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@@ -0,0 +1,996 @@
+schlably: A Python Framework for Deep
+Reinforcement Learning Based Scheduling
+Experiments
+Constantin Waubert de Puiseau, Jannik Peters, Christian Dörpelkus,
+Tobias Meisen
+Institute for Technologies and Management of the Digital Transformation
+University of Wuppertal
+Rainer-Gruenter-Str. 21, Wuppertal, 42119, NRW, Germany
+Abstract
+Research on deep reinforcement learning (DRL) based production schedul-
+ing (PS) has gained a lot of attention in recent years, primarily due to the
+high demand for optimizing scheduling problems in diverse industry settings.
+Numerous studies are carried out and published as stand-alone experiments
+that often vary only slightly with respect to problem setups and solution
+approaches. The programmatic core of these experiments is typically very
+similar. Despite this fact, no standardized and resilient framework for ex-
+perimentation on PS problems with DRL algorithms could be established so
+far. In this paper, we introduce schlably, a Python-based framework that
+provides researchers a comprehensive toolset to facilitate the development
+of PS solution strategies based on DRL. schlably eliminates the redundant
+overhead work that the creation of a sturdy and flexible backbone requires
+and increases the comparability and reusability of conducted research work.
+Keywords:
+Scheduling, Reinforcement Learning, JSSP, Framework
+Preprint submitted to SoftwareX
+January 12, 2023
+arXiv:2301.04182v1  [cs.LG]  10 Jan 2023
+
+Required Metadata
+Current code version
+Nr.
+Code metadata description
+Please fill in this column
+C1
+Current code version
+v0.1.0
+C2
+Permanent link to code/repository
+used for this code version
+https://github.com/tmdt-
+buw/schlably
+C4
+Legal Code License
+Apache License, 2.0 (Apache-2.0)
+C5
+Code versioning system used
+git
+C6
+Software code languages, tools, and
+services used
+Python,
+OpenAI
+Gym,
+RLlib,
+Weights and Biases
+C7
+Compilation requirements, operat-
+ing environments & dependencies
+Python 3.10
+C8
+If available Link to developer docu-
+mentation/manual
+https://github.com/tmdt-
+buw/schlably/tree/main/docs
+C9
+Support email for questions
+[email protected]
+Table 1: Code metadata (mandatory)
+1. Motivation and Significance
+Production scheduling (PS) is a challenging and highly researched prob-
+lem in operations research (OR) and optimization. It deals with allocating
+resources to production jobs over time to minimize criteria such as time,
+effort, and cost [1]. PS problems are considered NP-hard and therefore re-
+quire much computation effort to be solved sufficiently well with increasing
+problem sizes. In recent years, with increasingly powerful algorithms and
+computational hardware, deep reinforcement learning (DRL) has emerged as
+a promising tool to address PS problems [2, 3, 4, 5, 6, 7, 8]. DRL is a ma-
+chine learning paradigm, in which deep learning models are trained through
+interaction with an environment to autonomously derive solution strategies
+for sequential tasks [9].
+The young research field of DRL-based PS lies at the intersection of op-
+erations research and artificial intelligence and as such is characterized by
+a heterogeneous community with varying problem-solving approaches and
+technical skill sets. Yet, all empirical studies apply a very similar experi-
+mental setup consisting of the same main software components: An envi-
+ronment representing the production facility layout and logic, a scheduling
+problem generator, a DRL agent algorithm, and logging as well as evalua-
+tion tools. The difference between different experimental setups usually lies
+2
+
+within one or more of these components, for example by incorporating a new
+DRL algorithm [10, 11, 12], interaction logic between agent and environ-
+ment [6, 3], training procedure [13], learning objective [14, 12] or additional
+problem constraint [14, 15]. Regardless of large overlaps, all researchers im-
+plement their own individual experimentation framework with the following
+two consequences: Large initial ramp-up efforts when experimenting with
+new methodologies or custom problem settings, and scarcity of empirical
+comparisons to other works. In this paper, we address these shortcomings
+and present the software framework schlably for developing and evaluating
+DRL-based PS solutions. schlably provides the following contributions:
+• It is modular, so individual changes may be adapted without much
+overhead.
+• It works out of the box with functioning environments, data-generation
+scripts, agents, logging functions, training, and testing scripts.
+• It provides benchmark datasets for different scheduling problem classes
+and sizes.
+• It includes widely recognized benchmark algorithms and results.
+• It facilitates the application of algorithms designed for one problem
+class and size to other problems.
+schlably will accelerate the research area of DRL-based PS under real-
+world conditions by lowering the barrier of entry for researchers from different
+domains with different perspectives, levels of expertise, and objectives.
+2. Background and Related Work
+schlably started as a code base for experiments on a real-world inspired
+scheduling problem in the context of a university research project with in-
+dustrial partners. As such, several requirements became apparent early on
+that can be summarized in four general design goals. First, from the ap-
+plied industrial perspective, schlably has to offer the integration of DRL
+methods and heuristics which work out of the box. Second, it also has to
+cover different scheduling scenarios, e.g. including such bounded by resource
+constraints. Third, from the scientific research perspective, schlably should
+support detailed comparable evaluations of methods. Lastly, it has to be
+easy to interact with at the code level, to enable students with limited expe-
+rience to quickly understand the topic, concepts, and implementation. The
+implications of these design goals are discussed in this section.
+3
+
+In the following, we present an overview of related published experi-
+mental frameworks and compare them to our design goals in schlably. In
+the comparison, we include frameworks dedicated to being used by others
+[16, 17, 18, 19, 20] and frameworks published in a supplementary fashion
+along with research papers and projects [21, 22, 23, 24, 25, 26], because in
+practice both may serve as starting points for additional experiment designs.
+The frameworks were found via references in scientific publications and a
+search of "Scheduling Reinforcement Learning" on GitHub. We do not claim
+the list to be exhaustive but are not aware of any other popular frameworks
+at the time of writing this paper. Commercial or proprietary scheduling soft-
+ware was excluded because license fees and other accompanying challenges
+introduce a major barrier. However, we are not aware of any commercial
+software that provides the tight integration of DRL and PS. Table 2 provides
+an overview of the frameworks.
+In addition, we assessed them regarding
+their fulfillment of our design goals which are formally categorized into the
+following four groups.
+Pre-Implemented Benchmarks. Several frameworks either provide pre-
+implemented DRL agents and scripts for training or the easy integration of
+agents from popular DRL libraries, e.g. from StableBaselines [27] or RLlib
+[28]. Like [20], our goal is to enable and facilitate both, the manual extensions
+of basic DRL algorithms, like DQN and PPO, as well as the usage of powerful
+third-party libraries. Both options are important to empower users to choose
+the appropriate approach for their respective research interests. Additionally,
+for the sake of comparability, it is crucial to provide common benchmarks
+in the form of popular priority dispatching rules (PDRs), such as Shortest-
+Processing-Time-First, and a flexible optimal solver that can handle several
+scheduling problem types. Most frameworks only cover a few PDRs, often
+missing competitive ones, such as Most-Tasks-Remaining and even random
+baselines, where agent actions are sampled from a normal distribution.
+Scheduling Instance Generation. Generating new data with varying prob-
+lem cases is necessary to enable comprehensive training and testing of a DRL
+agent. Accordingly, a suitable framework must implement a flexible problem
+instance generator. This generator enables the user to create scheduling prob-
+lems of different popular categories (e.g. the Job Shop Scheduling Problem
+(JSSP) or the Flexible Job Shop Scheduling Problem (FJSSP) [1]) instances
+with any combination of instance variables, like the number of jobs, number
+of tasks, runtimes, and more. Moreover, its design simplifies the integration
+4
+
+[16] [17] [18] [21] [19] [20] [22] [23] [24] [25] [26] schlably
+B
+Implemented RL-agents
+�
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+Implemented opt. solver
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+Interface for RLLib
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+Flexible Data Generation
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+Resource constraint tool
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+L
+Log achieved results
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+Evaluate achieved results
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+Visualize Gantt-Chart
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+User manual in Readme
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+Legend:
+�not fulfilled
+�half-fulfilled
+�fulfilled
+Table 2: Overview of related frameworks and their fulfillment regarding pre-
+implemented benchmarks (B), scheduling instances (S), logging and evalua-
+tion (L), and code usability (C)
+5
+
+of additional scheduling problem types to encourage the implementation of
+individual, more complex, or more specific use cases. Finally, to the best
+of our knowledge, this is the first framework providing an optional resource
+constraint. Concretely, the user is able to specify a required tool per opera-
+tion.
+Logging and Evaluation. Logging results and evaluation metrics in a struc-
+tured manner is key for quick feedback during training runs, but also for
+identifying patterns in and drawing conclusions from large-scale experiments.
+Our objective with schlably is to provide extensive logging options that may
+be turned on and off, and where results and models may be shared to promote
+collaboration on projects. This design goal is met to a large degree by [20]
+from which we took much inspiration in this regard. All other frameworks
+do not address this design goal. For the evaluation of solutions generated
+by DRL agents, a comprehensive framework should apply the benchmarks
+mentioned above and provide an overview of the overall performance. Most
+of the reviewed frameworks lack functionality in this aspect. Moreover, to
+get a graphical overview and to visually support the tracking of very specific
+actions in a production schedule, a Gantt chart plotter is useful for human
+inspection. The Gantt chart should display all metadata of operations, e.g.
+the runtime or required tool, and has been found to be helpful for debugging
+and evaluating DRL agents. Many other frameworks, but not all, include a
+Gantt chart plotter.
+Code Usability. As usability is of utmost importance, a framework has to
+offer easy access through full documentation and must include a README,
+user application programming interface (API) manual, and formal functional-
+ity description. Within the reviewed frameworks, only [20] covers all criteria.
+To be usable with different skill sets our explicit goal is to enable users to
+start experimenting with small design decisions by only using the configura-
+tion files but at the same time facilitate substantial logical changes to routines
+and components by means of their own implementations. For that reason, we
+would favor comprehensibility over efficiency wherever a trade-off is unavoid-
+able. This requires a careful balance. In our opinion, all other frameworks
+overemphasize one side: [20] offers many functional changes through configu-
+ration files but at the cost of a comparatively complex software architecture.
+On the other hand, all other frameworks are much smaller and easier to get
+an overview of, at the expense of limited functionality. Lastly, a framework
+with a claim to widespread use should stick to conventional APIs. In the
+6
+
+context of DRL, the most commonly used API is the OpenAI Gym [29] API.
+Only half of the reviewed frameworks adhere to it.
+3. Software Architecture
+This chapter describes our framework with focus on implementation-
+specific details. We are providing a general overview of the code itself, fo-
+cusing on currently existing exemplary implementations while also pointing
+out open interfaces. Furthermore, we describe details regarding the main
+components of schlably to demonstrate the realization of the design goals, as
+introduced in Chapter 2, and to enable users to fit schlably to their needs.
+The overall structure is illustrated in Figure 1. We divided the code base
+into six main components, which are described below in detail. Following
+this component-oriented approach, and in combination with comprehensive
+code documentation, schlably adheres to the objective of the fourth design
+goal, which requires easy interaction and usability at the code level.
+agents
+code_tests
+environments
+utils
+visuals_generator
+data_generator
+GanttChartPlotter
++ VISUALS_DIRECTORY
++ get_gantt_chart_image
++ get_gantt_chart_gif_and_save
+EvaluationHandler
++ rewards
++ makespan
++ record_environment_episode
++ evaluate_test
+Logger
++ WANDB_PROJECT
++ LOG_MODE_DEFAULT
++ record
++ dump
++ dump_wandb
+ui_tools
+file_handler
+ConfigHandler
+DataHandler
+FileChooser
+ProgressBar
+MessageBox
+agent_tests
+data_generator_tests
+visuals_generator_tests
+Runner
+Env
++ num_jobs
++ num_machines
++ reset
++ step
+EnvIndirectAction
++ get_action_mask
+EnvironmentLoader
+ + ENVIRONMENT_MAPPER_DICT
++ load
++ check_environment_agent_compatibility
+InstanceFactory
+SPFactory
++ SP(Enum)
++ generate_instances
+Task
++ job_index
++ task_index
+heuristic
+reinforcement_learning
+solver
+dqn
+ppo
+ppo_masked
+intermediate_test
+test
+train
+Figure 1: Overview of schlably project and code structure.
+7
+
+Data generator. The general data structure of scheduling problems, as
+used in schlably, is represented by so-called instances. A user can generate
+infinite instances of a scheduling problem, however, each instance is a spe-
+cific configuration and entity. The specific configuration, contained within
+an instance, is given by a number of jobs, with a job simply being an encom-
+passing logical container consisting of individual tasks. The data_generator
+component incorporates the necessary classes to generate such an instance
+and the individual tasks. From the scheduling problem point-of-view, it is
+the centerpiece of the problem formulation and representation. The Task
+class is a specifically designed data class and its entities are the atomic
+units of a scheduling problem instance.
+Such an instance can be created
+via the SPFactory, which allows the generation of different types of schedul-
+ing problems that are given via the included Enum. If users would like to
+introduce a new type of scheduling problem into schlably, they would have
+to include their function in this class and add it to the Enum. Finally, the
+InstanceFactory enables high-level access to the problem factory class and
+manages the configuration-based creation of batches of instances. Thus, the
+data_generator component realizes the foundation of the second design
+goal, which requires the implementation and handling of different scheduling
+scenarios.
+Environment. An environment defines the observation space, action space,
+and reward strategy.
+Thus, it represents a simulation of the agent’s en-
+vironment and interaction dynamics and is the central piece of any DRL
+approach. All schlably environments are included in the environments com-
+ponent. Exemplary, we provide a simple scheduling Env as well as a derived
+version named EnvIndirectAction to showcase the expandability. All en-
+vironments adhere to the Gym API and are explicitly derived from a base
+Gym environment. The EnvironmentLoader class enables high-level access
+and management of the different environment types and appropriate algo-
+rithms, as not all algorithms are feasible for every environment. New envi-
+ronments have to be included in this component and added to the managing
+EnvironmentLoader. This encapsulated approach, in conjunction with the
+data_generator component, represents the implementation of the second
+design goal.
+Agent. The agents component combines the heuristic functionalities, the
+solver, and implementations of DRL algorithms as well as the train and test
+functions for the DRL approach. Users can to integrate functionalities from
+8
+
+other DRL frameworks, like more extensive training procedures, model types,
+and learning algorithms via pre-defined interfaces. As such, the agents com-
+ponent realizes the first design goal, to support and simplify the integration
+of out-of-the-box-methods as well as pre-implemented benchmarks.
+Visual generator. The component visuals_generator incorporates all
+classes and scripts which are used to create visualizations of the problem
+instances and generated solutions.
+These functionalities are intentionally
+isolated as different scheduling problem environments and still share the same
+visualization approach. schlably, for example, introduces a GanttChartPlotter
+that enables a user to generate individual Gantt chart images (see Figure 2b)
+or create a GIF of the scheduling progress. Thus, it is part of the implemen-
+tation of the third design goal.
+Utils. The utils component aggregates classes and functions which have a
+supporting character for the main functionalities of schlably. Specifically, it
+includes user interface components (ui_tools), data interface components
+(file_handler), e.g. to load and save data, and the high-level Logger class.
+Accordingly, the utils component realizes the third design goal, facilitating
+logging and evaluations for comparisons.
+Code tests. All code tests that ensure the crucial functionality of the de-
+scribed components are collected in the code_tests component. Up to this
+point, we included multiple unit tests with a central Runner. These are also
+intended as an example for users that plan to extend the code base.
+4. Illustrative Example
+To illustrate a typical use case, we consider a scenario in which an ML
+engineer wants to compare the learning behavior of two PPO agents. It is also
+part of our tutorial in the documentation. One agent is trained on 6x6 JSSP
+instances and receives a reward based on the change in the time to complete
+all tasks (i.e. the makespan) per step, as proposed in [3]. This setup is also the
+default setting delivered in the framework. The other one is trained on a 3x4
+tool-constrained JSSP instance and receives a zero reward per step with the
+exception of the last step, where the reward is equal to the overall achieved
+makespan. The remaining training parameters are kept constant. The second
+training requires only minimal manual changes to the base model. These
+include setting different configuration parameters, generating new data, and
+9
+
+Figure 2: Comparing agent runs in Weights&Biases (screenshot from the web interface
+shown on the left-hand side). a) Visualized training curves for interpreting the learning
+performance of the agent. b) Gantt chart depicting the solution of the trained agent on a
+selected test instance. c) Table providing evaluation results and comparison of the trained
+agents and benchmark methods on the test instances.
+changing the reward function in the base environment. Details may be found
+in the documentation. The integrated interface to Weights&Biases [30] makes
+it easy to compare the training curves and achieved results, as depicted in
+Figure 2.
+The described short example reflects several of our design goals. Figure
+2c) demonstrates that the agents’ performance is automatically compared
+to many other benchmarks and with respect to different dimensions such
+as the reward or the gap to the optimal solver.
+The continuous logging
+and graphical depiction are visible in Figure 2a) and b). The example also
+showcases our understanding of high code usability. The experiments could
+be defined by changing training parameters (only a few lines in configuration
+files) and minimal intended changes to the source code. Examples of the
+most common changes which are intended to be coded are explained in more
+detailed follow-along tutorials in the provided documentation.
+10
+
+wandb web interface
+agent_training/entropy_loss
+agent_training/policy_gradient_loss
+agent_training/value_loss
+a
+ still-river-17
+swept-blaze-15
+ still-river-17
+- swept-blaze-15
+ still-river-17
+swept-blaze-15
+400
+0.1
+0.08
+300
+agent_.training/loss
+1.42
+0.06
+200
+0.04
+1.44
+100
+1.46
+runs.summary["Final Evaluation Table"]
+Agent
+Mean Rewal
+Mean Tardin
+Tardiness M
+Mean Makesp
+MakespanS
+Tardiness ST
+Gap To Solv
+1 agent
+-81.333
+35.167
+20.667
+81.333
+13.237
+53.108
+17.167
+Ganttchart
+2 rand
+-94.667
+10.949
+19.117
+30.5
+b
+3EDD
+-180.5
+15.575
+68.966
+116.333
+4  SPT
+155.16
+24.423
+56.782
+91
+5 MTR
+-76.333
+23.5
+17.833
+76.333
+11.146
+31.261
+12.167
+C
+6 LTR
+-187.333
+255.5
+93.5
+187.333
+18.436
+52.677
+123.167
+- 6 of 14
+Export as CSV Columns...
+Reset Table
+lidden Panels 5. Impact
+schlably is useful for the entire community around PS with DRL. Com-
+pared to other frameworks, it is particularly useful to reduce the entry barrier
+for researchers from the OR or other related domains, who want to empir-
+ically explore a new methodology for scheduling problems, and for DRL
+researchers who want to test a new algorithm on a challenging and impactful
+problem domain. We believe that the seamless interchangeability of prob-
+lem settings offered by schlably will also encourage researchers in the domain
+of PS with DRL to try out methodologies applied to one particular prob-
+lem setting (e.g. 6x6 JSSP) on different problem settings (e.g. 11x11 tool-
+constrained JSSP). This has the potential to greatly speed up the transfer of
+research from academic problems to real-world problems.
+In several projects where our test partners and we have used schlably, it
+has significantly increased the throughput of experiments. This is achieved
+because new methodological ideas can be integrated more quickly and the
+results of experiments can be compared more easily. schlably facilitates the
+generation of new problem instances and the training and evaluation of cus-
+tom DRL agents. Due to the various pre-implementations in the framework,
+such as training and testing routines, well-known scheduling benchmarks,
+and visualization of logged results, it is much easier to conduct experimental
+research in DRL for PS. In addition, collaboration has become more effec-
+tive because design changes can be compared easily and the results of peers
+can be viewed online through Weights&Biases. We have further experienced
+a substantial increase in productivity in research projects, where new re-
+searchers and university students, who had no prior domain knowledge and
+little coding skills, had to conduct experiments on the PS domain. This, we
+mainly attribute to the code documentation and modular structure, but also
+to the fact that schlably is 100% written in Python and therefore runs on all
+relevant operation systems.
+6. Discussion and limitations
+In its current state, schlably serves as a useful framework for empiri-
+cal DRL-based PS research. It has reached a maturity level, at which it
+works out-of-the-box and, to the best of our knowledge, offers the broadest
+range of different easy-to-implement design choices compared to any pub-
+lished framework. schlably, on the one hand, is intended to be abstract and
+modular enough to offer different instance generation, training, and testing
+configurations without many lines of code. On the other hand, it is designed
+to not be too interwoven in its code structure to hinder the extension with
+11
+
+fundamentally different features experts might find desirable. As such, the
+development required a balancing act and certain compromises, which some
+may see as limitations.
+For example, one deliberate choice was made in
+favor of a class-based problem description as opposed to a vector representa-
+tion. The class-based description simplifies the search and usage of certain
+information about the current state of jobs and increases code readability
+compared to a vector problem representation. Hence, the choice was made
+between readability and computational efficiency in favor of the former.
+7. Conclusions
+In this paper, we introduced schlably, a software framework for research
+on DRL-based PS. With the release of the framework, we strive towards
+two main goals: the first is to lower the entry barrier for researchers, who
+have little experience with production scheduling, deep reinforcement learn-
+ing (DRL) and/or coding. The second goal is to encourage researchers al-
+ready active in the field to apply and test their methods on other problem
+settings, which is largely facilitated by schlably. Both goals aim at promoting
+the transfer of DRL methods to real-world scheduling applications. In the
+future, we plan to include more problem settings, such as the dynamic JSSP
+and stochastic properties of environments like machine breakdowns to get
+even closer to real-world scenarios.
+8. Conflict of Interest
+We wish to confirm that there are no known conflicts of interest associated
+with this publication and there has been no significant financial support for
+this work that could have influenced its outcome.
+Acknowledgements
+This research work was undertaken within the research project AlphaMES
+funded by the German Federal Ministry for Economic Affairs and Climate
+Action (BMWK).
+References
+[1] M. Pinedo, Scheduling:
+Theory, algorithms, and systems, fifth edi-
+tion Edition, Springer International Publishing, 2016. doi:10.1007/
+978-3-319-26580-3.
+12
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+[2] I. Bello, H. Pham, Q. Le V, M. Norouzi, S. Bengio, Neural combinatorial
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+1611.09940.
+URL http://arxiv.org/pdf/1611.09940v3
+[3] C. Zhang, W. Song, Z. Cao, J. Zhang, P. S. Tan, X. Chi, Learning
+to dispatch for job shop scheduling via deep reinforcement learning,
+Advances in Neural Information Processing Systems 33 (2020) 1621–
+1632.
+[4] A. Kuhnle, J.-P. Kaiser, F. Theiß, N. Stricker, G. Lanza, Designing
+an adaptive production control system using reinforcement learning,
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+doi:
+10.1007/s10845-020-01612-y.
+[5] T. van Ekeris, R. Meyes, T. Meisen, Discovering heuristics and meta-
+heuristics for job shop scheduling from scratch via deep reinforcement
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+gistics : CPSL 2021 1 (2021) 709–718. doi:10.15488/11231.
+[6] V. Samsonov, K. B. Hicham, T. Meisen, Reinforcement learning in man-
+ufacturing control: Baselines, challenges and ways forward, Engineering
+Applications of Artificial Intelligence vol. C (112) (2022).
+[7] C. W. de Puiseau, R. Meyes, T. Meisen, On reliability of reinforcement
+learning based production scheduling systems: a comparative survey,
+Journal of Intelligent Manufacturing 33 (4) (2022) 911–927. doi:10.
+1007/s10845-022-01915-2.
+[8] Sebastian Pol, Schirin Baer, Danielle Turner, Vladimir Samsonov, To-
+bias Meisen, Global reward design for cooperative agents to achieve
+flexible production control under real-time constraints, in: Proceedings
+of the 23rd International Conference on Enterprise Information Sys-
+tems - Volume 1: ICEIS„ INSTICC, SciTePress, 2021, pp. 515–526.
+doi:10.5220/0010455805150526.
+[9] R. S. Sutton, A. Barto, Reinforcement learning: An introduction, second
+edition Edition, Adaptive computation and machine learning, The MIT
+Press, Cambridge, Massachusetts, London, England, 2018.
+[10] A. Rinciog, C. Mieth, P. M. Scheikl, A. Meyer, Sheet-metal produc-
+tion scheduling using alphago zero, Proceedings of the Conference on
+Production Systems and Logistics :
+CPSL 2020 1 (2020) 342–352.
+doi:10.15488/9676.
+13
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+[11] M. Monaci, V. Agasucci, G. Grani, An actor-critic algorithm with deep
+double recurrent agents to solve the job shop scheduling problem (2021).
+doi:10.48550/ARXIV.2110.09076.
+URL https://arxiv.org/pdf/2110.09076
+[12] S. Luo, Dynamic scheduling for flexible job shop with new job insertions
+by deep reinforcement learning, Applied Soft Computing 91 (2020). doi:
+10.1016/j.asoc.2020.106208.
+[13] Z. Iklassov, D. Medvedev, R. Solozabal, M. Takac, Learning to gener-
+alize dispatching rules on the job shop scheduling, Advances in Neural
+Information Processing Systems 33 (2020) 1621–1632. doi:10.48550/
+ARXIV.2206.04423.
+URL https://arxiv.org/pdf/2206.04423
+[14] A. H. Sakr, A. Aboelhassan, S. Yacout, S. Bassetto, Simulation and
+deep reinforcement learning for adaptive dispatching in semiconductor
+manufacturing systems, Journal of Intelligent Manufacturing 1 (2021)
+1–14. doi:10.1007/s10845-021-01851-7.
+URL
+https://link.springer.com/article/10.1007/
+s10845-021-01851-7
+[15] P. C. Luo, H. Q. Xiong, B. W. Zhang, J. Y. Peng, Z. F. Xiong, Multi-
+resource constrained dynamic workshop scheduling based on proximal
+policy optimisation, International journal of production research 60 (19)
+(2022) 5937–5955. doi:10.1080/00207543.2021.1975057.
+[16] P.
+Tassel,
+M.
+Gebser,
+K.
+Schekotihin,
+Job_shop_scheduling_problem_with_reinforcement_learning,
+GitHub (2021).
+URL
+https://github.com/dmksjfl/Job_Shop_Scheduling_
+Problem_with_Reinforcement_Learning
+[17] L. Zheng, L. Zijun, Y. Dai, X. Li, B. Yuan, Gymjsp, GitHub (2022).
+URL https://github.com/yunhui1998/gymjsp
+[18] Dr-ilyassPHx, Auto-rl-competition: Dynamic job shop scheduling prob-
+lem challenge, GitHub (2022).
+URL https://github.com/Dr-ilyassPHx/Auto-RL-Competition
+[19] P. Tassel, P. Willms, Jssenv: An openai gym environment for the job
+shop scheduling problem., GitHub (2022).
+URL https://github.com/prosysscience/JSSEnv
+14
+
+[20] V. Samsonov, optimization-with-rl-in-manufacturing-control, GitHub
+(2021).
+URL
+https://github.com/v-samsonov/
+optimization-with-rl-in-manufacturing-control
+[21] tejaswini medi, Rl_scheduling_system, GitHub (2022).
+URL https://github.com/tejaswini-medi/RL_scheduling_system
+[22] D. Venturelli, D. Marchand, G. Rojo, job-shop-scheduling: Determine a
+schedule for running a set of jobs., GitHub (2015).
+URL https://github.com/dwave-examples/job-shop-scheduling
+[23] samy barrech, Flexible-job-shop-scheduling-problem, GitHub (2018).
+URL
+https://github.com/samy-barrech/
+Flexible-Job-Shop-Scheduling-Problem
+[24] C. Zhang, W. Song, Z. Cao, J. Zhan, P. Tan, X. Chi, L2d: Official
+implementation of paper "learning to dispatch for job shop scheduling
+via deep reinforcement learning", GitHub (2020).
