetuned with all train-
+ing data (100%) on the test set are listed in Table 1. The ImageNet pretrained ViT is ob-
+served to surpass the ImageNet pretrained ResNet-50 by a large margin (i.e., +4.8%), which
+demonstrates the superiority of ViT for medical image classification. Compared to the state-
+of-the-art self-supervised learning approaches (i.e., SimSam, BYOL and MoCo V3), our
+token-based BOLT achieves a higher ACC (80.8%). By using the difficulty-awareness loss
+(LDif f ), the ACC of BOLT can be further improved to 81.5%, which outperforms the runner-
+up (MoCo V3) by a margin of +1.2%.
+The goal of self-supervised learning approach primarily is to deal with the insufficient
+training data. Hence, to better verify the superiority of our BOLT approach, we conduct
+an experiment to assess the performance of BOLT pretrained ViTs with different numbers
+of labeled samples used for finetuning (i.e., 10% and 50% in Table 1). It can be observed
+that our BOLT can effectively tackle the situation with few labeled training samples—the
+proposed BOLT with difficulty-awareness loss achieves the best ACC under both 50% and
+10% settings.
+Knee Fatigue Grading. Consistent to the previous study [20], apart from classification ac-
+curacy, the F1 score is also adopted for performance evaluation. The experimental results on
+the uniform test set are listed in Table 2. As shown, the ViT pretrained with the proposed
+BOLT outperforms the ones using existing self-supervised learning approaches and the Ima-
+geNet pretrained weights, i.e., an ACC of 54.0% is achieved (+2.0% higher than the runner-
+
+STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES
+9
+Table 2: The accuracy (ACC and F1 score) presented in percentage (%) of different training
+strategies on knee fatigue grading and diabetic retinopathy grading tasks.
+Knee Fatigue Grading
+Diabetic Retinopathy Grading
+ACC
+F1
+ACC
+F1
+Train-from-scratch
+30.0
+23.1
+71.0
+65.3
+ImageNet Pretrained
+51.0
+49.4
+83.6
+83.2
+SimSam [5]
+52.0
+51.1
+84.5
+84.3
+BYOL [12]
+51.0
+50.2
+84.8
+84.7
+MoCo V3 [7]
+52.0
+51.2
+84.7
+84.3
+BOLT w./o. LDif f
+52.0
+51.2
+85.4
+85.3
+BOLT (ours)
+54.0
+53.6
+85.9
+85.8
+ImageNet Pretrained ResNet-50
+36.0
+31.7
+81.7
+82.0
+up). Similar trend to ISIC 2019 is observed—the ACC of ImageNet pretrained ViT (51%)
+is significantly higher than that of ImageNet pretrained ResNet-50 (36%), demonstrating
+the effectiveness of ViT backbone. We notice that the improvements to train-from-scratch
+yielded by pretraining are more obvious on our knee fatigue grading dataset (over +20%),
+compared to the skin lesion classification task. The reason may be that the target set of knee
+fatigue grading contains less training samples (around 1,000 X-ray images); thus, it is more
+difficult to well train the model from scratch, compared to the skin lesion classification task
+with a target set of 4,260 images.
+Diabetic Retinopathy Grading. Consistent to [22], we split the APTOS 2019 dataset into
+five folds for cross-validation and adopt the F1 score for performance evaluation. The grad-
+ing accuracy of models using different training strategies is shown in Table 2. The proposed
+BOLT pretrained ViT achieves the best ACC (85.9%) and F1 score (85.8%) among the listed
+approaches, which are +1.1% and +1.1% higher than the original BYOL, respectively.
+4
+Conclusion
+In this paper, a self-supervised learning approach, termed Boostrap Own Latent of Trans-
+former (BOLT), was proposed specifically for medical image classification with the vision
+Transformer backbone. The proposed BOLT involved online and target branches, which ex-
+tracted the self-supervised representation from raw data via contrastive learning. Concretely,
+the online network was trained to predict the target network representation of the same patch
+embedding tokens with a different perturbation. Furthermore, we proposed an auxiliary dif-
+ficulty ranking task to enable the vision Transformer to exploit diverse information from the
+limited medical data. The difference between the original patch embedding tokens and the
+perturbed ones was calculated as the difficulty measurement (i.e., the larger difference means
+more difficult for the vision Transformer to process), which was then adopted as the supervi-
+sion signal for self-supervised learning. The vision Transformer was trained to identify the
+branch (online/target) processing for the more difficult perturbed tokens, which enabled it
+to distill the transformation-invariant features from the perturbed tokens. The experimental
+results on three medical image classification tasks (i.e., skin lesion classification, knee fa-
+tigue fracture grading and dabetic retinopathy grading) demonstrated the effectiveness of the
+proposed BOLT. We notice several limitations of this study and plan to address them in the
+future works:
+Extension to Medical Image Segmentation Task. The proposed BOLT can be easily ex-
+
+10
+STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES
+tended to medical image segmentation in a similar way like [40], i.e., pretraining the encoder
+and using a random initialization for the decoder. Yet, the randomly initialized decoder may
+neutralize the performance improvement. Therefore, we plan to explore a more effective
+way extending our pretrained ViTs for medical image segmentation task in the future.
+Pretrained Weights for ViT Variants. Recently, many powerful ViT-based backbones, such
+as Swin Transformer [23], have been proposed. The weights of these ViT variants pretrained
+on our large-scale fundus image dataset will be continuously provided in the future.
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+arXiv:2301.08657v1  [cs.FL]  20 Jan 2023
+Certificates for Probabilistic Pushdown Automata
+via Optimistic Value Iteration
+Tobias Winkler and Joost-Pieter Katoen
+RWTH Aachen University, Germany
+Abstract. Probabilistic pushdown automata (pPDA) are a standard
+model for discrete probabilistic programs with procedures and recur-
+sion. In pPDA, many quantitative properties are characterized as least
+fixpoints of polynomial equation systems. In this paper, we study the
+problem of certifying that these quantities lie within certain bounds.
+To this end, we first characterize the polynomial systems that admit
+easy-to-check certificates for validating bounds on their least fixpoint.
+Second, we present a sound and complete Optimistic Value Iteration al-
+gorithm for computing such certificates. Third, we show how certificates
+for polynomial systems can be transferred to certificates for various quan-
+titative pPDA properties. Experiments demonstrate that our algorithm
+computes succinct certificates for several intricate example programs as
+well as stochastic context-free grammars with > 104 production rules.
+Keywords: Probabilistic Pushdown Automata · Probabilistic Model
+Checking · Certified Algorithms · Probabilistic Recursive Programs.
+1
+Introduction
+Complex software is likely to contain bugs. This applies in particular to model
+checking tools. This is a serious problem, as the possibility of such bugs com-
+promises the trust one can put in the verification results, rendering the process
+of formal modeling and analysis less useful. Ideally, the implementation of a
+model checker should be formally verified itself [15]. However, due to the great
+complexity of these tools, this is often out of reach in practice. Certifying algo-
+rithms [31] mitigate this problem by providing an easy-to-check certificate along
+with their regular output. This means that there exists a verifier that, given the
+input problem, the output, and the certificate, constructs a formal proof that the
+output is indeed correct. The idea is that the verifier is much simpler than the
+algorithm, and thus likely to be bug-free or even amenable to formal verification.
+This paper extends the recent line of research on probabilistic certifica-
+tion [19,23,24,40] to probabilistic pushdown automata [13,30] (pPDA). pPDA and
+related models have applications in, amongst others, pattern recognition [38],
+computational biology [28], and speech recognition [25]. They are moreover a
+natural operational model for programs with procedures, recursion, and (dis-
+crete) probabilistic constructs such as the ability to flip coins. With the advent
+
+2
+Tobias Winkler and Joost-Pieter Katoen
+X → a | XY Y
+x = 1
+2(1 + xy2)
+Y
+→ b | X | Y Y
+y = 1
+3(1 + x + y2)
+0.4
+0.6
+0.8
+1
+0.4
+0.6
+0.8
+1
+≈ (.66, .7)
+(1, 1)
+x
+y
+Fig. 1: Left: A stochastic context-free grammar (SCFG) and the associated pos-
+itive polynomial system (PPS) which encodes the termination probabilities of
+each non-terminal, assuming production rules are taken uniformly at random.
+Right: The curves defined by the two equations. The least fixpoint (lfp) is
+≈ (0.66, 0.70). The thin colored area to the top right of the lfp is the set of
+inductive, i.e., self-certifying upper bounds on the lfp.
+of probabilistic programming [32] as a paradigm for model-based machine learn-
+ing [6], such programs have received lots of attention recently. Moreover, several
+efficient algorithms such as Hoare’s quicksort with randomized pivot selection
+(e.g. [26]) are readily encoded as probabilistic recursive programs.
+A pPDA can be seen as a purely probabilistic variant of a standard pushdown
+automaton: Instead of reading an input word, it takes its transitions randomly
+based on fixed probability distributions over successor states. Quantities of inter-
+est in pPDA include reachability probabilities [13], expected runtimes [8], vari-
+ances [14], satisfaction probabilities of temporal logic formulas [44,41], and others
+(see [7] for an overview). pPDA are equivalent to recursive Markov chains [17].
+One of the difficulties of pPDA is that they induce infinite Markov chains.
+Despite this fact, many interesting quantitative properties are decidable, albeit
+with rather high complexity. Therefore, in the past two decades there have been
+significant research efforts on efficient approximative algorithms for pPDA, espe-
+cially a decomposed variant of Newton iteration [16,27,11,17,12,10,39] which pro-
+vides guaranteed lower, and occasionally upper [10,12] bounds on key quantities.
+However, even though implementations might be complex [43], these algorithms
+do not produce certificates for their results.
+Our technique for certificate generation is an adaption of Optimistic Value
+Iteration [22] (OVI) to the pPDA setting. In a nutshell, OVI computes some
+lower bound ⃗l on the solution—which can be done using an approximative iter-
+ative algorithm—and then optimistically guesses an upper bound ⃗u = ⃗l + ⃗ε and
+verifies that the guess was correct. Originally, OVI was formulated for Markov
+Decision Processes (MDP) where it is used to compute lower and upper bounds
+on minimal or maximal reachability probabilities and expected rewards. The up-
+per bounds computed by OVI have a special property: They are self-certifying
+(also called inductive in this paper). This means that, given the MDP and the
+upper bounds, one can check that the bounds are correct without the need for
+an additional certificate; and this check is conceptually and practically easier
+than finding the bounds in the first place.
+
+Certificates for Probabilistic Pushdown Automata via OVI
+3
+The analysis of pPDA, however, is more involved than that of MDP. In
+MDP, many quantitative properties are characterized as least fixpoints (lfp) of
+piece-wise linear equation systems and can be computed in PTIME via, e.g.,
+LP solving. In pPDA, on the other hand, the equation systems for the same
+properties may contain non-linear polynomials, and the best known complexity
+bounds are usually as high as PSPACE. An example of such a non-linear system
+is illustrated in Figure 1 which shows the translation of a stochastic context-free
+grammar (SCFG; special case of pPDA with a single state) to a polynomial
+equation system encoding termination probabilities. An important observation
+is that the polynomials arising in this context only have positive coefficients.
+Such systems are called positive polynomial systems (PPS) in this paper.
+Applications of PPS beyond the analysis of pPDA include the recent factor
+graph grammars [9] as well as obtaining approximate counting formulas for many
+classes of trees in the framework of analytic combinatorics [18].
+Contributions. In summary, this paper makes the following contributions:
+– We present an optimistic algorithm for computing inductive, self-certifying
+upper bounds of any desired precision ε > 0 on the lfp of a positive poly-
+nomial system. Compared to OVI from [22], the key innovation of our algo-
+rithm is to compute a certain direction ⃗v in which to guess, i.e., the guess is
+⃗u = ⃗l + ε⃗v rather than ⃗u = ⃗l + ⃗ε. This is to ensure that we eventually hit an
+inductive bound, even if the latter lie in a very “thin strip” as in Figure 1.
+– We prove that our algorithm is sound and complete in the sense that if a
+(non-trivial) inductive upper bound exists, then such a bound will be found.
+– We show how inductive bounds on the lfp of PPS can be used to certify
+various quantities of interest in pPDA and SCFG, such as non-termination
+or bounds on expected rewards/costs.
+– We implement our algorithm in the software tool pray and compare the new
+technique to an out-of-the-box approach based on SMT solving.
+Related Work. Certification of pPDA has not been addressed explicitly in the
+literature, but some existing technical results go in this direction. We mention
+[17, Prop. 8.7] which yields certificates for non almost-sure termination of SCFG.
+However, checking such certificates is not straightforward as it requires an SCC
+decomposition. The tool PReMo [43] implements iterative algorithms for lower
+bounds, but it supports neither certificates nor upper bounds.
+Beyond pPDA, OVI was recently generalized to stochastic games [1]. Farkas
+certificates for MDP [19] are verified by checking a set of linear constraints, which
+is in spirit similar to our certificates that requires checking a set of polynomial
+constraints. A deductive approach for verifying probabilistic recursive programs
+on the syntax level was studied in [35]. The same paper also includes inductive
+proof rules for verifying upper bounds just like we do. Recently, a higher-order
+generalization of pPDA called PHORS was introduced in [29], and an algorithm
+for finding upper bounds inspired by the Finite Elements method was proposed.
+
+4
+Tobias Winkler and Joost-Pieter Katoen
+Paper Outline. We review the relevant background information on PPS in Sec-
+tion 2. Section 3 presents our theoretical results on inductive upper bounds in
+PPS as well as the new Optimistic Value Iteration algorithm. In Section 4 we
+explain how inductive bounds in PPS are used to certify quantitative properties
+of pPPA. The experimental evaluation is in Section 5. We conclude in Section 6.
+2
+Preliminaries
+Notation for Vectors. All vectors in this paper are column vectors and are written
+in boldface, e.g., ⃗u = (u1, . . . , un)T . For vectors ⃗u, ⃗u′, we write ⃗u ≤ ⃗u′ if ⃗u is
+component-wise less than or equal to ⃗u′. Moreover, we write ⃗u < ⃗u′ if ⃗u ≤ ⃗u′
+and ⃗u ̸= ⃗u′, and ⃗u ≺ ⃗u′ if ⃗u is component-wise strictly smaller than ⃗u′. The zero
+vector is denoted ⃗0. The max norm of a vector ⃗u is ||⃗u||∞ = max1≤i≤n |ui|. We
+say that ⃗u is normalized if ||⃗u||∞ = 1.
+Positive Polynomial Systems (PPS). Let n ≥ 1 and ⃗x = (x1, . . . , xn)T be a
+vector of variables. An n-dimensional PPS is an equation system of the form
+x1 = f1(x1, . . . , xn)
+. . .
+xn = fn(x1, . . . , xn)
+where for all 1 ≤ i ≤ n, the function fi is a polynomial with non-negative real
+coefficients. An example PPS is the system x = 1
+2(1+xy2), y = 1
+3(1+x+y2) from
+Figure 1. We also use vector notation for PPS: ⃗x = ⃗f(⃗x) = (f1(⃗x), . . . , fn(⃗x))T .
+We write R≥0 = R≥0 ∪ {∞} for the extended non-negative reals. By conven-
+tion, for all a ∈ R≥0, a ≤ ∞, a + ∞ = ∞ + a = ∞, and a · ∞ = ∞ · a equals 0 if
+a = 0 and ∞ otherwise. For n ≥ 1, the partial order (R
+n
+≥0, ≤) is a complete lat-
+tice, i.e., all subsets of R
+n
+≥0 have an infimum and a supremum. In particular, there
+exists a least element ⃗0 and a greatest element ⃗∞ = (∞, . . . , ∞)T . Every PPS
+induces a monotone function ⃗f : R
+n
+≥0 → R
+n
+≥0, i.e., ⃗u ≤ ⃗v =⇒ ⃗f(⃗u) ≤ ⃗f(⃗v). By
+the Knaster-Tarski fixpoint theorem, the set of fixpoints of ⃗f is also a complete
+lattice, and thus there exists a least fixpoint (lfp) denoted by µ⃗f.
+In general, the lfp µ⃗f is a vector which may contain ∞ as an entry. For
+instance, this happens in the PPS x = x+1. A PPS ⃗f is called feasible if µ⃗f ≺ ⃗∞
+(or equivalently, µ⃗f ∈ Rn
+≥0), i.e., the lfp is a vector of real numbers. Besides
+existence of the lfp, the Knaster-Tarski theorem also implies the following:
+Lemma 1 (Inductive upper bounds). For all ⃗u ∈ R
+n
+≥0 it holds that
+⃗f(⃗u) ≤ ⃗u
+implies
+µ⃗f ≤ ⃗u .
+Such a vector ⃗u with ⃗u ≺ ⃗∞ is called inductive upper bound.
+Given a feasible PPS ⃗f, find an inductive upper bound ⃗u ≥ µ⃗f.
+Problem statement of this paper
+
+Certificates for Probabilistic Pushdown Automata via OVI
+5
+If ⃗f is feasible, then µ⃗f is obviously an inductive upper bound. In Section 3
+we show under which conditions there exist more useful inductive upper bounds.
+A PPS is called clean if µ⃗f ≻ ⃗0. Every PPS can be cleaned in linear time by
+identifying and removing the variables that are assigned 0 in the lfp [17,12].
+Given a PPS ⃗f and a point ⃗u ∈ Rn
+≥0, we define the Jacobi matrix of ⃗f at ⃗u
+as the n×n-matrix ∂ ⃗f(⃗u) with coefficients ∂ ⃗f(⃗u)1≤i,j≤n =
+∂
+∂xj fi(⃗u).
+Example 1. Consider the example PPS ⃗fex with variables ⃗x = (x, y)T :
+x = f1(x, y) = y + 0.1
+y = f2(x, y) = 0.2x2 + 0.8xy + 0.1 .
+The line and the hyperbola defined by these equations are depicted in Figure 2
+on Page 7. The fixpoints of ⃗fex are the intersections of these geometric objects;
+in this case there are two. In particular, ⃗fex is feasible and its lfp is
+µ⃗fex =
+�
+(27−
+√
+229)/50 , (22−
+√
+229)/50
+�T ≈ (0.237 , 0.137)T .
+Therefore, ⃗fex is clean as µ⃗fex ≻ ⃗0. The Jacobi matrix of ⃗fex is
+∂ ⃗fex(x, y) =
+�
+∂
+∂xf1
+∂
+∂yf1
+∂
+∂xf2
+∂
+∂yf2
+�
+=
+�
+0
+1
+0.4x + 0.8y 0.8x
+�
+.
+Note that the lfp µ⃗fex contains irrational numbers. However, we can still give ex-
+act expressions for these numbers (involving square roots) because the fixpoints
+of ⃗fex are the zeros of a quadratic polynomial. However, there are PPS whose lfp
+cannot be expressed using radicals, i.e., square roots, cubic roots, etc. [16]. This
+means that in general, there is no easy way to compute least fixpoints exactly.
+It is thus desirable to provide bounds, which we do in this paper.
+△
+Matrices and Eigenvectors. Let M be a real n×n-matrix. We say that M is non-
+negative (in symbols: M ≥ 0) if it has no negative entries. M is called irreducible
+if for all 1 ≤ i, j ≤ n there exists 0 ≤ k < n such that (M k)i,j ̸= 0. It is easy
+to show that M is irreducible iff the directed graph GM = ({1, . . . , n}, E) with
+(i, j) ∈ E iff Mi,j ̸= 0 is strongly connected. A maximal irreducible submatrix
+of M is a square submatrix induced by a strongly connected component of GM.
+The period of a strongly connected M is the length of the shortest cycle in GM.
+It is instructive to note that PPS ⃗x = ⃗f(⃗x) are generalizations of linear equation
+systems of the form ⃗x = M⃗x + ⃗c, with M ≥ 0 and ⃗c ≥ ⃗0. Moreover, note that
+for any PPS ⃗f it holds that ∂ ⃗f(⃗u) ≥ 0 for all ⃗u ≻ ⃗0.
+An eigenvector of an n×n-matrix M with eigenvalue λ ∈ C is a (complex)
+vector ⃗v ̸= ⃗0 satisfying M⃗v = λ⃗v. There are at most n different eigenvalues. The
+spectral radius ρ(M) ∈ R≥0 is the largest absolute value of the eigenvalues of
+M. The following is a fundamental theorem about non-negative matrices:
+Theorem 1 (Perron-Frobenius). Let M ≥ 0 be irreducible.
+
+6
+Tobias Winkler and Joost-Pieter Katoen
+(1) M has a strictly positive eigenvector ⃗v ≻ ⃗0 with eigenvalue ρ(M), the spectral
+radius of M, and all other eigenvectors ⃗v′ ≻ ⃗0 are scalar multiples of ⃗v.
+(2) The eigenvalues of M with absolute value ρ(M) are exactly the h numbers
+ρ(M), ξρ(M), . . . , ξh−1ρ(M), where ξ is a primitive hth root of unity.
+The unique eigenvector ⃗v ≻ ⃗0 with ||⃗v||∞ = 1 of an irreducible non-negative
+matrix M is called the Perron-Frobenius eigenvector of M.
+Strongly Connected Components. To each PPS ⃗f we associate a finite directed
+graph G ⃗f = ({x1, . . . , xn}, E), which, intuitively speaking, captures the depen-
+dency structure among the variables. Formally, (xi, xj) ∈ E if the polynomial fi
+depends on xj, i.e., xj appears in at least one term of fi with a non-zero coef-
+ficient. This is equivalent to saying that the partial derivative
+∂
+∂xj fi is not the
+zero polynomial. We say that ⃗f is strongly connected if G ⃗f is strongly connected,
+i.e., for each pair (xi, xj) of variables, there exists a path from xi to xj in G ⃗f.
+For instance, ⃗fex from Example 1 is strongly connected because the dependency
+graph has the edges E = {(x, y), (y, x), (y, y)}. Strong connectivity of PPS is a
+generalization of irreducibility of matrices; indeed, a matrix M is irreducible iff
+the PPS ⃗x = M⃗x is strongly connected. We often use the fact that ∂ ⃗f(⃗u) for
+⃗u ≻ ⃗0 is irreducible iff ⃗f is strongly connected.
+PPS are usually analyzed in a decomposed fashion by considering the sub-
+systems induced by the strongly connected components (SCCs) of G ⃗f in bottom-
+up order [16]. Here we also follow this approach and therefore focus on strongly
+connected PPS. The following was proved in [17, Lem. 6.5] and later generalized
+in [12, Thm. 4.1] (also see remark below [12, Prop. 5.4] and [17, Lem. 8.2]):
+Theorem 2 ([17,12]). If ⃗f is feasible, strongly connected and clean, then for
+all ⃗u < µ⃗f, we have ρ(∂ ⃗f(⃗u)) < 1. As a consequence, ρ(∂ ⃗f(µ⃗f)) ≤ 1.
+Theorem 2 partitions all PPS ⃗f which satisfy its precondition into two classes:
+Either (1) ρ(∂ ⃗f(µ⃗f)) < 1, or (2) ρ(∂ ⃗f(µ⃗f)) = 1. In the next section we show
+that ⃗f admits non-trivial inductive upper bounds iff it is in class (1).
+Example 2. Reconsider the PPS ⃗fex from Example 1. It can be shown that
+⃗v = (1, λ1)T where λ1 ≈ 0.557 is an eigenvector of ∂ ⃗fex(µ⃗fex) with eigenvalue λ1.
+Thus by the Perron-Frobenius Theorem, ρ(∂ ⃗fex(µ⃗fex)) = λ1 < 1. As promised,
+there exist inductive upper bounds as can be seen in Figure 2.
+△
+3
+Finding Inductive Upper Bounds in PPS
+In this section, we are concerned with the following problem: Given a feasible,
+clean, and strongly connected PPS ⃗f, find a vector ⃗0 ≺ ⃗u ≺ ⃗∞ such that
+⃗f(⃗u) ≤ ⃗u, i.e., an inductive upper bound on the lfp of ⃗f (see Lemma 1).
+
+Certificates for Probabilistic Pushdown Automata via OVI
+7
+0.2
+0.4
+0.6
+0.8
+0.2
+0.4
+0.6
+0.8
+µ⃗fex
+ε
+⃗v
+µ⃗˜fex
+x
+y
+x = y + 0.1
+y = 0.2x2 + 0.8xy + 0.1
+y = 0.2x2 + 0.8xy + 0.1916
+Fig. 2: The PPS ⃗fex corresponds to the solid red line and the solid blue curve. Its
+inductive upper bounds form the shaded area above the lfp µ⃗fex. Lemma 2(4)
+ensures that one can fit the gray “cone” pointing in direction of the Perron-
+Frobenius eigenvector ⃗v inside the inductive region. The PPS ⃗˜fex which com-
+prises the dashed curve and the solid line does not have any non-trivial inductive
+upper bounds. Note that the tangent lines at µ⃗˜fex are parallel to each other.
+3.1
+Existence of Inductive Upper Bounds
+An important first observation is that inductive upper bounds other than the
+exact lfp do not necessarily exist. As a simple counter-example consider the 1-
+dimensional PPS x = 1
+2x2 + 1
+2. If u is an inductive upper bound, then
+1
+2u2 + 1
+2 ≤ u
+=⇒
+u2 − 2u + 1 ≤ 0
+=⇒
+(u − 1)2 ≤ 0
+=⇒
+u = 1 ,
+and thus the only inductive upper bound is the exact lfp u = 1. Another example
+is the PPS ⃗˜fex from Figure 2. What these examples have in common is the
+following property: Their derivative evaluated at the lfp is not invertible. Indeed,
+we have
+∂
+∂x( 1
+2x2 + 1
+2 − x) = x − 1, and inserting the lfp x = 1 yields zero. The
+higher dimensional generalization of this property to arbitrary PPS ⃗f is that the
+Jacobi matrix of the function ⃗f − ⃗x evaluated at µ⃗f is singular; note that this
+is precisely the matrix ∂ ⃗f(µ⃗f) − I. Geometrically, this means that the tangent
+lines at µ⃗f are parallel, as can be seen in Figure 2 for the example PPS ⃗˜fex. It
+should be intuitively clear from the figure that inductive upper bounds only exist
+if the tangent lines are not parallel. The next lemma makes this more precise:
+Lemma 2 (Existence of inductive upper bounds).
+Let ⃗f be a feasible,
+clean, and strongly connected PPS. Then the following are equivalent:
+(1) The matrix I − ∂ ⃗f(µ⃗f) is non-singular.
+(2) The spectral radius of ∂ ⃗f(µ⃗f) satisfies ρ(∂ ⃗f(µ⃗f)) < 1.
+(3) There exists ⃗0 ≺ ⃗u ≺ ⃗∞ s.t. ⃗f(⃗u) < ⃗u (i.e. ⃗u is inductive but not a fixpoint).
+
+8
+Tobias Winkler and Joost-Pieter Katoen
+(4) The matrix ∂ ⃗f(µ⃗f) has a unique (normalized) eigenvector ⃗v ≻ ⃗0 and there
+exist numbers δmax > 0 and ε > 0 s.t.
+⃗f( µ���f + δ · ⃗˜v )
+≺
+µ⃗f + δ · ⃗˜v
+holds for all 0 < δ ≤ δmax and vectors ⃗˜v ≥ ⃗v with ||⃗v − ⃗˜v||∞ ≤ ε.
+The proof of Lemma 2 (see appendix) relies on a linear approximation of
+⃗f via Taylor’s familiar theorem as well as Theorems 1 and 2. Condition (4) of
+Lemma 2 means that there exists a “truncated cone”
+C(µ⃗f,⃗v, ε, δmax) = { µ⃗f + δ⃗˜v | 0 ≤ δ ≤ δmax,⃗˜v ≥ ⃗v, ||⃗˜v − ⃗v||∞ ≤ ε }
+which is entirely contained in the inductive region. This cone is located at the
+lfp µ⃗f and points in the direction of the Perron-Frobenius eigenvector ⃗v, as
+illustrated in Figure 2 (assuming δmax = 1 for simplicity). The length δmax
+and the radius ε of the cone depend quantitatively on ρ(∂ ⃗f(µ⃗f)), but for our
+purposes it suffices that they are non-zero. The idea of our Optimistic Value
+Iteration is to construct a sequence of guesses that eventually hits this cone.
+3.2
+The Optimistic Value Iteration Algorithm
+The basic idea of Optimistic Value Iteration (OVI) can be applied to monotone
+functions of the form ⃗φ: Rn
+≥0 → Rn
+≥0 (in [22], ⃗φ is the Bellman operator of an
+MDP). Kleene’s fixpoint theorem suggests a simple method for approximating
+the lfp µ⃗φ from below: Simply iterate ⃗φ starting at ⃗0, i.e., compute the sequence
+⃗l0 = ⃗0, ⃗l1 = ⃗φ(⃗l0), ⃗l2 = ⃗φ(⃗l1), etc.1 In the context of MDP, this iterative scheme
+is known as Value Iteration (VI). VI is easy to implement, but it is difficult
+to decide when to stop the iteration. In particular, standard stopping criteria
+such as small absolute difference of consecutive approximations are formally un-
+sound [20]. OVI and other algorithms [3,36] cope with this problem by computing
+not only a lower but also an upper bound on µ⃗φ. In the case of OVI, an upper
+bound with absolute error ≤ ε is obtained as follows (we omit some details):
+(1) Compute ⃗lk ≤ µ⃗φ such that ||⃗lk −⃗lk−1||∞ ≤ τ, for some (small) τ > 0.
+(2) Guess a candidate upper bound ⃗u = ⃗lk + ⃗ε.
+(a) If ⃗φ(⃗u) ≤ ⃗u holds, i.e., ⃗u is inductive, then return ⃗u.
+(b) If not, refine ⃗u (see [22] for details). If the refined ⃗u is still not inductive,
+then go back to step (1) and try again with 0 < τ ′ < τ.
+We present our variant of OVI for PPS as Algorithm 1. The main differences
+to the above scheme are that (i) we do not insist on Kleene iteration for obtaining
+the lower bounds ⃗l, and (ii) we approximate the eigenvector ⃗v from condition (4)
+of Lemma 2 and compute the “more informed” guesses ⃗u = ⃗l + ε⃗v, for various ε.
+Refining the guesses as original OVI does is not necessary (but see our remarks
+in Section 3.3 regarding floating point computations).
+1 In order for the Kleene seqence to converge to the lfp, i.e., limk→∞⃗lk = µφ, it suffices
+that ⃗φ is ω-continuous. This already implies monotonicity.
+
+Certificates for Probabilistic Pushdown Automata via OVI
+9
+Algorithm 1: Optimistic Value Iteration (OVI) for PPS
+input
+: strongly connected clean PPS ⃗f; maximum abs. error ε > 0
+output
+: a pair (⃗l, ⃗u) of real vectors s.t. ⃗l ≤ µ⃗f, ⃗f(⃗u) ≤ ⃗u (hence
+µ⃗f ≤ ⃗u), and ||⃗l − ⃗u||∞ ≤ ε
+termination : guaranteed if ⃗f is feasible and I − ∂ ⃗f(µ⃗f) is non-singular
+1 ⃗l ← ⃗0 ; N ← 0 ;
+2 τ ← ε ;
+/* τ is the current tolerance */
+3 while true do
+4
+⃗l′ ← improveLowerBound(⃗f,⃗l) ;
+/* e.g. Kleene or Newton update */
+/* guess and verify phase starts here
+*/
+5
+if ||⃗l − ⃗l′||∞ ≤ τ then
+6
+⃗v ← approxEigenvec(∂ ⃗f(⃗l), τ) ;
+/* recall ⃗v is normalized */
+7
+for k from 0 to N do
+8
+⃗u ← ⃗l + dkε · ⃗v ;
+/* optimistic guess, d ∈ (0, 1) */
+9
+if ⃗f(⃗u) ≤ ⃗u then
+10
+return (⃗l, ⃗u) ;
+/* guess was successful */
+11
+N ← N + 1 ;
+12
+τ ← c · τ ;
+/* decrease tolerance for next guess, c ∈ (0, 1) */
+13
+⃗l ← ⃗l′ ;
+The functions improveLowerBound and approxEigenvec used in Algorithm 1
+must satisfy the following contracts:
+– The sequence ⃗l0 = ⃗0, ⃗li+1 = improveLowerBound(⃗f,⃗li) is a monotonically
+increasing sequence converging to the lfp µ⃗f.
+– approxEigenvec must satisfy the following: Let M ≥ 0 be an irreducible
+matrix with (normalized) Perron-Frobenius eigenvector ⃗v ≻ ⃗0. Then for all
+ε > 0, we require that there exists τ > 0 such that ||approxEigenvec(M, τ)−
+⃗v||∞ ≤ ε. In words, approxEigenvec approximates ⃗v up to arbitrarily small
+absolute error if the tolerance τ is chosen sufficiently small.
+In practice, both the Kleene and the Newton [16,17,12] update operator can
+be used to implement improveLowerBound. We outline a possible implementa-
+tion of approxEigenvec further below in Section 3.3.
+Example 3. Consider the following PPS ⃗f: x = 1
+4x2 + 1
+8, y = 1
+4xy + 1
+4y + 1
+4. The
+table illustrates the execution of Algorithm 1 on ⃗f with ε = 0.1 and c = 0.5:
+# N
+τ
+⃗l
+⃗l′
+||⃗l −⃗l′||∞
+⃗v
+⃗u
+⃗f(⃗u) ≤ ⃗u
+1
+0
+0.1
+(0, 0)
+(0.4, 0.3)
+0.4
+2
+0
+0.1
+(0.4, 0.3)
+(0.5, 0.4)
+0.1
+(1.0, 0.8) (0.5, 0.38)
+✗
+3
+1
+0.05 (0.5, 0.4) (0.55, 0.41)
+0.05
+(1.0, 0.9) (0.6, 0.49)
+✓
+
+10
+Tobias Winkler and Joost-Pieter Katoen
+The algorithm has to improve the lower bound 3 times (corresponding to the
+3 lines of the table). After the second improvement, the difference between the
+current lower bound ⃗l2 and the new bound ⃗l′2 does not exceed the current tol-
+erance τ2 = 0.1 and the algorithm enters the optimistic guessing stage. The first
+guess ⃗u2 is not successful. The tolerance is then decreased to τ3 = c · τ2 = 0.05
+and the lower bound is improved to ⃗l′3. The next guess ⃗u3 is inductive.
+△
+Theorem 3. Algorithm 1 is correct: when invoked with a strongly connected
+clean PPS ⃗f and ε > 0, then (if it terminates) it outputs a pair (⃗l, ⃗u) s.t. ⃗l ≤ µ⃗f,
+⃗f(⃗u) ≤ ⃗u (and thus µ⃗f ≤ ⃗u), and ||⃗l − ⃗u||∞ ≤ ε. Moreover, if ⃗f is feasible and
+I − ∂ ⃗f(µ⃗f) is non-singular, then the algorithm terminates.
+The proof of Theorem 3 (see appendix) crucially relies on condition (4) of
+Lemma 2 that assures the existence of a “truncated cone” of inductive bounds
+centered around the Perron-Frobenius eigenvector of ∂ ⃗f(µ⃗f) (see Figure 2 for
+an illustration). Intuitively, since the lower bounds ⃗l computed by the algorithm
+approach the lfp µ⃗f, the eigenvectors of ∂ ⃗f(⃗l) approach those of ∂ ⃗f(µ⃗f). As a
+consequence, it is guaranteed that the algorithm eventually finds an eigenvector
+that intersects the cone. The inner loop starting on line 7 is needed because the
+“length” of the cone is a priori unknown; the purpose of the loop is to scale the
+eigenvector down so that it is ultimately small enough to fit inside the cone.
+3.3
+Considerations for Implementing OVI
+As mentioned above, there are at least two options for improveLowerBound:
+Kleene or Newton iteration. We now show that approxEigenvec can be effec-
+tively implemented as well. Further below we make some remarks on floating
+point arithmetic.
+Approximating the Eigenvector. A possible implementation of approxEigenvec
+relies on the power iteration method (e.g. [37, Thm. 4.1]). Given a square matrix
+M and an initial vector ⃗v0 with M⃗v0 ̸= ⃗0, power iteration computes the sequence
+(⃗vi)i≥0 such that for i > 0, ⃗vi = M⃗vi−1/||M⃗vi−1||∞.
+Lemma 3. Let M ≥ 0 be irreducible. Then power iteration applied to M + I
+and any ⃗v0 > ⃗0 converges to the Perron-Frobenius eigenvector ⃗v ≻ ⃗0 of M.
+The convergence rate of power iteration is determined by the ratio |λ2|/|λ1|
+where λ1 and λ2 are eigenvalues of largest and second largest absolute value,
+respectively. Each time approxEigenvec is called in Algorithm 1, the result of
+the previous call to approxEigenvec (if available) may be used as an initial
+approximation ⃗v0.
+
+Certificates for Probabilistic Pushdown Automata via OVI
+11
+Exact vs Floating Point Arithmetic. So far we have assumed exact arithmetic
+for the computations in Algorithm 1, but an actual implementation should use
+floating point arithmetic for efficiency. However, this may (and actually does)
+lead to unsound results. More specifically, the condition ⃗f(⃗u) ≤ ⃗u may hold in
+floating point arithmetic even though it is actually violated. As a remedy, we
+propose to nevertheless run the algorithm with floats, but then verify its output ⃗u
+with exact arbitrary-precision rational arithmetic. That is, we compute a rational
+number approximation ⃗uQ of ⃗u and check ⃗f(⃗uQ) ≤ ⃗uQ with exact arithmetic. If
+the check fails, we resort to the following refinement scheme which is an instance
+of the general k-induction principle for complete lattices from [5]: We iteratively
+check the conditions
+⃗f(⃗uQ ⊓ ⃗f(⃗uQ)) ≤ ⃗uQ ,
+⃗f(⃗uQ ⊓ ⃗f(⃗uQ ⊓ ⃗f(⃗uQ))) ≤ ⃗uQ ,
+and so on,
+where ⊓ denotes pointwise minimum. If one of the checks is satisfied, then µ⃗f ≤
+⃗uQ [5]. This scheme often works well in practice (see Section 5). The original
+OVI from [22] uses a similar technique to refine its guesses.
+4
+Certificates for Probabilistic Pushdown Automata
+This section shows how the results from Section 3 can be applied to pPDA.
+We introduce some additional notation. For finite sets A, D(A) denotes the
+set of probability distributions on A. We often denote tuples or triples without
+parentheses and separating commata when this causes no confusion, e.g., we may
+write ab rather than (a, b).
+Definition 1 (pPDA [13]). A probabilistic pushdown automaton (pPDA) is a
+triple ∆ = (Q, Γ, P) where Q ̸= ∅ is a finite set of states, Γ ̸= ∅ is a finite stack
+alphabet, and P : Q × Γ → D(Q × Γ ≤2) is a probabilistic transition function.
+In the following, we often write qZ
+p−→ rα instead of P(qZ)(rα) = p [13]. Intu-
+itively, qZ
+p−→ rα means that if the pPDA is in state q and Z is on top of the
+stack, then with probability p, the pPDA moves to state r, pops Z and pushes α
+on the stack. More formally, the semantics of a pPDA ∆ = (Q, Γ, P) is a count-
+ably infinite Markov chain with state space Q × Γ ∗ and transition probability
+matrix M such that for all q, r ∈ Q, Z ∈ Γ, α ∈ Γ ≤2, γ ∈ Γ ∗, we have
+M(qZγ, rαγ) = P(qZ)(rα) ,
+M(qε, qε) = 1 ,
+and all other transition probabilities are zero. This Markov chain, where the
+initial state is fixed to qZ, is denoted MqZ
+∆ (see Figure 3 for an example). As
+usual, one can formally define a probability measure PqZ
+∆ on the infinite runs of
+MqZ
+∆ via the standard cylinder construction (e.g., [2, Sec. 10]).
+Consider a triple qZr ∈ Q×Γ×Q. We define the return probability2 [qZr] as
+the probability of reaching rε in the Markov chain MqZ
+∆ , i.e., [qZr] = PqZ
+∆ (♦{rε}),
+where ♦{rε} is the set of infinite runs of MqZ
+∆ that eventually hit state rε.