+URL https://github.com/zcaicaros/L2D
+[25] V. Kumar, Jobschedulingrlenv:
+Reinforcement learning environment
+for job scheduling written in python., GitHub (2019).
+URL
+https://github.com/TimeTraveller-San/
+JobSchedulingRLenv
+[26] T. van Ekeris, jobshop: Deep reinforcement learning (drl) for jobshop
+scheduling problems (jsp) - an evaluation framework, GitLab (2020).
+URL https://gitlab.com/tvanekeris/jobshop
+[27] A. Raffin, A. Hill, A. Gleave, A. Kanervisto, M. Ernestus, N. Dor-
+mann, Stable-baselines3: Reliable reinforcement learning implementa-
+tions, Journal of Machine Learning Research 22 (268) (2021) 1–8.
+URL http://jmlr.org/papers/v22/20-1364.html
+[28] Eric Liang, Richard Liaw, Robert Nishihara, Philipp Moritz, Roy Fox,
+Ken Goldberg, Joseph Gonzalez, Michael Jordan, Ion Stoica, Rllib: Ab-
+stractions for distributed reinforcement learning, International Confer-
+ence on Machine Learning (2018) 3053–3062.
+URL https://proceedings.mlr.press/v80/liang18b.html
+[29] G. Brockman, V. Cheung, L. Pettersson, J. Schneider, J. Schulman,
+J. Tang, W. Zaremba, Openai gym (2016). arXiv:arXiv:1606.01540.
+15
+
+[30] L. Biewald, Experiment tracking with weights and biases (2020).
+URL https://www.wandb.com/
+Current executable software version
+Ancillary data table required for sub version of the executable software:
+(x.1, x.2 etc.)
+kindly replace examples in right column with the correct
+information about your executables, and leave the left column as it is.
+Nr.
+(Executable)
+software
+meta-
+data description
+Please fill in this column
+S1
+Current software version
+v0.1.0
+S2
+Permanent link to executables of
+this version
+https://github.com/tmdt-
+buw/schlably
+S3
+Legal Software License
+Apache License, 2.0 (Apache-2.0)
+S4
+Computing
+platforms/Operating
+Systems
+Python,
+OpenAI
+Gym,
+DRL-
+lib,Weights and Biases
+S5
+Installation requirements & depen-
+dencies
+Python 3.10
+S6
+If available, link to user manual - if
+formally published include a refer-
+ence to the publication in the refer-
+ence list
+https://github.com/tmdt-
+buw/schlably/docs
+S7
+Support email for questions
+[email protected]
+Table 3: Software metadata (optional)
+16
+

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+arXiv:2301.01213v1  [physics.optics]  3 Jan 2023
+Numerical study of magneto-optical binding between two dipolar particles under
+illumination by two counter-propagating waves
+Ricardo Mart´ın Abraham-Ekeroth
+Instituto de F´ısica Arroyo Seco, IFAS (UNCPBA), Tandil, Argentina and
+CIFICEN (UNCPBA-CICPBA-CONICET), Grupo de Plasmas Densos, Pinto 399, 7000 Tandil, Argentina∗
+The formation of a stable magneto plasmonic dimer with THz resonances is theoretically studied
+for the principal directions of the system. Unlike a recent report, our work provides a complete
+description of the full photonic coupling for arbitrary magnetic fields as, for instance, unbalanced
+particle spins. As an illustration, we consider two small, n-doped InSb nanoparticles under illumina-
+tion by two counter-propagating plane waves. Remarkably, when an external magnetic field exists,
+the symmetry in the system is broken, and a resonant radiation pressure for the dimer appears.
+Similarly, tunable inter-particle forces and spins are exerted on the non-reciprocal dimer. The sys-
+tem is also characterized when the magnetic field is absent. Moreover, we show how the mechanical
+observables truly characterize the dimer since their resonance dependency contains detailed informa-
+tion about the system. Unlike far-field observables like absorption, mechanical magnitudes depend
+on the system’s near-field. In addition, the nature of the particle spins is originally explained by
+the energy flow’s behavior around the dimer. This work constitutes a generalization of any previ-
+ous approach to optical binding between small nanoparticles. It paves the way for fully controlling
+optical matter and nano factory designs based on surface plasmon polaritons.
+Keywords:
+Magneto plasmonics,
+Spin torques,
+Dimers, Optical Binding, Photonics, Poynting field,
+Radiation Pressure, Optical Matter
+I.
+INTRODUCTION
+Optical matter (OM) consists of arrays of micro or
+nanoparticles that are somehow bound and controlled
+by light [10].
+An OM able to self-assemble at will
+to develop solid technology is a long-standing goal in
+photonics [38]. The background for OM is the particle
+manipulation by optical forces; the first results were
+applied to microparticles due to the lack of technology
+and the presence of thermal noise for smaller systems
+[4, 5].
+Generally, OM comprises multiple particles
+subjected to electromagnetic forces that come from
+their mutually scattered light. However, this multiple
+scattering phenomena can be very complex and bring
+about unusual effects such as “non-reciprocal” forces,
+torque opposite to the illumination angular momen-
+tum, and non-conservative forces [2, 38]. Appropri-
+ate control of OM by forces and torques could lead to
+programmable materials for optomechanical, rheologi-
+cal, and biological applications. In this respect, many
+works studied the optical binding between nanopar-
+ticles as a primary tool to develop OM. Some ap-
+proached specific combinations of optical beams like
+those with programmable phase [24, 31]. For example,
+light-induced rotation of objects holds potential for
+various applications such as sensing, cargo transporta-
+tion, drug delivery, and micro/nanosurgery [3, 7, 42].
+Optical traps use the combination of beams as a po-
+tent characterization tool for material science and bio-
+physics, as in Ref. [23], which uses electrostatic focus-
+ing to obtain the mass spectrum of SARS-CoV-2 and
+∗ [email protected]
+BoHV-1 virions. However, the high intensity at the
+focal spot may introduce laser heating, which is an
+issue for bio applications [31].
+On the other hand, recent advances in THz tech-
+nology call for new devices and materials that exhibit
+a non-reciprocal behavior for photonic networks and
+optical information processing [41].
+Non-reciprocal
+devices are a crucial component of modern commu-
+nication technology. They are nowadays required for
+miniaturized electronic and photonic devices [32]. One
+way to create optical non-reciprocity at the THz range
+is using magneto-optical (MO) systems like graphene,
+hexaferrites, and semiconductors [27]. For instance,
+Ref. [15] presented MO measurement of several sam-
+ples of InSb with different carriers and carrier con-
+centrations for low external magnetic field and room
+temperatures.
+With these advantages, dimers and
+trimers of InSb particles have been studied to enhance
+THz spectroscopy by forming electric and magnetic
+hotspots in the gap between them [6, 40]. Anisotropic
+materials like InSb in OM would make it strongly
+dependent on the beam combinations, allowing for
+countless possibilities [39].
+Recently, several efforts
+have focused on MO nanoparticle systems to shape
+OM and optical traps with a reasonable degree of con-
+trol and accuracy, besides other relevant applications
+[1, 18, 20, 22, 26]. on a broader sense, magnetoplas-
+monics relates the plasmonic behavior of nanoparticles
+with the presence of external magnetic fields.
+The
+modulation and tunability of plasmonic resonances
+offered by magnetoplasmonics results auspicious for
+ultra-sensitive sensors and active plasmonic devices
+[28].
+In particular, the formation of stable optical
+binding between two small magnetoplasmonic parti-
+cles has been lately studied [19]. Equilibrium binding
+distances were predicted and found tunable by the
+incoming wave’s polarization state and the magnetic
+field’s magnitude. However, the model developed in
+that report is valid only for relatively small magnetic
+
+2
+field values. Moreover, it predicts stable dimers only
+using alternating magnetic static fields and polariza-
+tion angles to remove azimuthal, unbalanced forces.
+More importantly, this work needs to discuss possible
+rotations of the particles due to angular momentum
+transfer in the multiple scattering scheme.
+In this paper, we study the formation of stable MO
+dimers for small nanoparticles in a complete frame-
+work involving all the possible optomechanical induc-
+tions. The dimer’s isotropic and anisotropic responses
+are assessed as a base for OM designs, i.e., under the
+presence/absence of an external magnetic field. This
+field can be of arbitrary magnitude in our model. The
+illumination consists of two counter-propagating plane
+waves with circular polarization, which simulates a
+simple optical trap in the vacuum.
+We found sev-
+eral possibilities to create stable dimers even when
+the magnetic field is off. The beams do not exert net
+forces for reciprocal dimers but may exert torques on
+them. On the contrary, there is a net radiation pres-
+sure and spin for the whole system when the static
+field is on, allowing complete control of the system’s
+movement. The results will enable one to infer that
+the mechanical variables can be used as near-field ob-
+servables to explore the content of unknown samples.
+Conversely, they can be used to accurately control the
+dimer’s creation/destruction and its mobility. Finally,
+the spins predicted are explained in terms of the en-
+ergy flows around the dimer, which constitutes a novel
+scattering-force effect for interacting particle arrays.
+II.
+MODEL
+In the following, we assume two equal particles of
+the same non-reciprocal material immersed in the vac-
+uum. Then the method of discrete dipoles (MDD) [17]
+simplifies considerably to
+p1 = ǫ0ˆαE0,1 + k2
+0 ˆα ˆGp2,
+(1)
+p2 = ǫ0ˆαE0,2 + k2
+0 ˆα ˆGp1,
+(2)
+where ˆα is the polarizability tensor representing the
+particles. The following definition automatically in-
+cludes the radiative corrections necessary to fulfill the
+optical theorem [21]
+ˆα =
+�
+ˆα−1
+0
+− ik3
+0 ˆI
+6π
+�−1
+(3)
+where ˆα0 is the so-called quasistatic polarizability,
+which can be given by
+ˆα−1
+0
+= 1
+V
+�
+ˆL + [ˆǫr − ˆI]−1�
+(4)
+being V the particles’ volume, ˆL = ˆI/3 is the elec-
+trostatic depolarization tensor specified for spheres or
+cubes, and ˆǫr is the relative dielectric tensor.
+The
+system of equations 1 can be solved straightforwardly,
+leading to
+p1 = ǫ0 ˆF
+�
+E0,1 + k2
+0 ˆα ˆGˆαE0,2
+�
+,
+(5)
+p2 = ǫ0 ˆF
+�
+E0,2 + k2
+0 ˆα ˆGˆαE0,1
+�
+,
+(6)
+where we define ˆF =
+�
+ˆα − k4
+0 ˆGˆα ˆG
+�−1
+. In this work,
+a counter-propagating configuration is assumed as a
+superposition of two left-handed circularly polarized
+(LCP) plane waves with the same intensity I0 [11, 18],
+see Fig. 1, namely,
+E0 = E0
+√
+2
+�
+(ˇx + iˇy) eik0z + (ˇx − iˇy) e−ik0z�
+.
+(7)
+This field is used in Eq. 5 to calculate the incident
+field at the particles’ positions E0,1 and E0,2.
+The absorption cross section of the system can be
+calculated once the dipole moments are known by
+σabs =
+k0
+ǫ0wtot
+E
+Im
+�
+p1 ·
+�
+ˆα−1
+0 p1
+�∗ + p2 ·
+�
+ˆα−1
+0 p2
+�∗�
+(8)
+where wtot
+E
+= ǫ0|E0|2. The i-component of the forces
+exerted on each particle can be obtained from the
+time-averaged force within the Rayleigh approxima-
+tion [13]. This is
+F1,i = 1
+2Re{pt
+1[∂iE∗(r, ω)|r=r1}
+(9)
+F2,i = 1
+2Re{pt
+2[∂iE∗(r, ω)|r=r2}
+(10)
+where the derivatives of the total field ∂iE(r, ω)|r=rn
+at the dipoles’ positions rn, n = {1, 2} can be obtained
+from [12]:
+∂iE(r, ω)|r=r1 = ∂iE0(r, ω)|r=r1+
++ k2
+0
+ǫ0
+(∂iG(r, r2))r=r1p2]}
+(11)
+∂iE(r, ω)|r=r2 = ∂iE0(r, ω)|r=r2+
++ k2
+0
+ǫ0
+(∂iG(r1, r))r=r2p1]}
+(12)
+The total force exerted on the dimer results from
+adding the force components for each particle, namely,
+Ftot,i = F1,i + F2,i. In particular, the net radiation
+pressure for the dimer under the illumination given
+by Eq. 7 is defined by taking i = 3, or the z compo-
+nents, as
+Ftot,z = F1,z + F2,z
+(13)
+Another useful mechanical variable is the binding
+force, which in the present case is defined as
+∆ = (F1 − F2) · ˇn
+(14)
+where ˇn =
+r2−r1
+|r2−r1| is the dimer’s versor. The optical
+torques can also be calculated, as given in Ref. [14]:
+Nspin,1 =
+1
+2ǫ0
+Re
+�
+p1 ×
+��
+ˆα−1
+0
+�∗ p∗
+1
+����
+(15)
+Norb,1 = r1 × F1
+(16)
+N1 = Nspin,1 + Norb,1
+(17)
+
+3
+FIG. 1.
+(Color online) Dimer configurations and in-
+cident
+waves
+treated
+in
+this
+work.
+Two
+counter-
+propagating waves with left circular polarization illumi-
+nate the magneto-optical dimer. The leading example con-
+sists of two n-doped InSb particles separated by a gap of
+a particle’s diameter, g = 2R. (a) Parallel [(b) perperdic-
+ular] configuration. The static magnetic field B is parallel
+to +z direction (green arrow). In (b), φ is the azimuthal
+angle of the dimer’s position.
+Nspin,2 =
+1
+2ǫ0
+Re
+�
+p2 ×
+��
+ˆα−1
+0
+�∗ p∗
+2
+��
+(18)
+Norb,2 = r2 × F2
+(19)
+N2 = Nspin,2 + Norb,2
+(20)
+The definitions of the orbital and spin torques were
+discussed previously in Refs. [14, 33, 34], among oth-
+ers. The spin torques are always defined with respect
+to the centers of the particles. Otherwise, the refer-
+ence system is located at the dimer’s center of mass,
+and orbital torques are set. Thus, the total torque
+exerted on the dimer is
+Norb = Norb,1 + Norb,2
+(21)
+Nspin = Nspin,1 + Nspin,2
+(22)
+Ntot = N1 + N2 = Norb + Nspin
+(23)
+In particular, this study simulates nanoparticles made
+of n-doped Indium antimonide (n-InSb) [37].
+In-
+dium antimonide (InSb) is one example of the most
+widely studied polar semiconductors for magnetoplas-
+monic applications because it can be easily doped
+for sizable magnetic-induced effects [15, 30]. As re-
+viewed in Refs. [30, 37], n-InSb is an exciting mate-
+rial that has two kinds of surface resonances in the
+absence of static field, namely, the phonon polari-
+ton (SPhP, higher-energy) and the plasmon polariton
+(SPP, lower-energy).
+Its model properties were de-
+scribed on Refs. [1, 18, 30], among others. Since we
+are interested in the near-field interactions between
+the particles, the study focuses on an example for
+which the interparticle’s gap equals one particle di-
+ameter (g = 2R); see Fig. 1. We add complementary
+examples for other values of the gap in the Supple-
+mentary Material (SM).
+III.
+RESULTS
+In this section, all the optical variables were scaled
+by the proper factors to make them adimensional.
+The following characteristic magnitudes, namely, wtot
+E ,
+Ap = πR2, Vp =
+4
+3πR3, and Vint =
+4
+3π (|r2 − r1|)3
+redefine the variables as Qabs = σabs
+2Ap for the absorp-
+tion efficiency, Frad =
+Ftot,z
+wtot
+E Ap and ∆′ =
+∆
+wtot
+E Ap for
+the radiation pressure and the binding forces, and
+N′
+spin =
+Nspin
+wtot
+E Vp for the spin torque.
+The variable
+N′
+orb =
+Norb
+wtot
+E Vint for the orbital torque is only shown
+by an example in the SM since it gave negligible re-
+sults unless the gap is minimal, see Figs. S1 and S2
+for details. We calculate the scaled Poynting vector
+as S =
+1
+2I0 Re{E × H∗} where the magnetic field H
+comes from an MDD equation similar to that for E
+[36]. The curl of S is calculated using an appropriate
+tridimensional mesh around the system’s near-field.
+A.
+Parallel Illumination.
+Fig. 2 shows the spectral results for parallel illumi-
+nation when the magnetic field is off (B = 0, black
+line) and on (B = 1 T, red line with squares). The
+absorption efficiencies, Fig. 2a, result independent of
+the direction of the dimer so that the same spectra
+remain for any other illumination configuration. The
+low-energy resonance (around 73µm) corresponds to
+an SPP, while the high-energy resonance (48.7µm)
+corresponds to an SPhP [30, 37]. Making use of the
+Plasmon Hybridization Model (PHM) for two dipo-
+lar particles, both kinds of surface modes show as
+bright antibonding modes for transverse electric fields
+according to the configuration shown in Fig. 1a, see
+Ref. [35] for details.
+When B is on, each isotropic surface mode splits
+into two modes due to degeneracy removal. The ab-
+sorption is the only far-field observable shown in this
+work since negligible scattering occurs for small sys-
+tems [8]. As it is illumination-independent, the spec-
+tra obtained remain invariant for all illumination di-
+rections concerning the dimer’s axis. Thus, the only
+available far-field observable is neither adequate to
+study the interactions occurring in the dimer nor valu-
+able to predict the dimer’s dynamics. In Fig. 2b, there
+is no net radiation pressure when B is off due to the
+high symmetry of both the system and incident field.
+On the other hand, there is a resonant pressure for the
+MO dimer when B is on, revealing the magnetoplas-
+monic resonances and directing the dimer upwards or
+downwards according to their energy. As a result of
+
+IKA
+LCP
+B
+KB
+B
+Z
+X
+X
+KA
+1
+-
+2R
+g
+LCP
+kB
+(a)
+(b)4
+the interaction, the radiation pressure identifies the
+modes by the sign of the force. Fig. 2c shows that
+the binding force leads to repulsion between the par-
+ticles for both cases, B = 0 and B = 1 T. In other
+words, there cannot be a stable dimer under this par-
+allel configuration. The response is still resonant but
+less sensitive than the radiation pressure.
+Remark-
+ably, the results agree with the interpretation given
+by the PHM.
+Notably, although we are dealing with a dimer sys-
+tem, our results agree with those reported in [18]
+for a single particle under the same illumination. In
+general, the absorption efficiency Qabs behaves like
+Re{α11}, both the radiation pressure Frad and the
+spin N ′
+spin,z on z behave like Im{α12}, and the bind-
+ing force ∆′ resembles −Re{α11}, being αij the carte-
+sian components of the polarizability tensor ˆα.
+In
+the case of the spins in Fig. 2d, these behave like
+±Im{α33} for each particle respectively (polarizabil-
+ities not shown here). These functional dependencies
+are due exclusively to the type of illumination; oth-
+erwise, other αij-terms would appear in the spectral
+variables [18].
+Following
+angular
+momentum’s
+conservation,
+Fig. 2d shows that the net spin for the system is zero
+when B is off (black line).
+To put it another way,
+the spins for each particle are equal and opposite,
+showing the resonant modes for the isotropic case
+(see red and blue lines with symbols).
+When B is
+on, however (Fig. 2e), there is a net spin for the
+system, black line, which is twice the spin for each
+particle (red line with squares). The spin resonates
+sensitively with the dimer’s modes, quite like the
+radiation pressure. Consequently, the spins become
+much stronger than those for B off, compare the
+scales of Fig. 2d and e. Thus, the radiation pressure
+and spins constitute the most sensitive observables
+in the near field, giving a common spectral shape
+(compare spectra in Figs. 2b and e).
+B.
+Perpendicular Illumination.
+Now we vary the dimer’s azimuthal angle since a de-
+pendency on the net polarization is expected. Figs. 3
+and 4 show maps as a function of the incident wave-
+length and azimuthal angle for B = 0 and B = 1 T,
+respectively. Similarly to that found in the previous
+subsection, there is no net radiation pressure when B
+is off (not shown). Yet this time, a different behav-
+ior is found for the binding force, Fig. 3a. The sys-
+tem offers a resonant spectral response but depends on
+the angle φ. Maxima [minima] of binding are found
+around φ = 90, 270 [φ = 0, 180] deg, meaning inter-
+particle attraction [repulsion]. This fact also defines
+stable positions for the dimer around the strongest
+optical resonance, namely the SPhP at 48.7µm, and
+around φ = 34, 145.6, 214.5, 325.7 deg for all wave-
+lengths when B is off (follow the black lines). A sim-
+ilar situation is found for the second resonant wave-
+length ≈ 72.6µm (SPP), where the variations are less
+pronounced. Regarding the spin, Fig. 3b shows a re-
+maining behavior for the whole system, which is res-
+onant with the surface modes and coordinated with
+the binding phenomenon. Remarkably, the spin gets
+its extremals (maxima or minima) when the dimer
+reaches its stable positions; namely, neither attraction
+nor repulsion, compare Figs. 3a and b. As mentioned
+above, the most sensitive resonance corresponds to the
+excitation of the SPhP.
+Noteworthy, our results are consistent with the in-
+terpretation of the PHM for isotropic, dipolar par-
+ticles [35].
+In particular, each value φ = nπ [φ =
+(n + 1/2)π] rad with n ∈ Z, the binding force shows
+repulsion [attraction] for both types of resonances,
+namely, the SPhP and SPP, see Fig. 3a. This outcome
+is due to the net polarization; the electric field is along
+ˇy, see the map for φ = 0 at the SPhP wavelength in
+Fig. 3c. Thus, φ = 0 corresponds to a transverse elec-
+tric field compared with the dimer’s direction, mean-
+ing an antibonding bright mode in the context of the
+PHM. Differently, φ = 90 deg corresponds to a parallel
+electric field compared with the dimer’s axis, meaning
+a bright bonding mode in the PHM (map not shown).
+In Fig. 4a, there is a remaining radiation pressure for
+the whole system due to the symmetry breaking that
+appears only at the resonances’ locations. This spec-
+trum results invariant with φ and follows the same
+resonances as in absorption in Fig. 2a when B is on
+(red line with squares). Thus the presence of a static
+magnetic field induces the dimer to move forward or
+backward in the illumination’s direction when the in-
+cident energy is that of a surface mode.
+Likewise,
+the system shows a resonant binding (Fig. 4b). As in
+Fig. 3a, the black lines follow the values of zero force.
+Note that the map strongly distorts by the presence
+of the resonances when B is on, making the dynam-
+ics more complex and even reducing the extremals of
+the binding force. However, the possibility to obtain
+stable binding enhances around the SPhP due to the
+overlapping of MO modes, which means more degree
+of control in the dimer’s creation and stability.
+In Fig. 4c, the system’s spin follows a trend simi-
+lar to that for the radiation pressure in Fig. 4a. This
+behavior is quite different from that found for B = 0.
+Note that spin is enhanced when B is on; the color-
+bar limits show values ≈ 6.2 − 7.5 times higher than
+those for B off, compare Figs. 3b and 4c. As a re-
+sult, the MO system could be readily identified in an
+experiment by observing the dimer’s dynamics at the
+resonance wavelengths.
+Fig. 4d shows the electric field around the dimer’s
+plane z = 0 for the SPhP found at 49.85µm. This
+wavelength corresponds to the most robust resonance
+when B is on. The rest of the configuration is equal to
+that given in Fig. 3c. The field hot spots are leaned on
+the right ≈ 65 deg from the x axis by the MO effect.
+Up to this point, we have explored a few examples
+of MO dimers to approach the idea of controlling the
+particle dynamics and ”photonic molecule” stability
+[25] in the presence/absence of a static magnetic field
+B.
+
+5
+FIG. 2. (Color online) Optical properties for parallel configuration. (a) Absorption efficiency, which is independent of
+the dimer’s orientation. (b) Radiation pressure (total force along z). (c) Binding force. Black line [red line with squares]
+for magnetic field B = 0 [B = 1] T. (d) [(e)] Spin torques for B = 0 [B = 1] T. In (d), the net spin torque is zero for all
+wavelengths.
+Below, we discuss the behavior of the dynamic ob-
+servables in terms of the information contained in
+the Poynting field. The reader is reminded that the
+particles’ photonic interaction matches a multiple-
+scattering framework [16, 29]. The near fields involve
+the evanescent waves, which play a crucial role in the
+particles’ interaction for surface modes and close par-
+ticles. This phenomenon can be seen through the en-
+ergy flows because they may have all the information
+of the near fields E and H.
+Generally, the magni-
+tudes obtained from far-field calculations lose some of
+the information about the system [36].
+C.
+Nature of the spins through an examination
+of the Poynting fields
+We explore the spins exerted on the system by show-
+ing a few calculations of the Poynting field around
+the dimer for perpendicular configuration. The paral-
+lel configuration is less interesting since it would only
+lead to repulsion states without dimer formation for
+any gap under both cases B = 0 and 1 T, see Fig. 2c
+for the example g = 2R. More clarifications can be
+found in the SM.
+Figs. 5a-d [e-f] consist of maps related to the en-
+ergy flow when B is off [on] upon different azimuthal
+angles. The wavelengths coincide with that for the
+strongest SPhP in each case. The left column (a-c-e)
+shows the Poynting field S when z = 0.
+Similarly,
+the right column (b-d-f) shows the z-component of
+∇ × S. The white arrows are rescaled to visualize the
+maps easily. Interestingly, Figs. 5a-b show an exam-
+ple of a repulsion state with zero spins when φ = 0,
+see Figs. 3a and b. Even though S aligns in a single
+direction, a resonant magnitude and a non-negligible
+curl appear near the surface of the particles. This res-
+onance is due to the excitation of the SPhP. However,
+the contributions to the spin cancel out due to high
+symmetry evidenced by these maps and zero net spin
+results for the system.
+Figs. 5c-d show the attrac-
+tion state with maximum positive spin when φ = 135
+deg, see Figs. 3a and b. This time, two hot spots of
+maximum magnitude face each other, and a kind of
+saddle point emerges in the gap region between the
+particles, Fig. 5c. As a result, the values of the curl
+clearly show a rotational state for light as the field
+spots have ”turbine-blade” shapes, Fig. 5d, explain-
+ing the net positive spin calculated for the system in
+Fig. 3b for φ = 135 deg. It is also evident from Fig. 5d
+that the two particles have the same spin, visually
+showing that the net spin is two times the spin of one
+particle. Finally, we show the example when B is on
+and φ = 67.2 deg, which coincides with the hot spot
+of maximum attraction at the resonance 49.85µm of
+the SPhP, see Fig. 4b.
+Notice in passing that this
+value for φ is close to the angle of the electric spots in
+Fig. 4d, namely, ≈ 65 deg. Remarkably, Fig. 5e illus-
+trates that the energy flow would make the particles
+spin counterclockwise. Moreover, it is also notewor-
+thy the vortex that appears in the field region between
+the particles with clockwise orientation, resembling a
+”gear” mechanism which coordinates field and parti-
+cles [9, 21, 38]. Consistently, the curl’s map shows a
+
+2.0
+0
+4.
+B=O T
+B=O T
+1.5
+11
+1.0
+×10-1
+3
+-2
+0.5
+1 0.0
+2
+Qabs
+-0.5
+-1.0
+B=O T
+(a)
+(b)
+(c)
+-1.5
+1T
+5
+-2.0
+20
+60
+80
+100
+20
+60
+80
+40
+40
+100
+20
+40
+60
+80
+100
+120
+120
+120
+Wavelength (μm)
+Wavelength (μm)
+Wavelength (μm)
+5
+321
+之
+23
+system
+ system
+-3
+45
+np. 1 = np. 2
+(d)
+(e)
+np. 2
+6-
+20
+40
+60
+100
+120
+20
+40
+60
+80
+100
+80
+120
+Wavelength (μm)
+Wavelength (μum)6
+FIG. 3.