+2 When modeling procedural programs with pPDA, [qZr] is the probability that a
+given procedure returns a specific value to its calling context. These probabilities
+
+12
+Tobias Winkler and Joost-Pieter Katoen
+q
+r
+(1/2, Z, ε)
+(1/4, Z, ZZ)
+(1/4, Z, ε)
+(1, Z, ε)
+qε
+qZ
+qZZ
+· · ·
+rε
+rZ
+rZZ
+· · ·
+1/4
+1/4
+1/2
+1/2
+1/2
+1/4
+1/4
+1/4
+1
+1
+1
+1
+1
+⟨qZq⟩ =
+1/4
+�
+⟨qZq⟩⟨qZq⟩ + ⟨qZr⟩⟨rZq⟩
+�
++ 1/2
+⟨rZq⟩ = 0
+⟨qZr⟩ =
+1/4
+�
+⟨qZq⟩⟨qZr⟩ + ⟨qZr⟩⟨rZr⟩
+�
++ 1/4
+⟨rZr⟩ = 1
+Fig. 3: Top left: The pPDA ∆ex = ({q, r}, {Z}, P) where P comprises the tran-
+sitions qZ
+1/4
+−−→ qZZ, qZ
+1/2
+−−→ qε, qZ
+1/4
+−−→ rε, rZ
+1−→ rε. Top right: A fragment of
+the infinite underlying Markov chain, assuming initial configuration qZ. Bottom:
+The associated equation system from Theorem 4.
+Theorem 4 (The PPS of return probabilities [13]). Let ∆ = (Q, Γ, P) be
+a pPDA and (⟨qZr⟩)qZr ∈ Q×Γ ×Q be variables. For each ⟨qZr⟩, define
+⟨qZr⟩
+=
+�
+qZ
+p−→sY X
+p ·
+�
+t∈Q
+⟨sY t⟩ · ⟨tXr⟩ +
+�
+qZ
+p−→sY
+p · ⟨sY r⟩ +
+�
+qZ
+p−→rε
+p
+and call the resulting PPS ⃗f∆. Then µ⃗f∆ = ([qZr])qZr ∈ Q×Γ ×Q.
+We refer to [30, Sec. 3] for an intuitive explanation of the equations in ⃗f∆.
+Example 4. Figure 3 shows a pPDA ∆ex and the associated PPS ⃗f∆ex. The
+least non-negative solution is ⟨qZq⟩ = 2 −
+√
+2 ≈ 0.586 and ⟨qZr⟩ =
+√
+2 − 1 ≈
+0.414 (and, of course, ⟨rZq⟩ = 0, ⟨rZr⟩ = 1). Thus by Theorem 4, the return
+probabilities are [qZq] = 2 −
+√
+2 and [qZr] =
+√
+2 − 1.
+△
+The PPS ⃗f∆ is always feasible (because µ⃗f∆ ≤ ⃗1). ⃗f∆ is neither necessarily
+strongly connected nor clean. Let ⃗ˆf∆ denote the cleaned up version of ⃗f∆.
+Proposition 1 (Basic Certificates for pPDA).
+A basic certificate for
+∆ = (Q, Γ, P) is a rational inductive upper bound ⃗u ∈ QQ×Γ ×Q
+≥0
+on the lfp of the
+return probabilities system ⃗f∆ (see Thm. 4). They have the following properties:
+– (Existence) ∀ε > 0 there exists a basic certificate ⃗u with ||µ⃗f∆ − ⃗u||∞ ≤ ε if
+all maximal irreducible submatrices M of ∂ ⃗ˆf∆(µ⃗ˆf∆) satisfy ρ(M) < 1.
+were called termination probabilities in previous works [12,7] but we believe this
+term is more appropriate for the numbers [qZ↓] = �
+r[qZr], i.e., the probability to
+eventually reach the empty stack from initial configuration qZ.
+
+Certificates for Probabilistic Pushdown Automata via OVI
+13
+– (Complexity) Let β be the maximum number of bits used to encode any of
+the numerators and denominators of the fractions occurring in ⃗u ∈ QQ×Γ ×Q
+≥0
+.
+Then checking ⃗f∆(⃗u) ≤ ⃗u, i.e., whether ⃗u is basic certificate for ∆, can be
+done in time polynomial in β and the size of ∆.
+Existence of basic certificates follows from Lemma 2 applied to each SCC of
+the cleaned-up version of ⃗f∆ individually. However, note that in order to merely
+check the certificate, i.e., verify the inequality ⃗f(⃗u) ≤ ⃗u, neither do SCCs need
+to be computed nor does the system has to be cleaned up.
+Example 5. Reconsider the example pPDA and its associated (non-strongly con-
+nected) system of return probabilities from Figure 3. We verify that ⃗uqZq = 3/5
+and ⃗uqZr = 1/2 (as well as ⃗urZq = 0, ⃗urZr = 1) is a basic certificate:
+1
+4
+�3
+5 · 3
+5 + 1
+2 · 0
+�
++ 1
+2 = 59
+100
+✓
+≤ 3
+5
+,
+1
+4
+�3
+5 · 1
+2 + 1
+2 · 1
+�
++ 1
+4 = 45
+100
+✓
+≤ 1
+2 .
+Note that [qZq] ≈ 0.586 ≤ 3/5 = 0.6 and [qZr] ≈ 0.414 ≤ 1/2 = 0.5.
+△
+In the following we outline how a variety of key quantities associated to pPDA
+can be verified using basic certificates. More details are in the appendix.
+Upper Bounds on Temporal Properties. We may use basic certificates to verify
+that a bad state rbad is reached with low probability, e.g., at most p = 0.01.
+To this end, we remove the outgoing transitions of rbad and add the transitions
+rbadZ
+1−→ rbadε for all Z ∈ Γ. Clearly, rbad is reached with probability at most p
+from initial configuration qZ iff [qZrbad] ≤ p. The results of [13] imply that this
+idea can be generalized to until-properties of the form C1 U C2, where C1 and C2
+are regular sets of configurations. (This requires a small extension of the basic
+certificates, but the overall idea stays the same).
+Certificates for the Output Distribution. Once a pPDA reaches the empty stack,
+we say that it has terminated. When modeling procedural programs, this cor-
+responds to returning from a program’s main procedure. Assuming initial con-
+figuration qZ, the probability sub-distribution over the possible return values is
+then given by the return probabilities {[qZr] | r ∈ Q}. Missing probability mass
+models the probability of non-termination. A basic certificate can thus be used
+immediately to verify a point-wise upper bound on the output distribution as
+well as to certify that a program is not almost-surely terminating (AST). If a
+pPDA ∆ is already known to be AST, then we can also certify a lower bound on
+the output distribution: Suppose that ⃗u is a basic certificate for ∆ and assume
+that ∆ is AST from initial configuration qZ. Define ε = �
+r∈Q ⃗uqZr − 1. Then
+for all r ∈ Q, we have ⃗uqZr − ε ≤ [qZr] ≤ ⃗uqZr.
+Example 6. The pPDA ∆ex from Figure 3 is AST from initial configuration qZ,
+as the transition qZ
+1/4
+−−→ rε is eventually taken with probability 1, and the stack
+is emptied certainly once r is reached. Using the basic certificate from Example 5
+we can thus (correctly) certify that 0.5 ≤ [qZq] ≤ 0.6 and 0.4 ≤ [qZr] ≤ 0.5.
+
+14
+Tobias Winkler and Joost-Pieter Katoen
+Certificates for Expected Rewards or Costs. Suppose we have equipped a pPDA
+with a state-based reward (or cost) function Q → R≥0. It was shown in [14] that
+the expected total reward accumulated during the run of a pPDA is the solution
+of a linear equation system where the return probabilities [qZr] appear as coef-
+ficients. Given a basic certificate ⃗u, we can replace each coefficient [qZr] by ⃗uqZr
+and thus obtain an equation system whose solution is an over-approximation of
+the true expected reward. We may extend the basic certificate ⃗u by the solution
+of this linear system to make verification straightforward. Note that a program’s
+expected runtime [8,35] is a special case of total expected reward.
+5
+Implementation and Experiments
+Our Tool: pray. We implemented our algorithm in the prototypical Java-tool
+pray (Probabilistic Recursion AnalYzer). It supports two input formats: (i)
+Recursive probabilistic programs in a Java-like syntax (e.g. Figure 4); these
+programs are automatically translated to pPDA. (ii) Explicit PPS in the same
+syntax used by the tool PReMo [43]. The output of pray is a rational inductive
+upper bound on the lfp of the return probability PPS of the input program’s
+pPDA model (a basic certificate), or on the lfp of the explicitly given PPS. The
+absolute precision ε is configurable. The implementation works as follows:
+(1) It parses the input and, if the latter was a program, constructs a pPDA
+model and the associated PPS of return probabilities.
+(2) It computes an SCC decomposition of the PPS under consideration using
+standard algorithms implemented in the jGraphT library [33].
+(3) It applies Algorithm 1 to the individual SCC in reverse topological order
+using floating point arithmetic. Algorithm 1 is instantiated with Kleene it-
+eration3, the power iteration for approximating eigenvectors as outlined in
+Section 3.3, and constants c = 0.1, d = 0.5. We allow ≤ 10 guesses per SCC.
+(4) If stage (3) is successful, the tool verifies the resulting floating point certifi-
+cate using exact rational number arithmetic as described in Section 3.3.
+Baselines. To the best of our knowledge, no alternative techniques for finding
+inductive upper bounds in PPS have been described explicitly in the literature.
+However, there is an (almost) out-of-the-box approach using an SMT solver:
+Given a PPS ⃗x = ⃗f(⃗x), compute some lower bound ⃗l ≤ µ⃗f using an iterative
+technique. Then query the SMT solver for a model (variable assignment) of the
+quantifier-free first-order logic formula ϕ⃗f(⃗x) = �n
+i=1 fi(⃗x) ≤ xi ∧⃗li ≤ xi ≤ ⃗li +ε
+in the (decidable) theory of polynomial real arithmetic with inequality (aka
+QF_NRA in the SMT community). If such a model ⃗u exists, then clearly µ⃗f ≤ ⃗u
+and ||⃗l − ⃗u||∞ ≤ ε. If no model exists, then improve ⃗l and try again. We have
+3 In fact, we use the slightly optimized Gauss-Seidel iteration (see [42, Sec. 5.2]) which
+provides a good trade-off between ease of implementation and efficiency [42].
+
+Certificates for Probabilistic Pushdown Automata via OVI
+15
+bool and() {
+prob {
+1//2: return
+(1//2: true | 1//2: false);
+1//2: {
+if(!or()) return false;
+else return or(); } } }
+bool or() {
+prob {
+1//2: return
+(1//2: true | 1//2: false);
+1//2: {
+if(and()) return true;
+else return and(); } } }
+Fig. 4: Program evaluating a random and-or tree [8]. The prob-blocks execute
+the contained statements with the respective probabilities (syntax inspired by
+Java’s switch). Our tool automatically translates this program to a pPDA and
+computes a basic certificate (Proposition 1) witnessing that calling and() returns
+true and false with probability ≤ 382/657 ≈ 0.58 and 391/933 ≈ 0.42, resp.
+implemented this approach using the state-of-the-art SMT solvers cvc5 [4] and
+z3 [34], the winners of the 2022 SMT-COMP in the category QF_NRA4.
+As yet another baseline, we have also implemented a variant of OVI for PPS
+which is closer to the original MDP algorithm from [22]. In this variant, called
+“standard OVI” from now on, we compute the candidate ⃗u based on the “relative”
+update rule ⃗u = (1 + ε)⃗l, where ⃗l is the current lower bound [22].
+Research Questions. We aim to shed some light on the following questions: (A)
+How well does our algorithm scale? (B) Is the algorithm suitable for PPS with dif-
+ferent characteristics, e.g., dense or sparse? (C) Is the requirement ρ(∂ ⃗f(µ⃗f)) < 1
+restrictive in practice? (D) How does our OVI compare to the baselines?
+Benchmarks. To answer the above questions we run our implementation on two
+sets of benchmarks (Table 3 and Table 2, respectively). The first set consists of
+various example programs from the literature as well as a few new programs,
+which are automatically translated to pPDA. This translation is standard and
+usually takes not more than a few seconds. The programs golden, and-or (see Fig-
+ure 4), virus, gen-fun are adapted from [35,8,41] and [32, Program 5.6], respec-
+tively. The source code of all considered programs is in the appendix. We have
+selected only programs with possibly unbounded recursion depth which induce
+infinite Markov chains. The second benchmark set comprises explicitly given
+PPS5. The instances brown, lemonde, negra, swbd, tiger, tuebadz, and wsj all en-
+code SCFG from the area of language processing (see [43] for details). random is
+the return probability system of a randomly generated pPDA.
+Summary of Experimental Results. We ran the experiments on a standard note-
+book. The approach based on cvc5 turns out to be not competitive (see Ap-
+pendix D). We thus focus on z3 in the following. Both pray and the z3 approach
+could handle most of the programs from Table 3 within a 10 minute time limit.
+The considered programs induce sparse PPS with 38 - 26,367 variables, and most
+4 https://smt-comp.github.io/2022/results
+5 These examples come with PReMo: https://cgi.csc.liv.ac.uk/~dominik/premo/
+
+16
+Tobias Winkler and Joost-Pieter Katoen
+Table 1: Experiments with PPS obtained from recursive probabilistic programs.
+Columns vars and terms display the number of variables and terms in the PPS.
+Columns sccs and sccmax indicate the number of non-trivial SCC and the size of
+the largest SCC. G is total number of guesses made by OVI (at least one guess per
+SCC). ttot is the total runtime excluding the time for model construction. tQ is
+the percentage of ttot spent on exact rational arithmetic. D is the average number
+of decimal digits of the rational numbers in the certificate. The timeout (TO)
+was set to 10 minutes. Timings are in ms. The absolute precision is ε = 10−3.
+benchmark
+|Q|
+|P|
+|Γ|
+vars terms sccs sccmax cert G D
+tQ
+ttot certz3 Dz3
+tz3 certstd Gstd Dstd
+tstd
+rw-0.499
+18
+29
+5
+38
+45
+1
+12
+✓
+5 5 17%
+163
+✓
+2
+11
+✓
+4
+5
+59
+rw-0.500
+18
+29
+5
+38
+45
+1
+12
+✗
+10
+-
+-
+7327
+✓
+2
+10
+✗
+10
+-
+8083
+rw-0.501
+18
+29
+5
+38
+45
+1
+12
+✓
+5 4
+6%
+36
+✓
+13
+12
+✓
+4
+5
+23
+geom-offspring
+24
+40
+5
+52
+80
+4
+24
+✓
+8 6 13%
+15
+✓
+9
+16
+✓
+8
+6
+14
+golden
+27
+49
+6
+81
+94
+1
+36
+✓
+1 5 30%
+10
+✓
+7
+14
+✓
+2
+4
+12
+and-or
+50
+90
+7
+149
+182
+1
+48
+✓
+2 4 26%
+19
+✓
+12
+15260
+✓
+2
+4
+19
+gen-fun
+85
+219
+7
+202
+327
+1
+16
+✓
+2 3 32%
+22
+✓
+15
+141
+✓
+2
+3
+21
+virus
+68
+149
+27
+341
+551
+1
+220
+✓
+1 5 38%
+40
+✓
+7
+139
+✓
+1
+6
+59
+escape10
+109
+174
+23
+220
+263
+1
+122
+✓
+1 4
+5%
+56
+✓
+7
+48
+✓
+1
+8
+71
+escape25
+258
+413
+53
+518
+621
+1
+300
+✓
+1 5 17%
+245
+✓
+7
+15958
+✓
+1
+9
+172
+escape50
+508
+813
+103
+1018
+1221
+1
+600
+✓
+1 7 23%
+653
+✓
+7
+410
+✗
+1
+-
+400
+escape75
+760 1215
+153
+1522
+1825
+1
+904
+✓
+2 9 10%
+3803
+✗
+-
+TO
+✗
+1
+-
+635
+escape100
+1009 1614
+203
+2020
+2423
+1
+1202
+✗
+5
+-
+-
+29027
+✓
+6
+939
+✗
+1
+-
+901
+escape200
+2008 3213
+403
+4018
+4821
+1
+2400
+✗
+6
+-
+-
+83781
+✗
+-
+TO
+✗
+1
+-
+2206
+sequential5
+230
+490
+39
+1017
+1200
+10
+12
+✓
+15
+4 26%
+103
+✓
+8
+1074
+✓
+15
+5
+204
+sequential7
+572 1354
+137
+3349
+3856
+14
+12
+✓
+21
+5 27%
+1049
+✓
+8
+12822
+✓
+20
+5
+1042
+sequential10
+3341 8666 1036 26367 29616
+20
+12
+✓
+30
+5
+2% 100613
+✓
+8 453718
+✓
+30
+6 101554
+mod5
+44
+103
+10
+296
+425
+1
+86
+✓
+1 5 39%
+28
+✓
+9
+34150
+✗
+2
+-
+178
+mod7
+64
+159
+14
+680
+1017
+1
+222
+✓
+1 6 69%
+172
+✓
+7
+443
+✗
+2
+-
+624
+mod10
+95
+244
+20
+1574
+2403
+1
+557
+✗
+1
+-
+-
+675
+✓
+7
+1245
+✗
+2
+-
+882
+of them have just a single SCC. Notably, the examples with greatest maximum
+SCC size were only solved by z3. pray and z3 need at most 95 and 31 seconds,
+respectively, for the instances where they succeed. In many cases (e.g., rw-5.01,
+golden, virus, brown, swbd), the resulting certificates formally disprove AST. For
+the explicit PPS in Table 2, pray solves all instances whereas z3 only solves
+3/8 within the time limit, and only finds the trivial solution ⃗1. Most of these
+benchmarks contain dense high-degree polynomials and our tool spends most
+time on performing exact arithmetic. pray never needs more than 6 guesses per
+SCC if it succeeds.
+Evaluation of Research Questions. (A) Scalability: Our algorithm succeeded on
+instances with maximum SCC size of up to 8,000 and number of terms over
+50,000. pray could solve all instances with a maximum SCC size of ≤ 1,000 in
+less than 2 minutes per instance. For the examples where our algorithm does
+not succeed (e.g., escape100) it is mostly because it fails converting a floating
+point to a rational certificate. (B) PPS with different flavors: The problems
+in Table 3 (low degree and sparse, i.e., few terms per polynomials) and Table 2
+(higher degree and dense) are quite different. A comparison to the SMT approach
+suggests that our technique might be especially well suited for dense problems
+with higher degrees. (C) Non-singularity: The only instance where our algorithm
+fails because of the non-singularity condition is the symmetric random walk rw-
+
+Certificates for Probabilistic Pushdown Automata via OVI
+17
+Table 2: Experiments with explicitly given PPS (setup as in Table 3).
+benchmark
+vars terms sccs sccmax cert G D
+tQ
+ttot certz3 Dz3
+tz3 certstd Gstd Dstd
+tstd
+brown
+37 22866
+1
+22
+✓
+2
+6 74%
+3212
+✗
+-
+TO
+✓
+2
+8
+9065
+lemonde
+121 32885
+1
+48
+✓
+2
+5 97% 40738
+✗
+-
+TO
+✓
+2
+5 38107
+negra
+256 29297
+1
+149
+✓
+2
+7 89% 10174
+✓
+1 37248
+✓
+1
+7
+8873
+swbd
+309 47578
+1
+243
+✓
+1
+7 93% 18989
+✗
+-
+TO
+✓
+1
+8 67314
+tiger
+318 52184
+1
+214
+✓
+2
+8 98% 94490
+✓
+1 17454
+✓
+1
+8 90801
+tuebadz
+196
+8932
+2
+168
+✓
+4
+9 85%
+2666
+✓
+1 15323
+✓
+3
+9
+2700
+wsj
+240 31170
+1
+194
+✓
+2
+9 96% 30275
+✗
+-
+TO
+✓
+2
+9 29038
+random
+10000 20129
+1
+8072
+✓
+3
+7
+5% 17585
+✗
+-
+TO
+✓
+4
+8 16357
+0.500. We therefore conjecture that this condition is often satisfied in practice.