+(Color online) Near-field observables for per-
+pendicular configuration in the absence of magnetic field,
+B = 0 T. (a-b) Maps of the mechanical variables as a
+function of wavelength and dimer’s azimuthal angle. (a)
+Binding force. The black lines correspond to zero force.
+(b) Spin torque for the system. In this case, the net radi-
+ation pressure is zero for all wavelengths (not shown). (c)
+Distribution of electric field around the dimer for z = 0 at
+the resonance wavelength 48.85µm for φ = 0.
+structure similar to that in Fig. 5d but this time en-
+hanced and possessing a structure in the gap region
+that contains the vortex indicated in Fig. 5e, see in-
+set in Fig. 5f. The inset zooms this region so that a
+connection between the curl hot spots is appreciated.
+FIG. 4. (Color online) Near-field observables for perpen-
+dicular configuration and B = 1 T. (a-c) Maps of the me-
+chanical variables as a function of wavelength and dimer’s
+azimuthal angle. (a) Radiation pressure for the system.
+(b) Binding force. The black lines correspond to zero force.
+(c) Spin torque for the system. (d) Distribution of electric
+field around the dimer for z = 0 at the resonant wave-
+length 49.85µm for φ = 0.
+
+360
+10.2
+(a)
+315
+0.14
+270
+0.09
+225
+0.03
+180
+-0.02
+135
+-0.08
+90
+45
+-0.14
+0.18
+20
+30
+40 
+50
+60
+70 
+80
+90
+100
+120
+Wavelength (μm)
+360
+0.79
+315
+(b)
+270
+0.5
+225
+0.25
+135
+0
+90
+45
+0.15
+0.32
+20
+30
+40
+50
+60
+70
+80
+90
+100110
+120
+Wavelength(μum)
+N'
+spin,z
+360
+(C)
+6
+315
+4
+270
+225
+2
+0
+135
+90
+45
+-4
+-5
+0
+20
+30
+40
+50
+60
+7080
+90
+100
+110120
+Wavelength (μum)
+[E
+Eo
+一
+1
+0.75
+6
+(d)
+0.5
+5
+0.25
+(wn)
+4
+0
+y
+-0.25
+3
+-0.5
+2
+-0.75
+1
+-1-0.75-0.5-0.2500.25 0.5 0.75
+x (μum)△(-)
+360
+1.2
+315
+1
+270
+0.7
+225
+0.5
+leg)
+0. 3 
+0180
+135
+106
+-0.2 
+45
+-0.4 
+0
+-0.6
+20
+30
+40
+50
+60
+70
+80
+90
+100
+110
+120
+Wavelength (μm)
+360
+0.8
+315
+(b)
+270
+0.4
+0
+135
+90
+-0.4 
+45
+-0.8
+0
+20
+30
+40
+50
+60
+70
+80
+90
+100
+110
+120
+Wavelength (μm)
+[E
+Eo
+(c)
+0.75
+6
+0.5
+5
+0.25
+(un)
+4
+0
+y
+3
+-0.25
+-0.5
+2
+-0.75
+-1-0.75-0.5-0.25 00.25 0.50.75
+x(μm)7
+FIG. 5. Scaled energy flows around the dimer for z = 0 under perpendicular configuration and at the strongest SPhP.
+The color scale corresponds to the magnitude; the white arrows show the Poynting flow (2× their original size). Left
+[right] column for the [z-component of the curl of the] Poynting field. (a-d) [(e-f)] Examples for B = 0 [B = 1] T; the
+wavelength is 48.85µm [49.85µm]. (a-b) For φ = 0 deg, (c-d) φ = 135 deg, and (e-f) 67.2 deg. The inset in (f) zooms the
+gap region up to a maximum of 0.05 in the colorbar.
+
+[Sxy]
+2RI(V × S)zl
+1o
+o
+0.8
+0.8
+0.4
+6
+(b)
+(a)
+0.6
+0.6
+5
+0.4
+0.4
+0.3
+00
+4
+0.2
+0.2
+(wn
+0
+0
+0.2
+-0.2
+2
+-0.4
+0.1
+-0.4
+-0.6
+-0.6
+-0.8
+-0.8
+0
+-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
+-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
+x (μm)
+x (μm)
+0.8
+0.8
+0.4
+(d)
+0.6
+0.6
+6
+0.4
+0.4
+0.3
+5
+0.2
+0.2
+(wn)
+(wn)
+0
+0.2
+0
+3 
+y
+-0.2
+2
+-0.4
+0.1
+-0.4
+-0.6
+-0.6
+-0.8
+0
+-0.8
+0
+-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
+-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
+x (μm)
+x (um)
+0.8
+0.8
+4
+(f)
+e)
+0.6
+0.6
+0.8
+0.4
+0.4
+3
+0.2
+0.2
+0.6
+(wn)
+(wn)
+0
+0
+0.4
+-0.2
+-0.4
+-0.4
+0.2
+-0.6
+-0.6
+-0.8
+-0.8
+-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
+-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
+x (μum)
+x (μum)8
+IV.
+CONCLUSIONS
+By a simple dipolar model, this work explores the
+behavior of small nanoparticle dimers when magneto-
+optical materials like n-doped InSb and moderate
+magnetic fields are used.
+Two counter-propagating
+waves with equal circular polarization are used as il-
+lumination to simulate a simple optical trap with nei-
+ther net gradient nor scattering forces. Our results
+show that the system can be thoroughly character-
+ized by observing its mechanical inductions, provided
+these latter depend on the near field.
+Besides, we
+found no possibility of forming stable dimers when
+the dimer is aligned with the illumination since the
+inter-particle force only leads to repulsion.
+On the
+contrary, under ”perpendicular” alignment, there are
+several ways to obtain stable dimers or inter-particle
+attraction, at least under this ”static” model for which
+the particles’ velocities and accelerations are not con-
+sidered. As the results strongly depend on the mag-
+netic field’s presence, this study constitutes a novel
+background to build OM or photonic molecule nano
+factories and control their movements, or conversely,
+to study samples containing this class of multimers.
+Finally, we give an original explanation for the ap-
+pearance of particle spins based on the energy flows.
+This interpretation offers satisfactory results from the
+”scattering” forces produced by the interaction be-
+tween the particles. Gradient forces were also inves-
+tigated but showed no appreciable influence on the
+spins’ appearances (results not shown in this work).
+ACKNOWLEDGMENTS
+The author would like to thank A. Garc´ıa-Mart´ın
+from IMN-CSIC and M.I. Marqu´es from Universidad
+Aut´onoma de Madrid for their valuable discussions
+during his postdoc in Spain.
+CONFLICTS OF INTEREST
+The authors declare that the research was con-
+ducted in the absence of any commercial or financial
+relationships that could be construed as a potential
+conflict of interest.
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+

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+Detection of Beyond-Quantum Non-locality based on Standard Local Quantum
+Observables
+Hayato Arai1, ∗ and Masahito Hayashi2, 3, 1, †
+1Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan
+2Shenzhen Institute for Quantum Science and Engineering,
+Southern University of Science and Technology, Nanshan District, Shenzhen, 518055, China
+3International Quantum Academy (SIQA), Shenzhen 518048, China
+Bell-CHSH inequality is one of the important ways to detect quantum non-locality because the
+whole protocol can be implemented by local observables. However, there are theoretically many
+types of beyond-quantum non-locality in General Probabilistic Theories. One important class is
+Entanglement Structures (ESs), which contain beyond-quantum non-local states even though their
+local systems are completely equivalent to standard quantum systems.
+It is known that Bell’s
+inequality cannot detect any beyond-quantum non-local states in ESs, and its detection based on
+local observables is open.
+This paper gives a way based on local observables to detect beyond-
+quantum non-local states in ESs, and especially, we give a way to detect beyond-quantum non-local
+states in two-qubit ESs by observing only spin observables on local systems.
+Introduction—Bell’s inequality [2] (or CHSH inequal-
+ity [3]) is one of the important ways to detect quantum
+non-locality in our physical systems. Bell-CHSH inequal-
+ity (hereinafter, CHSH inequality) consists of bipartite
+players and their local operations.
+It is especially im-
+portant that the protocol of CHSH inequality can be
+implemented by local observables.
+In other words, by
+implementing the protocol of CHSH inequality as a bi-
+partite communication task, we can experimentally de-
+tect quantum non-locality of our physical systems when
+Bell-CHSH inequality is violated. Actually, the violation
+of CHSH inequality is confirmed in physical experiments
+[4–9]. These remarkable results played an important role
+in the early studies of quantum physics and quantum
+information theory to ensure that our physical systems
+truly possess quantum non-locality.
+However, there are many other theoretical models with
+non-locality than quantum systems.
+Such models can
+be described as General Probabilistic Theories (GPTs)
+[11–33, 36]. GPT is a framework for general theoretical
+models with states and measurements, including classi-
+cal and quantum systems. PR-box [11–13] is a typical
+example of non-local models with beyond-quantum non-
+locality.
+Actually, the CHSH value in PR-box attains
+four even though the bound in quantum theory is given
+as 2
+√
+2, known as Tirelson’s bound [10]. In other words,
+the CHSH inequality can detect such beyond-quantum
+non-locality, including PR-box.
+On the other hand, a class in GPTs, called Entan-
+glement Structures (ESs) with local quantum subsys-
+tems, is not easily distinguished from the Standard En-
+tanglement Structure (SES), i.e., the standard quantum
+model defined by the tensor product. An ES is defined
+as a model of a composite system whose local systems
+are completely equivalent to standard quantum systems.
+Studies of GPTs have revealed that ESs are not uniquely
+determined as the SES, even if we impose operational
+postulates about local structures [24, 28–31, 36]. Some
+ESs have fewer non-local states than the SES [28, 30],
+and also, some ESs has beyond-quantum non-local states,
+i.e., non-local states that do not belong to the SES
+[29, 31, 33, 36].
+Preceding studies [21–23, 33] showed that some sym-
+metric condition derives the SES. However, such con-
+ditions are not operational and especially not observ-
+able even though the conditions are mathematically nat-
+ural; therefore, these results do not enable us to de-
+tect that our models truly obey the SES experimen-
+tally. On the other hand, CHSH inequality can be exper-
+imentally implemented. However, preceding studies [34–
+36] have revealed that all ESs satisfy Tirelson’s bound,
+i.e., CHSH inequality cannot distinguish the SES from
+beyond-quantum non-local states in any ES. Also, as an-
+other result about an experimental condition, the refer-
+ence [33] shows that verification of maximally entangled
+states cannot experimentally detect some ESs. In other
+words, such an experimental detection cannot deny the
+possibility of other ESs than the SES.
+In this way, it is an important problem to detect other
+ESs than the SES experimentally. Therefore, this paper
+gives a way to detect beyond-quantum non-local states by
+an experimental protocol. First, we give a criterion sep-
+arating an arbitrary given beyond-quantum state from
+all standard quantum states as an inequality defined by
+local observables (Theorem 5). Next, we give a bipartite
+protocol to implement the above criterion. Our proto-
+col consists of local operations by bipartite players Alice
+and Bob and classical communication by them. In the
+protocol, Alice and Bob detect whether a target state
+is beyond-quantum or not. If the target state is truly
+beyond-quantum, Alice and Bob conclude that the tar-
+get state is beyond-quantum with high probability.
+Our criterion and protocol are implemented by a com-
+plicated sequence of local observables in general. How-
+ever, in the 2-qubits case, i.e., in the 2 × 2-dimensional
+case, we give a simple detection of a beyond-quantum
+arXiv:2301.04196v1  [quant-ph]  10 Jan 2023
+
+2
+non-local state by observing Pauli’s spin observables in
+a specific order. It is known that maximally entangled
+states are detected when Alice and Bob observe Pauli’s
+spin observable σx, σy, σz in the same order with the se-
+quence of coefficients (1, 1, −1) [40, 41]. In contrast to
+this result, we clarify that the sequence of coefficients
+(1, 1, 1) detects a beyond-quantum non-local state (The-
+orem 6).
+Moreover, like Bell’s scenario, any beyond-
+quantum non-local “pure” state can be detected by se-
+quential local observables biased in the same way as
+σx, σy, σz (Theorem 8).
+As a result, we give a conve-
+nient detection for beyond-quantum non-local states like
+Bell’s inequality in the 2-qubits case.
+The setting of GPTs and entanglement structures—A
+model of GPTs is defined as follows.
+Definition 1 (A Model of GPTs). A model of GPTs is
+defined by a tuple G = (V, ⟨ , ⟩, C, u), where (V, ⟨ , ⟩), C,
+and u are a real-vector space with inner product, a proper
+cone, and an order unit of C∗, respectively.
+For a model of GPTs G, state space and measurement
+space are defined as follows.
+Definition 2 (State Space of GPTs). Given a model of
+GPTs G = (V, ⟨ , ⟩, C, u), the state space of G is defined
+as
+S(G) := {ρ ∈ V|⟨ρ, u⟩ = 1} .
+(1)
+Here, we call an element ρ ∈ S(G) a state of G.
+Definition 3 (Measurements of GPTs). Given a model
+of GPTs G = (V, ⟨ , ⟩, C, u), we say that an family
+{Mi}i∈I is a measurement if Mi ∈ C∗ and �
+i∈I Mi = u.
+Besides, the index i is called an outcome of the mea-
+surement. Here, the set of all measurements is denoted
+as M(G). Especially, the set of measurements with n-
+number of outcomes is denoted as Mn(G).
+In this setting, state space and measurement space are
+always convex. We call an element pure when the element
+is extremal. Also, when a state ρ ∈ S(G) is measured by
+a measurement {Mi} ∈ M(G), the probability pi to get
+an outcome i is given as
+pi := ⟨ρ, Mi⟩.
+(2)
+Quantum theory is a typical example of a model of
+GPTs, i.e., quantum theory is given as a model G =
+(LH(H), Tr, L+
+H(H)), I), where LH(H) and L+
+H(H) be the
+set of Hermitian matrices and the set of positive semi-
+definite matrices on a finite dimensional Hilbert space
+H, respectively. In this model, the state space is the set
+of density matrices, also the measurement space is the
+set of Positive Operator Valued Measures (POVMs).
+In order to discuss Bell scenario, we introduce the
+model of composite systems in GPTs. Given two mod-
+els in GPTs GA = (VA, ⟨ , ⟩A, CA, uA) and GB =
+(VB, ⟨ , ⟩B, CB, uB), a model G = (V, ⟨ , ⟩, C, u) of
+composite system of two models is defined as follows
+[13]: The vector space V is defined as the tensor prod-
+uct V = VA ⊗ VB. The inner product ⟨ , ⟩ is defined
+as the induced inner product by the tensor product,
+i.e., the inner product ⟨x1, x2⟩ is defined as ⟨x1, x2⟩ =
+�
+i,j⟨a(i)
+1 , a(j)
+2 ⟩A⟨b(i)
+1 , b(j)
+2 ⟩B for elements x1 = �
+i a(i)
+1
+⊗
+b(i)
+1
+∈ V and x2 = �
+j a(j)
+2
+⊗ b(j)
+2
+∈ V. The cone C is
+chosen as a cone satisfying the inequality
+CA ⊗ CB ⊂ C ⊂ (C∗
+A ⊗ C∗
+B)∗ ,
+(3)
+where the tensor product CA ⊗ CB is defined as
+CA ⊗ CB :=
+��
+i
+ai ⊗ bi
+�����ai ∈ CA, bi ∈ CB
+�
+.
+(4)
+The order unit u is defined as u = uA ⊗ uB ∈ V.
+Especially, in this paper, we mainly consider Entan-
+glement Structures (ESs) with local quantum subsystems
+(hereinafter, we simply call them ESs), i.e., models of
+composite systems G whose local systems GA and GB
+are equivalent to quantum theory.
+By modifying the
+above conditions of the definition of composite systems,
+we define an ES as follows.
+Definition 4 (Entanglement Structure [13, 24, 29, 33]).
+We say that a model G = (LH(HA ⊗ HB), Tr, C, I) is an
+entanglement structure if C satisfies
+SEP(A; B) ⊂ C ⊂ SEP(A; B)∗,
+(5)
+where the proper cone SEP(A; B) is defined as
+SEP(A; B) := L+
+H(HA) ⊗ L+
+H(HB).
+(6)
+By definition, an ES is given by a positive cone with
+(5), and therefore, an ES is denoted as the corresponding
+positive cone C.
+The inclusion relation (5) implies that a model of quan-
+tum composite system is not uniquely determined. On
+the other hand, standard quantum systems obey the only
+model SES(A; B) := L+
+H(HA⊗HB). In this paper, we call
+this model the Standard Entanglement Structure (SES).
+The set SEP(A; B)∗ contains non-positive Hermitian ma-
+trices, and therefore, a non-positive state is available in
+an ES C ̸⊂ SES(A; B).
+We call a non-positive state,
+i.e., a state ρ ∈ S(SEP∗(A; B))\S(SES(A; B)) a beyond-
+quantum state. Our interest is how we detect beyond-
+quantum states if they exist.
+Observables and CHSH inequality in GPTs—Next, we
+introduce observables and CHSH inequality in GPTs.
+In a model G, an observable O is defined as a pair of
+a measurement {Mi(O)}i∈I ∈ M(G) and its outcomes
+oi(O)
+(i ∈ I). In this paper, we consider a bipartite
+composite model whose local models are given by GA
+and GB. When Alice’s and Bob’s local observables are
+
+3
+given as OA = {M A
+i , oA
+i }i∈I and OB = {M B
+j , oB
+j }j∈J,
+respectively, the total observable is given as OA ⊗ OB =
+{M A
+i ⊗ M B
+j , oA
+i oB
+j }i∈I,j∈J.
+Given an observable O = {Mi, oi}i∈I and a state ρ, an
+expectation value of the outcome of O with ρ is given as
+⟨O⟩ρ :=
+�
+i∈I
+oi ⟨ρ, Mi⟩ .
+(7)
+In a composite model G of local modesl GA and GB,
+CHSH inequality is defined by the following function
+B(ρ; OA
+1 , OA
+2 , OB
+1 , OB
+2 )
+:=
+���
+�
+OA
+1 ⊗ OB
+1
+�
+ρ +
+�
+OA
+1 ⊗ OB
+2
+�
+ρ
++
+�
+OA
+2 ⊗ OB
+1
+�
+ρ −
+�
+OA
+2 ⊗ OB
+2
+�
+ρ
+���,
+(8)
+where ρ ∈ S(G) is a global state and OA
+1 , OA
+2 ∈ M2(GA),
+OB
+1 , OB
+2 ∈ M2(GB) are observables with two outcomes
+±1. The general CHSH inequality is defined as the bound
+of the function B(ρ; OA
+1 , OA
+2 , OB
+1 , OB
+2 ). For example, in
+the model SES(A; B), B(ρ; OA
+1 , OA
+2 , OB
+1 , OB
+2 ) ≤ 2
+√
+2,
+known as Tirelson’s bound.
+However, the reference [35] shows that the inequal-
+ity B(G) ≤ 2
+√
+2 holds whenever the local systems GA
+and GB are equivalent to quantum theory.
+This im-
+plies that any ES G satisfies Tirelson’s bound, i.e.,
+B(ρ; OA
+1 , OA
+2 , OB
+1 , OB
+2 ) ≤ 2
+√
+2.
+In this way, CHSH in-
+equality can not detect whether an entanglement struc-
+ture has beyond-quantum state or not.
+Criterion to Detect Beyond-Quantum State and Its
+Implementation—In this way, CHSH inequality is not
+enough to detect beyond-quantum states in ESs. In con-
+trast to CHSH inequality, we give a detection of an arbi-
+trary given beyond-quantum state in ESs.
+As a preliminary, we consider a standard quantum ob-
+servable whose measurement is given by a Projection-
+Valued Measure (PVM) {Mi}i∈I. In this case, we regard
+an observable O as a Hermitian matrix �
+i∈I oiMi, where
+oi is an outcome of the observable.
+Conversely, when
+a Hemitian matrix T is written as T = �
+i∈I oiMi by
+spectral decomposition, the matrix T is regarded as an
+observable O = {Mi, oi}i∈I with PVM. For example, the
+following Hermitian matrices are regarded as standard
+Pauli’s spin observables in the qubit-system:
+σx :=
+�
+0 1
+1 0
+�
+,
+σy :=
+�
+0 −i
+i
+0
+�
+,
+σz :=
+�
+1
+0
+0 −1
+�
+.
+(9)
+Due to this correspondence, the expectation value of an
+observable O defined by a Hermitian matrix TO with a
+state ρ is given as
+⟨O⟩ρ = Tr ρTO.
+(10)
+Hereinafter, we consider only observables defined as a
+Hermitian matrix, and an observable is denoted as a Her-
+mitian matrix.
+Now, we consider a detection of beyond-quantum
+states in ESs. First, as the following theorem, we give
+a criterion detecting beyond-quantum state based on lo-
+cal quantum observables.
+Theorem
+5.
+Given
+an
+arbitrary
+state
+σ
+∈
+S(SEP∗(A; B)) \ S(SES(A; B)), there exist a family
+of observables {OA
+i
+⊗ OB
+i }m
+i=1 and a real number α
+satisfying the following two properties:
+1. �m
+i=1
+�
+OA
+i ⊗ OB
+i
+�
+ρ ≤ α for any ρ ∈ S(SES(A; B)).
+2. �m
+i=1
+�
+OA
+i ⊗ OB
+i
+�
+σ > α.
+Theorem 5 states that the value �m
+i=1
+�
+OA
+i ⊗ OB
+i
+�
+ρ
+separates the beyond-quantum state σ from standard
+quantum states.
+As the proof of Theorem 5, we give
+the following deterministic way to find the observables
+{OA
+i ⊗ OB
+i }m
+i=1 and a real number α for any given state
+σ ∈ S(SEP∗(A; B)) \ S(SES(A; B)).
+At first, because the state space S(SES(A; B)) is a
+closed convex set and the relation σ ̸∈ S(SES(A; B)),
+hyperplane separation theorem [42] ensures the existence
+of a hyperplane that separates σ from the convex set
+S(SES(A; B)).
+In other words, there exist an element
+x ∈ LH(HA ⊗ HB) and a real number α ∈ R such that
+Tr xσ > α and Tr xρ ≤ α for any ρ ∈ S(SES(A; B)). In
+practical situation, we need to find such a Hermitian ma-
+trix x by an analytical way, but x = σ separates σ and
+S(SES(A; B)) as follows. Any ρ ∈ S(SES(A; B)) satisfies
+the following inequality:
+|Tr σρ|
+(a)
+< ∥σ∥2∥ρ∥2
+(b)
+≤ ∥σ∥2.
+(11)
+The inequality (a) is shown by Schwarz inequality and
+its equality condition. The equality condition of Schwarz
+inequality holds only when ρ is proportional to σ, which
+never holds because ρ is positive semi-definite and σ is
+not positive semi-definite. The inequality (b) is shown by
+∥ρ∥ ≤ 1 for any ρ ∈ S(SES(A; B)). On the other hand,
+the equation Tr σ2 = ∥σ∥2 holds by definition, therefore
+we find at least one element x = σ separating σ and
+S(SES(A; B)). Due to the latter discussion, we consider
+general separation x here.
+Next, we formulate the element x as a tensor product
+form. Because the element x belongs to the vector space
+LH(HA ⊗ HB) = LH(HA) ⊗ LH(HB), the element x is
+written as x = �m
+i=1 xA
+i ⊗ xB
+i , where xA
+i ∈ LH(HA) and
+xB
+i ∈ LH(HB). As seen in the latter discussion, the Her-
+mitian matrices xA
+i and xB
+i
+can be regarded as observ-
+ables OA
+i and OB
+i , respectively. Finally, the observables
+
+4
+OA
+i and OB
+i satisfy the equation
+m
+�
+i=1
+�
+OA
+i ⊗ OB
+i
+�
+ρ = Tr xρ
+(12)
+for any ρ ∈ S(SEP∗(A; B)).
+This criterion is implemented as the following detec-
+tion protocol on bipartite scenario (Figure 1).
+• Aim and Strategy
+– Alice and Bob aim to determine whether a
+given target global state ρ is beyond-quantum
+or not.
+– Alice and Bob choose {OA
+i ⊗ OB
+i }m
+i=1 and α
+given in Theorem 5 based on their prediction
+that the target state ρ is close to a beyond-
+quantum state σ.
+– Alice and Bob repeat the following protocol
+by nm-times for sufficiently large n.
+• The Whole Protocol
+1. Set Up: Alice and Bob prepare a generator
+of the target state ρ. The generator always
+transmits the same global state ρ.
+2. i-th Round:
+The generator transmits the
+state ρ to the composite system of Alice’s and
+Bob’s systems. Alice and Bob measure their
+local observables OA
+j and OB
+j with i = kn + j
+(1 ≤ k ≤ m). As a result, they get outcomes
+oA
+i and oB
+i , respectively.
+3. Determination: Alice and Bob share their
+outcomes with classical communication. They
+then calculate the value
+1
+n
+�nm
+i=1 oA
+i oB
+i .
+If
+the inequality
+1
+n
+�nm
+i=1 oA
+i oB
+i > α holds, Alice
+and Bob conclude that the target state ρ is
+beyond-quantum.
+• Justification
+– On the limit n → ∞, the value 1
+n
+�nm
+i=1 oA
+i oB
+i
+approximates
+the
+expectation
+value
+�m
+i=1
+�
+OA
+i ⊗ OB
+i
+�
+ρ.
+– If n is sufficiently larger and the inequality
+1
+n
+�nm
+i=1 oA
+i oB
+i > α holds, Theorem 5 ensures
+that the target state ρ is beyond-quantum
+state with sufficiently large probability.
+– If 1
+n
+�nm
+i=1 oA
+i oB
+i ≤ α holds for sufficiently large
+n, their prediction σ is sufficiently different
+from the target state ρ.
+In this way, any beyond-quantum state can be detected
+by a finite number of local quantum observables with
+large probability. In general dimensional cases, a certain
+family of observables is not applied to general beyond-
+quantum states but only to certain beyond-quantum
+states, and the number of observables might be large in
+Generator
+Alice’s System
+Bob’s System
+Classical 
+Communication
+Local Observable
+Local Observable
+Outcome
+Target State
+Outcome
+FIG. 1.
+The Detection Protocol of Criterion Given in The-
+orem 5.
+A generator of global state ρ is given.
+Alice and
+Bob aim to detect whether ρ is beyond-quantum. Alice and
+Bob prepare only their local observables with a certain order
+given in Theorem 5, and they estimate the expectation value
+in Theorem 5 as average of all outcomes gotten in nm-rounds
+of observation.
+general. However, in the 2 × 2 dimensional case, our cri-
+terion detects any pure beyond-quantum states based on
+three Pauli’s spin observables as the following discussion.
+Detection
+of
+beyond-quantum
+state
+on
+2 × 2-
+dimensional system by Pauli’s spin observables—Now,
+we
+consider
+an
+arbitrary
+bipartite
+model
+G
+=
+(T (H), Tr, C, I) of two local quantum systems with di-
+mension 2,
+i.e.,
+we consider the case dim(HA)
+=
+dim(HB) = 2.
+First, we introduce the following function A(ρ) as
+A(ρ) :=
+�
+i=x,y,z
+⟨σi ⊗ σi⟩ρ = Tr
+�
+�
+�
+�
+1
+0
+0
+0
+0 −1
+2
+0
+0
+2
+−1 0
+0
+0
+0
+1
+�
+�
+�
+� ρ. (13)
+Now, we see how the function A(ρ) detects beyond-
+quantum states. The following theorem holds.
+Theorem 6. The function A(ρ) satisfies the following
+two properties:
+1. A(ρ) ≤ 1 for any ρ ∈ S(SES(A; B)).
+2. A(ρ0) = 3, where ρ0 ∈ S(SEP ∗(A; B)) is defined
+as
+ρ0 =
+�
+�
+�
+�
+1
+2 0 0 0
+0 0
+1
+2 0
+0
+1
+2 0 0
+0 0 0
+1
+2
+�
+�
+�
+�
+(14)
+Proof of Theorem 6 lays in Appendix. Due to The-
+orem 6, we detect the beyond-quantum state ρ0 by the
+criterion based on the function A(ρ). Besides, because
+the observables (9) are available in standard quantum
+theory, we detect the beyond quantum states sufficiently
+close to ρ0.
+
+OB2B25
+Moreover, the function A(ρ) satisfies the following
+property.
+Theorem 7. The function A(ρ) satisfies
+max{A(ρ) | ρ ∈ S(SEP∗(A; B))} = 3
+(15)
+and attains the maximum only on ρ = ρ0.
+Proof of Theorem 7 lays in Appendix. Due to The-
+orem 7, the function A(ρ) distinguishes ρ0 not only
+from all standard quantum states but also from all other
+beyond-quantum states.
+Next, we give a more efficient criterion.
+Similar to
+Bell’s scenario, we generalize the function A by replacing
+an arbitrary three observables Oi (i = x, y, z) biased in
+ths same way as σx, σy, σz. In other words, we consider
+the situation Oi = U †σiU in (13). We define the gen-
+eralized function of A(ρ) for unitary matrices UA, UB on
+HA and HB as
+A(ρ; UA, UB) := A
+�
+(UA ⊗ UB) ρ
+�
+U †
+A ⊗ U †
+B
+��
+.
+(16)
+The function A(ρ; UA, UB) can also be experimentally
+obtained by the shared global state ρ, local observables
+σx, σy, σz, and local quantum operations UA, UB. Then,
+we define the maximization of A(ρ; UA, UB) as
+Amax(ρ) := max
+�
+A(ρ; UA, UB)
+���
+UA, UB : unitary matrices on HA and HB
+�
+.
+(17)
+The function Amax(ρ) separates any beyond-quantum
+pure state from standard quantum states more efficiently
+as the following theorem.
+Theorem 8. The function Amax(ρ) satisfies the follow-
+ing two properties:
+1. Amax(ρ) ≤ 1 for any ρ ∈ S(SES(A; B)).
+2. Amax(σ) > 1 for any beyond-quantum pure state
+σ ∈ S(SEP∗(A; B)).
+Proof of Theorem 8 lays in Appendix.
+Theorem 8
+states that the function Amax(ρ) completely detects any
+beyond-quantum pure state in S(SEP ∗(A; B)). In other
+words, if a target state ρ is beyond-quantum pure, there
+exists a pair of unitary matrices UA and UB such that
+A(ρ; UA, UB) detects the target state ρ from all standard
+quantum states. The criterion given in Theorem 8 is ap-
+proximately implemented by a reiteration of the protocol
+in Figure 1 with several unitary operations.
+Conclusion—In this paper, we have discussed the de-
+tection of beyond-quantum states in ESs of GPTs. Even
+though local systems of ESs are equivalent to standard
+quantum systems, CHSH inequality cannot separate any
+beyond-quantum states in ESs from the standard quan-
+tum states in the SES. In contrast to CHSH inequality,
+we have given a criterion to separate an arbitrary given
+beyond-quantum state in any ES from the SES based
+on local standard quantum observables. Also, we have
+given an experimental implementation of the criterion
+as a bipartite protocol. As an example of the detection
+in the 2-qubits case, we have given a simple detection
+based on Pauli’s spin observables. Moreover, like Bell’s
+scenario, we have clarified that all beyond-quantum non-
+local pure states can be detected by a sequential local
+observables biased by the same way as Pauli’s spin ob-
+servables σx, σy, σz.
+Finally, we give some open problems. First, we have
+not given a strict estimation of the error probability of
+the detection protocol. In order to discuss how surely
+our models contains beyond-quantum non-local states,
+we need to discuss the protocol in the context of hypoth-
+esis testing. This is an open proble. Second, our criterion
+cannot detect beyond-quantum state efficiently in general
+dimension. For a practical situation, it is important that
+we detect beyond-quantum states based on a small num-
+ber of observables.
+In other words, it is important to
+estimate how many observables we need for the criterion
+given in Theorem 5 This is another open problem.
+ACKNOWLEDGMENTS
+HA is supported by a JSPS Grant-in-Aids for JSPS
+Research Fellows No.
+JP22J14947.
+MH is supported
+in part by the National Natural Science Foundation of
+China (Grant No. 62171212) and Guangdong Provincial
+Key Laboratory (Grant No. 2019B121203002).
+∗ [email protected]
+† [email protected]
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+APPENDIX
+Proof of Theorem 6
+Proof. STEP1: Proof of the statement 1.
+At first, the matrix given in (13) is calculated as
+A :=
+�
+�
+�
+�
+1
+0
+0
+0
+0 −1
+2
+0
+0
+2
+−1 0
+0
+0
+0
+1
+�
+�
+�
+� = I −
+�
+�
+�
+�
+0
+0
+0
+0
+0
+2
+−2 0
+0 −2
+2
+0
+0
+0
+0
+0
+�
+�
+�
+� . (18)
+the second matrix in right-hand-side is positive semi-
+definite with rank 1. Therefore, the maximum eigenvalue
+of A is 1., which implies that A(ρ) ≤ 1 for any positive
+semi-definite matrix ρ with Tr ρ = 1.
+STEP2: Proof of the statement 2.
+
+7
+This is shown by the following simple calculation.
+A(ρ0) = Tr
+�
+�
+�
+�
+1
+0
+0
+0
+0 −1
+2
+0
+0
+2
+−1 0
+0
+0
+0
+1
+�
+�
+�
+�
+�
+�
+�
+�
+1
+2 0 0 0
+0 0
+1
+2 0
+0
+1
+2 0 0
+0 0 0
+1
+2
+�
+�
+�
+� = 3. (19)
+As the above, Theorem 6 has been proven.
+Proof of Theorem 7
+In order to show Theorem 7, we apply the following
+proposition.
+Proposition 9 (essentially shown in [37, Prop 5.6] or
+[38]). In 2 × 2-dimensional case, the set of all extremal
+points of S(SEP∗(A; B)) is given as
+Ext(SEP∗(A; B))
+={Γ(ρ) | ρ ∈ S(SES(A; B)) ρ is entangled pure}
+∪ {ρ | ρ ∈ S(SES(A; B)) ρ is pure}.
+(20)
+Actually, the reference [37, Prop 5.6] shows that the
+all extremal points of the set of decomposable elements
+in S(SEP∗(A; B)) is given as (20). It is known that all
+elements in SEP∗(A; B) are decomposable [39] in 2 × 2-
+dimensional case, and therefore, Proposition 9 holds.
+Proof of Theorem 7. Because of Proposition 9, the func-
+tion of A(ρ) is attained by an element in Ext(A; B). Also,
+because of Theorem 6, the function of A(ρ) is attained
+by an element Γ(ρ), where ρ is positive semi-definite with
+rank 1. Then, the statement is obtained as follows:
+A(Γ(ρ)) + 1
+(a)
+= Tr
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+0
+0
+0
+0 −1
+2
+0
+0
+2
+−1 0
+0
+0
+0
+1
+�
+�
+�
+� + I
+�
+�
+�
+�
+�
+�
+�
+Γ(ρ)
+= Tr Γ
+�
+�
+�
+�
+2 0 0 2
+0 0 0 0
+0 0 0 0
+2 0 0 2
+�
+�
+�
+� Γ(ρ) = Tr
+�
+�
+�
+�
+2 0 0 2
+0 0 0 0
+0 0 0 0
+2 0 0 2
+�
+�
+�
+� ρ
+(b)
+≤ 4.
+(21)
+The equation (a) holds because Tr Γ(ρ) = 1. The inequal-
+ity (b) holds because Φ and ρ are positive semi-definite
+with rank 1.
+Proof of Theorem 8
+Proof. The statement (1) is similarly shown by Theo-
+rem 8 because any unitary matrix does not change the
+trace. We will show the statement (2).
+Let
+σ
+be
+a
+beyond-quantum
+pure
+state
+in
+S(SEP∗(A; B), i.e., σ is written as Γ(ρ),
+where ρ
+is entangled positive semi-definite with trace 1 by
+Proposition 9. Then, we obtain the following equation:
+Amax(Γ(ρ)) + 1 = A((UA ⊗ UB) (Γ(ρ) + I)
+�
+U †
+A ⊗ U †
+B
+�
+= Tr
+�
+�
+�
+�
+2 0 0 0
+0 0 2 0
+0 2 0 0
+0 0 0 2
+�
+�
+�
+� (UA ⊗ UB) Γ(ρ)
+�
+U †
+A ⊗ U †
+B
+�
+= Tr 4Γ(Φ) (UA ⊗ UB) Γ(ρ)
+�
+U †
+A ⊗ U †
+B
+�
+= Tr 4Γ(Φ)Γ (UA ⊗ U ′
+B) ρ
+�
+U †
+A ⊗ U ′†
+B
+�
+= Tr 4Γ(Φ)Γ(ρ0) = Tr 4Φρ0,
+(22)
+where U ′
+B := U ⊤
+B and ρ0 := (UA ⊗ U ′
+B) ρ
+�
+U †
+A ⊗ U ′†
+B
+�
+.
+Because ρ0 is entangled and Φ is maximally entangled
+in the 2 × 2-dimensional bipartite quantum system, the
+inequality Tr Φρ0 > 1
+2 holds, which implies the statement
+(2).
+As the above, Theorem 8 has been proven.
+

4dE0T4oBgHgl3EQfvQGV/content/tmp_files/2301.02616v1.pdf.txt

@@ -0,0 +1,259 @@
+On the Width of the Regular n-Simplex
+Sariel Har-Peled∗
+Eliot W. Robson†
+January 9, 2023
+Abstract
+Consider the regular n-simplex ∆n – it is formed by the convex-hull of n+1 points in Euclidean
+space, with each pair of points being in distance exactly one from each other. We prove an exact
+bound on the width of ∆n which is ≈
+�
+2/n. Specifically, width(∆n) =
+�
+2
+n+1 if n is odd, and
+width(∆n) =
+�
+2(n+1)
+n(n+2) if n is even. While this bound is well known [GK92, Ale77], we provide a
+self-contained elementary proof that might (or might not) be of interest.
+1. The width of the regular simplex
+A regular n-simplex ∆n is a set of n + 1 points in Euclidean space such that every pair of points is
+in distance exactly 1 from each other. For simplicity, it is easier to work with the simplex Dn formed
+by the convex-hull of e1, . . . , en+1 ∈ Rn+1, where ei is the ith standard unit vector1. Observe that
+∥eiej∥ =
+√
+2, which implies that Dn =
+√
+2∆n.
+All the vertices of Dn lie on the hyperplane h ≡
+�n+1
+i=1 xi =
+�
+(x1, . . . , xn+1), 1
+�
+= 1, where 1 = (1, 1, . . . , 1). In particular, the point cn = 1/(n + 1) ∈ Dn
+is the center of Dn, and is in equal distance
+�
+Rn = ∥cnei∥ =
+�
+n
+1
+(n + 1)2 +
+�
+1 −
+1
+n + 1
+�2
+=
+�
+1 −
+2
+n + 1 +
+n + 1
+(n + 1)2 =
+�
+n
+n + 1.
+from all the vertices of Dn. Note that the largest ball one can place in h, which is still contained in Dn,
+is centered at cn. Furthermore, it has radius
+�rn =
+��cn−(0, cn−1)
+�� =
+�
+1
+(n + 1)2 + n
+�1
+n −
+1
+n + 1
+�2
+=
+�
+n + 1
+n(n + 1)2 =
+1
+�
+n(n + 1)
+.
+For a unit vector u, and a set P ⊆ Rn+1, let
+ω(u, P) = max
+p∈P ⟨u, p⟩ − min
+p∈P ⟨u, p⟩
+denote the projection width of P in the direction of u. The width of P is the minimum projection
+width over all directions.
+∗Department of Computer Science; University of Illinois; 201 N. Goodwin Avenue; Urbana, IL, 61801, USA;
+[email protected]; http://sarielhp.org/.
+Work on this paper was partially supported by a NSF AF award
+CCF-1907400.
+†Department of Computer Science; University of Illinois; 201 N. Goodwin Avenue; Urbana, IL, 61801, USA;
+[email protected]; https://eliotwrobson.github.io/.
+1That is, ei is 0 in all coordinates except the ith coordinate where it is 1.
+1
+arXiv:2301.02616v1  [cs.CG]  6 Jan 2023
+
+Energy of points.
+For a vector v = (v1, . . . , vn+1) ∈ Rn+1, let µ(v) = �
+i vi/(n + 1), and let
+�v = v − µ(v)1,
+be the translation of v by its centroid µ(v)1 so that ⟨�v, 1⟩ = 0. The energy of v is σ(v) = ∥�v∥2. The
+energy is the minimum 1-mean clustering price of the numbers v1, . . . , vn+1. We need the following
+standard technical claim, which implies that if we move a value away from the centroid, the 1-mean
+clustering price of the set goes up.
+Claim 1.1. Let v = (v1, . . . , vn+1) ∈ Rn+1 be a point, and let u be a point that is identical to v in
+all coordinates except the ith one, where ui > vi ≥ µ(v).
+Then σ(u) > σ(v).
+The same holds if
+ui < vi < µ(v).
+Proof: Let δi = (ui − vi)/(n + 1), and observe that µ(u) = µ(v) + δi. As such, we have
+σ(u) =
+�
+j(uj − µ(u))2 =
+�
+j(vj − µ(v) + δi)2 − (vi − µ(v) + δi)2 + (ui − µ(v) + δi)2.
+Since �
+j(vj − µ(v)) = 0, we have that
+�
+j(vj − µ(v) + δi)2 =
+�
+j(vj − µ(v))2 +
+�
+2δi
+�
+j(vj − µ(v))
+�
++
+�
+j δ2
+i = σ(v) + (n + 1)δ2
+i .
+Rearranging the above and using that ui > vi ≥ µ(v), we have
+σ(u) − σ(v) = (n + 1)δ2
+i + (ui − µ(v) + δi)2 − (vi − µ(v) + δi)2 > (ui − vi)(ui + vi − 2µ(v) + 2δi) > 0.
+Lemma 1.2. For n odd, the width of Dn is 2/√n + 1, and this is realized by they projection width of
+all the directions in H =
+�
+v/√n + 1
+�� v ∈ {−1, +1}n+1 and ⟨v, 1⟩ = 0
+�
+(and no other direction).
+Proof: Consider a unit vector z that realizes the minimum width of Dn – here, in addition to ∥z∥ = 1,
+we also require that ⟨z, 1⟩ = 0, as one has to consider only directions that are parallel to the hyperplane
+containing Dn. To this end, let β = maxi zi and α = mini zi and observe that
+width(Dn) = ω(z, Dn) = max
+i
+⟨z, ei⟩ − min
+i
+⟨z, ei⟩ = β − α.
+Next, Consider the point u, where for all i we set
+ui =
+�
+α
+zi < 0
+β
+zi ≥ 0.
+A careful repeated application of Claim 1.1, implies that σ(u) > σ(z) if any coordinate of z is not
+already either α or β. But then the point �u has (i) “width” β − α, (ii)∥�u∥ > 1, and (iii) ⟨�u, 1⟩ = 0. But
+this implies that the projection width of Dn on �u/∥�u∥ is (β − α)/∥�u∥ < β − α, which is a contradiction
+to the choice of z.
+Thus, it must be that all the coordinates of z are either α or β. Let t be the number of coordinates
+of z that are α, and observe that
+σ(z) =∥z∥2 = tα2 + (n + 1 − t)β2 = 1
+and
+⟨z, 1⟩ = tα + (n + 1 − t)β = 0.
+2
+
+This implies that β = −
+t
+n+1−tα, and thus
+tα2 + (n + 1 − t)t2
+(n + 1 − t)2 α2 = 1
+=⇒
+t(n + 1 − t) + t2
+n + 1 − t
+α2 = 1
+=⇒
+α = −
+�
+n + 1 − t
+t(n + 1) .
+Thus, the width of Dn is β −α =
+�
+1+
+t
+n+1−t
+��
+n+1−t
+t(n+1) =
+�
+n+1
+t(n+1−t). The last quantity is minimized when
+the denominator is maximized, which happens for t = (n + 1)/2. Namely, the width of Dn is 2/√n + 1.
+We have that tα + tβ = 0, which implies that α = −β and thus β = 1/√n + 1. It follows that
+z ∈ H.
+Lemma 1.3. For n even, the width of Dn is 2
+�
+n+1
+n(n+2).
+Proof: The proof of Lemma 1.2 goes through with minor modifications. The minimum value is realized
+by t = n/2. This implies that
+α = −
+�
+n + 1 − t
+t(n + 1) = −
+�
+n + 2
+n(n + 1)
+and
+β = −
+t
+n + 1 − tα =
+�
+n
+(n + 1)(n + 2).
+As a sanity check, observe that tα2 + (n + 1 − t)β2 = n
+2
+n+2
+n(n+1) + n+2
+2
+n
+(n+1)(n+2) = 1. Thus, the width of
+Dn is
+β − α =
+�
+n
+(n + 1)(n + 2) +
+�
+n + 2
+n(n + 1) =
+2(n + 1)
+�
+n(n + 1)(n + 2)
+= 2
+�
+n + 1
+n(n + 2).
+Using that ∆n = Dn/
+√
+2 and rescaling the above bounds, we get the following.
+Corollary 1.4. The width of the regular n-simplex ∆n is
+�
+2/(n + 1) if n is odd. The width is
+�
+2(n+1)
+n(n+2)
+if n is even.
+The inradius (i.e., radius of largest ball inside ∆n) is rn = 1/
+�
+2n(n + 1), and the
+circumradius (i.e., radius of minimum ball enclosing ∆n is Rn =
+�
+n
+2(n+1).
+References
+[Ale77]
+R. Alexander. The width and diameter of a simplex. Geometriae Dedicata, 6(1), 1977.
+[GK92]
+P. Gritzmann and V. Klee. Inner and outer j-radii of convex bodies in finite-dimensional
+normed spaces. Discret. Comput. Geom., 7: 255–280, 1992.
+3
+

4dE0T4oBgHgl3EQfvQGV/content/tmp_files/load_file.txt

@@ -0,0 +1,122 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf,len=121
+page_content='On the Width of the Regular n-Simplex  Sariel Har-Peled∗  Eliot W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Robson†  January 9, 2023  Abstract  Consider the regular n-simplex ∆n – it is formed by the convex-hull of n+1 points in Euclidean  space, with each pair of points being in distance exactly one from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' We prove an exact  bound on the width of ∆n which is ≈  �  2/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Specifically, width(∆n) =  �  2  n+1 if n is odd, and  width(∆n) =  �  2(n+1)  n(n+2) if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' While this bound is well known [GK92, Ale77], we provide a  self-contained elementary proof that might (or might not) be of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' The width of the regular simplex  A regular n-simplex ∆n is a set of n + 1 points in Euclidean space such that every pair of points is  in distance exactly 1 from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' For simplicity, it is easier to work with the simplex Dn formed  by the convex-hull of e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' , en+1 ∈ Rn+1, where ei is the ith standard unit vector1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Observe that  ∥eiej∥ =  √  2, which implies that Dn =  √  2∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  All the vertices of Dn lie on the hyperplane h ≡  �n+1  i=1 xi =  �  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' , xn+1), 1  �  = 1, where 1 = (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' In particular, the point cn = 1/(n + 1) ∈ Dn  is the center of Dn, and is in equal distance  �  Rn = ∥cnei∥ =  �  n  1  (n + 1)2 +  �  1 −  1  n + 1  �2  =  �  1 −  2  n + 1 +  n + 1  (n + 1)2 =  �  n  n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  from all the vertices of Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Note that the largest ball one can place in h, which is still contained in Dn,  is centered at cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Furthermore, it has radius  �rn =  ��cn−(0, cn−1)  �� =  �  1  (n + 1)2 + n  �1  n −  1  n + 1  �2  =  �  n + 1  n(n + 1)2 =  1  �  n(n + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  For a unit vector u, and a set P ⊆ Rn+1, let  ω(u, P) = max  p∈P ⟨u, p⟩ − min  p∈P ⟨u, p⟩  denote the projection width of P in the direction of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' The width of P is the minimum projection  width over all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  ∗Department of Computer Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' University of Illinois;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' 201 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Goodwin Avenue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Urbana, IL, 61801, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  sariel@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' http://sarielhp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Work on this paper was partially supported by a NSF AF award  CCF-1907400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  †Department of Computer Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' University of Illinois;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' 201 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Goodwin Avenue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Urbana, IL, 61801, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  erobson2@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' https://eliotwrobson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  1That is, ei is 0 in all coordinates except the ith coordinate where it is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='02616v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='CG]  6 Jan 2023  Energy of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  For a vector v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' , vn+1) ∈ Rn+1, let µ(v) = �  i vi/(n + 1), and let  �v = v − µ(v)1,  be the translation of v by its centroid µ(v)1 so that ⟨�v, 1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' The energy of v is σ(v) = ∥�v∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' The  energy is the minimum 1-mean clustering price of the numbers v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' , vn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' We need the following  standard technical claim, which implies that if we move a value away from the centroid, the 1-mean  clustering price of the set goes up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Let v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' , vn+1) ∈ Rn+1 be a point, and let u be a point that is identical to v in  all coordinates except the ith one, where ui > vi ≥ µ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Then σ(u) > σ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  The same holds if  ui < vi < µ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Proof: Let δi = (ui − vi)/(n + 1), and observe that µ(u) = µ(v) + δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' As such, we have  σ(u) =  �  j(uj − µ(u))2 =  �  j(vj − µ(v) + δi)2 − (vi − µ(v) + δi)2 + (ui − µ(v) + δi)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Since �  j(vj − µ(v)) = 0, we have that  �  j(vj − µ(v) + δi)2 =  �  j(vj − µ(v))2 +  �  2δi  �  j(vj − µ(v))  �  +  �  j δ2  i = σ(v) + (n + 1)δ2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Rearranging the above and using that ui > vi ≥ µ(v), we have  σ(u) − σ(v) = (n + 1)δ2  i + (ui − µ(v) + δi)2 − (vi − µ(v) + δi)2 > (ui − vi)(ui + vi − 2µ(v) + 2δi) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' For n odd, the width of Dn is 2/√n + 1, and this is realized by they projection width of  all the directions in H =  �  v/√n + 1  �� v ∈ {−1, +1}n+1 and ⟨v, 1⟩ = 0  �  (and no other direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Proof: Consider a unit vector z that realizes the minimum width of Dn – here, in addition to ∥z∥ = 1,  we also require that ⟨z, 1⟩ = 0, as one has to consider only directions that are parallel to the hyperplane  containing Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' To this end, let β = maxi zi and α = mini zi and observe that  width(Dn) = ω(z, Dn) = max  i  ⟨z, ei⟩ − min  i  ⟨z, ei⟩ = β − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Next, Consider the point u, where for all i we set  ui =  �  α  zi < 0  β  zi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  A careful repeated application of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='1, implies that σ(u) > σ(z) if any coordinate of z is not  already either α or β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' But then the point �u has (i) “width” β − α, (ii)∥�u∥ > 1, and (iii) ⟨�u, 1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' But  this implies that the projection width of Dn on �u/∥�u∥ is (β − α)/∥�u∥ < β − α, which is a contradiction  to the choice of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Thus, it must be that all the coordinates of z are either α or β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Let t be the number of coordinates  of z that are α, and observe that  σ(z) =∥z∥2 = tα2 + (n + 1 − t)β2 = 1  and  ⟨z, 1⟩ = tα + (n + 1 − t)β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  2  This implies that β = −  t  n+1−tα, and thus  tα2 + (n + 1 − t)t2  (n + 1 − t)2 α2 = 1  =⇒  t(n + 1 − t) + t2  n + 1 − t  α2 = 1  =⇒  α = −  �  n + 1 − t  t(n + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Thus, the width of Dn is β −α =  �  1+  t  n+1−t  ��  n+1−t  t(n+1) =  �  n+1  t(n+1−t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' The last quantity is minimized when  the denominator is maximized, which happens for t = (n + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Namely, the width of Dn is 2/√n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  We have that tα + tβ = 0, which implies that α = −β and thus β = 1/√n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' It follows that  z ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' For n even, the width of Dn is 2  �  n+1  n(n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Proof: The proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='2 goes through with minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' The minimum value is realized  by t = n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' This implies that  α = −  �  n + 1 − t  t(n + 1) = −  �  n + 2  n(n + 1)  and  β = −  t  n + 1 − tα =  �  n  (n + 1)(n + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  As a sanity check, observe that tα2 + (n + 1 − t)β2 = n  2  n+2  n(n+1) + n+2  2  n  (n+1)(n+2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Thus, the width of  Dn is  β − α =  �  n  (n + 1)(n + 2) +  �  n + 2  n(n + 1) =  2(n + 1)  �  n(n + 1)(n + 2)  = 2  �  n + 1  n(n + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Using that ∆n = Dn/  √  2 and rescaling the above bounds, we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' The width of the regular n-simplex ∆n is  �  2/(n + 1) if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' The width is  �  2(n+1)  n(n+2)  if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  The inradius (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=', radius of largest ball inside ∆n) is rn = 1/  �  2n(n + 1), and the  circumradius (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=', radius of minimum ball enclosing ∆n is Rn =  �  n  2(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  References  [Ale77]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' The width and diameter of a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Geometriae Dedicata, 6(1), 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  [GK92]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Gritzmann and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Klee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Inner and outer j-radii of convex bodies in finite-dimensional  normed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Discret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content=', 7: 255–280, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}
+page_content='  3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfvQGV/content/2301.02616v1.pdf'}

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+ 
+ 
+FEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN 
+SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE 
+TOMAS CABEZON PEDROSO1 and JINMO RHEE2 and DARAGH 
+BYRNE3 
+1,2,3Carnegie Mellon University, USA. 
+[email protected], 0000-0002-5483-2676 
+[email protected], 0000-0003-4710-7385 
+[email protected], 0000-0001-7193-006X 
+ 
+Abstract. This paper compares the parametric design space with a 
+feature space generated by the extraction of design features using 
+deep learning (DL) as an alternative way for design space 
+exploration. In this comparison, the parametric design space is 
+constructed by creating a synthetic dataset of 15.000 elements using 
+a parametric algorithm and reducing its dimensions for visualization. 
+The feature space — reduced-dimensionality vector space of 
+embedded data features — is constructed by training a DL model on 
+the same dataset. We analyze and compare the extracted design 
+features by reducing their dimension and visualizing the results. We 
+demonstrate that parametric design space is narrow in how it 
+describes the design solutions because it is based on the combination 
+of individual parameters. In comparison, we observed that the 
+feature design space can intuitively represent design solutions 
+according to complex parameter relationships. Based on our results, 
+we discuss the potential of translating the features learned by DL 
+models to provide a mechanism for intuitive design exploration 
+space and visualization of possible design solutions. 
+Keywords. Deep Learning, VAE, Design Space, Feature Design Space, 
+Parametric Design Space, Design Exploration. 
+1. Introduction  
+ 
+Parametric modeling has acquired widespread acceptance among creative 
+practitioners as it allows the synthesis of various design options and solutions. 