+(D) Comparison to SMT: There is no clear winner. Some instances can only be
+solved by one tool or the other (e.g. escape100 and brown). However, pray often
+delivers more succinct certificates, i.e., the rational numbers have less digits.
+Overall, z3 behaves less predictably than pray.
+6
+Conclusion and Future Work
+We have proposed using inductive bounds as certificates for various properties in
+probabilistic recursive models. Moreoever, we have presented the first dedicated
+algorithm for computing inductive upper bounds. While our algorithm already
+scales to non-trivial problems, the main bottleneck is the generation of an exact
+rational bound from a floating point approximation. This might be improved
+using appropriate rounding modes as in [21]. Additional future work includes
+further certificates for pPDA, especially for lower bounds and termination.
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+19
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+
+20
+Tobias Winkler and Joost-Pieter Katoen
+A
+Full Proofs
+A.1
+Proof of Lemma 2
+Lemma 2 (Existence of inductive upper bounds).
+Let ⃗f be a feasible,
+clean, and strongly connected PPS. Then the following are equivalent:
+(1) The matrix I − ∂ ⃗f(µ⃗f) is non-singular.
+(2) The spectral radius of ∂ ⃗f(µ⃗f) satisfies ρ(∂ ⃗f(µ⃗f)) < 1.
+(3) There exists ⃗0 ≺ ⃗u ≺ ⃗∞ s.t. ⃗f(⃗u) < ⃗u (i.e. ⃗u is inductive but not a fixpoint).
+(4) The matrix ∂ ⃗f(µ⃗f) has a unique (normalized) eigenvector ⃗v ≻ ⃗0 and there
+exist numbers δmax > 0 and ε > 0 s.t.
+⃗f( µ⃗f + δ · ⃗˜v )
+≺
+µ⃗f + δ · ⃗˜v
+holds for all 0 < δ ≤ δmax and vectors ⃗˜v ≥ ⃗v with ||⃗v − ⃗˜v||∞ ≤ ε.
+We now explain the proof of Lemma 2. The proof heavily relies on a linear
+approximation of ⃗f around the lfp µ⃗f. Intuitively, this is where the Jacobi matrix
+∂ ⃗f(µ⃗f) comes into play. This is formalized via Taylor’s familiar theorem.
+Lemma 4 (Taylor’s Theorem; cf. [12, Lem. 2.3]). Let ⃗f be a feasible PPS.
+Then for all vectors ⃗u ≥ ⃗0, we have
+⃗f(µ⃗f + ⃗u)
+=
+µ⃗f + ∂ ⃗f(µ⃗f)⃗u + R⃗u⃗u
+where R⃗u is a matrix that depends on ⃗u such that lim⃗u→⃗0 R⃗u = 0. More specifi-
+cally, it holds that ⃗0 ≤ R⃗u⃗u ≤
+�
+∂ ⃗f(µ⃗f + ⃗u) − ∂ ⃗f(µ⃗f)
+�
+⃗u.
+Proof (Proof of Lemma 2). “(1) =⇒ (2)”: By Theorem 2 we have ρ(∂ ⃗f(µ⃗f)) ≤
+1. Towards contradiction assume that ρ(∂ ⃗f(µ⃗f)) = 1. By the Perron-Frobenius
+Theorem, 1 is an eigenvalue of ∂ ⃗f(µ⃗f), which means that there exists ⃗u ̸= ⃗0 such
+that ∂ ⃗f(µ⃗f)⃗u = ⃗u. This ⃗u is in the kernel of I − ∂ ⃗f(µ⃗f), which contradicts the
+assumption that I − ∂ ⃗f(µ⃗f) is non-singular.
+“(2) =⇒ (1)”: It is a well-known result that for an arbitrary real matrix M
+the series �∞
+k=0 M k converges iff ρ(M) < 1. The limit of the series is the inverse
+of I − M because
+(I − M)
+∞
+�
+k=0
+M =
+∞
+�
+k=0
+M k −
+∞
+�
+k=1
+M k = M 0 = I .
+“(2)
+=⇒
+(4)”: Let ρ(∂ ⃗f(µ⃗f)) =: λ < 1. By the Perron-Frobenius Theo-
+rem, the Jacobi matrix ∂ ⃗f(µ⃗f) has a unique normalized eigenvector ⃗v ≻ ⃗0 wrt.
+eigenvalue λ:
+∂ ⃗f(µ⃗f)⃗v = λ⃗v ≺ ⃗v .
+(1)
+
+Certificates for Probabilistic Pushdown Automata via OVI
+21
+Our goal is to define the values ε and δmax whose existence we claimed in
+Lemma 2(4). Let cmin > 0 be the smallest component of (1−λ)⃗v ≻ ⃗0. We define
+ε :=
+cmin
+3||∂ ⃗f(µ⃗f)||∞
+,
+(2)
+where ||∂ ⃗f(µ⃗f)||∞ = max||⃗y||∞=1 ||∂ ⃗f(µ⃗f)⃗y||∞ is the maximum row sum of
+∂ ⃗f(µ⃗f). Note that || · ||∞ is the operator norm induced by the maximum norm.
+Then it holds for all ⃗ε with ||⃗ε||∞ ≤ ε that
+||∂ ⃗f(µ⃗f)⃗ε||∞ ≤ ||∂ ⃗f(µ⃗f)||∞||⃗ε||∞ ≤ ||∂ ⃗f(µ⃗f)||∞
+cmin
+3||∂ ⃗f(µ⃗f)||∞
+= 1
+3cmin .
+(3)
+The first inequality in (3) is a property of operator norms (which is straightfor-
+ward in the case of the maximum norm). Since cmin was the smallest component
+of (1 − λ)⃗v, (3) implies
+∂ ⃗f(µ⃗f)⃗ε ≤ 1
+3(1 − λ)⃗v .
+(4)
+We now define δmax as follows:
+δmax := sup {δ > 0 | ∀⃗ε ≥ ⃗0 s.t. ||⃗ε||∞ ≤ ε: Rδ(⃗v+⃗ε)(⃗v + ⃗ε) ≤ 1
+2(1 − λ)⃗v} ,
+(5)
+where Rδ(⃗v+⃗ε) is the matrix from Lemma 4 which satisfies
+⃗f(µ⃗f + δ(⃗v + ⃗ε)) = µ⃗f + δ∂ ⃗f(µ⃗f)(⃗v + ⃗ε) + δRδ(⃗v+⃗ε)(⃗v + ⃗ε) .
+We now argue that δmax > 0. This is not immediately obvious because of the
+∀-quantification in (5). Let δ > 0 be arbitrary. Further, let ⃗ε ≥ ⃗0 be such that
+||⃗ε||∞ ≤ ε. In the following, we write ⃗ε′ = (ε . . . ε). We have
+Rδ(⃗v+⃗ε)(⃗v + ⃗ε)
+= 1
+δ Rδ(���v+⃗ε)δ(⃗v + ⃗ε)
+≤ 1
+δ
+�
+∂ ⃗f(µ⃗f + δ(⃗v + ⃗ε)) − ∂ ⃗f(µ⃗f)
+�
+δ(⃗v + ⃗ε)
+(Lemma 4)
+=
+�
+∂ ⃗f(µ⃗f + δ(⃗v + ⃗ε)) − ∂ ⃗f(µ⃗f)
+�
+(⃗v + ⃗ε)
+≤
+�
+∂ ⃗f(µ⃗f + δ(⃗v + ⃗ε′)) − ∂ ⃗f(µ⃗f)
+�
+(⃗v + ⃗ε′)
+(Jacobi matrix is monotonic)
+=: Mδ(⃗v + ⃗ε′)
+Note that Mδ does not depend on ⃗ε and limδ→0 Mδ = 0. We can therefore find a
+specific δ∗ > 0 such that Mδ∗(⃗v+⃗ε′) ≤ 1
+2(1−λ)⃗v. On the other hand, we have just
+
+22
+Tobias Winkler and Joost-Pieter Katoen
+shown for all ⃗ε ≥ ⃗0 with ||⃗ε||∞ ≤ ε and all δ > 0 that Rδ(⃗v+⃗ε)(⃗v+⃗ε) ≤ Mδ(⃗v+⃗ε′).
+So we have in particular for all ⃗ε ≥ ⃗0 with ||⃗ε||∞ ≤ ε that
+Rδ∗(⃗v+⃗ε)(⃗v + ⃗ε) ≤ Mδ∗(⃗v + ⃗ε′) ≤ 1
+2(1 − λ)⃗v .
+Hence δmax ≥ δ∗ > 0.
+Finally, let 0 < δ ≤ δmax and ⃗˜v ≥ ⃗v with ||⃗v − ⃗˜v||∞ ≤ ε, i.e., ⃗˜v = ⃗v + ⃗ε for
+some ⃗ε ≥ ⃗0 with ||⃗ε||∞ ≤ ε. Then
+⃗f(µ⃗f + δ(⃗v + ⃗ε))
+= µ⃗f + δ∂ ⃗f(µ⃗f)(⃗v + ⃗ε) + δRδ(⃗v+⃗ε)(⃗v + ⃗ε)
+(by Taylor’s Theorem (Lemma 4))
+= µ⃗f + δλ⃗v + δ∂ ⃗f(µ⃗f)⃗ε + δRδ(⃗v+⃗ε)(⃗v + ⃗ε)
+(by (1))
+≤ µ⃗f + δλ⃗v + δ 1
+3(1 − λ)⃗v + δRδ(⃗v+⃗ε)(⃗v + ⃗ε)
+(by (4))
+≤ µ⃗f + δλ⃗v + δ 1
+3(1 − λ)⃗v + δ 1
+2(1 − λ)⃗v
+(by (5))
+≺ µ⃗f + δλ⃗v + δ 1
+2(1 − λ)⃗v + δ 1
+2(1 − λ)⃗v
+(because δ(1 − λ)⃗v ≻ ⃗0)
+= µ⃗f + δ⃗v
+(simplification)
+≤ µ⃗f + δ(⃗v + ⃗ε)
+(because ⃗ε ≥ ⃗0)
+“(4) =⇒ (3)”: Trivial.
+“(3) =⇒ (2)”: By (3) there exists ⃗u such that ⃗f(⃗u) < ⃗u. By Lemma 1 this
+implies that µ⃗f < ⃗u, so we can write ⃗u = µ⃗f + ⃗v for some ⃗v > ⃗0.
+Using Taylor’s Theorem (Lemma 4), it follows that
+⃗f(µ⃗f + ⃗v) = µ⃗f + ∂ ⃗f(µ⃗f)⃗v + R⃗v⃗v < µ⃗f + ⃗v .
+(6)
+Using that R⃗v⃗v ≥ ⃗0, (6) implies that
+∂ ⃗f(µ⃗f)⃗v < ⃗v .
+(7)
+The claim now follows by applying the following lemma to the matrix ∂ ⃗f(µ⃗f)
+and the vector ⃗v:
+Lemma 5. Let M ≥ 0 be an irreducible n× n-matrix. If there exists ⃗u > ⃗0 such
+that M⃗u < ⃗u, then ⃗u ≻ ⃗0, M n⃗u ≺ ⃗u and ρ(M) < 1.
+Proof. First observe that since multiplication by M is monotone we have for all
+0 ≤ k1 ≤ k2 that
+⃗0 ≤ M k2⃗u ≤ M k1⃗u ≤ ⃗u .
+We first show that ⃗u ≻ ⃗0, which is essentially [12, Lemma 5.3]. Since ⃗u > ⃗0,
+there must be 1 ≤ i ≤ n such that ⃗ui > 0. Now let 1 ≤ j ≤ n be arbitrary. Since
+
+Certificates for Probabilistic Pushdown Automata via OVI
+23
+M is irreducible there exists 0 ≤ k < n such that M k
+j,i > 0. This implies that
+(M k⃗u)j > 0. By monotonicty, ⃗u ≥ M k⃗u, and thus ⃗uj ≥ (M k⃗u)j > 0. Since j
+was arbitrary, ⃗u ≻ ⃗0.
+Next we show M n⃗u ≺ ⃗u. Since M⃗u < ⃗u holds by assumption, there exists
+1 ≤ i ≤ n such that (M⃗u)i < ⃗ui. Let 1 ≤ j ≤ n be a arbitrary. Since M is
+irreducible, there exists 0 ≤ k < n such that (M k)j,i > 0. We now show that
+(M n⃗u)j < uj which implies that M n⃗u ≺ ⃗u as j was chosen arbitrarily:
+(M n⃗u)j
+≤ (M kM⃗u)j
+(by monotonicity, and because k + 1 ≤ n)
+= (M k)j,i(M⃗u)i +
+�
+l̸=i
+(M k)j,l(M⃗u)l
+(Def. matrix-vector product)
+< (M k)j,i⃗ui +
+�
+l̸=i
+(M k)j,l(M⃗u)l
+(because (M⃗u)i < ⃗ui and (M k)j,i > 0)
+≤ (M k)j,i⃗ui +
+�
+l̸=i
+(M k)j,l⃗ul
+(because (M⃗u)l ≤ ⃗ul)
+= (M k⃗u)j ≤ ⃗uj
+It remains to show that ρ(M) < 1. We do this by showing that the powers
+of M (i.e., the sequence (M k)k≥0) converge to the zero matrix. Since M n⃗u ≺ ⃗u,
+we can choose c < 1 such that M n⃗u ≤ c⃗u. Then for all m ≥ 1 it holds that
+M nm⃗u ≤ cm⃗u, so we have
+lim
+k→∞ M k⃗u = ⃗0 .
+Recall from above that we already know ⃗u ≻ ⃗0. Thus limk→∞ M k⃗u = ⃗0 means
+that a positive linear combination of the entries of each individual row of M k
+converges to zero, i.e., for all 1 ≤ i ≤ n we have limk→∞
+�
+j M k
+i,j⃗uj = 0, and
+thus for all 1 ≤ j ≤ n, limk→∞ M k
+i,j = 0. Thus limk→∞ M k = 0, which completes
+the proof.
+⊓⊔
+A.2
+Proof of Theorem 3
+Theorem 3. Algorithm 1 is correct: when invoked with a strongly connected
+clean PPS ⃗f and ε > 0, then (if it terminates) it outputs a pair (⃗l, ⃗u) s.t. ⃗l ≤ µ⃗f,
+⃗f(⃗u) ≤ ⃗u (and thus µ⃗f ≤ ⃗u), and ||⃗l − ⃗u||∞ ≤ ε. Moreover, if ⃗f is feasible and
+I − ∂ ⃗f(µ⃗f) is non-singular, then the algorithm terminates.
+Proof. Correctness is obvious, so we only show termination assuming that ⃗f is
+feasible and I − ∂ ⃗f(µ⃗f) is non-singular. Clearly, the algorithm terminates iff it
+eventually finds a ⃗u in line 8 which is inductive.
+Assume towards contradiction that the algorithm never terminates, i.e., it
+never finds an inductive ⃗u. For all i ≥ 1 let ⃗li, ⃗vi, τi be the values of the variables
+⃗l, ⃗v and τ at the ith time the inner loop at line 7 is reached (note that we
+then have N = i − 1). Clearly, limi→∞ τi = 0. By the contract satisfied by
+
+24
+Tobias Winkler and Joost-Pieter Katoen
+improveLowerBound, we have limi→∞ ∂ ⃗f(⃗li) = ∂ ⃗f(µ⃗f). Since the eigenvectors
+of ∂ ⃗f(µ⃗f) depend continuously on those of the matrices ∂ ⃗f(⃗li), and because of
+the contract satisfied by approxEigenvec, the sequence ⃗v1,⃗v2, . . . converges to
+the true unique normalized Perron-Frobenius eigenvector ⃗vtrue of ∂ ⃗f(µ⃗f).
+We now apply condition (4) of Lemma 2. The condition ensures that the cone
+C(µ⃗f,⃗vtrue, ε′, δmax) = { µ⃗f + δ⃗˜v | 0 ≤ δ ≤ δmax, ||⃗˜v − ⃗vtrue||∞ ≤ ε′ }
+which is located at µ⃗f, points in direction ⃗vtrue and has radius ε′ and length
+δmax contains only inductive points. For the sake of illustration suppose that the
+algorithm already knows δmax and computes ⃗ui = ⃗li +δ⃗vi for some 0 < δ < δmax
+instead of executing the loop starting at line 7. But then the sequence (⃗ui)i≥1
+converges to µ⃗f + δ⃗vtrue, which is a point that lies inside the interior of C, so
+there must be some i ≥ 1 such that ⃗ui ∈ C, i.e., ⃗ui is inductive.
+The remaining difficulty is that δmax is of course unknown in practice. We
+handle this using the inner loop that starts at line 7. Eventually, the variable
+N is sufficiently large such that dkε < δmax for some k ≤ N. Termination then
+follows by applying the argument in the previous paragraph to δ = dkε.
+⊓⊔
+A.3
+Proof of Lemma 3
+Lemma 3. Let M ≥ 0 be irreducible. Then power iteration applied to M + I
+and any ⃗v0 > ⃗0 converges to the Perron-Frobenius eigenvector ⃗v ≻ ⃗0 of M.
+Proof. Consider the following conditions for an irreducible matrix M ≥ 0 and a
+vector M⃗v0 with M⃗v0 ̸= ⃗0:
+1. M has a unique dominant eigenvalue |λ1| > |λ2| ≥ . . . ≥ |λn|.
+2. λ1 is semisimple, i.e., its algebraic multiplicity6 equals its geometric multi-
+plicity7.
+3. ⃗v0 is not orthogonal to the eigenspace {⃗v | M⃗v = λ1⃗v}.
+It is known that if all these conditions are satisfied, then the power iteration
+sequence (⃗vi)i∈N converges to a (normalized) eigenvector ⃗v with eigenvalue λ1
+(e.g. [37, Theorem 4.1]).
+We now show that these conditions are satisfied for the irreducible matrix
+M + I ≥ 0 and every initial vector ⃗v0 > ⃗0. The eigenvectors of M and M + I
+are exactly the same but the eigenvalues are all shifted by +1. Indeed, if ⃗v is
+some eigenvector of M with eigenvalue λ, then (M + I)⃗v = λ⃗v + ⃗v = (λ + 1)⃗v.
+However, unlike M, the matrix M +I always has period 1, and so it has a unique
+dominant eigenvalue λ1 by Theorem 1(2). Therefore the first of the above three
+conditions is satisfied by the matrix M + I.
+6 The algebraic multiplicity is the multiplicity of a given eigenvalue as a root of the
+characteristic polynomial.
+7 The geometric multiplicity is the dimension of the eigenspace associated with a
+particular eigenvalue.
+
+Certificates for Probabilistic Pushdown Automata via OVI
+25
+Next, by Theorem 1(1) it holds that the geometric multiplicity of λ1 is 1. As
+the algebraic multiplicity is bounded by the geometric multiplicity, it must also
+be 1 and thus the matrix M + I satisfies the second condition as well.
+Finally, the third condition is satisfied for any ⃗v0 > ⃗0 because the scalar
+product ⃗v0 · ⃗v is non-zero (either strictly positive or strictly negative) for all
+non-zero eigenvectors ⃗v of λ1 by Theorem 1(1).
+⊓⊔
+A.4
+Proof of Proposition 1
+Proposition 1 (Basic Certificates for pPDA). A basic certificate for ∆ =
+(Q, Γ, P) is a rational inductive upper bound ⃗u ∈ QQ×Γ ×Q
+≥0
+on the lfp of the
+return probabilities system ⃗f∆ (see Thm. 4). They have the following properties:
+– (Existence) ∀ε > 0 there exists a basic certificate ⃗u with ||µ⃗f∆ − ⃗u||∞ ≤ ε if
+all maximal irreducible submatrices M of ∂ ⃗ˆf∆(µ⃗ˆf∆) satisfy ρ(M) < 1.
+– (Complexity) Let β be the maximum number of bits used to encode any of
+the numerators and denominators of the fractions occurring in ⃗u ∈ QQ×Γ ×Q
+≥0
+.
+Then checking ⃗f∆(⃗u) ≤ ⃗u, i.e., whether ⃗u is basic certificate for ∆, can be
+done in time polynomial in β and the size of ∆.
+Proof. This proof closely follows the general idea of decomposed analysis of
+PPS [16].