+Changing the parameters in this modeling process, either manually or randomly, 
+can rapidly create a vast set of design variations (Toulkeridou, 2019). Navigating 
+the resulting parametric design space — where the design variants are 
+topologically placed by their parameters — is part of the design exploration 
+process — a crucial step in the development of new alternatives and design 
+solutions. Exploration of the parametric design space allows creative practitioners  
+ 
+ 
+
+T. CABEZON PEDROSO, J. RHEE AND D. BYRNE 
+  
+ 
+many benefits: to reach satisfying solutions, better define design problems, and 
+understand the opportunities and limitations of the possible solutions. 
+Despite these benefits, design exploration is laborious within the parametric space 
+and challenged along two fronts: comparison and selection (Fuchkina et al., 2018). 
+Parametric design exploration is an iterative process that focuses on the variation 
+of these individual parameters, rather than on the relationship among them 
+(Yamamoto and Nakakoji, 2005). Hence, comparing one design solution with 
+others by their parameters alone does not always result in a superior solution; for 
+example, the variants generated by the local combination of parameters might not 
+match the design requirements. Moreover, infinite alternative design solutions can 
+be generated by inputting new parameter values. Thus, the parametric design 
+space consists of a huge amount of design variants that cannot be fully or 
+sufficiently explored.  
+ 
+We propose an alternative way to construct and examine the design space, by 
+extracting features from a DL model. By comparing and analyzing how the DL 
+feature design space differs from the parametric design space, we illustrate the 
+potential of feature design space for design practitioners during the design 
+exploration process and provide a new way to compare, examine and select the 
+design alternatives based on the exploration of a properly constrained design 
+space.  
+ 
+No previous approach to compare the parametric design space and feature design 
+space as design exploration tools has been found. To demonstrate how the feature 
+space compares to the parametric space, we designed an experiment to construct 
+both a parametric design space and a feature design space using the same dataset. 
+The dataset consists of 15.000 synthetic 3D models produced by a parametric 
+algorithm with five parameters. This parametric design space consists of five axes; 
+each axis corresponds to each of the parameters that are used as inputs of the 
+parametric algorithm. Subsequently, this same dataset is used to train a DL model 
+to compress the data into a feature vector of 128 dimensions. Both the parametric 
+space (five-axes) and the feature space (128 axes) are not directly visualizable due 
+to their high dimensionality. Nevertheless, as visual feedback plays an important 
+role in design exploration (Bradner, Iorio and Davis, 2014), we employ a 
+dimensionality reduction algorithm (t-SNE) to the design space. We are able to 
+illustrate the design exploration space, showing how the data is distributed across 
+both the parametric and feature design spaces. 
+ 
+In the next section, we describe the generation of the dataset, as well as the 
+construction of parametric design space and its visualization. In Section 3., we 
+illustrate how training a DL model resulted in a feature space for design 
+exploration and comparison with the parametric approach. Then, in Section 4., we 
+will compare, contrast, and discuss the characteristics of the DL feature space and 
+the parametric space. (Figure 1.) 
+
+FEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN 
+SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE 
+ 
+ 
+ 
+ 
+Figure 1. The overall process of comparing parametric design space and feature space from deep 
+learning 
+ 
+2. Constructing Parametric Design Space 
+2.1. DATASET GENERATION 
+ 
+To conduct a design space comparison, a simple parametric modeling system was 
+designed: a parametric algorithm for generating different styles of vessels. As with 
+handcraft of pottery wheel throwing, a simple Bezier curve with three control 
+points was turned around an axis to generate each 3D digital vessels; the form of 
+each vessel is specified by the five parameters that were used as inputs. These 
+parameters, as can be seen in Figure 2, are: the height of the vessel, the width of 
+the base, the width of the top opening, and the horizontal and vertical coordinates 
+of the central control point of the Bezier curve that are used to create the curve of 
+the form. The five parameters are represented as a vector, and each vector 
+corresponds to a specific 3D model of a vessel.  
+ 
+Using this system, we created a 3D vessel dataset by randomly generating a total 
+of 15.000 different vessels. The total shape of the parametric representation of the 
+vessel dataset is [15.000, 5], however, as it will be explained in the next section, 
+
+Parametric
+5
+algorithm
+Parametric
+Data
+parameters
+Design Space
+Comparison
+Feature
+VAE
+DesignSpace
+ENCODER
+DECODERT. CABEZON PEDROSO, J. RHEE AND D. BYRNE 
+  
+ 
+only 3.000 vessels were used for the space exploration and visualization, so this 
+will be a design space of size [3.000, 5]. 
+ 
+ 
+ 
+Figure 2. Upper: An illustration of the dataset parameters. Lower: Three illustrative examples from 
+the dataset with the parameters and the resulting 3D form side-by-side. 
+ 
+2.2. DIMENSIONALITY REDUCTION 
+ 
+As a five-dimensional space makes it hard to compare models and to visualize 
+and compare the characteristics, we employed a dimensionality reduction process  
+to reduce the space to two-dimensions and enable the objects to be plotted and 
+compared to one another. Figure 3. shows the overall process of visualizing the 
+space using t-Distributed Stochastic Neighbour Embedding (t-SNE) algorithm 
+(van der Maaten and Hinton, 2008). t-SNE is a popular dimensionality-reduction 
+algorithm for visualizing high-dimensional data. The hyper-parameters used for 
+this reduction are: perplexity: 30; learning rate: 200; and iterations: 1.000. 
+ 
+Figure 3.  Illustration of the dimensional reduction process for the 3D vessel dataset, and the 
+construction of a parametric design space. 
+ 
+After dimensional  reduction, each point in the plot represents the corresponding 
+embedding of a vessel in the parametric design space. Each point is expressed as 
+
+1. Pot top width
+2. Pot bottom width
+3. Control point horizontal coordinate
+4. Control point vertical coordinate
+5. Pot height
+2TopLevelApp (400 x 400)
+Parameters:
+bottomWidth: 0.452
+topWidth: 0.943
+height: 0.593
+thickness: 0.050
+bMp_x: 0.797
+bMp_y: 0.093
+bLp_x: 0.943
+≤69'0 :K_d7q
+Figure 1
+Press:
+d, to display the mesh
+s, to savethe,sti file
+100
+75
+50
+25
+0
+25
+50
+75
+100
+100
+50
+100
+50
+0
+50
+50
+100
+100
+x=-93.6943,y=126.5305,z=148.6895TopLevelApp(400x400)
+Parameters:
+bottomWidth: 0.652
+topWidth: 0.676
+height: 0.595
+thickness: 0.050
+bMp_x: 0.675
+bMp_y: 0.019
+bLp_xc 0.676
+bLp_y: 0.595
+Figure1
+Press:
+d, to display the mesh
+bezier
+s, to save the,sti flle
+100
+50
+0
+-50
+-100
+100
+50
+-100
+0
+-50
+50
+0
+50
+-100
+100
+x=-71.5373,y=146.2703,z=159.6157TopLevelApp(400x400)
+Parameters:
+bottomWidth: 0.273
+height: 0.755
+thickness: 0.050
+bMp_x: 0.719
+bMp_y: 0.524
+bLp_xc 0.904
+bLp_y: 0.755
+Figure1
+Press:
+d, to display the mesh
+bezier
+s, to save the,sti flle
+75
+50
+25
+0
+-25
+50
+75
+100
+75
+50
+25
+-100
+0
+-75
+_50
+-25
+-25
+0
+50
+25
+75
+50
+75
+100
+100[3k,5]
+[3k,2]
+dimension 1
+PLOT
+...
+t-SNE
+x
+Parametric
+Dimensional
+dimension 0
+Design Space
+ReductionFEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN 
+SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE 
+a 2D image of the profile cut section of the corresponding vessel. Figure 4. 
+represents the reduced parametric design space of the dataset.  
+ 
+Figure 4. A 2D visualization of the parametric design space of the vessel dataset. Inset image: a 
+detailed section for a subset of the models. 
+3. Constructing the Feature Space 
+ 
+To construct the design space based on the features and not the parameters, we 
+used a Variational Autoencoder (VAE) as a tool for extracting the morphological 
+features of the vessels. VAEs (Kingma and Welling, 2013) are a type of generative 
+deep neural network used for statistical inference problems as they generalize a 
+probabilistic distribution of the given dataset and synthesize new data samples 
+from that distribution. VAEs are composed of two modules: encoder and decoder. 
+The encoder abstracts the input data into smaller dimensional vectors, latent 
+vectors, and the decoder reconstructs the latent vector back into a 3D shape. 
+During the encoding process, the network captures and extracts the features of the 
+input data. These features can be topologically placed in the data space, namely, 
+latent space. In the latent space, the distance between two data points represents 
+the degree of resemblance of data: the closer points, the more resembled.  We 
+translate this latent space as the feature space for an alternative way to explore the 
+design space. 
+3.1. DATA PRE-PROCESSING 
+ 
+Different representations of 3D data have been used in DL research, like point 
+clouds (Achlioptas et al., 2018), meshes (Ranjan et al., 2018), or voxels (Wu et al., 
+2017). As resolutions of the data is not key for our purpose rather than the 
+extracted features of it; and because we will implement a VAE for this experiment 
+that needs fixed space inputs for the Convolutional Neural Networks (CNNs), we 
+will be representing our 3D data with voxels. Voxels are discretized three-
+dimensional grids containing a binary value of volumetric occupancy of an object; 
+they distinguish between the elements on the grid that are filled with material and 
+those that are empty. The size of the voxel will determine the number of divisions 
+
+-
+-
+1
+-
+4
+I
+--
+I
+-
+-
+1
+1
+1
+1
+1
+-
+I
+!
+1
+I
+1
+1
+1
+I
+1
+1
+1
+I
+1
+I
+=T. CABEZON PEDROSO, J. RHEE AND D. BYRNE 
+  
+ 
+of the grid, consequently, the resolution at which we represent our 3D models; the 
+larger size, the more detailed 3D models. In this experiment, we used 32-sized 
+voxels so that a 3D vessel model is represented by 32x32x32 grid, the shape of 
+the entire dataset is [15.000, 32, 32, 32]. Finally, the dataset was then divided into 
+two groups: 80% of the dataset (12.000 vessels) was used for training the DL 
+model, and the remaining 20% (3.000 vessels) was used for testing the model and 
+the parametric and feature space analysis and comparison.  
+3.2. TRAINING 
+ 
+For training the model, we adopted the VAE architecture implemented in 
+‘Adversarial Generation of Continuous Implicit Shape Representations’ 
+(Kleineberg, Fey and Weichert, 2020). The encoder consists of four residual 
+blocks. Each residual block is composed of a 3D convolution layer, followed by 
+a batch normalization and a Leaky ReLu activation layer. The decoder, on the 
+contrary, comprises four residual blocks. Each block starts with a batch 
+normalization, followed by a Leaky ReLu activation layer, and finally a 3D 
+transposed convolution layer. The following hyper-parameters are used for 
+training the VAE with the voxelized vessel dataset: batch size 32,  Adam 
+optimizer (Kingma and Ba, 2015), learning rate 5e-. The model was trained in 
+Google Colab Pro using the Nvidia Tesla T4 GPU. 
+ 
+Figure 5. Training process losses. 
+ 
+The model was trained for a total of 240 epochs. We early stopped the model 
+before the model started to overfit, Figure 5. The loss function used during 
+training was a combination of two losses. The first one, is the Kullback–Leibler 
+divergence (KLD) loss (Kullback and Leibler, 1951), with a weight in the total 
+loss formula of 1. This function is a measurement of the difference between two 
+statistical distributions. The second loss is the Minimum Square Distance (MSE) 
+loss (Sammut and Webb, 2010). It is used as the reconstruction loss and measures 
+the error between the input voxels and the reconstructed output. Figure 6. shows 
+the reasonable quality of the reconstruction of the training result after 240 epochs. 
+To ensure the performance of the model, it was evaluated using the test set and 
+showed that the model maintained the accuracy with the new dataset, which shows 
+
+0.07
+IMSE Loss
+0.06
+KLD Loss
+Total Loss
+Total Loss
+0.05
+0.04
+0.03
+0.02
+0.01
+0
+50
+100
+150
+200
+250
+EpochsFEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN 
+SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE 
+that the model generalizes well to new data and is able to encode never seen before 
+3D vessels.  
+ 
+ 
+Figure 6. Two examples of reconstructions from the trained VAE: the section slides and 3D voxels 
+of the ground truth (the top row of each example) and the reconstruction (bottom of each example). 
+3.3. DIMENSIONALITY REDUCTION 
+ 
+Figure 7. Feature space generation and visualization diagram. 
+ 
+Once the VAE is trained, the encoder is used to extract the features of each vessel 
+in the test dataset from 32.768 dimensions, the size of each voxelized vessel, into 
+128-dimensional vectors, the latent vectors. Consequently, the entire test dataset 
+of the vessels is represented into vectors whose total shape is [3.000, 128]. 
+ 
+Like in the parametric case, 128 dimensions are non-visualizable so the same 
+process as in Section 2 is followed. t-SNE algorithm is used to reduce the 
+dimensionality of each vector and plot the resulting two dimensions in an image 
+with the section of each of the vessels (Figure 7.). The hyper-parameters used for 
+this reduction are: perplexity: 50; learning rate: 700; and iterations: 3. Figure 8. 
+shows the results of distributed feature vectors in the reduced dimensional space, 
+the feature space. 
+
+1- Input (Ground truth)
+10
+10
+10
+20
+30
+05
+20
+20
+20
+20
+0.0
+0.0
+0.5
+1.0
+Output
+10
+to
+10
+O1
+1o
+20
+20
+20
+20
+20
+20
+30
+20
+0
+20
+20
+20
+20
+20
+0.0 
+0.0
+20
+0.5
+1.0
+2- Input (Ground truth)
+10 
+10
+10
+10
+10
+20
+20 
+20
+20
+30
+20
+20
+20
+20
+20
+0.0
+0.0
+0.5
+1.0
+Output
+10
+OT
+10
+10
+10
+10
+.6
+20
+20
+20
+20
+20
+20
+20
+30
+20
+20
+20
+20
+0
+20
+o
+20
+0
+20
+0.0-
+0.0
+0.5
+1.0[3k,128]
+[3k,2]
+dimension
+ENCODER
+PLOT
+voxel.npy
+t-SNE
+:
+dimension 0
+Latent
+Dimensional
+object.stl
+Space
+ReductionT. CABEZON PEDROSO, J. RHEE AND D. BYRNE 
+  
+ 
+ 
+Figure 8. A 2D visualization of the feature design space of the vessel dataset. Inset image: a detailed 
+section for a subset of the models. 
+4. Comparison Between the spaces 
+ 
+Figure 8. shows that similar vessels have been clustered together. Thinner vessels 
+are located at the top right of the image, in contrast to the opposite lower bottom 
+corner with the bigger vessels. The figure illustrates how the VAE model is able 
+to understand the relationship between the parameters and their influence on the 
+output morphological shape.  
+ 
+On the contrary, in the parametric space (Figure 4.), we can see how concave 
+vessels were gathered at the bottom of the image, however, if the height of the 
+vessels is considered, we can see that this parameter was not considered when 
+clustering the vessels. Parametric space is based on each parameter independently, 
+and not on the relationship among them. Therefore, we observe that parametric 
+design space insufficiently expresses the final form characteristics of the vessels 
+by the combinations of the parameters. On the contrary, in Figure 8., the feature 
+space, a gradual change in the shape or concavity as well as height or width is 
+observed.  
+ 
+To further examine and compare the characteristics of both design spaces, we used 
+a clustering, algorithm: a Density-Based Spatial Clustering of Applications with 
+Noise (DBSCAN) (Ester M et al., 1996). It is one of the most common clustering 
+algorithms that finds core samples of high density and expands clusters with them. 
+Figure 9. shows the results of this clustering.  
+ 
+The parametric design space has a total of seven clusters: three of them large, and 
+four of them small. It shows how the parametric design space doesn’t provide 
+enough information to intuitively compare the design variants locally, this space 
+shows extreme changes in vessel forms even in the same cluster. 
+
+"FEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN 
+SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE 
+The feature design space, on the contrary, has a total of nine clusters: six main big 
+clusters, and three smaller ones. In the feature design space, we can trace smooth 
+changes in the forms as we move through the different clusters (local changes) 
+and along the whole image (global changes). Shorter vessels are located on the 
+top, while taller ones are on the bottom. If we move on the horizontal axis, the 
+curve that generates the vessels goes from a concave shape on the right to a convex 
+shape on the left. This gives the designer the ability to locally compare similar 
+design alternatives. 
+ 
+    Parametric Design Space:                        Feature Design Space:  
+ 
+ 
+Figure 9. Final visualization and clusters of the parametric and feature design spaces with 
+representative vessels of each group. 
+5. Conclusion and Future work 
+ 
+We constructed the parametric and the feature design spaces using a custom 
+synthetic dataset and a VAE model. By comparing the parametric and feature 
+design spaces, we observed improved distributions of design alternatives in the 
+later. When the multi-dimensional parametric design space is projected into a 2D 
+space (Figures 4. and 9. left), the clusters are insufficiently relevant to the 
+morphological characteristics. On the other hand, when the multi-dimensional 
+feature space is projected into a 2D space (Figures 8. and 9. right), the clusters 
+show sufficient relevance to the features of the data they represent. 
+ 
+Based on this comparison, we conclude that combination of individual parameters 
+in the parametric design space is limited in representing the morphological 
+characteristics of the shapes. However, we showed that DL models can be used to 
+extract design features from 3D models and that the extracted features are more 
+complex than the combinations of individual parameters. Hence, we conclude that 
+the extracted features, that include information of the relationships between the 
+parameters, can construct a well-distributed design space. For that reason, we 
+propose feature design space as a tool for design space exploration that creative 
+practitioners can use as a new way for looking at objects beyond the parametric 
+design space. 
+
+T. CABEZON PEDROSO, J. RHEE AND D. BYRNE 
+  
+ 
+Our results and implications are limited to a single dataset and DL model, however 
+the results seem promising. Future work will expand on this study with more 
+diverse datasets generated by more complex parametric algorithms. Accordingly, 
+to perform the feature extraction, we would like to train other types of DL models 
+to investigate different potentials of DL in design. 
+6. References  
+ 
+Achlioptas, P. et al. (2018) ‘Learning Representations and Generative Models for 3D Point 
+Clouds’, In International conference on machine learning (pp. 40-49). PMLR. 
+Bradner, E., Iorio, F. and Davis, M. (2014) ‘Parameters Tell the Design Story: Ideation and 
+Abstraction in Design Optimization’, In Proceedings of the symposium on simulation for 
+architecture & urban design (Vol. 26) 
+Ester M et al. (1996) ‘A Density-Based Algorithm for Discovering Clusters in Large Spatial 
+Databases with Noise.’, KDD’96 Proceedings of the Second International Conference on 
+Knowledge Discovery and Data Mining, 96, pp. 226–231. 
+Fuchkina, E. et al. (2018) ‘Design Space Exploration Framework’, p. 10. 
+Kingma, D.P. and Ba, J.L. (2015) ‘Adam: A method for stochastic optimization’, in 3rd 
+International Conference on Learning Representations, ICLR 2015 - Conference Track 
+Proceedings. International Conference on Learning Representations, ICLR. Available at: 
+https://arxiv.org/abs/1412.6980v9 (Accessed: 30 May 2021). 
+Kingma, D.P. and Welling, M. (2013) ‘Auto-Encoding Variational Bayes’. Available at: 
+https://doi.org/10.48550/arXiv.1312.6114. 
+Kleineberg, M., Fey, M. and Weichert, F. (2020) ‘Adversarial Generation of Continuous 
+Implicit Shape Representations’. arXiv. Available at: 
+https://doi.org/10.48550/arXiv.2002.00349. 
+Kullback, S. and Leibler, R.A. (1951) ‘On Information and Sufficiency’, The Annals of 
+Mathematical Statistics, 22(1), pp. 79–86. Available at: 
+https://doi.org/10.1214/aoms/1177729694. 
+van der Maaten, L. and Hinton, G. (2008) ‘Viualizing data using t-SNE’, Journal of Machine 
+Learning Research, 9, pp. 2579–2605. 
+Ranjan, A. et al. (2018) ‘Generating 3D Faces using Convolutional Mesh Autoencoders’, in. 
+Proceedings of the European Conference on Computer Vision (ECCV), pp. 704–720. 
+Available at: 
+https://openaccess.thecvf.com/content_ECCV_2018/html/Anurag_Ranjan_Generating_3
+D_Faces_ECCV_2018_paper.html (Accessed: 7 December 2022). 
+Sammut, C. and Webb, G.I. (eds) (2010) ‘Mean Squared Error’, in Encyclopedia of Machine 
+Learning. Boston, MA: Springer US, pp. 653–653. Available at: 
+https://doi.org/10.1007/978-0-387-30164-8_528. 
+Toulkeridou, V. (2019) ‘Steps towards AI augmented parametric modeling systems for 
+supporting design exploration’, in Blucher Design Proceedings. 37 Education and 
+Research in Computer Aided Architectural Design in Europe and XXIII Iberoamerican 
+Society of Digital Graphics, Joint Conference (N. 1), Porto, Portugal: Editora Blucher, pp. 
+81–92. Available at: https://doi.org/10.5151/proceedings-ecaadesigradi2019_602. 
+Wu, J. et al. (2017) ‘Learning a Probabilistic Latent Space of Object Shapes via 3D 
+Generative-Adversarial Modeling’. arXiv. Available at: http://arxiv.org/abs/1610.07584 
+(Accessed: 7 December 2022). 
+Yamamoto, Y. and Nakakoji, K. (2005) ‘Interaction design of tools for fostering creativity in 
+the early stages of information design’, International Journal of Human-Computer 
+Studies, 63(4–5), pp. 513–535. Available at: https://doi.org/10.1016/j.ijhcs.2005.04.023. 
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+A gallery of diagonal stability conditions with
+structured matrices (and review papers)
+Zhiyong Sun
+Control Systems Group, Department of EE, TU Eindhoven, the Netherlands
+Abstract
+This note presents a summary and review of various conditions and characterizations
+for matrix stability (in particular diagonal matrix stability) and matrix stabilizability.
+Keywords:
+Matrix stability, matrix stabilizability, diagonal stability.
+1. Definitions and notations
+• A square real matrix is a Z-matrix if it has nonpositive off-diagonal elements.
+• A Metzler matrix is a real matrix in which all the off-diagonal components are
+nonnegative (equal to or greater than zero).
+• A Z-matrix with positive principal minors is an M-matrix.
+– Note: There are numerous equivalent characterizations for M-matrix (Fiedler
+and Ptak, 1962; Plemmons, 1977). A more commonly-used condition is the
+following: A matrix A ∈ Rn×n is called an M-matrix, if its non-diagonal
+entries are non-positive and its eigenvalues have positive real parts.
+• A square matrix A is (positive) stable if all its eigenvalues have positive parts.
+Equivalently, a square matrix A is (positive) stable iff there exists a positive
+definite matrix D such that AD + DAT is positive definite.
+– Note: in control system theory we often define stable matrix as the set of
+square matrices whose eigenvalues have negative real parts (a.k.a. Hurwitz
+matrix). The two definitions of stable matrices will be distinguished in the
+context.
+• A square complex matrix is a P-matrix if it has positive principal minors.
+• A square complex matrix is a P +
+0 -matrix if it has nonnegative principal minors
+and at least one principal minor of each order is positive.
+• A real square matrix A is multiplicative D-stable (in short, D-stable) if DA is
+stable for every positive diagonal matrix D.
+Preprint submitted to Nowhere
+January 4, 2023
+arXiv:2301.01272v1  [eess.SY]  1 Jan 2023
+
+• A square matrix A is called totally stable if any principal submatrix of A is
+D-stable.
+• A real square matrix A is said to be additive D-stable if A + D is stable for
+every nonnegative diagonal matrix D.
+• A real square matrix A is said to be Lyapunov diagonally stable if there exists
+a positive diagonal matrix D such that AD + DAT is positive definite.
+– Note: Lyapunov diagonally stable matrices are often referred to as just di-
+agonally stable matrices or as Volterra–Lyapunov stable, or as Volterra
+dissipative in the literature (see e.g., (Logofet, 2005)).
+• A matrix A = {aij} ∈ Rn×n is generalized row-diagonally dominant, if there
+exists x = (x1, x2, · · · , xn) ∈ Rn with xi > 0, ∀i, such that
+|aii|xi >
+n
+�
+j=1,j̸=i
+|aij|xj, ∀i = 1, 2, · · · , n.
+(1)
+• A matrix A = {aij} ∈ Rn×n is generalized column-diagonally dominant, if
+there exists x = (x1, x2, · · · , xn) ∈ Rn with xi > 0, ∀i, such that
+|ajj|xj >
+n
+�
+i=1,i̸=j
+|aij|xi, ∀j = 1, 2, · · · , n.
+(2)
+– Note: the set of generalized column-diagonally dominant matrices is equiv-
+alent to the set of generalized row-diagonally dominant matrices (Varga,
+1976; Sun et al., 2021). They are also often referred to as quasi-diagonally
+dominant matrices (Kaszkurewicz and Bhaya, 2012).
+• For a real matrix A = {aij} ∈ Rn×n, we associate it with a comparison matrix
+MA = {mij} ∈ Rn×n, defined by
+mij =
+�
+|aij|,
+if
+j = i;
+−|aij|,
+if
+j ̸= i.
+A given matrix A is called an H-matrix if its comparison matrix MA is an M-
+matrix.
+– The set of H-matrix is equivalent to the set of quasi-diagonally dominant
+matrices (Kaszkurewicz and Bhaya, 2012; Sun et al., 2021).
+• A square matrix A is diagonally stabilizable if there exists a diagonal matrix D
+such that DA is stable.
+Note: Many definitions above for real matrices also carry over to complex matrices;
+the distinction between real and complex matrices will be made clear in the context.
+2
+
+2. Conditions for diagonally stabilizable matrices
+A key motivating question: Given a square matrix A, can we find a diagonal ma-
+trix D such that the matrix DA is stable?
+Fisher and Fuller (Fisher and Fuller, 1958) proved the following result:
+Theorem 1. (Fisher and Fuller, 1958) If P is a real n × n matrix fulfilling the condi-
+tion:
+• (A): P has at least one sequence of non-zero principal minors Mk of every order
+k = 1, 2, · · · , n, such that Mk−1 is one of the k first principal minors of Mk;
+then there exists a real diagonal matrix D such that the characteristic equation of DP
+is stable.
+The Fisher-Fuller theorem is also formulated as the following alternative version:
+Theorem 2. Let P be an n × n real matrix all of whose leading principal minors are
+positive. Then there is an n × n positive diagonal matrix D such that all the roots of
+DP are positive and simple.
+Fisher later gave a simple proof for a similar yet stronger result (Fisher, 1972).
+Theorem 3. (Fisher, 1972) If P is an n × n real matrix that has at least one nested
+set of principal minors, Mk, such that (−1)kMk > 0, ∀k = 1 · · · , n, then there exists
+a real diagonal matrix D with positive diagonal elements such that the characteristic
+roots of DP are all real, negative, and distinct.
+Remark 1. Some remarks on the conditions of diagonally stabilizable matrices are in
+order.
+• The above theorems involve determining the sign of (at least) one nested set
+of principal minors. In (Johnson et al., 1997), sufficient conditions are deter-
+mined for an n-by-n zero-nonzero pattern to allow a nested sequence of nonzero
+principal minors. In particular, a method is given to sign such a pattern so
+that it allows a nested sequence of k-by-k principal minors with sign(−1)k for
+k = 1, · · · , n.
+• The condition in the Fisher-Fuller theorem appears as a sufficient condition for
+matrix diagonal stabilizability. A necessary condition for matrix diagonal stabi-
+lizability is: for each order k = 1 · · · , n, at least one k ×k principal minor of P
+is non-zero. It is unclear what would be the necessary and sufficient condition.
+Ballantine (Ballantine, 1970) extended the above Fisher-Fuller theorem to the com-
+plex matrix case.
+Theorem 4. (Ballantine, 1970) Let A be an n×n complex matrix all of whose leading
+principal minors are nonzero. Then there is an n × n complex diagonal matrix D such
+that all the roots of DA are positive and simple.
+3
+
+Remark 2. It is shown in (Hershkowitz, 1992) the above Ballantine theorem cannot be
+strengthened by replacing “complex diagonal matrix D” by “positive diagonal matrix
+D”. A counterexample is shown in (Hershkowitz, 1992) involving a 2 × 2 complex
+matrix A with positive leading principal minors that there exists no positive diagonal
+matrix D such that the eigenvalues of DA are positive.
+A related problem to characterize diagonal stabilizable matrix is the Inverse Eigen-
+value Problem (IEP), and Friedland (Friedland, 1977) proved the following theorem.
+Theorem 5. (Friedland, 1977) Let A be a given n×n complex valued matrix. Assume
+that all the principal minors of A are different from zero. Then for any specified set
+λ = {λ, · · · , λn} ∈ Cn there exists a diagonal complex valued matrix D such that the
+spectrum of AD is the set λ. The number of such D is finite and does not exceed n!.
+Moreover, for almost all λ the number of the diagonal matrices D is exactly n!.
+Remark 3. The Friedland theorem of the IEP problem in the complex matrix case
+cannot be directly carried over to the real case. Further, it is easy to show with a
+counterexample of a 2×2 matrix that eigenvalue positionability in the real case cannot
+always be guaranteed, even with nonzero principal minors.
+In (Hershkowitz, 1992) the following two theorems are proved.
+Theorem 6. (Hershkowitz, 1992) Let A be a complex square matrix with positive
+leading principal minors, and let ϵ be any positive number. Then there exists a positive
+diagonal matrix D such that the eigenvalues of DA are simple, and the argument of
+every such eigenvalue is less in absolute value than ϵ.
+Theorem 7. (Hershkowitz, 1992) Let A be a complex square matrix with real principal
+minors and positive leading principal minors. Then there exists a positive diagonal
+matrix D such that DA has simple positive eigenvalues.
+Remark 4. The above theorems all present certain sufficient conditions to characterize
+diagonally stabilizable matrix and the IEP problem, and they are not necessary. A
+necessary condition for the diagonal matrix D to exist is that for each order i, at
+least one i × i principal minor of A is nonzero. However, a full characterization (with
+necessary and sufficient condition) for diagonally stabilizable matrix still remains an
+open problem.
+A variation of the diagonal matrix stabilization problem is the following:
+• Problem (*): Given a real square matrix G, find a real diagonal matrix D such
+that the product GD is Hurwitz together with all its principal submatrices.
+Surprisingly, a necessary and sufficient condition exists for solving the above problem
+as shown in (Locatelli and Schiavoni, 2012). Let M := {1, 2, · · · , m} and F :=
+{f|f ⊂ M}. Further, for any m×m matrix ∆, denote by ∆(f) the principal submatrix
+obtained from it after removing the rows and columns with indexes in f, f ∈ F. The
+main result of (Locatelli and Schiavoni, 2012) proves the following:
+4
+
+Theorem 8. (Locatelli and Schiavoni, 2012) Problem (*) admits a solution if and only
+if
+det(G(f))det(GD(f)) > 0, ∀f ∈ F
+(3)
+where GD = diag{gii}. Moreover, if the above condition is satisfied, then there exists
+¯ϵ > 0 such that, for any given ϵ ∈ (0, ¯ϵ), the matrix
+D := GDZ(ϵ), Z(ϵ) := −diag{ϵi}
+(4)
+solves the stabilization problem (*).
+5
+
+3. Conditions for diagonally stable matrices
+We give a short summary of available conditions for diagonally stable matrices
+(excerpts from (Barker et al., 1978), (Cross, 1978) and (Hershkowitz, 2006)).
+• (Barker et al., 1978) Lyapunov diagonally stable matrices are P-matrices.
+• (Barker et al., 1978) A matrix A being Lyapunov diagonally stable is equivalent
+to that there exists a positive diagonal matrix D such that xT DAx > 0 for all
+nonzero vectors x.
+• (Barker et al., 1978) A 2 × 2 real matrix is Lyapunov diagonally stable if and
+only if it is a P-matrix.
+• (Cross, 1978) For a given Lyapunov diagonally stable matrix P, all principal
+submatrices of P are Lyapunov diagonally stable.
+• (Barker et al., 1978) A real square matrix A is Lyapunov diagonally stable if
+and only if for every nonzero real symmetric positive semidefinite matrix H the
+matrix HA has at least one positive diagonal element.
+– Note: this result is termed the BBP theorem, and is proved again in (Shorten
+et al., 2009) with a simpler proof.
+• (Cross, 1978) The set of Lyapunov diagonally stable matrices is a strict subset of
+multiplicative D-stable matrices.
+• (Cross, 1978) The set of Lyapunov diagonally stable matrices is a strict subset of
+additive D-stable matrices.
+– Note: Multiplicative D-stable and additive D-stable matrices are not neces-
+sarily diagonally stable.
+• A Z-matrix is Lyapunov diagonally stable if and only if it is a P -matrix (that is,
+it is an M-matrix).
+• A non-singular H-matrix with nonnegative diagonal parts is Lyapunov diago-
+nally stable.
+• A quasi-diagonal dominant matrix with nonnegative diagonal parts is Lyapunov
+diagonally stable. Note the equivalence of Hurwitz H-matrix and quasi-diagonal
+dominant matrix (Sun et al., 2021).
+The following facts are shown in (Cross, 1978) and (Kaszkurewicz and Bhaya, 2012):
+• For normal matrices and within the set Z, D-stability, additive D-stability, and
+diagonal stability are all equivalent to matrix stability.
+• If a matrix A is Hurwitz stable, D-stable, or diagonally stable, then the matrices
+AT and A−1 also have the corresponding properties.
+6
+
+In (Shorten and Narendra, 2009) Shorten and Narendra showed the following nec-
+essary and sufficient condition for matrix diagonal stability (an alternative proof of the
+theorem of Redheffer via the KYP lemma):
+Theorem 9. (Shorten and Narendra, 2009) and (Redheffer, 1985) Let A ∈ Rn×n be a
+Hurwitz matrix with negative diagonal entries. Let An−1 denote the [n − 1 × n − 1]
+leading sub-matrix of A, and Bn−1 denote the corresponding block of A−1. Then,
+the matrix A is diagonally stable, if and only if there is a common diagonal Lyapunov
+function for the LTI systems ΣAn−1 and ΣBn−1.
+The above theorem involves finding a common diagonal Lyapunov function for a
+set of LTI systems, which may be restrictive and computationally demanding in practi-
+cal applications especially when the dimension of the matrix A is large.
+7
+
+Figure 1: Relations of matrix stability under different matrix types: the main theorem in (Berman and
+Hershkowitz, 1983)
+4. Relations of matrix stability and diagonal stability
+The paper (Berman and Hershkowitz, 1983) characterizes the relations of certain
+special matrices for matrix diagonal stability. They define
+• A = {A : there exists a positive definite diagonal matrix D such that AD +
+DAT is positive definite};
+i.e., A denotes the set of diagonally stable matrices;
+• L = {A : there exists a positive definite matrix D such that AD+DAT is positive definite};
+i.e., L denotes the set of (positive) stable matrices;
+• P = {A : the principle minors of A are positive};
+i.e., P denotes the set of P-matrices;
+• S = {A : there exists a positive vector x such that Ax is positive};
+i.e., S denotes the set of semipositive matrices.
+The main result of (Berman and Hershkowitz, 1983) is cited and shown in Fig. 1.
+In general, these different sets of structured matrices are not equivalent, and the set A
+is a subset of the other sets. However, for Z-matrices, these sets are equivalent. In
+particular, for the case of Z-matrices, the characterizations of these sets give equiva-
+lent conditions for M-matrices (upon a sign change). Note there are yet many more
+conditions to characterize M-matrices; see e.g., (Plemmons, 1977).
+The review paper (Hershkowitz, 1992) presents the implication relations between
+matrix stability conditions, and the equivalent relations of matrix stabilities for Z-
+matrices, as cited in Figs. 2 and 3. Again, as shown in Figs. 3, for Z-matrices, all
+the stability types are equivalent.
+8
+
+THEOREM 1.
+a. In general
+b. For Z-matrices, i.e., matrices with nonpositive off-diagonal entries,
+c. For symmetric matrices
+d. For triangular matrices
+e. for normal matrices
+The absence of an implication in the above relations means that a counterexample exists.Figure 2: The implication relations between matrix stability conditions, cited from (Hershkowitz, 1992)
+.
+Figure 3: For Z-matrices, all the stability types are equivalent. Cited from (Hershkowitz, 1992)
+.
+9
+
+A is Lyapunov diagonally stable
+A is a P-matrix
+A is D-stable
+A is additive D-stable
+A is stable
+A is a Pt -matrixA is Lyapunov diagonally stable
+A is a P-matrix
+A is D-stable
+A is additive D-stable
+A is a nonsingular M-matrix
+A is stableFigure 4: Relations among matrix stabilities. Cited from (Logofet, 2005, Fig.2)
+.
+The survey paper (Logofet, 2005) presents some beautiful flower-shaped character-
+izations of the relations among matrix stabilities, as cited in Figs. 4 and 5.
+10
+
+STABLE
+D-STABLE
+TOTALLYSTABLE
+DISSIPATIVE
+QUASI-
+DOMINANT
+M-MATRICES
+?NORMAL
+D-STABLE
+STABLE
+STABLEFigure 5: Petals of sign-stable matrices within the Flower. Cited from (Logofet, 2005, Fig.4)
+.
+5. Stability conditions with submatrices and Schur complement
+Stability conditions of ‘structured’ matrices often involve stability properties of
+submatrices, which employ block submatrices and their Schur complements to deter-
+mine stability.
+In (Narendra and Shorten, 2010), Narendra and Shorten presented necessary and
+sufficient conditions to characterize if a given Metzler matrix is Hurwitz, based on the
+fact that a Hurwitz Metzler matrix is diagonally stable. These conditions are general-
+ized in (Souza et al., 2017). We recall some main stability criteria from (Souza et al.,
+2017).
+Lemma 1. Let A ∈ Rn×n be a Metzler matrix partitioned in blocks of compatible
+dimensions as A = [A11, A12; A21, A22] with A11 and A22 being square matrices.
+Then the following statements are equivalent.
+• A is Hurwitz stable.
+• A11 and its Schur complement A/A11 := A22 −A21A−1
+11 A12 are Hurwitz stable
+Metzler matrices.
+• A22 and its Schur complement A/A22 := A11 −A12A−1
+22 A21 are Hurwitz stable
+Metzler matrices
+Remark 5. Some remarks are in order.
+11
+
+STABLE
+D-STABLE
+TOTALLYSTABLE
+DISSIPA-
+QUAS
+TIVE
+DOMINANT
+M-
+SIGN
+NORMAL
+STABLE
+D-STABLE
+QUASI-RECESSIVE
+STABLE
+OUAST-RECESSIVE
+STABLE• For a structured matrix, the property that its Schur complements also preserve
+the same stability and structure properties is termed Schur complement stabil-
+ity property. Other types of structured matrices that have Schur complement
+stability property include symmetric matrices, triangular matrices, and Schwarz
+matrices. See (Souza et al., 2017).
+• The result on M-matrix in Lemma 1 can be generalized to H-matrix: Let A be a
+H-matrix partitioned in blocks of compatible dimensions as A = [A11, A12; A21, A22]
+with A11 and A22 being square matrices. If A is Hurwitz stable, then A11 and
+its Schur complement A/A11 := A22 −A21A−1
+11 A12 are Hurwitz stable H matri-
+ces, or A22 and its Schur complement A/A22 := A11−A12A−1
+22 A21 are Hurwitz
+stable H matrices.
+• Schur complement and its closure property for several structured matrices (in-
+cluding diagonal matrices, triangular matrices, symmetric matrices, P-matrices,
+diagonal dominant matrices, M-matrices etc.) are discussed in (Zhang, 2006,
+Chap. 4).
+12
+
+6. Application examples of matrix diagonal stability conditions
+The Fisher-Fuller theorem on diagonal matrix stabilizability (Theorem 1 and its
+variations) has been rediscovered several times by the control system community, and
+has been applied in solving distributed stabilization and decentralized control problems
+in practice. This section reviews two application examples.
+6.1. Conditions for decentralized stabilization
+In (Corfmat and Morse, 1973) Corfmat and Morse solved the following problem:
+• For given and fixed real matrices A and P, find (if possible) a non-singular di-
+agonal matrix D such that I + ADP is Schur stable (i.e., all eigenvalues of
+I + ADP are located within the unit circle in the complex plane.
+To solve the above problem they proved the following:
+Theorem 10. If A is an n × n strongly non-singular matrix, then there exists a diag-
+onal matrix D such that (I + DA) is Schur stable.
+Note: in (Corfmat and Morse, 1973) a matrix is termed strongly non-singular, if
+its all n leading principal minors are nonzero.
+Theorem 11. If A is a fixed non-singular matrix, then there exists a permutation matrix
+P such that PA is strongly non-singular.
+Solution to decentralized stabilization: the non-singularity of A is a necessary and
+sufficient condition for the existence of a permutation matrix P and a non-singular
+diagonal matrix D such that (I + ADP) is Schur stable.
+6.2. Distributed stabilization of persistent formations
+In (Yu et al., 2009), the problem on persistent formation stabilization involves
+studying the stabilizability of the following differential equation
+˙z = ∆Az
+where ∆ is a diagonal or possibly block diagonal matrix, and A is a rigidity-like matrix
+on formation shapes. To solve the formation stabilization problem in (Yu et al., 2009)
+the following result is employed ((Yu et al., 2009, Theorem 3.2)):
+Theorem 12. Suppose A is an m × m non-singular matrix with every leading prin-
+cipal minor nonzero. Then there exists a diagonal D such that the real parts of the
+eigenvalues of DA are all negative.
+We remark that this is a reformulation of the Fisher-Fuller theorem.
+13
+
+7. A selection of key review papers and books on matrix stability and diagonal
+stability conditions
+• The survey paper (Hershkowitz, 1992) that presents a summary of relevant ma-
+trix stability results and the developments, up until 1992.
+• The paper (Bhaya et al., 2003) that presents comprehensive discussions and char-
+acterizations for various classes of matrix stability conditions.
+• The paper (Hershkowitz and Keller, 2003) that studies the relations between pos-
+itivity of principal minors, sign symmetry and stability of matrices.
+• The review paper (Hershkowitz, 2006) that presents an concise overview on ma-
+trix stability and inertia.
+• The book (Kaszkurewicz and Bhaya, 2012) on matrix diagonal stability in sys-
+tems and computation.
+• The summary paper (Logofet, 2005) that presents a review and some beautiful
+connections/relations on different matrix stabilities.
+• The very long survey paper (Kushel, 2019) that provides a unifying viewpoint
+on matrix stability, and its historical development.
+• The recent book (Johnson et al., 2020) on positive matrix, P-matrix and inverse
+M-matrix.
+References
+Ballantine, C., 1970. Stabilization by a diagonal matrix. Proceedings of the American
+Mathematical Society 25, 728–734.
+Barker, G., Berman, A., Plemmons, R.J., 1978.
+Positive diagonal solutions to the
+Lyapunov equations. Linear and Multilinear Algebra 5, 249–256.
+Berman, A., Hershkowitz, D., 1983. Matrix diagonal stability and its implications.
+SIAM Journal on Algebraic Discrete Methods 4, 377–382.
+Bhaya, A., Kaszkurewicz, E., Santos, R., 2003. Characterizations of classes of stable
+matrices. Linear algebra and its applications 374, 159–174.
+Corfmat, J., Morse, A., 1973. Stabilization with decentralized feedback control. IEEE
+Transactions on Automatic Control 18, 679–682.
+Cross, G., 1978. Three types of matrix stability. Linear algebra and its applications 20,
+253–263.
+Fiedler, M., Ptak, V., 1962. On matrices with non-positive off-diagonal elements and
+positive principal minors. Czechoslovak Mathematical Journal 12, 382–400.
+14
+
+Fisher, F.M., 1972. A simple proof of the Fisher–Fuller theorem, in: Mathematical
+Proceedings of the Cambridge Philosophical Society, Cambridge University Press.
+pp. 523–525.
+Fisher, M.E., Fuller, A., 1958. On the stabilization of matrices and the convergence
+of linear iterative processes, in: Mathematical Proceedings of the Cambridge Philo-
+sophical Society, Cambridge University Press. pp. 417–425.
+Friedland, S., 1977. Inverse eigenvalue problems. Linear Algebra and Its Applications
+17, 15–51.
+Hershkowitz, D., 1992. Recent directions in matrix stability. Linear Algebra and its
+Applications 171, 161–186.
+Hershkowitz, D., 2006. Matrix stability and inertia, in: Handbook of Linear Algebra.
+Chapman and Hall/CRC, pp. 19–1.
+Hershkowitz, D., Keller, N., 2003. Positivity of principal minors, sign symmetry and
+stability. Linear algebra and its applications 364, 105–124.
+Johnson, C.R., Maybee, J.S., Olesky, D., Van den Driessche, P., 1997. Nested se-
+quences of principal minors and potential stability. Linear Algebra and its Applica-
+tions 262, 243–257.
+Johnson, C.R., Smith, R.L., Tsatsomeros, M.J., 2020. Matrix positivity. volume 221.
+Cambridge University Press.
+Kaszkurewicz, E., Bhaya, A., 2012. Matrix diagonal stability in systems and compu-
+tation. Springer Science & Business Media.
+Kushel, O.Y., 2019. Unifying matrix stability concepts with a view to applications.
+SIAM Review 61, 643–729.
+Locatelli, A., Schiavoni, N., 2012. A necessary and sufficient condition for the stabil-
+isation of a matrix and its principal submatrices. Linear algebra and its applications
+436, 2311–2314.
+Logofet, D.O., 2005. Stronger-than-Lyapunov notions of matrix stability, or how “flow-
+ers” help solve problems in mathematical ecology. Linear Algebra and its Applica-
+tions 398, 75–100.
+Narendra, K.S., Shorten, R., 2010. Hurwitz stability of metzler matrices. IEEE Trans-
+actions on Automatic Control 55, 1484–1487.
+Plemmons, R.J., 1977. M-matrix characterizations. I–nonsingular M-matrices. Linear
+Algebra and its Applications 18, 175–188.
+Redheffer, R., 1985. Volterra multipliers ii. SIAM Journal on Algebraic and Discrete
+Methods 6, 612–623.
+15
+
+Shorten, R., Mason, O., King, C., 2009. An alternative proof of the Barker, Berman,
+Plemmons (BBP) result on diagonal stability and extensions. Linear Algebra and its
+Applications 430, 34–40.
+Shorten, R., Narendra, K.S., 2009. On a theorem of Redheffer concerning diagonal
+stability. Linear algebra and its applications 431, 2317–2329.
+Souza, M., Wirth, F.R., Shorten, R.N., 2017. A note on recursive schur complements,
+block hurwitz stability of metzler matrices, and related results. IEEE Transactions
+on Automatic Control 62, 4167–4172.
+Sun, Z., Rantzer, A., Li, Z., Robertsson, A., 2021. Distributed adaptive stabilization.
+Automatica 129, 109616.
+Varga, R.S., 1976. On recurring theorems on diagonal dominance. Linear Algebra and
+its Applications 13, 1–9.
+Yu, C., Anderson, B.D.O., Dasgupta, S., Fidan, B., 2009. Control of minimally per-
+sistent formations in the plane.
+SIAM Journal on Control and Optimization 48,
+206–233.
+Zhang, F., 2006. The Schur complement and its applications. volume 4. Springer
+Science & Business Media.