+We first address existence. Note that ⃗f∆ is guaranteed to be feasible, in fact
+⃗0 ≤ µ⃗f∆ ≤ ⃗1. For all qZr with (µ⃗f∆)qZr = 0 we set ⃗uqZr = 0. By removing
+these variables from ⃗f∆ we obtain the clean PPS ⃗ˆf∆ with ⃗0 ≺ µ⃗ˆf∆.
+Now consider the decomposition of ⃗ˆf∆ into the subsystems induced by the
+strongly connected components of the graph G ⃗ˆf∆: ⃗ˆf 1
+∆, . . . , ⃗ˆf m
+∆ . Note that in these
+subsystems, some variables might only appear on the right hand sides but not on
+the left (e.g. x1 = 0.5x1+0.5x2, x2 = 0.5x1+0.5x3). Since µ⃗ˆf∆ ≻ ⃗0, there is a 1 - 1
+correspondence of these subsystems and the maximal irreducible submatrices Mi
+of ∂ ⃗ˆf∆(µ⃗ˆf∆). More specifically, Mi = ∂ ⃗ˆf i
+∆(µ⃗ˆf∆)8. By assumption, ρ(Mi) < 19.
+Now assume w.l.o.g. that ⃗ˆf 1
+∆ is a bottom SCC (i.e., in the dependency graph
+G ⃗ˆ
+f∆ there is no path from the variables in ⃗ˆf 1
+∆ to any variable not in ⃗ˆf 1
+∆). Then
+⃗ˆf 1
+∆ is a strongly connected PPS with ∂ ⃗ˆf 1
+∆(µ⃗ˆf∆) = ∂ ⃗ˆf 1
+∆(µ⃗ˆf 1
+∆) and we can apply
+Lemma 2(4) to obtain a rational ⃗u1 with ⃗ˆf 1
+∆(⃗u1) ≤ ⃗u1 and ||µ⃗ˆf 1
+∆ − ⃗u1||∞ ≤ ε (in
+fact, we can do this for any ε > 0).
+Suppose we have done the above for all bottom SCCs and now start traversing
+the DAG of SCCs bottom-up, i.e., in reverse topological order. Let ⃗u be the
+8 The Jacobi matrix of a sub-PPS with n′ < n equations is an n′ × n′ matrix where
+all variables that occur only on the right hand sides are considered constants.
+9 The spectral radius of the zero matrix is zero.
+
+26
+Tobias Winkler and Joost-Pieter Katoen
+bound we have constructed to far (i.e., ⃗u contains ⃗u1 and the bounds from
+the other bottom SCC as subvectors and is zero elsewhere). Note that we can
+always make ⃗u smaller while retaining the inductivity property. W.l.o.g. suppose
+that subsystem ⃗ˆf 2
+∆ is one of the first non-bottom SCCs in the reverse topological
+order. The idea is now to modify ⃗ˆf 2
+∆ to a strongly connected PPS ˜⃗f 2
+⃗u by replacing
+all variables that occur only in right hand sides by their value in ⃗u. Clearly,
+lim⃗u→µ⃗ˆ
+f∆ ∂ ˜⃗f 2
+⃗u(µ ˜⃗f 2
+⃗u) = ∂ ⃗ˆf 2
+∆(µ⃗ˆf∆). This means we can choose ⃗u sufficiently close
+to µ⃗ˆf∆ such that the spectral radius of ∂ ˜⃗f 2
+⃗u(µ ˜⃗f 2
+⃗u) is strictly smaller than 1. We
+can then apply Lemma 2(4) to ˜⃗f 2
+⃗u to obtain a rational ⃗u2 with ˜⃗f 2
+⃗u(⃗u2) ≤ ⃗u2 to
+enlarge our current ⃗u with.
+We can repeat this scheme for all finitely many subsystems until we have
+constructed a rational ⃗u with ˜⃗f i
+⃗u(⃗u) ≤ ⃗u for all i. Clearly, this ⃗u also satisfies
+⃗ˆf∆(⃗u) ≤ ⃗u. Finally, we may extend ⃗u by zero entries corresponding to the vari-
+ables that are assigned zero in the lfp of the (not necessarily clean) ⃗f∆. This
+yields an inductive upper bound for ⃗f∆. We stress that in order to verify this
+bound, we neither have to clean ⃗f∆ nor do we have to compute the SCCs.
+For complexity observe that ⃗f∆ is cubic in the size of ∆ and that all polyno-
+mials in ⃗f∆ have degree at most 2. Since multiplication and addition of rational
+numbers can be done in polynomial time in the number of their bits, evaluat-
+ing a polynomial of fixed maximum degree can also be done in polynomial time
+in the size of the polynomial and the number of bits representing the rationals
+where the polynomial is to be evaluated. Note that this is not true for arbitrary
+polynomials where exponents are encoded in binary: For instance, evaluating the
+polynomial x2n (which can be represented with O(n) bits) at x = 2 yields 22n,
+a number that needs O(2n) bits. This means that in order to verify certificates
+efficiently with exact rational arithmetic, it is important that the polynomials in
+the PPS do not have very high degrees. Fortunately, this is the case for pPDA.
+
+Certificates for Probabilistic Pushdown Automata via OVI
+27
+B
+Certificates for Expected Rewards
+We can certify upper bounds on the expected value of rewards collected during
+the run of a pPDA. To simplify the presentation, in this section we assume
+w.l.o.g. that qZ
+p−→ rα with p > 0 implies |α| ∈ {0, 2}, i.e., all transitions either
+decrease or increase the stack height by 1. Let R: Q → R≥0 be a state-based
+reward function. Consider the following PPS ⃗f∆,R with variables {⟨EqZr⟩ | qZr ∈
+Q × Γ × Q}:
+⟨EqZr⟩ =
+�
+qZ
+p−→sY X
+p ·
+�
+t∈Q
+[sY t] · [tXr] · KqZ,sY X +
+�
+qZ
+p−→rε
+p · R(r) ,
+where KqZ,sY X = R(r) + ⟨EsY t⟩ + ⟨EtXr⟩. Note that ⃗f∆,R is linear but uses
+the return probabilities which are themselves characterized as the lfp of the
+non-linear system ⃗f R
+∆ from Theorem 4 as coefficients.
+Suppose that in the lfp µ⃗f∆,R, each variable EqZr is assigned the quantity
+EqZr ∈ R≥0. It follows from the results of [14] that EqZr equals the expected
+value of the following random variable V r
+R under the probability measure PqZ
+∆ :
+V r
+R(q0γ0, q1γ1, . . .) =
+firstHit(rε)
+�
+i>0
+R(qi)
+where firstHit(rε) is the minimum integer k such that qkγk = rε, or 0 if no such
+k exists. In words, EqZr is the expected reward accumulated on the runs from
+qZ to rε, where it is assumed that runs which never reach rε contribute zero
+reward. Consequently, E(qZ) = �
+r∈Q EqZr is the expected reward accumulated
+on all terminating runs.
+Example 7. Setting R = 1 we can characterize the expected runtime of pPDA.
+Reconsider Example 4. The equation system for expected runtimes becomes
+⟨EqZq⟩ =1
+4([qZq]2(1+2⟨EqZq⟩) + [qZr][rZq](1+⟨EqZr⟩+⟨ErZq⟩)) + 1
+2
+⟨EqZr⟩ =1
+4([qZq][qZr](1+⟨EqZq⟩+⟨EqZr⟩)+[qZr][rZr](1+⟨EqZr⟩+⟨ErZr⟩)) + 1
+4
+as well as ⟨ErZq⟩ = 0 and ⟨ErZr⟩ = 1. The solution is ⟨EqZq⟩ = 2063/2624 ≈
+0.786 and ⟨EqZr⟩ = 59/82 ≈ 0.712, so the total expected runtime is E(qZ) ≈
+1.506.
+△
+C
+Benchmark Programs
+
+28
+Tobias Winkler and Joost-Pieter Katoen
+void f() {
+if flip(p) {
+f();
+f();
+}
+}
+# main block
+{
+f();
+}
+(a) rw-p
+void f() {
+if flip(1//2) {
+f();
+f();
+f();
+}
+}
+# main block
+{
+f();
+}
+(b) golden
+void offspring() {
+while flip(2//5) {
+offspring();
+while flip(3//5) {
+offspring();
+}
+}
+}
+# main block
+{
+offspring();
+}
+(c) geom-offspring
+void gen_operator() {
+uniform(4);
+}
+void gen_expression() {
+prob {
+4//10: uniform(10);
+3//10: { }
+3//10: {
+gen_operator();
+gen_expression();
+gen_expression();
+}
+}
+}
+void gen_function() {
+gen_operator();
+gen_expression();
+gen_expression();
+}
+# main block
+{
+gen_function();
+}
+(d) gun-fun
+void young() {
+int y = uniform(4);
+while(y > 0) {
+young();
+y = y-1;
+}
+int e = uniform(3);
+while(e > 0) {
+elder();
+e = e-1;
+}
+}
+void elder() {
+int y = uniform(2);
+while(y > 0) {
+young();
+y = y-1;
+}
+int e = uniform(5);
+while(e > 0) {
+elder();
+e = e-1;
+}
+}
+# main block
+{
+young();
+}
+(e) virus
+bool f() {
+prob {
+1//2:
+return flip(1//2);
+1//2:
+if f()
+{
+return f();
+} else {
+return false;
+}
+}
+}
+# main blcok
+{
+bool res1 = f();
+...
+bool resN = f();
+}
+(f) sequentialN
+
+Certificates for Probabilistic Pushdown Automata via OVI
+29
+int f(int n, int m) {
+prob {
+(n+1)//(n+2) : {
+f((n + 1) % m, m);
+f((n + 1) % m, m);
+return 0;
+}
+1//(n+2) :
+return 0;
+}
+}
+# main block
+{
+f(0, N);
+}
+(a) escapeN
+void f(int n) {
+while(n > 0) {
+prob {
+2//3: f(n-1);
+1//3: f((n+1) % N);
+}
+n = n-1;
+}
+}
+# main block
+{
+f(1);
+}
+(b) modN
+
+30
+Tobias Winkler and Joost-Pieter Katoen
+D
+Z3 vs CVC5
+Table 3: Comparison of the SMT-approach (see §Baselines in Section 5) using z3
+and cvc5 on SCFG given as explicit PPS (right), and on programs automatically
+translated to pPDA (left).
+benchmark
+certz3
+tz3
+certcvc5
+tcvc5
+rw-0.499
+✓
+11
+✓
+92
+rw-0.500
+✓
+10
+✓
+87
+rw-0.501
+✓
+12
+✓
+104
+geom-offspring
+✓
+16
+✓
+4687
+golden
+✓
+14
+✓
+1097
+and-or
+✓
+15260
+✗
+TO
+gen-fun
+✓
+141
+✗
+TO
+virus
+✓
+139
+✓
+163727
+escape10
+✓
+48
+✓
+12031
+escape25
+✓
+15958
+✗
+TO
+escape50
+✓
+410
+✗
+TO
+escape75
+✗
+TO
+✗
+TO
+escape100
+✓
+939
+✗
+TO
+escape200
+✗
+TO
+✗
+TO
+sequential5
+✓
+1074
+✗
+TO
+sequential7
+✓
+12822
+✗
+TO
+sequential10
+✓
+453718
+✗
+TO
+mod5
+✓
+34150
+✗
+TO
+mod7
+✓
+443
+✗
+TO
+mod10
+✓
+1245
+✗
+TO
+benchmark
+certz3
+tz3
+certcvc5
+tcvc5
+brown
+✗
+TO
+✗
+TO
+lemonde
+✗
+TO
+✗
+TO
+negra
+✓
+37248
+✓
+10144
+swbd
+✗
+TO
+✗
+Error
+tiger
+✓
+17454
+✓
+16118
+tuebadz
+✓
+15323
+✓
+5534
+wsj
+✗
+TO
+✗
+TO
+random
+✗
+TO
+✗
+TO
+

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@@ -0,0 +1,788 @@
+RMSim: Controlled Respiratory Motion Simulation
+on Static Patient Scans
+Donghoon Lee, Ellen Yorke, Masoud Zarepisheh,
+Saad Nadeem*, Yu-Chi Hu*
+Department of Medical Physics, Memorial Sloan Kettering Cancer Center, New York,
+NY, USA
+E-mail: {leed10,yorkee,zarepism,nadeems,huj}@mskcc.org
+*Corresponding Authors
+Objective:
+This work aims to generate realistic anatomical deformations from
+static
+patient
+scans.
+Specifically,
+we
+present
+a
+method
+to
+generate
+these
+deformations/augmentations via deep learning driven respiratory motion simulation
+that provides the ground truth for validating deformable image registration (DIR)
+algorithms and driving more accurate deep learning based DIR.
+Approach: We present a novel 3D Seq2Seq deep learning respiratory motion simulator
+(RMSim) that learns from 4D-CT images and predicts future breathing phases given
+a static CT image. The predicted respiratory patterns, represented by time-varying
+displacement vector fields (DVFs) at different breathing phases, are modulated through
+auxiliary inputs of 1D breathing traces so that a larger amplitude in the trace results in
+more significant predicted deformation. Stacked 3D-ConvLSTMs are used to capture
+the spatial-temporal respiration patterns.
+Training loss includes a smoothness loss
+in the DVF and mean-squared error between the predicted and ground truth phase
+images.
+A spatial transformer deforms the static CT with the predicted DVF to
+generate the predicted phase image. 10-phase 4D-CTs of 140 internal patients were
+used to train and test RMSim. The trained RMSim was then used to augment a public
+DIR challenge dataset for training VoxelMorph to show the effectiveness of RMSim-
+generated deformation augmentation.
+Main results:
+We validated our RMSim output with both private and public
+benchmark datasets (healthy and cancer patients).
+The structure similarity index
+measure (SSIM) for predicted breathing phases and ground truth 4D CT images was
+0.92±0.04, demonstrating RMSim’s potential to generate realistic respiratory motion.
+Moreover, the landmark registration error in a public DIR dataset was improved from
+8.12±5.78mm to 6.58mm±6.38mm using RMSim-augmented training data.
+Significance: The proposed approach can be used for validating DIR algorithms as
+well as for patient-specific augmentations to improve deep learning DIR algorithms.
+The code, pretrained models, and augmented DIR validation datasets will be released
+at https://github.com/nadeemlab/SeqX2Y. The supplementary video can be found
+at https://youtu.be/xIx8B_Q_R9o.
+arXiv:2301.11422v1  [cs.CV]  26 Jan 2023
+
+RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+2
+1. Introduction
+Respiratory motion hampers accurate diagnosis as well as image-guided therapeutics.
+For example, during radiotherapy, it may lead to poor local tumor control and increased
+radiation toxicity to the normal organs [1]. It can also exhibit itself as motion artifacts
+in the acquired images, making it difficult to differentiate nodule/tumor morphology
+changes from those induced by respiratory motion.
+This also makes the image
+registration task across different breathing phases as well as across different time points
+challenging. To validate the image registration accuracy/performance for commissioning
+solutions available in clinical commercial systems, the American Association of
+Physicists in Medicine(AAPM) TG-132 [2] recommended independent quality checks
+using digital phantoms. Current commercial solutions such as ImSimQA allow creation
+of synthetic deformation vector fields (DVFs) by user-defined transformations with only
+a limited degree of freedom.
+These monotonic transformations can not capture the
+realistic respiratory motion.
+For modeling respiration motion, an intuitive representation of motion is time-
+varying displacement vector fields (DVFs) obtained by deformable image registrations
+(DIR) in 4D images, acquired in a breathing cycle. Surrogate-driven approaches [3]
+employ DVF as a function of the surrogate breathing signal. However, an exact and
+direct solution in the high-dimensional space of DVFs is computationally intractable.
+Still, motion surrogates have been widely studied in the field of radiotherapy for
+building models establishing the relationship between surrogates and respiratory motion
+estimated from the image data [3].
+For example, the 1D diaphragm displacement
+has been reported as a reliable surrogate for tumor motion model [4] as well as for
+PCA (principle component analysis) respiratory motion model to correct CT motion
+artifacts [5].
+Recently,
+Romaguera et al. [6] used a 2D sequence-to-sequence (Seq2Seq)
+network [7] to predict 2D in-plane motion for a single future time point.
+Krebs et
+al. [8] applied a similar encoder-decoder network in a conditional variational autoencoder
+(cVAE) framework in which network parameters were learned to approximate the
+distribution of deformations in low-dimensional latent space with the encoder and decode
+the latent features for 2D motion prediction with the decoder. Romaguera et al. [9]
+integrated Voxelmorph [10] for assisting the VAE encoder to map deformations in latent
+space conditioned on anatomical features from 3D images. Temporal information of 2D
+surrogate cine images from a 2D Seq2Seq network was used to predict 3D DVF at a
+single future time point.
+In this paper, we present a novel deep learning respiratory motion simulator
+(RMSim) that learns to generate realistic patient-specific respiratory motion represented
+by time-varying DVFs at different breathing phases from a static 3D CT image. For the
+first time, we also allow modulation of this simulated motion via arbitrary 1D breathing
+traces as auxiliary input to create large variations. This in turn creates diverse patient-
+specific data augmentations while also generating ground truth for DIR validation.
+
+RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+3
+Our work has several differences and advantages over the aforementioned deep learning
+approaches: (1) we used 3D Seq2Seq architecture for the first time which has never been
+attempted before for predicting deformations due to GPU memory limitations, (2) we
+did not use VoxelMorph in its entirety but only the Spatial Transform module to train
+our model end-to-end, and (3) as opposed to predicting just a single future time point,
+we can predict 9 future time point breathing phases simultaneously (learnt from 4D-CT
+images with 10 3D CT breathing phases) along with their 3D DVFs. We have thoroughly
+validated our RMSim output with both private and public benchmark datasets (healthy
+and cancer patients) and demonstrated that adding our patient-specific augmentations
+to training data can improve performance/accuracy of state-of-the-art deep learning DIR
+algorithms. We also showcase breathing trace-modulated respiratory motion simulations
+for public static radiology scans (in the accompanying supplementary video). The
+code, pretrained models, and augmented DIR validation datasets will be released at
+https://github.com/nadeemlab/SeqX2Y.
+Figure 1. The schematic image for the proposed deep learning model. The Seq2Seq
+encoder-decoder framework was used as the backbone of the proposed model. The
+model was built with 3D convolution layers for feature encoding and output decoding
+and 3D convolutional Long Short-Term Memory (3D ConvLSTM) layers for spatial-
+temporal correlation between time points. The last layer of the decoder was a spatial
+transform layer to warp the initial phase image with the predicted Deformation Vector
+Field (DVF). To modulate the respiratory motions the 1D breathing trace was given
+as input along with the initial phase image. The dimension of image volume was 128
+× 128 × 128 and the input feature to 3D ConvLSTM is 64 × 64 × 64 × 96 (Depth ×
+Width × Height × Channel)
+
+1D Respiratory Signal
+D
+128×128×128x3
+Phase 1
+64×64×64×96
+3D Convolution
+ConvLSTM3D
+Multiplication
+T
+Spatial transform
+Phase 1
+Phase 2
+Phase k
+DVF
+128×128×128RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+4
+2. Materials and Methods
+2.1. Datasets
+We used an internal lung 4D-CT dataset retrospectively collected and de-identified
+from 140 non-small cell lung cancer (NSCLC) patients receiving radiotherapy in our
+institution. The helical and cine mode 4D-CTs were acquired using Philips Brilliance
+Big Bore or GE Advantage respectively and binned into 10 phases using the vendor’s
+proprietary software with breathing signals from bellows or external fiducial markers.
+The x-ray energy for the CT image was 120 kVp and tube current varies case by case
+according to vendor-specific tube current modulations based on patient size. The mAs
+range is [100, 400] for GE and [500, 800] for Philips. The image slice dimension was
+512x512, while the number of image slices varied patient by patient. We used the 100:40
+split for training:testing.
+We used 20 cases of the Lung Nodule Analysis (LUNA) challenge dataset [11]
+containing 3D radiology CTs for lung tumor screening to show that our RMSim
+model trained with the internal dataset can be effectively applied to an external
+radiology/diagnostic dataset to generate realistic respiration motions (see accompanying
+supplementary video). For quantitative evaluation of the model generality on an
+external data set, we used POPI [12] dataset which contains 6 10-phase 4D-CTs
+with segmented lung masks as well as annotated landmarks on the vessel and airway
+bifurcations.