+16
+

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+page_content='A gallery of diagonal stability conditions with  structured matrices (and review papers)  Zhiyong Sun  Control Systems Group, Department of EE, TU Eindhoven, the Netherlands  Abstract  This note presents a summary and review of various conditions and characterizations  for matrix stability (in particular diagonal matrix stability) and matrix stabilizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Keywords:  Matrix stability, matrix stabilizability, diagonal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Definitions and notations  A square real matrix is a Z-matrix if it has nonpositive off-diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A Metzler matrix is a real matrix in which all the off-diagonal components are  nonnegative (equal to or greater than zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A Z-matrix with positive principal minors is an M-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  – Note: There are numerous equivalent characterizations for M-matrix (Fiedler  and Ptak, 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Plemmons, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' A more commonly-used condition is the  following: A matrix A ∈ Rn×n is called an M-matrix, if its non-diagonal  entries are non-positive and its eigenvalues have positive real parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A square matrix A is (positive) stable if all its eigenvalues have positive parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Equivalently, a square matrix A is (positive) stable iff there exists a positive  definite matrix D such that AD + DAT is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  – Note: in control system theory we often define stable matrix as the set of  square matrices whose eigenvalues have negative real parts (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Hurwitz  matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' The two definitions of stable matrices will be distinguished in the  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A square complex matrix is a P-matrix if it has positive principal minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A square complex matrix is a P +  0 -matrix if it has nonnegative principal minors  and at least one principal minor of each order is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A real square matrix A is multiplicative D-stable (in short, D-stable) if DA is  stable for every positive diagonal matrix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Preprint submitted to Nowhere  January 4, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='01272v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='SY]  1 Jan 2023  A square matrix A is called totally stable if any principal submatrix of A is  D-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A real square matrix A is said to be additive D-stable if A + D is stable for  every nonnegative diagonal matrix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A real square matrix A is said to be Lyapunov diagonally stable if there exists  a positive diagonal matrix D such that AD + DAT is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  – Note: Lyapunov diagonally stable matrices are often referred to as just di-  agonally stable matrices or as Volterra–Lyapunov stable, or as Volterra  dissipative in the literature (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', (Logofet, 2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A matrix A = {aij} ∈ Rn×n is generalized row-diagonally dominant, if there  exists x = (x1, x2, · · · , xn) ∈ Rn with xi > 0, ∀i, such that  |aii|xi >  n  �  j=1,j̸=i  |aij|xj, ∀i = 1, 2, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  (1)  A matrix A = {aij} ∈ Rn×n is generalized column-diagonally dominant, if  there exists x = (x1, x2, · · · , xn) ∈ Rn with xi > 0, ∀i, such that  |ajj|xj >  n  �  i=1,i̸=j  |aij|xi, ∀j = 1, 2, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  (2)  – Note: the set of generalized column-diagonally dominant matrices is equiv-  alent to the set of generalized row-diagonally dominant matrices (Varga,  1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' They are also often referred to as quasi-diagonally  dominant matrices (Kaszkurewicz and Bhaya, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  For a real matrix A = {aij} ∈ Rn×n, we associate it with a comparison matrix  MA = {mij} ∈ Rn×n, defined by  mij =  �  |aij|,  if  j = i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  −|aij|,  if  j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A given matrix A is called an H-matrix if its comparison matrix MA is an M-  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  – The set of H-matrix is equivalent to the set of quasi-diagonally dominant  matrices (Kaszkurewicz and Bhaya, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A square matrix A is diagonally stabilizable if there exists a diagonal matrix D  such that DA is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Note: Many definitions above for real matrices also carry over to complex matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  the distinction between real and complex matrices will be made clear in the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Conditions for diagonally stabilizable matrices  A key motivating question: Given a square matrix A, can we find a diagonal ma-  trix D such that the matrix DA is stable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Fisher and Fuller (Fisher and Fuller, 1958) proved the following result:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' (Fisher and Fuller, 1958) If P is a real n × n matrix fulfilling the condi-  tion:  (A): P has at least one sequence of non-zero principal minors Mk of every order  k = 1, 2, · · · , n, such that Mk−1 is one of the k first principal minors of Mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  then there exists a real diagonal matrix D such that the characteristic equation of DP  is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The Fisher-Fuller theorem is also formulated as the following alternative version:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Let P be an n × n real matrix all of whose leading principal minors are  positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Then there is an n × n positive diagonal matrix D such that all the roots of  DP are positive and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Fisher later gave a simple proof for a similar yet stronger result (Fisher, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' (Fisher, 1972) If P is an n × n real matrix that has at least one nested  set of principal minors, Mk, such that (−1)kMk > 0, ∀k = 1 · · · , n, then there exists  a real diagonal matrix D with positive diagonal elements such that the characteristic  roots of DP are all real, negative, and distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Some remarks on the conditions of diagonally stabilizable matrices are in  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The above theorems involve determining the sign of (at least) one nested set  of principal minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' In (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 1997), sufficient conditions are deter-  mined for an n-by-n zero-nonzero pattern to allow a nested sequence of nonzero  principal minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' In particular, a method is given to sign such a pattern so  that it allows a nested sequence of k-by-k principal minors with sign(−1)k for  k = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The condition in the Fisher-Fuller theorem appears as a sufficient condition for  matrix diagonal stabilizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' A necessary condition for matrix diagonal stabi-  lizability is: for each order k = 1 · · · , n, at least one k ×k principal minor of P  is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' It is unclear what would be the necessary and sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Ballantine (Ballantine, 1970) extended the above Fisher-Fuller theorem to the com-  plex matrix case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' (Ballantine, 1970) Let A be an n×n complex matrix all of whose leading  principal minors are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Then there is an n × n complex diagonal matrix D such  that all the roots of DA are positive and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  3  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' It is shown in (Hershkowitz, 1992) the above Ballantine theorem cannot be  strengthened by replacing “complex diagonal matrix D” by “positive diagonal matrix  D”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' A counterexample is shown in (Hershkowitz, 1992) involving a 2 × 2 complex  matrix A with positive leading principal minors that there exists no positive diagonal  matrix D such that the eigenvalues of DA are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A related problem to characterize diagonal stabilizable matrix is the Inverse Eigen-  value Problem (IEP), and Friedland (Friedland, 1977) proved the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' (Friedland, 1977) Let A be a given n×n complex valued matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Assume  that all the principal minors of A are different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Then for any specified set  λ = {λ, · · · , λn} ∈ Cn there exists a diagonal complex valued matrix D such that the  spectrum of AD is the set λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' The number of such D is finite and does not exceed n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='.  Moreover, for almost all λ the number of the diagonal matrices D is exactly n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='.  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' The Friedland theorem of the IEP problem in the complex matrix case  cannot be directly carried over to the real case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Further, it is easy to show with a  counterexample of a 2×2 matrix that eigenvalue positionability in the real case cannot  always be guaranteed, even with nonzero principal minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  In (Hershkowitz, 1992) the following two theorems are proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' (Hershkowitz, 1992) Let A be a complex square matrix with positive  leading principal minors, and let ϵ be any positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Then there exists a positive  diagonal matrix D such that the eigenvalues of DA are simple, and the argument of  every such eigenvalue is less in absolute value than ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' (Hershkowitz, 1992) Let A be a complex square matrix with real principal  minors and positive leading principal minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Then there exists a positive diagonal  matrix D such that DA has simple positive eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' The above theorems all present certain sufficient conditions to characterize  diagonally stabilizable matrix and the IEP problem, and they are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' A  necessary condition for the diagonal matrix D to exist is that for each order i, at  least one i × i principal minor of A is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' However, a full characterization (with  necessary and sufficient condition) for diagonally stabilizable matrix still remains an  open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A variation of the diagonal matrix stabilization problem is the following:  Problem (*): Given a real square matrix G, find a real diagonal matrix D such  that the product GD is Hurwitz together with all its principal submatrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Surprisingly, a necessary and sufficient condition exists for solving the above problem  as shown in (Locatelli and Schiavoni, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Let M := {1, 2, · · · , m} and F :=  {f|f ⊂ M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Further, for any m×m matrix ∆, denote by ∆(f) the principal submatrix  obtained from it after removing the rows and columns with indexes in f, f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' The  main result of (Locatelli and Schiavoni, 2012) proves the following:  4  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' (Locatelli and Schiavoni, 2012) Problem (*) admits a solution if and only  if  det(G(f))det(GD(f)) > 0, ∀f ∈ F  (3)  where GD = diag{gii}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Moreover, if the above condition is satisfied, then there exists  ¯ϵ > 0 such that, for any given ϵ ∈ (0, ¯ϵ), the matrix  D := GDZ(ϵ), Z(ϵ) := −diag{ϵi}  (4)  solves the stabilization problem (*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Conditions for diagonally stable matrices  We give a short summary of available conditions for diagonally stable matrices  (excerpts from (Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 1978), (Cross, 1978) and (Hershkowitz, 2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  (Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 1978) Lyapunov diagonally stable matrices are P-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  (Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 1978) A matrix A being Lyapunov diagonally stable is equivalent  to that there exists a positive diagonal matrix D such that xT DAx > 0 for all  nonzero vectors x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  (Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 1978) A 2 × 2 real matrix is Lyapunov diagonally stable if and  only if it is a P-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  (Cross, 1978) For a given Lyapunov diagonally stable matrix P, all principal  submatrices of P are Lyapunov diagonally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  (Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 1978) A real square matrix A is Lyapunov diagonally stable if  and only if for every nonzero real symmetric positive semidefinite matrix H the  matrix HA has at least one positive diagonal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  – Note: this result is termed the BBP theorem, and is proved again in (Shorten  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2009) with a simpler proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  (Cross, 1978) The set of Lyapunov diagonally stable matrices is a strict subset of  multiplicative D-stable matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  (Cross, 1978) The set of Lyapunov diagonally stable matrices is a strict subset of  additive D-stable matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  – Note: Multiplicative D-stable and additive D-stable matrices are not neces-  sarily diagonally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A Z-matrix is Lyapunov diagonally stable if and only if it is a P -matrix (that is,  it is an M-matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A non-singular H-matrix with nonnegative diagonal parts is Lyapunov diago-  nally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A quasi-diagonal dominant matrix with nonnegative diagonal parts is Lyapunov  diagonally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Note the equivalence of Hurwitz H-matrix and quasi-diagonal  dominant matrix (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The following facts are shown in (Cross, 1978) and (Kaszkurewicz and Bhaya, 2012):  For normal matrices and within the set Z, D-stability, additive D-stability, and  diagonal stability are all equivalent to matrix stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  If a matrix A is Hurwitz stable, D-stable, or diagonally stable, then the matrices  AT and A−1 also have the corresponding properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  6  In (Shorten and Narendra, 2009) Shorten and Narendra showed the following nec-  essary and sufficient condition for matrix diagonal stability (an alternative proof of the  theorem of Redheffer via the KYP lemma):  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' (Shorten and Narendra, 2009) and (Redheffer, 1985) Let A ∈ Rn×n be a  Hurwitz matrix with negative diagonal entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Let An−1 denote the [n − 1 × n − 1]  leading sub-matrix of A, and Bn−1 denote the corresponding block of A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Then,  the matrix A is diagonally stable, if and only if there is a common diagonal Lyapunov  function for the LTI systems ΣAn−1 and ΣBn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The above theorem involves finding a common diagonal Lyapunov function for a  set of LTI systems, which may be restrictive and computationally demanding in practi-  cal applications especially when the dimension of the matrix A is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  7  Figure 1: Relations of matrix stability under different matrix types: the main theorem in (Berman and  Hershkowitz, 1983)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Relations of matrix stability and diagonal stability  The paper (Berman and Hershkowitz, 1983) characterizes the relations of certain  special matrices for matrix diagonal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' They define  A = {A : there exists a positive definite diagonal matrix D such that AD +  DAT is positive definite};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', A denotes the set of diagonally stable matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  L = {A : there exists a positive definite matrix D such that AD+DAT is positive definite};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', L denotes the set of (positive) stable matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  P = {A : the principle minors of A are positive};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', P denotes the set of P-matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  S = {A : there exists a positive vector x such that Ax is positive};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', S denotes the set of semipositive matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The main result of (Berman and Hershkowitz, 1983) is cited and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  In general, these different sets of structured matrices are not equivalent, and the set A  is a subset of the other sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' However, for Z-matrices, these sets are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' In  particular, for the case of Z-matrices, the characterizations of these sets give equiva-  lent conditions for M-matrices (upon a sign change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Note there are yet many more  conditions to characterize M-matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', (Plemmons, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The review paper (Hershkowitz, 1992) presents the implication relations between  matrix stability conditions, and the equivalent relations of matrix stabilities for Z-  matrices, as cited in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Again, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' 3, for Z-matrices, all  the stability types are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  8  THEOREM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' In general  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' For Z-matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', matrices with nonpositive off-diagonal entries,  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' For symmetric matrices  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' For triangular matrices  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' for normal matrices  The absence of an implication in the above relations means that a counterexample exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='Figure 2: The implication relations between matrix stability conditions, cited from (Hershkowitz, 1992)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Figure 3: For Z-matrices, all the stability types are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Cited from (Hershkowitz, 1992)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  9  A is Lyapunov diagonally stable  A is a P-matrix  A is D-stable  A is additive D-stable  A is stable  A is a Pt -matrixA is Lyapunov diagonally stable  A is a P-matrix  A is D-stable  A is additive D-stable  A is a nonsingular M-matrix  A is stableFigure 4: Relations among matrix stabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Cited from (Logofet, 2005, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The survey paper (Logofet, 2005) presents some beautiful flower-shaped character-  izations of the relations among matrix stabilities, as cited in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  10  STABLE  D-STABLE  TOTALLYSTABLE  DISSIPATIVE  QUASI-  DOMINANT  M-MATRICES  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='NORMAL  D-STABLE  STABLE  STABLEFigure 5: Petals of sign-stable matrices within the Flower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Cited from (Logofet, 2005, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='4)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Stability conditions with submatrices and Schur complement  Stability conditions of ‘structured’ matrices often involve stability properties of  submatrices, which employ block submatrices and their Schur complements to deter-  mine stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  In (Narendra and Shorten, 2010), Narendra and Shorten presented necessary and  sufficient conditions to characterize if a given Metzler matrix is Hurwitz, based on the  fact that a Hurwitz Metzler matrix is diagonally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' These conditions are general-  ized in (Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' We recall some main stability criteria from (Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=',  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Let A ∈ Rn×n be a Metzler matrix partitioned in blocks of compatible  dimensions as A = [A11, A12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' A21, A22] with A11 and A22 being square matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Then the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A is Hurwitz stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A11 and its Schur complement A/A11 := A22 −A21A−1  11 A12 are Hurwitz stable  Metzler matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  A22 and its Schur complement A/A22 := A11 −A12A−1  22 A21 are Hurwitz stable  Metzler matrices  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Some remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  11  STABLE  D-STABLE  TOTALLYSTABLE  DISSIPA-  QUAS  TIVE  DOMINANT  M-  SIGN  NORMAL  STABLE  D-STABLE  QUASI-RECESSIVE  STABLE  OUAST-RECESSIVE  STABLE• For a structured matrix, the property that its Schur complements also preserve  the same stability and structure properties is termed Schur complement stabil-  ity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Other types of structured matrices that have Schur complement  stability property include symmetric matrices, triangular matrices, and Schwarz  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' See (Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The result on M-matrix in Lemma 1 can be generalized to H-matrix: Let A be a  H-matrix partitioned in blocks of compatible dimensions as A = [A11, A12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' A21, A22]  with A11 and A22 being square matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' If A is Hurwitz stable, then A11 and  its Schur complement A/A11 := A22 −A21A−1  11 A12 are Hurwitz stable H matri-  ces, or A22 and its Schur complement A/A22 := A11−A12A−1  22 A21 are Hurwitz  stable H matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Schur complement and its closure property for several structured matrices (in-  cluding diagonal matrices, triangular matrices, symmetric matrices, P-matrices,  diagonal dominant matrices, M-matrices etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=') are discussed in (Zhang, 2006,  Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  12  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Application examples of matrix diagonal stability conditions  The Fisher-Fuller theorem on diagonal matrix stabilizability (Theorem 1 and its  variations) has been rediscovered several times by the control system community, and  has been applied in solving distributed stabilization and decentralized control problems  in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' This section reviews two application examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Conditions for decentralized stabilization  In (Corfmat and Morse, 1973) Corfmat and Morse solved the following problem:  For given and fixed real matrices A and P, find (if possible) a non-singular di-  agonal matrix D such that I + ADP is Schur stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', all eigenvalues of  I + ADP are located within the unit circle in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  To solve the above problem they proved the following:  Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' If A is an n × n strongly non-singular matrix, then there exists a diag-  onal matrix D such that (I + DA) is Schur stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Note: in (Corfmat and Morse, 1973) a matrix is termed strongly non-singular, if  its all n leading principal minors are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' If A is a fixed non-singular matrix, then there exists a permutation matrix  P such that PA is strongly non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Solution to decentralized stabilization: the non-singularity of A is a necessary and  sufficient condition for the existence of a permutation matrix P and a non-singular  diagonal matrix D such that (I + ADP) is Schur stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Distributed stabilization of persistent formations  In (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2009), the problem on persistent formation stabilization involves  studying the stabilizability of the following differential equation  ˙z = ∆Az  where ∆ is a diagonal or possibly block diagonal matrix, and A is a rigidity-like matrix  on formation shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' To solve the formation stabilization problem in (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2009)  the following result is employed ((Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2009, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='2)):  Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Suppose A is an m × m non-singular matrix with every leading prin-  cipal minor nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Then there exists a diagonal D such that the real parts of the  eigenvalues of DA are all negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  We remark that this is a reformulation of the Fisher-Fuller theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  13  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' A selection of key review papers and books on matrix stability and diagonal  stability conditions  The survey paper (Hershkowitz, 1992) that presents a summary of relevant ma-  trix stability results and the developments, up until 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The paper (Bhaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2003) that presents comprehensive discussions and char-  acterizations for various classes of matrix stability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The paper (Hershkowitz and Keller, 2003) that studies the relations between pos-  itivity of principal minors, sign symmetry and stability of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The review paper (Hershkowitz, 2006) that presents an concise overview on ma-  trix stability and inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The book (Kaszkurewicz and Bhaya, 2012) on matrix diagonal stability in sys-  tems and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The summary paper (Logofet, 2005) that presents a review and some beautiful  connections/relations on different matrix stabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The very long survey paper (Kushel, 2019) that provides a unifying viewpoint  on matrix stability, and its historical development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  The recent book (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', 2020) on positive matrix, P-matrix and inverse  M-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  References  Ballantine, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
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+page_content=' Stabilization by a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Proceedings of the American  Mathematical Society 25, 728–734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Barker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', Berman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', Plemmons, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
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+page_content='  Positive diagonal solutions to the  Lyapunov equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Linear and Multilinear Algebra 5, 249–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
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+page_content=', Hershkowitz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
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+page_content=' Matrix diagonal stability and its implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  SIAM Journal on Algebraic Discrete Methods 4, 377–382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content='  Bhaya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', Kaszkurewicz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=', Santos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
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+page_content=' Characterizations of classes of stable  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
+page_content=' Linear algebra and its applications 374, 159–174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
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+page_content=' Stabilization with decentralized feedback control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
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+page_content=' Positivity of principal minors, sign symmetry and  stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfUvyD/content/2301.01272v1.pdf'}
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HdE2T4oBgHgl3EQfTwd_/content/tmp_files/2301.03806v1.pdf.txt

@@ -0,0 +1,974 @@
+Impurity induced Modulational instability in Bose-Einstein condensates
+Ishfaq Ahmad Bhat, Bishwajyoti Dey
+Department of Physics, Savitribai Phule Pune University, Pune 411007, Maharashtra, India
+Abstract
+By means of linear stability analysis (LSA) and direct numerical simulations of the coupled Gross-Pitaevskii (GP)
+equations, we address the impurity induced modulational instability (MI) and the associated nonlinear dynamics
+in Bose-Einstein condensates (BECs). We explore the dual role played by the impurities within the BECs —the
+instigation of MI and the dissipation of the initially generated solitary waves. Because of the impurities, the repulsive
+BECs are even modulationally unstable and this tendency towards MI increases with increasing impurity fraction and
+superfluid-impurity coupling strength. However, the tendency of a given BEC towards the MI decreases with the
+decreasing mass of the impurity atoms while the sign of the superfluid-impurity interaction plays no role. The above
+results are true even for attractive BECs except for a weak superfluid-impurity coupling, where the MI phenomenon
+is marginally suppressed by the presence of impurities. The dissipation of the solitons reduce their lifetime and is
+eminent for a larger impurity fraction and strong superfluid-impurity strength respectively.
+Keywords: Modulational Instability, solitons, Bose-Einstein Condensates, coupled Gross-Pitaevskii Equation
+1. Introduction
+Modulational instability (MI) is the most prevalent phenomenon in nonlinear systems [1] wherein a continuous
+wave background becomes unstable against the growth of weak perturbations. As a result, localized solitary waves
+are formed due to the interplay between the intrinsic nonlinearity and dispersion. Although MI has been studied in
+nonlinear fibre optics [2], fluid dynamics [3], and plasma [4], the MI in BECs has recently received great attention
+both theoretically [5, 6, 7, 8, 9] and experimentally [10, 11]. A variety of BEC applications, such as quantum phase
+transitions [12], matter-wave amplification [13], atom interferometry [14] and atom lasers [15] result from the efficient
+manipulation of the matter wave dynamics of an ultracold BEC by external fields. The MI in BECs and its associated
+dynamics is well described by the mean-field Gross-Pitaevskii (GP) equation which is nonlinear in nature. Scalar
+BECs are governed by the single component GP equations while the multicomponent BECs are described by coupled
+GP equations. The nonlinearity in a BEC is introduced via the s-wave interactions that are manipulated precisely in
+experiments by the optical [16, 17] and magnetic [18, 19] Feshbach resonances. In addition to accounting for the
+MI in BECs, the GP equation exhibits different types of nonlinear structures, including solitons [20, 21] and vortices
+[22, 23].
+The presence of impurities in BECs offer a platform to investigate mass-imbalanced multicomponent systems
+[24, 25, 26, 27]. The impurity-superfluid interactions in these imbalanced BEC systems realize Bose polarons which
+under strong interactions result into impurity-solitons [28]. The impurity-superfluid systems are also central towards
+the understanding of Kondo effect [29], spin transport [30], spin-charge separation [31] and synthetic superfluid
+chemistry [32]. While the number of impurities in a BEC can be fixed either by the controlled doping [33] or by
+transferring coherently a fraction of the condensate atoms from one hyperfine level to another by means of a radio-
+frequency (rf) pulse [34].
+The earlier works addressing MI in single component BECs [6, 35, 36] dictate that MI is a natural precursor to the
+formation of bright solitons. In a single component BEC, MI is possible only for a focusing (attractive) nonlinearity
+[37, 38]. By tuning the atomic interactions in a BEC from repulsive to attractive, MI results in the formation of bright
+matter-wave solitons [20, 10, 39]. In a binary BEC, MI is possible even for repulsive interactions whereby dark-bright
+solitary wave complexes are formed if the self-repulsion of each component is outbalanced by the cross-repulsion
+Preprint submitted to Physics Letters A
+January 11, 2023
+arXiv:2301.03806v1  [cond-mat.quant-gas]  10 Jan 2023
+
+between the two components [40, 7, 9]. The dark-bright soliton complexes are coherent structures consisting of a
+bright soliton in one component effectively trapped by a dark soliton in the second component. This results in the
+formation of domain walls which realize the phase separation in the immiscible binary BEC [7, 9, 41, 42]. The MI
+phenomenon has also been studied extensively in BECs with synthetic gauge potentials [43, 44]. In presence of gauge
+potentials, MI occurs even in the miscible binary BECs. Recently, the MI analysis has been extended to dipolar
+[45, 46] and density-dependent BECs [47]. In dipolar BECs, the MI results in the formation of quantum droplets
+while in density-dependent BECs unidirectionally moving chiral solitons are obtained.
+In the framework of the mean-field approach, this paper addresses the MI and the associated nonlinear dynamics
+of solitary wave formation in an effectively one-dimensional superfluid-impurity system. We consider a binary BEC
+system in which the condensate (superfluid) interacts with a dilute fraction (0-6%) of impurity atoms. When the
+impurity fraction in a BEC exceeds 0.2%, the impurity atoms are also Bose-condensed [34]. The MI analysis in
+a binary BEC has usually been carried out for an equal distribution of atoms in the two components [7, 12, 43].
+However, by varying the impurity fraction and their interaction with the superfluid, we study the effect of mass
+imbalance on the MI phenomena in the BECs. Moreover, the impurity atoms can be either heavier or lighter or of
+the same mass as the superfluid atoms. A unit mass ratio can be achieved by transferring coherently a small fraction
+of superfluid atoms into another hyperfine state. Such a situation is neatly modelled by the experiments of Aycock
+[34] and Jørgensen [25]. By considering different superfluid-to-impurity atomic mass ratios, we additionally study its
+effect on the impurity-induced MI in BECs.
+The subsequent material is structured as follows. Section 2 introduces the model and the corresponding coupled
+GP equations. The dispersion relation produced by the linear stability analysis (LSA) is derived and discussed. Section
+3 reports the results of numerical simulations of the system under consideration. The paper is concluded by Sec. 4.
+2. Model and MI Analysis
+We consider a one-dimensional (1D) BEC with atomic mass m1 and consisting of N1 atoms in a state ψ1 colli-
+sionally coupled to N2 Bose-condensed impurity atoms of mass m2 in the state ψ2. Such a projected BEC system is
+described by the following coupled GP equations [32, 34]:
+iℏ∂ψ1
+∂t =
+�
+− ℏ2
+2m1
+∂2
+∂x2 + u1|ψ1|2 + u12|ψ2|2�
+ψ1
+(1a)
+iℏ∂ψ2
+∂t =
+�
+− ℏ2
+2m2
+∂2
+∂x2 + u12|ψ1|2�
+ψ2
+(1b)
+The nonlinear coefficient u1 = 2ℏ2a1/(m1a2
+⊥) specifies the interaction between the superfluid atoms in state ψ1 and
+u12 = 2ℏ2a12/m12(a2
+⊥ + b2
+⊥) with m−1
+12 = m−1
+1 + m−1
+2 , determines the coupling between the superfluid and impurity
+atoms. Here a1 and a12 are the characteristic scattering lengths while a⊥ = √ℏ/m1ω⊥ and b⊥ = √ℏ/m2ω⊥ are the
+harmonic oscillator lengths such that ω⊥ is the transverse trapping frequency. The number of atoms in each state
+is conserved by the normalization,
+�
+dx|ψi(x)|2 = Ni and N = N1 + N2 is the total number of atoms. Further, the
+impurity fraction N2/N1 is kept vey low (within a few percent) in experiments and is varied by tuning the rf-amplitude
+[34]. Accordingly, we neglect the interactions between the impurity atoms. By means of spatiotemporal scaling [48],
+t = ω−1
+⊥ t′, x = a⊥x′, and ψi = ψ′
+i/ √|a⊥|, Eqs. (1a) and (1b) can be recast in the following form:
+i∂ψ1
+∂t =
+�
+− 1
+2
+∂2
+∂x2 + g1|ψ1|2 + g12|ψ2|2�
+ψ1
+(2a)
+i∂ψ2
+∂t =
+�
+− ρ
+2
+∂2
+∂x2 + g12|ψ1|2�
+ψ2
+(2b)
+where ρ = m1/m2 denotes the mass ratio. In this study, we consider the mass ratios, ρ = {1/2, 1, 2}. These correspond
+to the experimentally relevant cases of 14K−87Rb, 87Rb−87Rb and 87Rb−14K superfluid-impurity systems respectively.
+The higher mass ratios of ρ = 3 and 4 apply respectively to the 133Cs −41 K and 87Rb −23 Li mixtures. In the resulting
+dimensionless GP equations the primes have been omitted and have the same structure as above, but with ℏ = m = 1.
+The scaled coupling coefficients are now defined by, g1 = 2a1/a⊥ and g12 = 4a12/a⊥.
+2
+
+In the context of LSA of Eqs. (2a) and (2b), we examine the MI in a BEC system with n10 and n20 as uniform
+densities of the superfluid and impurity atoms i.e., ψ j = √nj0e−iµjt for j = 1, 2. In terms of the equilibrium densities,
+chemical potentials, µ1 = g1n10 + g12n20 and µ2 = g12n20. For the perturbed wave functions of the form, ψ j =
+( √nj0 + δψj)e−iµ jt, the linearized equations for the perturbations are:
+i∂δψ1
+∂t
+= −1
+2
+∂2δψ1
+∂x2
++ g1n10(δψ1 + δψ∗
+1) + g12
+√n10n20(δψ2 + δψ∗
+2)
+(3a)
+i∂δψ2
+∂t
+= −ρ
+2
+∂2δψ2
+∂x2
++ g12
+√n10n20(δψ1 + δψ∗
+1)
+(3b)
+where “∗” denotes the complex conjugate. We consider the perturbations in the form of plane waves, δψ j = ajCos(kx−
+Ωt) + bjSin(kx − Ωt) with real wavenumber k, complex eigenfrequency Ω and j = 1, 2. Substituting this in Eqs. (3a)
+and (3b) results in the following dispersion relation for eigenfrequency, Ω:
+Ω4 − Ω2k2
+�k2
+4
+�
+ρ2 + 1
+�
++ g1n10
+�
++ k4ρ
+4
+�k4ρ
+4 + g1n10ρ − 4n10n20g2
+12
+�
+= 0
+(4)
+Quartic Eq. (4) for the eigenfrequency, Ω can be solved for its roots and results in:
+Ω2
+± = k2
+2
+��������
+k2 �
+ρ2 + 1
+�
+4
++ g1n10
+��������1 ±
+�
+1 + 4δg2δn −
+k2∆
+2g1n10
++
+k4∆2
+16g2
+1n2
+10
+��������
+����������
+(5)
+where δg = g12/g1 is the interaction ratio, δn = n20/n10 represents the mass imbalance and ∆ = ρ2 − 1. Eq. (5)
+which governs the propagation of perturbations on the top of continuous wave solutions may be positive or negative
+depending on the nature of interactions and mass imbalance. The continuous wave solutions are stable if Ω2
+± > 0 for
+all real k. On the other hand, the instability gain is defined as ξ = |Im(Ω±)|. In case of scalar BECs without an impurity
+(δg = δn = ∆ = 0), Eq. (5) reproduces the well-known results of MI in dilute BECs where it occurs for an attractive
+interaction (g1 < 0) in the wavenumber range 0 < k < 2
+�
+|g1|n10 [2, 6, 35]. The maximum MI gain, ξmax = n10|g1| is
+attained for kmax =
+�
+2n10|g1|.
+The effect of the impurities on the MI in BECs can be analyzed in the framework of Eq. (5) whereby it is evident
+that the BEC is modulationally unstable even for g1 > 0. This situation is of peculiar importance since the scalar
+(δg = 0) [6] and miscible binary BECs (δg < 1) [9] with repulsive g1 interactions are modulationally stable. A plot
+of Eq. (5) shown in Figs. 1 (a)-(c) display the variation of the MI gain with the nature and concentration of the
+impurities in a repulsive BEC with weak superfluid-impurity coupling (δg < 1). Here the instability is accounted for
+by Ω− perturbations only and occurs in the wavenumber range 0 < k <
+�
+2g1n10
+�
+−1 +
+�
+1 + 4δg2δn/ρ
+�
+. It is evident
+from Fig. 1(a)-(c) that for a fixed interaction, δg and atomic mass, ρ ratios, the MI gain and bandwidth increases with
+the fraction of impurities in the BEC. In the simplest case of ρ = 1, the largest MI gain,
+ξmax = Im
+��������
+�
+g2
+1n2
+10(−1 − 2δg2δn +
+�
+1 + 4δg2δn)/2
+��������
+(6)
+is attained at,
+kmax =
+�
+−g1n10 + g1n10
+�
+1 + 4δg2δn.
+(7)
+Moreover, the MI gain and the instability bandwidth decreases with the mass ratio, ρ as shown in Fig 2. Consequently,
+the values of kmax also decrease with ρ. It is worth mentioning that the MI in such BEC systems are also independent
+of the sign of g12, though the the tendency of the system against the growth of the perturbations increase together with
+the ratio δg. By comparing Figs. 1(a,b,c) and 1 (d,e,f) respectively, it is evident that the MI gain for a given fraction
+of impurities are considerably larger for a stronger superfluid-impurity coupling ( δg > 1).
+In an attractive BEC with g1 < 0, the modulational instability is accounted for by Ω+ perturbations and occurs in
+the wavenumber range 0 < k <
+�
+2g1n10
+�
+−1 −
+�
+1 + 4δg2δn/ρ
+�
+as shown in Fig. 3. It is evident from Fig. 3(a)-(c)
+3
+
+a) ρ=1/2
+n20
+0
+0.02
+0.04
+0.06
+0
+2
+4
+6
+0
+1
+2
+3
+4
+k
+ξ=|Im(Ω-)|
+b) ρ=1
+n20
+0
+0.02
+0.04
+0.06
+0
+2
+4
+6
+0
+1
+2
+3
+4
+k
+ξ=|Im(Ω-)|
+c) ρ=2
+n20
+0
+0.02
+0.04
+0.06
+0
+2
+4
+6
+0
+1
+2
+3
+4
+k
+ξ=|Im(Ω-)|
+d) ρ=1/2
+n20
+0
+0.02
+0.04
+0.06
+0
+2
+4
+6
+0
+1
+2
+3
+4
+k
+ξ=|Im(Ω-)|
+e) ρ=1
+n20
+0
+0.02
+0.04
+0.06
+0
+2
+4
+6
+0
+1
+2
+3
+4
+k
+ξ=|Im(Ω-)|
+f) ρ=2
+n20
+0
+0.02
+0.04
+0.06
+0
+2
+4
+6
+0
+1
+2
+3
+4
+k
+ξ=|Im(Ω-)|
+Figure 1: The MI gain corresponding to the Ω− perturbation branch for g1 = 50 and different impurity concentrations, n20. The superfluid-impurity
+coupling is fixed at δg = 0.95 in (a)-(c) and δg = 1.2 in (d)-(f) respectively, while n10 + n20 = 1. Note that the repulsive scalar BECs with n20 = 0
+are always modulationally stable.
+a) δg= 0.95
+ρ
+1.0
+2.0
+3.0
+4.0
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+k
+ξ=|Im(Ω-)|
+b) δg= 1.2
+ρ
+1.0
+2.0
+3.0
+4.0
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+k
+ξ=|Im(Ω-)|
+Figure 2: Variation of the MI gain with atomic mass ratio, ρ in a BEC with a) weak (δg < 1) and b) strong (δg > 1) superfluid-impurity interactions,
+The impurity fraction, n20 = 0.06 in each case.
+that the MI gain is marginally suppressed with the impurity concentration if δg < 1 while for δg > 1, the largest
+MI gain and the instability bandwidth increases with increasing impurity fraction in the attractive BECs as shown in
+Fig. 3(d)-(f). These modifications to the already existing MI in an attractive BEC due to the impurity fraction can be
+further understood by considering ρ = 1. In such a case, the largest MI gain,
+ξmax = Im
+��������
+�
+−g2
+1n2
+10(1 + 2δg2δn +
+�
+1 + 4δg2δn)/2
+��������
+(8)
+is attained at,
+kmax =
+�
+−g1n10 − g1n10
+�
+1 + 4δg2δn.