+To validate the effectiveness of data augmentation using synthetic respiratory
+motion images generated from our RMSim model in the deformable registration task,
+we used the Learn2Reg 2020 challenge dataset [13]. The Learn2Reg dataset consists of
+30 subjects (20 for the training / 10 for the testing) with 3D CT thorax images taken
+in inhale and exhale phases. For each Learn2Reg 20 inhale/exhale pairs, we generated
+other phases of images using our RMSim model which was trained with the internal
+dataset, therefore increasing the sample size to 200 in total to augment the training of
+a well-known unsupervised deep learning DIR method, VoxelMorph [10]. Unfortunately
+the inhale-exhale landmarks are not publicly available in Learn2Reg dataset to assess
+the registration accuracy. For the landmarks evaluation in registration task, we used the
+POPI dataset. Brief description/purpose of all the datasets used in this study is given in
+Table 1. All datasets used in this study were cropped to eliminate the background and
+resampled to 128×128×128 with 2mm voxel size due to the GPU memory constrains.
+2.2. Realistic Respiratory Motion Simulation
+Sequence-to-Sequence (Seq2Seg) is a many-to-many network architecture originally
+developed for natural language processing tasks such as language translation.
+Inspired by Seq2Seq, the proposed RMSim, illustrated in Figure 1, is a novel deep
+learning encoder-decoder architecture that comprises three main parts including 3D
+convolution, ConvLSTM3D (3D Convolutional Long-Short Term Memory), and spatial
+
+RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+5
+Table 1. Datasets used in this study.
+Dataset
+Size
+Description
+Purpose
+Evaluation
+Internal 4D-CTs
+140 (100 train-
+ing, 40 testing)
+10-phase
+radiother-
+apy 4D-CTs
+Training and testing RM-
+Sim
+Image similar-
+ity
+LUNA
+20
+Radiology CTs for
+lung nodule detec-
+tion
+Testing model generality
+Visualization
+and qualitative
+POPI 4D-CTs
+6
+10-phase
+4D-CTs
+with landmarks
+Testing
+model
+general-
+ity (evaluating DVF accu-
+racy)
+Target
+Regis-
+tration
+Error
+(TRE)
+of
+landmarks
+Learn2Reg
+30 (20 training,
+10 testing)
+Inspiration-
+expiration
+thorax
+CT pairs with lung
+segmentations
+Training
+and
+testing
+RMSim-augmented deep
+learning
+deformable
+image registration (Vox-
+elmorph)
+Lung
+segmen-
+tation
+(Dice
+score) and im-
+age similarity
+Figure 2. Respiration motion surrogate extraction using a diaphragm point that has
+the maximum superior-inferior displacement across the phases. LDDMM was used to
+register the phase 1 (fixed) image to other phases (moving) to get the DVFs. The
+diaphragm point’s trajectory in z-axis (shown in red) across the phases was considered
+as the breathing trace. Yellow line shows the diaphragm position at phase 1.
+transformation layer (adapted from VoxelMorph [10]).
+The 3D convolution in the
+encoder is used to reduce the matrix dimension and extract salient features from images.
+We used 3×3×3 kernel size and 2×2×2 stride size to reduce the matrix dimension
+to 1/8. The number of channels for 3D convolution layer is 96. LSTM has a more
+complex cell structure than a neuron in classical recurrent neural network (RNN).
+Apart from the cell state, it contains gate units to decide when to keep or override
+information in and out of memory cells to better handle the gradient vanishing problem
+in recurrent neural network. This helps in learning long term dependencies. ConvLSTM
+[14] replaces Hadamard product with convolution operators in the input as well as the
+state transitions to capture the spatial pattern of the feature representations aggregated
+from different time points. We implemented ConvLSTM in 3D for handling the 3D
+phase images from the 4D-CT. We used two stacked ConvLSTM3D layers to make the
+network deeper, adding levels of abstraction to input observations similar to the typical
+deep neural network. The hidden state output from ConvLSTM3D was fed to both the
+
+Phase 1
+Phase 2
+Phase 3
+Phase 4
+Phase 5
+Phase 6
+Phase7
+Phase 8
+Phase9
+Phase 10
+Fixed
+Moving1
+Moving3
+Moving4
+Moving5
+Moving6
+Moving7
+Moving8
+Moving2
+Moving9
+DVF1
+DVF2
+DVF3
+DVF4
+DVF5
+DVF6
+DVF7
+DVF8
+DVF9RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+6
+next layer in the same stack and the next timepoint ConvLSTM3D layer. The output
+of ConvLSTM3D in the decoder at each predicted time point was up-sampled to the
+original input resolution and output channels were reduced via 3D convolution, resulting
+in the 3D DVF for the final output. The initial phase CT image was then deformed to
+a predicted phase image at different breathing phase using spatial transformation layer
+and the predicted 3D DVFs.
+Moreover, to modulate the predicted motion with a patient-specific pattern, we
+used an auxiliary input of 1D breathing trace.
+In this paper, we considered the
+amplitude of diaphragm apex motion as the surrogate of the respiratory signal [4].
+The 1D breathing trace for each training case was extracted using DVF obtained
+from large deformation diffeomorphic metric mapping (LDDMM) DIR provided by
+ANTs (Advanced Normalization Tools). Specifically, using the DVF, the apex point
+in diaphragm was propagated from the phase at the end of inhalation to other phases
+to generate the 1D displacement trace. The apex of the diaphragm was determined by
+finding the lung surface voxel with the maximum superior-inferior (z-axis) displacement
+among the DVFs. The z-axis displacement of the apex voxel at each phase resembles
+the 1D breathing trace. Figure 2 describes the process of preparing the 1D respiratory
+signal. Feature-wise transformations, e.g. addition or multiplication, are simple and
+effective mechanisms to incorporate conditioning information from another data source
+to the features learned in the network. In this paper, the hidden state of ConvLSTM at
+each phase is modulated by a simple element-wise multiplication of the phase-amplitude
+of the trace:
+m(Ht, bt) = btHt,
+(1)
+where Ht is the hidden state encoded from the sequence of phase images up to phase t
+and bt is the amplitude of the breathing trace at phase t,
+The loss function for training includes the mean-squared error of ground truth
+phase image and predicted phase image, and the regularization on the gradient of DVF
+by promoting smoothness of DVF:
+Loss =
+�
+t>0
+[(Yt − T(X0, φt))2 + ||∇φt||2],
+(2)
+where X0 is the initial phase image (phase 1 in this paper), T is the spatial transform
+(adapted from VoxelMorph), φt is the predicted DVF for phase t and Yt is the ground
+truth phase image at phase t.
+We developed RMSim using the PyTorch library (version 1.2.0). We used Adam
+for optimization and set learning rate to be 0.001 (as done in the original Seq2Seq
+paper [14]). Due to the large data size of 4D image sequence (10 3D CT phase images
+constituting a single 4D-CT), the batch size was limited to 1 and the number of feature
+channels was 96, considering GPU memory and training time. The model was trained
+and tested on an internal high performance computing cluster with 4 NVIDIA A40
+GPUs with 48GB memory each. Our model consumed 35.2 GB GPU memory and the
+
+RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+7
+training time was approximately 72 hours. The inference time for 9 phases and 40 total
+test cases from the internal dataset was less than 3 minutes.
+2.3. Data augmentation by RMSim
+Since RMSim can generate a series of realistic respiratory motion-induced images from
+a single 3D CT, one of its use cases is data augmentation for training DIR algorithms.
+For each of the 20 training cases in the Learn2Reg Grand Challenge dataset [13], we
+randomly selected a 1D breathing trace from our internal dataset to modulate the motion
+on the Learn2Reg inhalation image to generate 9 additional phase images, increasing
+the training size 10-fold. We chose a popular deep learning DIR method, VoxelMorph,
+suitable for unsupervised training for the propose of validating effectiveness of data
+augmentation. We first trained a VoxelMorph model with the original 20 inhalation-
+to-exhalation image pairs in the Learn2Reg training set.
+We then trained another
+VoxelMorph model with the augmented data including 200 pairs of inhalation-to-phase
+images. We compared the registrations from the two VoxelMorph models for validating
+the effectiveness of data augmentation.
+2.4. Evaluation Metrics
+For image similarity, we used structure similarity index measure (SSIM) [15] which
+measures the similarity of two given images based on the degradation of structural
+information, including luminance, contrast and structure. The closer the SSIM value
+is to 1, the more similarity between the two images. SSIM was used for comparing
+RMSim-predicted phase images and ground truth phase images in the internal test cases.
+SSIM was also used for comparing deformable registration results from VoxelMorph to
+validate data augmentation effectiveness in Learn2Reg test cases, which additionally
+were evaluated with the provided lung segmentation using Dice score to compare the
+ground truth lung contours and propagated lung contours.
+For landmark comparison in the POPI dataset, we used Target Registration
+Error (TRE), defined as the Euclidean distance between a landmark position spatially
+transformed and the target position.
+3. Results
+For each test case in the internal 4D-CT dataset, we generated 9 simulated phase images
+from the ground truth phase 1 image by deforming the phase 1 image using the predicted
+DVF at each phase. We calculated SSIM to measure the image similarity (SSIMsim)
+between the simulated phase image and the ground truth phase image. For comparison,
+we also calculated the SSIM (SSIMgnd) between the ground truth phase 1 image and the
+rest of the ground truth phase images. The average SSIMsim was 0.92±0.04, compared
+to 0.86±0.08 of SSIMgnd (p < 0.01.)
+
+RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+8
+We also measured the diaphragm displacement between the reference respiratory
+signal and the predicted signal (see Figure 3).
+As can be seen, the error increased
+from inhale to exhale phases. This is because prediction accuracy decreases at later
+time points. However, the overall displacement error was within 3 mm. Adding more
+realistic respiratory data for training can further reduce this displacement error.
+Figure 3. The error between reference respiratory signal (diaphragm displacement in
+mm) and predicted signal.
+To demonstrate the modulation flexibility of the 1D breathing traces, we applied
+different breathing traces to the same 3D CT image to generate different motion
+simulations, as shown in Figure 4. The plot on the top illustrates the two 1D breathing
+traces used for modulation.
+The breathing trace 1 (BT1), denoted by orange color
+line, represents the original respiratory signal for the case. BT2 denoted by gray line
+is a trace from another patient that was used to generate the simulated images. The
+white horizontal line indicates the position of the apex of the diaphragm in the initial
+phase (the first column). It is used as a reference to show the relative positions of the
+diaphragm at different phases. The diaphragm in images on the upper row clearly shows
+the more significant movement as BT2 has higher amplitudes in the trace.
+The amplitude range in our internal dataset was 0.14 – 40 mm. To validate the
+prediction performance on out-of-range displacement, we predicted additional sequences
+using a 5 times larger respiratory amplitude. The prediction results using a 5 times
+larger respiratory signal achieve a higher diaphragm level which means the predicted
+respiratory has larger fluctuation than the original respiratory signal but it was not
+proportional to the respiratory signal that was used for inference (see Figure 5).
+The results of propagating anatomical structures using the predicted DVFs are also
+shown in Figure 4. We propagated the lung, heart, esophagus, and tumor from the initial
+phase image. The propagated contours are well-matched with the predicted image and
+the motion of structures looks very realistic. We also provided the supplementary
+video of the simulated 4D-CT along with the ground truth 4D-CT and the 3D
+
+9
+8
+7
+6
+Error
+5
+4
+3
+X
+X
+X
+2
+X
+X
+1
+0RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+9
+Figure 4.
+Two different breathing traces, BT1 and BT2 shown in the plot, were
+used to simulate the respiration motion of an internal case, resulting in 2 series of
+modulated phase images according to the breathing traces. The diaphragm has larger
+displacement in images simulated with BT2 (upper row) than the displacement in
+images simulated with shallower BT1 (bottom row.) The white horizontal line indicates
+the position of the apex of the left diaphragm at the initial phase (left-most column.)
+We also overlay the propagated lung(in yellow), heart(in red), esophagus(in blue) and
+tumor(in green) contours using predicted DVFs.
+volume-rendered visualizations. Specifically, the 3D volume-rendered visualizations on
+LUNA challenge datasets as well as internal lung radiotherapy datasets with structure
+propagation are included in the accompanying supplementary video with chained
+predictions for 60-phase predictions for LUNA challenge (radiology lung nodule) and
+30-phase predictions for the lung radiotherapy datasets.
+In POPI dataset, there is only one case which contains lung segmentations on all
+the phases. For this case, we extracted 1D breathing trace from the lung segmentations
+as we did for our internal dataset. RMSim trained with our internal dataset predicted
+the remaining phases from the inhale phase with the modulation from the 1D breathing
+trace. The average TRE (Target Registration Error) of landmarks propagated with our
+predicted DVFs in this case was 0.92±0.64mm, showing that RMSim can accurately
+predict the patient-specific motion from the patient’s 1D breathing trace.
+Figure 6
+shows the TRE results for all predicted phases in this case. For the three other 4D-CT
+cases in POPI there were no lung segmentation masks so we performed semi-automatic
+lung segmentation for extracting the 1D breathing traces and the results are shown in
+Figure 7.
+Additionally, we used the RMSim for augmenting the Learn2Reg Challenge dataset.
+The Dice score of lung segmentation of 10 Learn2Reg testing cases using the VoxelMorph
+without augmentation was 0.96 ± 0.01 while the model trained with RMSim data
+
+15
+10
+mm
+5
+0
+2
+3
+5
+6
+7
+8
+9
+BT1---BT2RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+10
+Figure 5. The predicted phase 5 images using different 1D respiratory signal. Blue
+line is original respiratory signal, orange line is 3 times amplitude and green line is 5
+times amplitude.
+Figure 6. TRE results of all 9 phases from the 4DCT case in POPI. RMSim trained
+with the internal dataset were able to achieve sub-mm accuracy in this external case.
+augmentation was 0.97 ± 0.01 (p < 0.001 using the paired t-test). The SSIM between
+the warped images and the ground truth images was 0.88 ± 0.02 for the model without
+augmentation and 0.89 ± 0.02 (p < 0.001) for the model with augmentation.
+To validate the improvement of DIR using VoxelMorph with augmentation, we
+propagated the landmark points from the inhale phase to the exhale phase for the 6
+
+Phase
+Respiratory signal
+140
+Amplitude(mm)
+120
+100
+016.00
+Without Prediction
+14.00
+With Prediction
+12.00
+TRE (mm)
+10.00
+8.00
+6.00
+X
+4.00
+X
+2.00
+0.00
+P.2
+P.3
+P.4
+P.5
+P.6
+P.7
+P.8
+P.9
+P.10RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+11
+Figure 7. Three other 4D-CT POPI cases including 10 phases with landmarks on each
+phase (TRE plots for the three cases given below). For each case, we show original and
+predicted phase images overlaid with the difference with respect to original phase 1
+input. In original DIR Validation 03 phase difference image, the diaphragm in the left
+lung (viewer’s right) did not move due to the large tumor but it does in our prediction
+(shown in red bounding boxes). This case does not deflect from the goals of this paper,
+i.e. data augmentation and DIR validation. The difference in Case #1 appears minor
+because the breathing is shallower (less diaphragm movement) and Case #2 and Case
+#3 have larger differences due to deeper breathing.
+cases available in POPI dataset and computed the TRE. On average, pre-DIR TRE
+was 8.05±5.61mm, VoxelMorph w/o augmentation was 8.12±5.78mm compared to
+6.58±6.38mm for VoxelMorph with augmentation (p < 3e-48). The TRE comparison
+of all 6 cases are shown in Figure 8.
+
+Phase2
+Phase3
+Phase4
+Phase5
+Phase6
+Phase7
+Phase8
+Phase9
+Phase10
+ Original
+DIR_Validation_01
+Prediction
+ Original
+Validation_02
+Prediction
+DIR
+Original
+DIR_Validation_03
+Prediction
+WithoutPrediction
+WithPrediction
+12
+35
+20
+30
+10
+5
+(mm)
+15
+20
+TRE
+15
+10
+10
+5
+0
+P2 P3 P4 P5P6 P7 P8 P9 P10
+P2 P3 P4 P5 P6 P7 P8 P9 P10
+P2 P3 P4 P5P6 P7 P8 P9 P10
+DIR Validation01
+DIR Validation 02
+DIRValidation 03RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+12
+Figure 8. TRE results of POPI dataset. VoxelMorph with RMSim augmentation
+outperformed the VoxelMorph w/o augmentation in all 6 cases.
+4. Discussion
+In this work, we presented a 3D Seq2Seq network, referred to as RMSim, to predict
+patient-specific realistic motion induced/modulated with 1D breathing trace.
+We
+successfully validated our RMSim output with both private and public benchmark
+datasets (healthy and cancer patients) and demonstrated that adding our patient-
+specific augmentations to training data can improve performance/accuracy of state-
+of-the-art deep learning DIR algorithms. We also showcased breathing trace-modulated
+respiratory motion simulations for public static radiology scans.
+In this work, we
+predicted the motion in one breathing cycle.
+In the future, we will fine-tune our
+current model to predict multiple cycles in one-shot. Possible solutions include making
+our model bi-directional and using cross-attention to improve temporal dynamics in a
+long sequence. Further research is needed to investigate the impact of training data
+augmentation on different image modalities such as 4D-MRI.
+Another application of our work is in external radiotherapy treatment planning.
+RMSim simulated 4D-CT can be used to delineate the internal target volume (ITV)
+which is the union of the target volumes in all respiratory phases. The entire ITV is
+irradiated in radiation therapy to ensure all regions of tumor receive enough radiation.
+There is a more sophisticated alternative to ITV, referred to as robust treatment
+planning, where the key idea is to model the motion and directly incorporate it into the
+planning [16]. This typically can be done by assuming a probability density function
+(PDF) for the position of the target and doing plan optimization based on that [17, 18].
+It is also possible to assume a set of possible motion PDFs to account for uncertainty in
+breathing and plan accordingly [19, 20]. The simulated 4D-CT can be used to extract
+
+40
+Vanilla VoxelMorph
+35
+Pre_DIR
+30
+VoxelMorph + Augmentation
+TRE (mm)
+25
+20
+15
+10
+5
+0
+#1
+#2
+#3
+#4
+#5
+#6RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+13
+the motion PDF or a set of motion PDFs from varied breathing patterns exhibited by
+the patient.
+Additional interesting future direction is the extension of our earlier work in
+exhaustively simulating physics-based artifacts in CT and CBCT images for more robust
+cross-modal deep learning translation, segmentation, and motion-correction algorithms
+[21, 22, 23], available via our Physics-ArX library (https://github.com/nadeemlab/
+Physics-ArX).
+Specifically, in our previous work we presented a proof-of-concept
+pipeline for physics-based motion artifact simulation in CT/CBCT images using 4D-
+CT phases [22]. Using the method proposed in the current paper, we can generate and
+modulate large/diverse 4D-CT phases from any static 3D CT scan using the 1D RPM
+signal. These generated 4D-CT variations can then be used to produce large realistic
+motion-artifact variations via our earlier pipeline[22].
+Limitations: For simplicity, we used the maximal displacement on the diaphragm
+as the surrogate of clinical breathing trace to drive the modulation.
+We assume
+(1) the breathing pattern is regular since we extracted the diaphragm displacements
+from amplitude-binned 4D-CT, and (2) regional DVFs are linearly scaled according to
+diaphragm motion. Note 1D breathing trace might not represent the actual cardiac
+motion.
+Because of the GPU memory constraints, our input and output dimension
+was limited to 128x128x128.
+Nevertheless, the precise estimation of motion is not
+required for providing realistic motion-induced ground truth DVFs for the validation
+of the DIR algorithms and data augmentation for training DIR algorithms, as shown in
+this work. To extend our work to tumor tracking during radiation treatment, we will
+use the signals from the actual external real-time motion management (RPM) device
+to drive the modulation more precisely. We will also explore incorporating 2D MV/kV
+projections acquired during the treatment to infer more realistic cardiac/tumor motion.
+Acknowledgements
+This work was supported partially by NCI/NIH P30 CA008748.
+Conflict of interest
+We have no conflict of interest to declare.
+Code Availability Statement
+The code, pretrained models, and augmented DIR validation datasets will be released
+at https://github.com/nadeemlab/SeqX2Y.
+Data Availability Statement
+The public datasets used in this study and their urls are as follows: (1) Learn2Reg
+Challenge Lung CT dataset (Empire10 Challenge Dataset): https://drive.google.
+
+RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans
+14
+com/drive/folders/1yHWLQEK9c1xzggkCC4VX0X4To7BBDqu5,
+(2)
+LUNA
+challenge
+dataset (subset0.zip):
+https://zenodo.org/record/3723295, (3) DIR Validation
+POPI Dataset (6 4D CT patients with landmarks): https://www.creatis.insa-lyon.
+fr/rio/dir_validation_data, and (4) POPI model dataset (one 4D CT patient
+dataset with landmarks on all phases as well as lung segmentation mask): https:
+//www.creatis.insa-lyon.fr/rio/popi-model_original_page.
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+

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+arXiv:2301.04014v1  [quant-ph]  9 Jan 2023
+Ozawa’s Intersubjectivity Theorem as
+objection to QBism individual agent
+perspective
+Andrei Khrennikov
+Linnaeus University, International Center for Mathematical Modeling
+in Physics and Cognitive Sciences V¨axj¨o, SE-351 95, Sweden
+January 11, 2023
+Abstract
+QBism’s foundational statement that “the outcome of a measure-
+ment of an observable is personal” is in the straight contraversion with
+Ozawa’s Intersubjectivity Theorem (OIT). The latter (proven within
+the quantum formalism) states that two observers, agents within the
+QBism terminology, performing joint measurements of the same ob-
+servable A on a system S in the state ψ should get the same outcome
+A = x. In Ozawa’s terminology, this outcome is intersubjective and it
+can’t be treated as personal. This is the strong objection to QBism
+which can’t survive without updating its principles.
+The essential
+aspect in understanding of the OIT-impact on QBism’s foundations
+takes the notion of quantum observable. This paper comprises the
+complementary discussion highlighting the difference between the ac-
+curate, von Neumann, and inaccurate, noisy, quantum observables
+which are represented by PVMs and POVMs respectively. Moreover,
+we discuss the OIT-impact on the Copenhagen interpretation of quan-
+tum mechanics.
+1
+Introduction
+In this paper I move ahead my critical analysis of QBism’s founda-
+tions (see, e.g., [1]–[4] for QBism basics). This paper, as well as my
+two previous articles [5, 6], straightly critiques the individual agent
+perspective on measurement’s outcomes [7]. My previous appraisal
+1
+
+convinced QBists to specify the level of agent’s individuality. In con-
+trast to the general subjective probability theory, the class of agents
+should be restricted, at least to agents who were educated in basics
+of quantum theory. So, Ivan who lives in a Siberian village, a busy
+hunter, can’t be treated as a QBism’s agent.
+Now I have an intention to offense QBism by using Ozawa’s Inter-
+subjectivity Theorem (OIT) [8]. Qbism’s statement that “the outcome
+of a measurement of an observable is personal” is in the straight con-
+traversion with OIT. This theorem is not so widely known and one
+of the present paper’s intention is the theorem’s advertizement. OIT
+states that two observers, agents within the QBism terminology, per-
+forming joint measurements of the same observable A on a system S in
+the state ψ should register the same outcome A = x with probability
+one. Hence, the outcome is intersubjective [8], and it’s unnatural to
+consider outcomes of quantum observations as agent’s personal expe-
+riences.
+OIT is proven within the quantum formalism, it is the rigorous
+mathematical statement. But, as many theorems having the quan-
+tum foundational impact, its interpretation is not straightforward.
+The analysis of the OIT-impact onto QBism is coupled to the foun-
+dations of quantum measurement theory and especially the notion of
+quantum observable. Therefore, this paper comprises the complemen-
+tary discussion, highlighting the difference between the accurate, von
+Neuman, and inaccurate, noisy, quantum observables, mathematically
+represented by projection valued measures (PVMs) and positive oper-
+ator valued measures (POVMs), respectively. QIT is about the agents
+who are able to perform the joint accurate measurements. For such
+agents, measurement’s outcome loses its personalization, in favour of
+intersubjectivity.
+The conclusion of our analysis is that QBism should update its
+ideology by taking in consideration OIT. But, how? See section 6.
+Thus, I am in line with the criticism of QBism presented in article
+[8]. However, I depart from its conclusion that OIT contradicts to the
+Copenhagen interpretation; in contrast, OIT peacefully coexist with
+this interpretation. It is relevant to recall here that QBism fundamen-
+tally differs from the Copenhagen interpretation [2].
+Right away we initiate with the mathematical formulation of OIT
+and its proof. We set out to make the presentation very shortly (see
+[8] for details). The indirect measurement scheme is the heart of OIT.
+We go ahead with the recollection of the notion of quantum observ-
+able, namely, Hermitian operator or PVM, and generalized quantum
+observable (POVM) and the indirect measurements scheme for their
+generation.
+2
+
+2
+Quantum observables vs.
+general-
+ized quantum observables
+In quantum mechanics’ axiomatics, von Neumann [9] introduced quan-
+tum observables as Hermitian operators acting in complex Hilbert
+space H, the state space of a system.1 The spectral decomposition is
+the essential part in this framework.
+We restrict considerations to observables represented by the oper-
+ators with totally discrete spectra X ⊂ R. Here
+A =
+�
+x
+xEA(x),
+(1)
+where EA(x) is projection on the eigensubspace corresponding to the
+eigenvalue x; these projectors form the resolution of unity:
+I =
+�
+x
+EA(x).
+(2)
+The Born rule determines the probabilities of the outcomes of mea-
+surements for a system S in the state ψ,
+P(A = x|ψ) = ⟨ψ|EA(x)|ψ⟩.
+(3)
+Later generalized quantum observables were invented. Such ob-
+servables are represented by POVMs. We restrict considerations to
+POVMs with a discrete domain of definition X. POVM is a map
+x → Π(x) : for each x ∈ X, Π(x) is a positive contractive self-adjoint
+operator (i.e., 0 ≤ Π(x) ≤ I) (called an effect), and effects form the
+resolution of unity
+�
+x
+Π(x) = I.
+(4)
+This map defines an operator valued measure on algebra of all subsets
+of set X. For O ⊂ X,
+Π(O) =
+�
+x∈O
+Π(x).
+The condition (4) is the operator-measure counterpart of the condition
+normalization by 1 for usual probability measures.
+1Why did he select the Hermitian operators for mathematical representation of observ-
+ables in quantum theory? Moreover, he considered only such observables as the genuine
+quantum observables. I guess that he followed Schr¨odinger’s quantization rule for the
+position and momentum observables which are realized by Hermitian operators in L2-
+space. This rule implies that each classical observable given by the real-valued function
+A = A(q, p) on the phase space is represented as a Hermitian operator in L2-space.
+3
+
+POVM Π represents statistics of measurements for observable A
+with the following generalization of the Born’s rule:
+P(Π = x|ψ) = ⟨ψ|Π(x)|ψ⟩.
+(5)
+We remark that equality (4) implies that
+�
+x
+P(A = x|ψ) = 1.
+Any quantum observable A can also be represented as POVM of the
+special type – PVM EA = (EA(x)).
+Quantum observables given by PVMs were interpreted by von Neu-
+mann [9] as describing accurate measurements. And generalized ob-
+servables given by POVMs which are not PVMs are interpreted as
+representing inaccurate measurements. In von Neumann’s [9], the no-
+tion of measurement’s precision was not completely formalized. Only
+recently the consistent formalization of this notion was presented in
+[11].
+We shall keep firmly the expression “quantum observable” for ob-
+servable axiomatically introduced by von Neumann [9] and represented
+by PVMs and the expression “generalized quantum observable” for
+POVMs.
+3
+Generalized quantum observables from
+the indirect measurement scheme
+The indirect measurement scheme involves the following components
+• the states spaces H and K of the systems S and the apparatus
+M for measurement of some observable A;
+• the evolution operator U = U(t) representing the interaction-
+dynamics for the system S + M;
+• the meter observable M giving outputs of the pointer of the
+apparatus M.
+Here the quantum observables A and M can be represented as PVMs,
+EA = (EA(x)), EM = (EM(x)), where EA(x), EM(x) are projections
+in Hilbert spaces H and K respectively. It is assumed that the com-
+pound system’s evolution is driven by the Schr¨odinger equation, so
+the evolution operator is unitary.
+Formally, an indirect measurement model for an observable A, in-
+troduced in [10] as a “measuring process”, is a quadruple
+(K, |ξ⟩, U, M)
+4
+
+where |ξ⟩ ∈ K represents the apparatus state.
+We explore the Heisenberg picture. To describe meter’s evolution,
+we represent it in the state space of the compound system, i.e., as
+I ⊗ M. The meter observable evolves as
+M(t) = U ⋆(t)(I ⊗ M)U(t).
+(6)
+By the Born rule
+P(M(t) = x|ψξ) = ⟨ψξ|EM(t)(x)|ψξ⟩.
+(7)
+This is the probability distribution for the outputs of measure-
+ments done by the apparatus and given by the meter. In principle,
+one can ignore the representation of the measurement process as the
+system-apparatus interaction and operate solely with system’s states.
+In this picture one proceeds with generalized observables given by
+POVMs. The meter observable generates the POVM Π = (Π(x))
+Π(x) = ⟨ξ|EM(T)(x)|ξ⟩,
+(8)
+where T is the time needed to complete the experiment.
+The probability distribution of the generalized observable given by
+a POVM is determined by (5).
+Generally the probability distribution generated by a measurement
+process does not coincide with the probability distribution of the quan-
+tum observable A for which this process was constructed, i.e., generally
+P(Π = x|ψ) = ⟨ψ|Π(x)|ψ⟩ ̸= P(A = x|ψ) = ⟨ψ|EA(x)|ψ⟩,
+(9)
+We remark that, as was proven by Ozawa [10], any generalized
+observable (POVM) can be generated via the indirect measurement
+scheme.
+Typically one operates solely with generalized observables
+by ignoring the indirect measurement scheme. This simplifies consid-
+erations, but it can lead to misunderstanding of the foundations the
+quantum measurement theory.
+4
+Probability reproducibility condition
+Definition. A measurement process (K, |ξ⟩, U, M) reproduces the prob-
+ability distribution for quantum observable A (accurate von Neumann
+observable) if
+P(A = x|ψ) = P(M(T) = x|ψξ).
+(10)
+In this case
+⟨ψξ|EM(T)(x)|ψξ⟩ = ⟨ψ|E(x)|ψ⟩.
+(11)
+5
+
+or
+⟨ψ|Π(x)|ψ⟩ = ⟨ψ|E(x)|ψ⟩,
+(12)
+and hence,
+Π(x) = E(x),
+Proposition. Probability reproducibility condition for a measure-
+ment process is equivalent to the representation of the corresponding
+generalized observable by the PVM EA of measured quantum observ-
+able A.
+5
+Intersubjectivity of outcomes of quan-
+tum observables
+Following [8], consider two remote observers O1 and O2 who perform
+joint measurements on a system S, in mathematical terms it means
+that the meter quantum observables of the corresponding measure-
+ment processes commute,
+[M1(t), M2(t)] = 0.
+Here each apparatus has its own state space, i.e., K = K1 ⊗ K2. We
+call such measurements local. In this situation the joint probability
+distribution is well defined
+P(M1(t) = x, M1(t) = y|ψξ1ξ2) = ⟨ψξ1ξ2|EM1(t)(x)EM1(t)(y)|ψξ1ξ2⟩
+(13)
+Suppose that both observers perform the accurate measurements
+of the quantum observable A given by PVM EA = (EA(x)). Then the
+corresponding POVMs Πj, j = 1, 2, coincide with EA :
+Π1(x) = Π2(x) = EA(x).
+(14)
+This equality implies:
+Theorem. (OIT [8]) Two observers performing the joint local and
+probability reproducible measurements of the same quantum observable
+A on the system S should get the same outcome with probability 1:
+P(M1(T) = x, M1(T) = y|ψξ1ξ2) = δ(x − y)P(E = x|ψ) = ∥E(x)ψ∥2.
+(15)
+6
+
+6
+Intersubjectivity challenges QBism
+We start with the following citation of Fuchs and Schack [2]:
+“The fundamental primitive of QBism is the concept of experience.
+According to QBism, quantum mechanics is a theory that any agent
+can use to evaluate her expectations for the content of her personal
+experience. ...
+In QBism, a measurement is an action an agent takes to elicit an
+experience. The measurement outcome is the experience so elicited.
+The measurement outcome is thus personal to the agent who takes the
+measurement action. In this sense, quantum mechanics, like probabil-
+ity theory, is a single user theory. A measurement does not reveal a
+pre-existing value. Rather, the measurement outcome is created in the
+measurement action.”
+However, OIT implies that, for accurate local observables, mea-
+surement’s outcome is intersubjective which is the strong objection to
+QBism. There is nothing concerning personal experiences and QBists
+should response to this objection. My suggestion (see also [7]) is to fol-
+low Brukner’s work [12] where he proceeds not with individual agents
+and their personal experiences, but with a universal agent. I remark
+that consideration of universal agents is common in general theory of
+decision making. However, for QBists, such solution seems to be un-
+acceptable, since it would destroy consistency of the QBism’s private
+agency perspective. It would move QBism closer to Zeilinger-Brukner
+information interpretation of quantum mechanics [13, 14, 15].
+This objection to QBism is foundationally interesting and gen-
+erates the discussion on the notion of quantum observable. Due to
+efforts Helstrom, Holevo, and Ozawa [16]–[19], [10], generalized quan-
+tum observables which are mathematically represented by POVMs
+became one of the basic tools of quantum information theory. Nowa-
+days the special role of accurate observables represented by PVMs is
+not emphasized. In particular, the notion of observables in QBism is
+identified with generalized quantum observable given by POVM. How-
+ever, the clash between QBism and OIT stimulates highlighting of the
+accurate PVM- as the genuine quantum observables, and treating the
+generalized quantum observables which are not accurate POVM as
+imprecise and noisy ones. Of course, it is a well known fact, but the
+clash between OIT and QBism is good occasion to emphasize this
+difference.
+What does this difference between accurate PVM and noisy POVM
+observables mean for QBism?
+I have the following picture of the situation. OIT holds only for the
+accurate PVM-observables; for generalized quantum observables, it
+7
+
+can be violated and generally it is impossible to assign the same value
+for measurements’ outcomes for observers O1 and O2. Thus, QBism
+ideology of the personal experiences of observers (agents) can still be
+kept for such generalizad observables. But, where does individuality
+come from? The personal experiences come from noise! So, different
+observers performing inaccurate measurements are coupled to different
+noisy environments. This is just my personal view on consequences of
+IOT for QBism.
+In conclusion, QBism might response to the OIT-challenge by con-
+sidering the universal agent who is able to perform accurate measure-
+ments; individuality of agents’ experience is reduced to individuality
+of noise generated in the process of measurement.
+7
+Intersubjectivity and Copenhagen in-
+terpretation
+By the Copenhagen interpretation (at least by its Bohr’s version2)
+measurements’ outcomes cannot be treated as the objective properties
+of a system S. They are results of the complex process of interaction
+of a system and an apparatus, see Bohr [21]:
+“This crucial point ... implies the impossibility of any sharp sep-
+aration between the behaviour of atomic objects and the interaction
+with the measuring instruments which serve to define the conditions
+under which the phenomena appear. In fact, the individuality of the
+typical quantum effects finds its proper expression in the circumstance
+that any attempt of subdividing the phenomena will demand a change
+in the experimental arrangement introducing new possibilities of inter-
+action between objects and measuring instruments which in principle
+cannot be controlled. Consequently, evidence obtained under different
+experimental conditions cannot be comprehended within a single pic-
+ture, but must be regarded as complementary in the sense that only the
+totality of the phenomena exhausts the possible information about the
+objects.”
+The indirect measurement scheme matches perfectly with the Copen-
+hagen interpretation. Therefore it is surprising that OIT contradicts
+to it. The clash between OIT and the the Copenhagen interpretation
+was highlighted in the conclusion section of OIT-article [8]:
+2As was stressed by Plotnitsky [20], one should recognize the diversity of views on the
+Copenhagen interpretation. He suggested to speak about interpretations in the spirit of
+Copenhagen. Even Bohr changed the views a few times during his life [20].
+8
+
+“Schr¨odinger [22] argued that a measurement does not ascertain
+the pre-existing value of the observable and is only required to be re-
+peatable. Since the inception of quantum mechanics, this view has long
+been supported as one of the fundamental tenets of quantum mechan-
+ics. In contrast, we have shown that any probability reproducible mea-
+surement indeed ascertains the value that the observable has, whether
+the repeatability is satisfied or not.”
+I disagree with the author of [8]. The seed of this misunderstand-
+ing is in ignoring the two level structure of physical theories, ontic
+and epistemic [23, 24, 25]. The former is about reality as it is and
+the latter is about knowledge about reality.
+Bohr and Schr¨odinger
+wrote about the ontic reality, about impossibility to assign to quan-
+tum systems preexisting values and here “preexisting” is the synonym
+for “objective”, “ontic”. But OIT is not about such values, it is about
+epistemic reality, reality of knowledge about the possible outcome of
+measurement.
+Hence, in my opinion OIT can peacefully coexist with the Copen-
+hagen interpretation.
+But, as was stressed, OIT is a challenge for QBism which operates
+at the epistemic level of scientific description of quantum phenom-
+ena. This is the good place to recall that QBism should be sharply
+separated from the Copenhagen interpretation, see again Fuchs and
+Schack [2]:
+“According to QBism, quantum mechanics can be applied to any
+physical system. QBism treats all physical systems in the same way,
+including atoms, beam splitters, Stern-Gerlach magnets, preparation
+devices, measurement apparatuses, all the way to living beings and
+other agents. In this, QBism differs crucially from various versions
+of the Copenhagen interpretation.”
+Acknowledgments
+This paper was written on the basis of the long discussions with
+Masanao Ozawa and I would like to thank him; Arkady Plotnitsky
+told me a lot about the Copenhagen interpretation and Bohr’s views
+and I would like to thank him; Christopher Fuchs ignited my inter-
+est to QBism at the second V¨axj¨o conference (in 2001) and I am
+sorry if this paper would disturb QBists; I am also thankful to Harald
+Atmanspacher who introduced me into ontic-epistemic approach to
+scientific representation of natural phenomena.
+9
+
+References
+[1] Fuchs, C. A. and Schack, R. (2011). A Quantum-Bayesian Route
+to Quantum-State Space, Found. Phys. 41, p. 345.
+[2] Fuchs, C. A. and Schack, R. (2014). QBism and the Greeks:
+why a quantum state does not represent an element of physical
+reality, Phys. Scr., 90, 015104.
+[3] Fuchs, C. A., Mermin, N. D. and Schack, R. (2014). An In-
+troduction to QBism with an Application to the Locality of
+Quantum Mechanics, Am. J. Phys. 82, p. 749.
+[4] DeBrota, J. B., Fuchs, C. A., Pienaar, J. L., and Stacey, B. C.
+(2021). Born’s rule as a quantum extension of Bayesian coher-
+ence. Physical Review A, 104(2), 022207.
+[5] Khrennikov,
+A. (2018).
+External Observer Reflections on
+QBism, Its Possible Modifications, and Novel Applications, In:
+Quantum Foundations, STEAM-H: Science, Technology, Engi-
+neering, Agriculture, Mathematics & Health; Khrennikov A. and
+Toni B. Eds.; Springer, Cham, pp. 93–118.
+[6] Khrennikov, A. (2018). Towards better understanding QBism,
+Found. Sc., 23 (1), 181–195.
+[7] Khrennikov, A. Reflections on Zeilinger–Brukner Information
+Interpretation of Quantum Mechanics. Found Phys 46, 836–844
+(2016).
+[8] Ozawa, M. (2019). Intersubjectivity of outcomes of quantum
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+[9] von Neuman, J. (1955). Mathematical foundations of quan-
+tum mechanics (Princeton Univ. Press, Princenton) [Originally
+published: Mathematische Grundlagen der Quanten-mechanik,
+Springer, Berlin, 1932].
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+square errors. Quant. Inf. 5, Article number: 1 (2019)
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+C.
+On
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+measurement
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+https://arxiv.org/abs/1507.05255.
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+Found. Phys. 29, 31–641 (1999)
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+10
+
+[15] Brukner, C., Zeilinger, A.: Information invariance and quantum
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+Academic, New York, 1976.
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+ficient to describe quantum systems exhaustively. In: Sympo-
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+quantum realities. In: Time, quantum and information (pp. 301-
+321). Springer, Berlin, Heidelberg.
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+dom fields. Annals of Physics, 377, 147-163.
+11
+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf,len=305
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='04014v1  [quant-ph]  9 Jan 2023  Ozawa’s Intersubjectivity Theorem as  objection to QBism individual agent  perspective  Andrei Khrennikov  Linnaeus University, International Center for Mathematical Modeling  in Physics and Cognitive Sciences V¨axj¨o, SE-351 95, Sweden  January 11, 2023  Abstract  QBism’s foundational statement that “the outcome of a measure-  ment of an observable is personal” is in the straight contraversion with  Ozawa’s Intersubjectivity Theorem (OIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The latter (proven within  the quantum formalism) states that two observers, agents within the  QBism terminology, performing joint measurements of the same ob-  servable A on a system S in the state ψ should get the same outcome  A = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In Ozawa’s terminology, this outcome is intersubjective and it  can’t be treated as personal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' This is the strong objection to QBism  which can’t survive without updating its principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  The essential  aspect in understanding of the OIT-impact on QBism’s foundations  takes the notion of quantum observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' This paper comprises the  complementary discussion highlighting the difference between the ac-  curate, von Neumann, and inaccurate, noisy, quantum observables  which are represented by PVMs and POVMs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Moreover,  we discuss the OIT-impact on the Copenhagen interpretation of quan-  tum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  1  Introduction  In this paper I move ahead my critical analysis of QBism’s founda-  tions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', [1]–[4] for QBism basics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' This paper, as well as my  two previous articles [5, 6], straightly critiques the individual agent  perspective on measurement’s outcomes [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' My previous appraisal  1  convinced QBists to specify the level of agent’s individuality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In con-  trast to the general subjective probability theory, the class of agents  should be restricted, at least to agents who were educated in basics  of quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' So, Ivan who lives in a Siberian village, a busy  hunter, can’t be treated as a QBism’s agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Now I have an intention to offense QBism by using Ozawa’s Inter-  subjectivity Theorem (OIT) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Qbism’s statement that “the outcome  of a measurement of an observable is personal” is in the straight con-  traversion with OIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' This theorem is not so widely known and one  of the present paper’s intention is the theorem’s advertizement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' OIT  states that two observers, agents within the QBism terminology, per-  forming joint measurements of the same observable A on a system S in  the state ψ should register the same outcome A = x with probability  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Hence, the outcome is intersubjective [8], and it’s unnatural to  consider outcomes of quantum observations as agent’s personal expe-  riences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  OIT is proven within the quantum formalism, it is the rigorous  mathematical statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' But, as many theorems having the quan-  tum foundational impact, its interpretation is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  The analysis of the OIT-impact onto QBism is coupled to the foun-  dations of quantum measurement theory and especially the notion of  quantum observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Therefore, this paper comprises the complemen-  tary discussion, highlighting the difference between the accurate, von  Neuman, and inaccurate, noisy, quantum observables, mathematically  represented by projection valued measures (PVMs) and positive oper-  ator valued measures (POVMs), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' QIT is about the agents  who are able to perform the joint accurate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' For such  agents, measurement’s outcome loses its personalization, in favour of  intersubjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  The conclusion of our analysis is that QBism should update its  ideology by taking in consideration OIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' But, how?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' See section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Thus, I am in line with the criticism of QBism presented in article  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' However, I depart from its conclusion that OIT contradicts to the  Copenhagen interpretation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' in contrast, OIT peacefully coexist with  this interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' It is relevant to recall here that QBism fundamen-  tally differs from the Copenhagen interpretation [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Right away we initiate with the mathematical formulation of OIT  and its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' We set out to make the presentation very shortly (see  [8] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The indirect measurement scheme is the heart of OIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  We go ahead with the recollection of the notion of quantum observ-  able, namely, Hermitian operator or PVM, and generalized quantum  observable (POVM) and the indirect measurements scheme for their  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  2  2  Quantum observables vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  general-  ized quantum observables  In quantum mechanics’ axiomatics, von Neumann [9] introduced quan-  tum observables as Hermitian operators acting in complex Hilbert  space H, the state space of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='1 The spectral decomposition is  the essential part in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  We restrict considerations to observables represented by the oper-  ators with totally discrete spectra X ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Here  A =  �  x  xEA(x),  (1)  where EA(x) is projection on the eigensubspace corresponding to the  eigenvalue x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' these projectors form the resolution of unity:  I =  �  x  EA(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (2)  The Born rule determines the probabilities of the outcomes of mea-  surements for a system S in the state ψ,  P(A = x|ψ) = ⟨ψ|EA(x)|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (3)  Later generalized quantum observables were invented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Such ob-  servables are represented by POVMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' We restrict considerations to  POVMs with a discrete domain of definition X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' POVM is a map  x → Π(x) : for each x ∈ X, Π(x) is a positive contractive self-adjoint  operator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', 0 ≤ Π(x) ≤ I) (called an effect), and effects form the  resolution of unity  �  x  Π(x) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (4)  This map defines an operator valued measure on algebra of all subsets  of set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' For O ⊂ X,  Π(O) =  �  x∈O  Π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  The condition (4) is the operator-measure counterpart of the condition  normalization by 1 for usual probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  1Why did he select the Hermitian operators for mathematical representation of observ-  ables in quantum theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Moreover, he considered only such observables as the genuine  quantum observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' I guess that he followed Schr¨odinger’s quantization rule for the  position and momentum observables which are realized by Hermitian operators in L2-  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' This rule implies that each classical observable given by the real-valued function  A = A(q, p) on the phase space is represented as a Hermitian operator in L2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  3  POVM Π represents statistics of measurements for observable A  with the following generalization of the Born’s rule:  P(Π = x|ψ) = ⟨ψ|Π(x)|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (5)  We remark that equality (4) implies that  �  x  P(A = x|ψ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Any quantum observable A can also be represented as POVM of the  special type – PVM EA = (EA(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Quantum observables given by PVMs were interpreted by von Neu-  mann [9] as describing accurate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' And generalized ob-  servables given by POVMs which are not PVMs are interpreted as  representing inaccurate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In von Neumann’s [9], the no-  tion of measurement’s precision was not completely formalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Only  recently the consistent formalization of this notion was presented in  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  We shall keep firmly the expression “quantum observable” for ob-  servable axiomatically introduced by von Neumann [9] and represented  by PVMs and the expression “generalized quantum observable” for  POVMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  3  Generalized quantum observables from  the indirect measurement scheme  The indirect measurement scheme involves the following components  the states spaces H and K of the systems S and the apparatus  M for measurement of some observable A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  the evolution operator U = U(t) representing the interaction-  dynamics for the system S + M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  the meter observable M giving outputs of the pointer of the  apparatus M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Here the quantum observables A and M can be represented as PVMs,  EA = (EA(x)), EM = (EM(x)), where EA(x), EM(x) are projections  in Hilbert spaces H and K respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' It is assumed that the com-  pound system’s evolution is driven by the Schr¨odinger equation, so  the evolution operator is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Formally, an indirect measurement model for an observable A, in-  troduced in [10] as a “measuring process”, is a quadruple  (K, |ξ⟩, U, M)  4  where |ξ⟩ ∈ K represents the apparatus state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  We explore the Heisenberg picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' To describe meter’s evolution,  we represent it in the state space of the compound system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', as  I ⊗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The meter observable evolves as  M(t) = U ⋆(t)(I ⊗ M)U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (6)  By the Born rule  P(M(t) = x|ψξ) = ⟨ψξ|EM(t)(x)|ψξ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (7)  This is the probability distribution for the outputs of measure-  ments done by the apparatus and given by the meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In principle,  one can ignore the representation of the measurement process as the  system-apparatus interaction and operate solely with system’s states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  In this picture one proceeds with generalized observables given by  POVMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The meter observable generates the POVM Π = (Π(x))  Π(x) = ⟨ξ|EM(T)(x)|ξ⟩,  (8)  where T is the time needed to complete the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  The probability distribution of the generalized observable given by  a POVM is determined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Generally the probability distribution generated by a measurement  process does not coincide with the probability distribution of the quan-  tum observable A for which this process was constructed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', generally  P(Π = x|ψ) = ⟨ψ|Π(x)|ψ⟩ ̸= P(A = x|ψ) = ⟨ψ|EA(x)|ψ⟩,  (9)  We remark that, as was proven by Ozawa [10], any generalized  observable (POVM) can be generated via the indirect measurement  scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Typically one operates solely with generalized observables  by ignoring the indirect measurement scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' This simplifies consid-  erations, but it can lead to misunderstanding of the foundations the  quantum measurement theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  4  Probability reproducibility condition  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' A measurement process (K, |ξ⟩, U, M) reproduces the prob-  ability distribution for quantum observable A (accurate von Neumann  observable) if  P(A = x|ψ) = P(M(T) = x|ψξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (10)  In this case  ⟨ψξ|EM(T)(x)|ψξ⟩ = ⟨ψ|E(x)|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (11)  5  or  ⟨ψ|Π(x)|ψ⟩ = ⟨ψ|E(x)|ψ⟩,  (12)  and hence,  Π(x) = E(x),  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Probability reproducibility condition for a measure-  ment process is equivalent to the representation of the corresponding  generalized observable by the PVM EA of measured quantum observ-  able A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  5  Intersubjectivity of outcomes of quan-  tum observables  Following [8], consider two remote observers O1 and O2 who perform  joint measurements on a system S, in mathematical terms it means  that the meter quantum observables of the corresponding measure-  ment processes commute,  [M1(t), M2(t)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Here each apparatus has its own state space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', K = K1 ⊗ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' We  call such measurements local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In this situation the joint probability  distribution is well defined  P(M1(t) = x, M1(t) = y|ψξ1ξ2) = ⟨ψξ1ξ2|EM1(t)(x)EM1(t)(y)|ψξ1ξ2⟩  (13)  Suppose that both observers perform the accurate measurements  of the quantum observable A given by PVM EA = (EA(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Then the  corresponding POVMs Πj, j = 1, 2, coincide with EA :  Π1(x) = Π2(x) = EA(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (14)  This equality implies:  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (OIT [8]) Two observers performing the joint local and  probability reproducible measurements of the same quantum observable  A on the system S should get the same outcome with probability 1:  P(M1(T) = x, M1(T) = y|ψξ1ξ2) = δ(x − y)P(E = x|ψ) = ∥E(x)ψ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (15)  6  6  Intersubjectivity challenges QBism  We start with the following citation of Fuchs and Schack [2]:  “The fundamental primitive of QBism is the concept of experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  According to QBism, quantum mechanics is a theory that any agent  can use to evaluate her expectations for the content of her personal  experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  In QBism, a measurement is an action an agent takes to elicit an  experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The measurement outcome is the experience so elicited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  The measurement outcome is thus personal to the agent who takes the  measurement action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In this sense, quantum mechanics, like probabil-  ity theory, is a single user theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' A measurement does not reveal a  pre-existing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Rather, the measurement outcome is created in the  measurement action.”  However, OIT implies that, for accurate local observables, mea-  surement’s outcome is intersubjective which is the strong objection to  QBism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' There is nothing concerning personal experiences and QBists  should response to this objection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' My suggestion (see also [7]) is to fol-  low Brukner’s work [12] where he proceeds not with individual agents  and their personal experiences, but with a universal agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' I remark  that consideration of universal agents is common in general theory of  decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' However, for QBists, such solution seems to be un-  acceptable, since it would destroy consistency of the QBism’s private  agency perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' It would move QBism closer to Zeilinger-Brukner  information interpretation of quantum mechanics [13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  This objection to QBism is foundationally interesting and gen-  erates the discussion on the notion of quantum observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Due to  efforts Helstrom, Holevo, and Ozawa [16]–[19], [10], generalized quan-  tum observables which are mathematically represented by POVMs  became one of the basic tools of quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Nowa-  days the special role of accurate observables represented by PVMs is  not emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In particular, the notion of observables in QBism is  identified with generalized quantum observable given by POVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' How-  ever, the clash between QBism and OIT stimulates highlighting of the  accurate PVM- as the genuine quantum observables, and treating the  generalized quantum observables which are not accurate POVM as  imprecise and noisy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Of course, it is a well known fact, but the  clash between OIT and QBism is good occasion to emphasize this  difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  What does this difference between accurate PVM and noisy POVM  observables mean for QBism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  I have the following picture of the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' OIT holds only for the  accurate PVM-observables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' for generalized quantum observables, it  7  can be violated and generally it is impossible to assign the same value  for measurements’ outcomes for observers O1 and O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Thus, QBism  ideology of the personal experiences of observers (agents) can still be  kept for such generalizad observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' But, where does individuality  come from?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The personal experiences come from noise!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' So, different  observers performing inaccurate measurements are coupled to different  noisy environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' This is just my personal view on consequences of  IOT for QBism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  In conclusion, QBism might response to the OIT-challenge by con-  sidering the universal agent who is able to perform accurate measure-  ments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' individuality of agents’ experience is reduced to individuality  of noise generated in the process of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  7  Intersubjectivity and Copenhagen in-  terpretation  By the Copenhagen interpretation (at least by its Bohr’s version2)  measurements’ outcomes cannot be treated as the objective properties  of a system S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' They are results of the complex process of interaction  of a system and an apparatus, see Bohr [21]:  “This crucial point .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' implies the impossibility of any sharp sep-  aration between the behaviour of atomic objects and the interaction  with the measuring instruments which serve to define the conditions  under which the phenomena appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In fact, the individuality of the  typical quantum effects finds its proper expression in the circumstance  that any attempt of subdividing the phenomena will demand a change  in the experimental arrangement introducing new possibilities of inter-  action between objects and measuring instruments which in principle  cannot be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Consequently, evidence obtained under different  experimental conditions cannot be comprehended within a single pic-  ture, but must be regarded as complementary in the sense that only the  totality of the phenomena exhausts the possible information about the  objects.”  The indirect measurement scheme matches perfectly with the Copen-  hagen interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Therefore it is surprising that OIT contradicts  to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The clash between OIT and the the Copenhagen interpretation  was highlighted in the conclusion section of OIT-article [8]:  2As was stressed by Plotnitsky [20], one should recognize the diversity of views on the  Copenhagen interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' He suggested to speak about interpretations in the spirit of  Copenhagen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Even Bohr changed the views a few times during his life [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  8  “Schr¨odinger [22] argued that a measurement does not ascertain  the pre-existing value of the observable and is only required to be re-  peatable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Since the inception of quantum mechanics, this view has long  been supported as one of the fundamental tenets of quantum mechan-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In contrast, we have shown that any probability reproducible mea-  surement indeed ascertains the value that the observable has, whether  the repeatability is satisfied or not.”  I disagree with the author of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The seed of this misunderstand-  ing is in ignoring the two level structure of physical theories, ontic  and epistemic [23, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The former is about reality as it is and  the latter is about knowledge about reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Bohr and Schr¨odinger  wrote about the ontic reality, about impossibility to assign to quan-  tum systems preexisting values and here “preexisting” is the synonym  for “objective”, “ontic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' But OIT is not about such values, it is about  epistemic reality, reality of knowledge about the possible outcome of  measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Hence, in my opinion OIT can peacefully coexist with the Copen-  hagen interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  But, as was stressed, OIT is a challenge for QBism which operates  at the epistemic level of scientific description of quantum phenom-  ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' This is the good place to recall that QBism should be sharply  separated from the Copenhagen interpretation, see again Fuchs and  Schack [2]:  “According to QBism, quantum mechanics can be applied to any  physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' QBism treats all physical systems in the same way,  including atoms, beam splitters, Stern-Gerlach magnets, preparation  devices, measurement apparatuses, all the way to living beings and  other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In this, QBism differs crucially from various versions  of the Copenhagen interpretation.”  Acknowledgments  This paper was written on the basis of the long discussions with  Masanao Ozawa and I would like to thank him;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Arkady Plotnitsky  told me a lot about the Copenhagen interpretation and Bohr’s views  and I would like to thank him;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Christopher Fuchs ignited my inter-  est to QBism at the second V¨axj¨o conference (in 2001) and I am  sorry if this paper would disturb QBists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' I am also thankful to Harald  Atmanspacher who introduced me into ontic-epistemic approach to  scientific representation of natural phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  9  References  [1] Fuchs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' and Schack, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' A Quantum-Bayesian Route  to Quantum-State Space, Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 41, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [2] Fuchs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' and Schack, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' QBism and the Greeks:  why a quantum state does not represent an element of physical  reality, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', 90, 015104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [3] Fuchs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', Mermin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' and Schack, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' An In-  troduction to QBism with an Application to the Locality of  Quantum Mechanics, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 82, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [4] DeBrota, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', Fuchs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', Pienaar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', and Stacey, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Born’s rule as a quantum extension of Bayesian coher-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Physical Review A, 104(2), 022207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [5] Khrennikov,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  External Observer Reflections on  QBism, Its Possible Modifications, and Novel Applications, In:  Quantum Foundations, STEAM-H: Science, Technology, Engi-  neering, Agriculture, Mathematics & Health;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Khrennikov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' and  Toni B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Springer, Cham, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 93–118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [6] Khrennikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Towards better understanding QBism,  Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', 23 (1), 181–195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [7] Khrennikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Reflections on Zeilinger–Brukner Information  Interpretation of Quantum Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Found Phys 46, 836–844  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [8] Ozawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Intersubjectivity of outcomes of quantum  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='org/abs/1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='10893  [9] von Neuman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Mathematical foundations of quan-  tum mechanics (Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Press, Princenton) [Originally  published: Mathematische Grundlagen der Quanten-mechanik,  Springer, Berlin, 1932].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [10] Ozawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Quantum measuring processes for continuous  observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 25, 79–87  [11] Ozawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Soundness and completeness of quantum root-mean-  square errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 5, Article number: 1 (2019)  [12] Brukner,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  On  the  quantum  measurement  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='org/abs/1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='05255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [13] Zeilinger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=': A foundational principle for quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 29, 31–641 (1999)  [14] Brukner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', Zeilinger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=': Malus’ law and quantum informa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Acta Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Slovava 49, 647–652 (1999)  10  [15] Brukner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=', Zeilinger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=': Information invariance and quantum  probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Helstrom, Quantum Detection and Estimation Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Academic, New York, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Holevo, Probabilistic and Statistical Aspects of Quantum  Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' North-Holland, Amsterdam, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Ozawa, Optimal measurements for general quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 18, 1980, 11–28  [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Ozawa,  Realization of measurement and the standard  quantum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Squeezed and Nonclassical Light, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Tombesi  and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Pike, Plenum, New York, 1989,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 263–286,  arXiv:1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='01083 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [20] Plotnitsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Niels Bohr and complementarity: An in-  troduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Berlin and New York: Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [21] Bohr, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=': (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The Philosophical Writings of Niels Bohr, 3  vols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (Ox Bow Press, Woodbridge, CT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [22] Schr¨odinger, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' The present situation in quantum mechanics:  A translation of Schr¨odinger’s “Cat Paradox” paper (by: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  Trimmer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 124, 323–338 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' [Orig-  inally published: Die gegenw¨artige Situation in der Quanten-  mechanik, Naturwissenschaften 23, 807–812, 823–828, 844– 849  (1935)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [23] Atmanspacher, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Is the ontic/epistemic-distinction suf-  ficient to describe quantum systems exhaustively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In: Sympo-  sium on the Foundations of Modern Physics (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 15-32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [24] Atmanspacher, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' and Primas, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Epistemic and ontic  quantum realities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' In: Time, quantum and information (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' 301-  321).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Springer, Berlin, Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  [25] Khrennikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Quantum epistemology from subquan-  tum ontology: Quantum mechanics from theory of classical ran-  dom fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content=' Annals of Physics, 377, 147-163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}
+page_content='  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'}

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