+(9)
+A plot of Eq. (9) in Fig. 4(a) shows the variation of kmax with the impurity fraction, n20 for both δg < 1 and δg > 1.
+Notice that kmax and hence ξmax (not shown in Fig. 4) decreases linearly together with n20 in case δg < 1 while it
+4
+
+a) ρ=1/2
+n20
+0
+0.02
+0.04
+0.06
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.00
+0.05
+0.10
+0.15
+k
+ξ=|Im(Ω+)|
+b) ρ=1
+n20
+0
+0.02
+0.04
+0.06
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.00
+0.05
+0.10
+0.15
+k
+ξ=|Im(Ω+)|
+c) ρ=2
+n20
+0
+0.02
+0.04
+0.06
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.00
+0.05
+0.10
+0.15
+k
+ξ=|Im(Ω+)|
+d) ρ=1/2
+n20
+0
+0.02
+0.04
+0.06
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.00
+0.05
+0.10
+0.15
+k
+ξ=|Im(Ω+)|
+e) ρ=1
+n20
+0
+0.02
+0.04
+0.06
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.00
+0.05
+0.10
+0.15
+k
+ξ=|Im(Ω+)|
+f) ρ=2
+n20
+0
+0.02
+0.04
+0.06
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.00
+0.05
+0.10
+0.15
+k
+ξ=|Im(Ω+)|
+Figure 3:
+The MI gain corresponding to the Ω+ perturbation branch for g1 = −0.1 and different impurity concentrations, n20. The superfluid-
+impurity coupling is fixed at δg = 0.05 in (a)-(c) and δg = 4.0 in (d)-(f) respectively, while n10 + n20 = 1. Note that scalar BECs with attractive
+interactions and n20 = 0 are always modulationally unstable.
+increases non-linearly when δg > 1. The same is true for higher values of ρ as shown in Fig. 4(b), except that the kmax
+values are lower for the corresponding values of impurity fraction. Like in case of BECs with repulsive interactions,
+the MI gain and instability bandwidth decrease with ρ for a fixed impurity fraction.
+a) ρ=1
+δg
+0.05
+4.0
+0.00 0.01 0.02 0.03 0.04 0.05 0.06
+0.44
+0.46
+0.48
+0.50
+0.52
+0.54
+0.56
+n20
+kmax
+b) ρ=2
+δg
+0.05
+4.0
+0.00 0.01 0.02 0.03 0.04 0.05 0.06
+0.44
+0.46
+0.48
+0.50
+0.52
+0.54
+0.56
+n20
+kmax
+Figure 4: Variation of kmax with the concentration of impurities, n20 for g1 = −0.1, (a) ρ = 1 and (b) ρ = 2 .
+3. Numerics
+In this section, we discuss the nonlinear dynamics caused by impurity-induced MI in the BECs. We Solve Eqs.
+(2a) and (2b) numerically by employing FFTW [49] mediated split-step fast-Fourier method [50]. Throughout the
+simulations, we set the time step ∆t = 0.001 and the spatial grid size ∆x = L/Ns with Ns = 2048. The domain size,
+L = 200 and 100 is respectively chosen for repulsive (g1 > 0) and attractive (g1 < 0) BECs. Initial continuous wave
+background with impurity fraction ranging from 0-6% is studied for its MI by seeding weak and random perturbations
+for different strengths of g1 and δg in the following subsections.
+5
+
+3.1. g1 > 0 and δg < 1
+The MI analysis discussed in the previous section shows that impurities within a BEC make it vulnerable towards
+the growth of perturbations (MI) even for repulsive g1 interactions and δg < 1. In this regard, Fig. 5(a)-(c) shows the
+time development of the initially perturbed ρ = 1 continuous wave density for 2%, 4% and 6% impurity concentrations
+respectively. For a given value of g1 = 50 and δg = 0.95, it is evident that the BEC realizes phase separation through
+the formation of dark solitary waves. This is because of the repulsive superfluid-impurity coupling and the density
+modulation grows out of phase between the two components. This density disturbance in the superfluid increases
+with the increasing concentration of the impurities. Consequently, the number of phase-separated domains or solitary
+waves increases proportionately with the concentration of the impurities such that for a 2% impurity, a pair of parallel
+dark-solitons appear at tMI > 900. The critical time at which MI-induced solitary waves start appearing decreases
+with the increasing concentration of impurities. Though the impurities felicitate the MI phenomena in BECs, these
+are also responsible for the dissipation and Brownian motion of the generated solitary waves [34]. Fig. 5(c) shows that
+one of the several generated solitary waves in the superfluid executes zig-zag motion during its evolution and finally
+disappears due to the dissipation by impurities. Since, δg < 1, the dissipation and the Brownian motion of the solitary
+waves due to the impurities is not prominent, though it increases with the concentration of the impurities. Similar
+outcomes are observed in the lower panel (Fig. 5(d)-(f)) where ρ = 2. However, in accordance with the analytical
+results, the number of initially generated dark solitons is less than those produced in the corresponding simulations
+with ρ = 1.
+Figure 5: The evolution of the condensate density |ψ1|2 due to the development of the MI by a) 2%, b) 4%, and c) 6% impurities respectively in the
+coordinate space. The upper panel (a)-(c) corresponds to ρ = 1 while the lower panel (d)-(f) represent the corresponding densities with ρ = 2. The
+superfluid-impurity coupling is maintained at δg = 0.05. The number of solitons increases together with the impurity fraction and decreases with
+the mass ratio, ρ. The dissipation of the generated solitons is negligible for a weaker superfluid-impurity coupling.
+3.2. g1 > 0 and δg > 1
+The condition δg > 1 refers when superfluid-impurity coupling outbalances the interaction between the superfluid
+atoms. From Eq. (5) it is clear that the MI gain increases with the increase in δg for a fixed value of ρ and impurity
+concentration. By comparing the corresponding plots in Fig. 5 and Fig. 6, it is clear that the number of generated
+6
+
+a)
+b)
+c)
+1:2
+1100
+0.8
+0.6
+550
+0.4
+0.2
+0
+0
+d)
+e)
+f)
+1100
+t
+550
+0
+-50 0 50
+-50 0 50
+-50 0 50
+x
+x
+xsolitary waves is more for a larger superfluid-impurity coupling (δg). Moreover, when the superfluid-impurity inter-
+action is large, the dissipation due to the impurity atoms within the BEC is also large. Fig. 6 shows that dark solitons
+within the superfluid component dissipate after executing the zig-zag motion. As the number of impurities in a BEC
+increases, more solitons in the superfluid dissipate during their time evolution. This consequently reduces the lifetime
+of the generated solitons. Further, by comparing the corresponding plots in the upper (a-c) and lower (d-f) panels of
+Fig. 6, it is evident that the number of generated solitons decrease with increasing values of mass ratio, ρ.
+Figure 6: The evolution of the condensate density |ψ1|2 due to the development of the MI by a) 2%, b) 4%, and c) 6% impurities respectively in the
+coordinate space. The upper panel (a)-(c) corresponds to ρ = 1 while the lower panel (d)-(f) represent the corresponding densities with ρ = 2. The
+superfluid-impurity coupling is maintained at δg = 4. Both the number of solitons and their dissipation are large for a larger superfluid-impurity
+coupling.
+3.3. g1 < 0 and δg < 1
+It is well known that MI in BECs with attractive self-interactions lead to the formation of bright matter-wave
+solitons. Eq. (5) and Fig. 3 shows that the BEC is modulationally unstable for g1 = −0.1. However, the presence
+of impurities changes the MI in the system only slightly if δg < 1. Fig. 7 shows the spatiotemporal evolution of the
+initially perturbed continuous wave density for different impurity concentrations and parameters, g = −0.1, δg = 0.05
+and ρ = 1. It shows the generation of a train of solitons at tMI ≥ 205 in the absence of impurities. In accordance
+with Eq. (9), the wavenumber corresponding to the initially excited mode is kmax ≈ 0.45 as shown in 7(d). This
+corresponds to λmax = 2π/kmax ≈ 14.19 and hence, ns = L/λmax ≈ 7 solitons. As the concentration of impurities is
+increased, it is evident that the critical time, tMI at which MI-induced bright solitons start appearing increases slightly
+due to the slight decrease in the largest MI gain, ξmax. With an impurity concentration of 2% and 6% respectively, the
+largest wavenumber, kmax ≈ 0.44 and 0.43 correspond to the initially excited modes as shown in Figs. 7(e) and 7(f)
+respectively. However, due to the slight changes in the MI parameters, kmax and ξmax, the number of initially generated
+solitons remains the same in all cases. Moreover, due to the weak superfluid-impurity coupling the dissipation is even
+negligible.
+7
+
+a)
+b)
+c)
+1:2
+1100
+1
+0.8
+0.6
+550
+0.4
+0.2
+0
+0
+(p
+e)
+f)
+1100
+t
+550
+0
+-50 0 50
+-50 0 50
+-50 0 50
+x
+x
+xFigure 7: The evolution of the condensate density |ψ1|2 due to the development of the MI by a) 0%, b) 2%, and c) 6% impurities respectively in
+the coordinate space. The lower panel (d)- (f) represent the corresponding densities in k- space. The superfluid-impurity coupling is maintained at
+δg = 0.95. For δg < 1, the critical time for the generation of bright solitons increases with the impurity fraction.
+3.4. g1 < 0 and δg > 1
+In the previous subsection 3.3, we showed that the impurities do not play a significant role in the MI-associated
+nonlinear dynamics in attractive BECs for a weak superfluid-impurity coupling (δg < 1). The same is true for a larger
+range of δg > 1. In order to explain the effects on the MI due to the impurities, we have here chosen a much stronger
+superfluid-impurity coupling, δg = 4. Fig. 8 shows the spatiotemporal evolution of the initially perturbed continuous
+wave density with 0%, 2% and 6% impurity concentrations respectively. The critical time tMI decreases with the
+concentration of impurities. This is due to the increase in the largest MI gain, ξmax given by Eq. (8). With an impurity
+concentration of 0%, 2% and 6% respectively, the largest wavenumber, kmax ≈ 0.45, 0.49 and 0.55 correspond to
+the initially excited modes as shown in Fig. 8(d)-(f) respectively. Consequently, the number of initially generated
+8
+
+8
+360
+7
+6
+240
+5
+4
+3
+120
+2
+1
+0
+0
+-30 0 30 -30 0 30 -30 0 30
+x
+x
+x
+10
+360
+9
+8
+7
+240
+6
+5 104
+4
+120
+3
+2
+1
+0
+0
+0
+0
+1
+k
+k
+kFigure 8: The evolution of the condensate density |ψ1|2 due to the development of the MI by a) 0%, b) 2%, and c) 6% impurities respectively in
+the coordinate space. The lower panel (d)- (f) represent the corresponding densities in k- space. The superfluid-impurity coupling is maintained at
+δg = 1.2. For δg > 1, the critical time for the generation of bright solitons decreases with the impurity fraction and there is significant dissipation.
+solitons, ns = 7, 8 and 9 respectively. Moreover, due to the strong superfluid-impurity coupling, the dissipation of the
+generated solitons is prominent. This reduces the lifetime of the solitons and hence the number of solitons decreases
+with time evolution. The number of initially generated solitons and the critical time for MI development as a function
+of impurity concentration and the interaction parameters in the numerical simulations for ρ = 1 is summarized in Fig.
+9. The MI time always decreases with the impurity concentration and the strength of superfluid-impurity coupling
+in BECs with repulsive g1 nonlinearity while in attractive BECs, the MI time either decreases or increases with the
+impurity fraction, depending on the strength of the superfluid-impurity coupling. In the latter case with δg < 1, the
+MI time increases while if δg > 1 MI time decreases.
+9
+
+8
+360
+7
+6
+240
+5
+4
+3
+120
+2
+1
+0
+0
+-30 0 30 -30 0 30 -30 0 30
+x
+x
+x
+10
+360
+9
+8
+7
+240
+6
+5 104
+4
+120
+3
+2
+1
+0
+0
+0
+0
+1
+k
+k
+k 0
+ 500
+ 1000
+ 1500
+ 2
+ 5
+ 8
+ 11
+tMI
+a) g1 = 50
+ns
+δg = 0.95
+δg = 1.2  
+ 0
+ 500
+ 1000
+ 1500
+ 2
+ 5
+ 8
+ 11
+ 150
+ 175
+ 200
+ 225
+ 250
+0
+1
+2
+3
+4
+5
+6
+ 6
+ 8
+ 10
+tMI
+b) g1 = -0.1
+ns
+n20%
+δg = 0.05
+δg = 4.0  
+ 150
+ 175
+ 200
+ 225
+ 250
+0
+1
+2
+3
+4
+5
+6
+ 6
+ 8
+ 10
+Figure 9: Variation of the MI time, tMI (plot-markers with solid lines) and the number of initially generated solitons, ns (plot-markers with dashed
+lines) with impurity concentration, n20 in BECs with ρ = 1.
+4. Conclusion
+In summary, we investigated the MI by employing linear stability analysis and direct numerical simulations in
+a BEC coupled with a dilute fraction of Bose-condensed impurities. Being dilute, the coupling among the impurity
+atoms is neglected. It is well established that single-component BECs with repulsive interactions are always mod-
+ulationally stable. Moreover, the binary BECs with an equal proportion of the two components are modulationally
+unstable iff the cross-phase mediated repulsion between the components outbalances their self-repulsion. However,
+in the presence of dilute impurities, BECs are always modulationally unstable and is independent of the sign of the
+superfluid-impurity interaction. The MI induces spatial pattern formation in a repulsive BEC and the generated do-
+mains have a solitary wave structure. The tendency of the BEC towards MI increases with the increasing impurity
+fraction for a fixed superfluid-impurity coupling strength. Consequently, the number of domains increases together
+with the fraction of impurities in the BEC. Moreover, the MI gain in a given BEC, the instability bandwidth, the
+number of solitons as well as kmax, the value of the wave vector k at which the MI gain attains maximum, decreases
+with the decreasing mass of the impurity atoms. Despite increasing the tendency of a repulsive BEC towards MI, the
+impurities are also responsible for the Brownian motion of the solitons and their dissipation. This reduces the lifetime
+of the solitary wave structures. Naturally, the dissipation is prominent for a greater impurity fraction and increasing
+superfluid-impurity coupling strength.
+As concerns the presence of impurities in attractive BECs slightly affect the preexisting MI phenomena for a weak
+superfluid-impurity coupling (δg < 1). The modulational instability time only slightly increases with the impurity
+fraction. However, for a strong superfluid-impurity coupling (δg > 1) the modulational instability time decreases and
+the number of generated solitons increases with the impurity fraction.
+5. Acknowledgements
+I A Bhat acknowledges CSIR, Government of India, for funding via CSIR Research Associateship (09/137(0627)/2020
+EMR-I). BD thanks Science and Engineering Research Board, Government of India for funding through research
+10
+
+project CRG/2020/003787.
+References
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+12
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+Memership Inference Attacks Against Latent Factor 
+Model 
+Dazhi Hu 
+College of Information Science and Technology, Donghua University, Shanghai China 
+[email protected] 
+Abstract.  
+The advent of the information age has led to the problems of information over-
+load and unclear demands. As an information filtering system, personalized 
+recommendation systems predict users' behavior and preference for items and 
+improves users' information acquisition efficiency. However, recommendation 
+systems usually use highly sensitive user data for training. In this paper, we use 
+the latent factor model as the recommender to get the list of recommended 
+items, and we representing users from relevant items Compared with the tradi-
+tional member inference against machine learning classifiers. We construct a 
+multilayer perceptron model with two hidden layers as the attack model to 
+complete the member inference. Moreover,a shadow recommender is estab-
+lished to derive the labeled training data for the attack model. The attack model 
+is trained on the dataset generated by the shadow recommender and tested on 
+the dataset generated by the target recommender. The experimental data show 
+that the AUC index of our attack model can reach 0.857 on the real dataset 
+MovieLens, which shows that the attack model has good performance. 
+Keywords: membership inference, recommender systems, collaborative filter-
+ing, LFM. 
+1 
+Introduction 
+A recommender system is an information screening system that can predict a user's 
+evaluation or preference for an item. With the advent of the Internet age, human be-
+ings have moved from "information shortage" to "information power". Consumers 
+hope to find content they are interested in in the massive information, and it is diffi-
+cult for information producers to differentiate their products and attract people's atten-
+tion, which leads to the problem of "information overload". As a tool to link users 
+with information, recommender systems can better deal with the problems of "infor-
+mation overload" and "long tail", so as to provide users with personalized services. 
+There are three common recommendation methods: 1) Content-based recommenda-
+tion algorithms. 2) Collaborative filtering recommendation algorithm. 3) Hybrid rec-
+ommendation algorithm. 
+However, the success of recommender systems is based on large-scale user data, 
+which is very likely to leak users' privacy, such as users' facial information, social 
+
+2 
+information, location information, medical information, etc. For example, as one of 
+the biological features of the human body, face information, like fingerprints and iris, 
+has become a means of proving one's identity. Face recognition technology, which 
+performs identity verification based on human facial features, has been widely used, 
+such as mobile phone unlocking, registration verification of various APPs, lending 
+and so on. According to security industry sources, using tools such as PS, you can 
+create a face image with a background, and then through dynamic video software, the 
+face photo can realize actions such as blinking, nodding, etc., combined with the now 
+very mature AI face changing Technology, with some audio simulation technology 
+and personal privacy information, will be used by criminals for video chat fraud. This 
+very confusing fraud method is not easy for ordinary people to identify, so it brings 
+serious information security risks. It is true that the wide application of machine 
+learning has brought people a lot of convenience, but it has also brought about the 
+problem of user privacy protection [1,44]. 
+Therefore, this paper attempts to study the member speculation attack mecha-
+nism in the implicit semantic recommendation system. The attacker's goal [20]is to 
+determine whether the target recommender uses the user's data for training, and to 
+evaluate the performance of the attack model through indicators such as AUC, so as 
+to gain a more comprehensive understanding of the recommendation system. There-
+fore, defense measures such as popular randomization are proposed to achieve the 
+goal of privacy protection. 
+In this paper, a latent semantic recommendation system based on matrix decom-
+position is used to generate a user-item matrix, and a stochastic gradient descent algo-
+rithm is used to minimize the loss function and improve the recommendation perfor-
+mance. In the past, most of the member speculation attacks of machine learning clas-
+sifiers [30]were at the sample level, while this paper focuses on the user-level mem-
+ber speculation attacks, that is, to determine whether the target recommender uses the 
+user's data for training, based on the user's interaction with the item and the recom-
+mendation system. For user's recommendation, we extract the user's feature vector as 
+the input of the attack model. Compared with the previous work of most members 
+speculating on the attack, the attacker can only observe the recommended items from 
+the recommender system, not as the posterior probability. Recommended results. 
+User-level member speculation attack has wider application fields, and it can also 
+make us better understand the mechanism of member speculation attack. 
+2 
+Recommender system related background knowledge 
+2.1 
+Recommendation system definition and value 
+People are looking for what they need in the massive information, and at the same 
+time, the massive information is also looking for the right person. Two important 
+preconditions for the emergence of the system arise, namely information overload and 
+unclear requirements. Recommendation systems are widely used now. The most 
+common example is Taobao's recommendation algorithm, which can recommend 
+
+3 
+things that you may be interested in based on your basic information, such as gender, 
+region, browsing. The recommender system essentially solves the problem of match-
+ing users, items and environments, and helps to establish connections between users 
+and items. The recommendation system is a branch application of machine learning. 
+The recommendation system uses a lot of machine learning technology, and uses 
+various algorithms to build recommendation models to improve the accuracy, sur-
+prise, coverage, etc. of recommendations, and even try to feedback changes in users' 
+interests, such as Today’s Toutiao APP pulls down to display new news, and feedback 
+changes in users’ interests in real time. The recommender system well meets the 
+needs of the "target" provider, the platform, and the user. Taking Taobao shopping as 
+an example, the provider of the "subject matter" is Taobao's thousands of store own-
+ers, the platform is Taobao, and the user is the natural person or enterprise shopping 
+on Taobao. Through the recommendation system, products can be better exposed to 
+users who need to buy, and the allocation efficiency of social resources can be im-
+proved. 
+2.2 
+Existing Recommender System Algorithms  
+The recommendation algorithm is the core of the recommendation system. With the 
+continuous development of the recommendation system, the types of recommendation 
+algorithms are also increasing, and more and more attention has been paid by the 
+academic and industrial circles. Existing recommendation algorithms [21]can be di-
+vided into content-based recommendation, collaborative filtering recommendation 
+and hybrid recommendation. Collaborative filtering recommendation can be divided 
+into neighborhood-based recommendation and model-based recommendation. In 
+neighborhood-based recommendation, user-based recommendation and item-based 
+recommendation are its two categories, and in model-based recommendation, latent 
+semantics, latent semantic model and graph model are the main research hotspot 
+models [16,40-43]. 
+2.3 
+Latent Semantic Model Recommendation System Algorithm 
+The recommendation algorithm adopted in this work is the implicit semantic model 
+algorithm. The latent semantic model algorithm was first proposed in the text field to 
+find the hidden semantics of the text. In 2006, it was used for recommendation. Its 
+core idea is to link user interests and items through implicit features, find potential 
+topics and categories based on user behavior, and then automatically cluster items into 
+different categories or theme. 
+
+4 
+3 
+Member Speculation Attack Design in Recommender Systems 
+3.1 
+Member Speculation Attack Definition 
+At present, most machine learning systems are designed with only a weak threat mod-
+el in mind. When faced with natural input, the system performance can be better re-
+flected, but when encountering malicious users, the performance of these systems 
+may be greatly threatened. 
+A common definition of a membership inference attack is that, given access to a 
+data record and a target model, an attacker needs to determine whether that record is 
+in the model's training dataset. Specifically, it will directly reveal their private infor-
+mation if it is known that the user's data is part of the recommender system's training 
+data. The essence of the member speculation attack is a binary classifier. The perfor-
+mance of machine learning models on their training data and the test set data they 
+encounter for the first time is often different. The attacker's goal is to build an attack 
+model that can distinguish the behavioral differences of the target model, and use this 
+attack model to identify the target model. members and non-members. 
+After synthesizing the shadow dataset 𝐷′, the attacker starts to simulate the tar-
+get model. If the target model is trained on a machine learning service platform, the 
+attacker can use the same service to train a simulated model. The purpose of this step 
+is to train a simulated model with the same or similar predictive ability as the target 
+model. In this way, the attacker can clearly grasp the information of the training data 
+and test data of the simulated model, thereby providing training data for the attacking 
+model. Based on the obtained attack model data, the attacker uses machine learning 
+models such as Naive Bayes, decision tree, and neural network to train the attack 
+model 
+3.2 
+Member speculates attacking adversary information 
+enemy attack target 
+The adversary target means the degree of damage caused by the attacker's target, 
+which can be divided into: (1) Confidentiality attacks. Attackers try to steal structural 
+parameters about the target model, as well as confidential information such as training 
+data that contains sensitive information. (2) Integrity attack. Attackers try to induce 
+model output, affect the integrity of the training process, or affect the output of the 
+model's prediction phase. (3) Availability attacks. Attackers attempt to affect model 
+performance or quality of service, hinder or interrupt normal user requests for the 
+model, and make it untrustworthy in the target environment. 
+Adversary Background Information 
+The knowledge of the adversary inferred by the members of the attack is divided 
+into three categories: black-box knowledge, gray-box knowledge, and white-box 
+knowledge. Black-box knowledge: When the attacker does not have any expertise 
+
+5 
+about the training data, the attacker has black-box knowledge at this time. Grey-box 
+knowledge: When the attacker knows part of the expertise about the training data, the 
+attacker has grey-box knowledge. White-box knowledge: An attacker with white-box 
+knowledge can obtain a certain version of the real data of the training data, and can 
+train the corresponding mirror model according to this version, or can use all the 
+knowledge of the network structure and parameters to attack the model. 
+4 
+Evaluation of Member Speculation Attack Performance in 
+Recommender Systems 
+4.1 
+Experimental setup 
+data set 
+The data used in this experiment is the MovieLens dataset provided by 
+GroupLens, which is a public dataset. The MovieLens dataset has been widely used as 
+an effective dataset that can be widely used and validated for algorithms. This dataset 
+provides a good source for recommender system scientists and researchers, and many 
+recommendation algorithms have been developed and validated with this dataset. The 
+dataset has been denoised and necessary processing and can be used directly. The 
+dataset used in this paper contains 100,836 ratings data for 9,742 movies by 610 users 
+during the period from 1996.3.29 to 2018.9.24. Users are randomly selected. All se-
+lected users have rated at least 20 movies. Demographics are not included, each user 
+is represented by an id, and no other information is provided. Rating scores ranging 
+from 1 to 5 indicate how much the user likes the movie. The dataset is randomly di-
+vided into two disjoint subsets, namely the shadow dataset and the target dataset. 
+Recommended method 
+Recommendation algorithms output recommended items based on the infor-
+mation learned from the input. In this paper, the latent semantic algorithm is adopted, 
+the stochastic gradient descent algorithm is used to update the parameters with a 
+learning rate of 0.01, and the model is constructed with a regularization coefficient of 
+0.01 to enhance the generalization ability of the model. The adversary can observe the 
+list of recommended items from the recommender system and employ a matrix factor-
+ization method to project user movies into a shared latent space. 
+4.2 
+Experimental results 
+To evaluate the attack performance, it is assumed that the adversary has a shadow 
+dataset from the same distribution as the target recommender's training data and 
+knows the target recommender's algorithm, for this paper, both the target recommend-
+er and the shadow recommender are Use latent semantic models. Experimental results 
+
+6 
+show that our attack model has good performance. Plotting the ROC curve, we get the 
+following image: 
+ 
+Fig.1. MLP model experiment ROC curve 
+Next, we try to explore the influence of the length k of the vector on the attack per-
+formance. We evaluate the attack model performance from 10 to 100 different values, 
+as shown in Figure 4-2, the abscissa is the vector length, and the ordinate is the value 
+of AUC. According to the experimental results, we get the following conclusions: 
+when the vector length is less than 50, the attack performance improves with the in-
+crease of k. When the value of k exceeds 50, the attack performance will theoretically 
+be improved, because a larger length vector can provide more dimensional perspec-
+tives, however, when the vector length is large enough, the performance of the attack 
+model tends to be stable. 
+
+ROC
+10
+0.8 
+0.2 
+Val AUC = 0.857
+0.0
+0.0
+0.2
+0.4
+0.6
+80
+1.0
+False Positive Rate7 
+ 
+Fig. 2. Influence curve of vector length on attack performance 
+ 
+Fig. 3. Model attack performance under different learning rates 
+5 
+Conclusion and Outlook 
+5.1 
+Conclusion of the work of this paper 
+Personalized recommender systems have received more and more attention with the 
+deepening of research and the emergence of the needs of the Internet industry. Not 
+only traditional electronic websites need the functions and services of recommender 
+systems, but some emerging fields also introduce recommender systems. Music, mov-
+ies, news are all recommended objects. This paper introduces several widely used 
+recommendation algorithms and the related work of member speculation attack, in-
+troduces its basic principles, and proposes the recommendation algorithm and mem-
+ber speculation attack model used in this experiment. In this paper, a personalized 
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+k=10 k=20 k=30 k=40 k=50 k=60 k=70 k=80 k=90 k=100
+AUC
+vector length
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1E-5
+1E-4
+1E-3
+1E-2
+1E-1
+AUC
+learning rate
+
+8 
+recommendation implicit semantic model is used to generate a recommendation list, 
+and then the adversary extracts the user's feature vector, that is, the difference be-
+tween the interaction and the center vector of the recommendation set, as the user's 
+feature vector. The user feature vector and labels are then used as input to the attack 
+model, which is trained on the shadow dataset and tested on the target dataset. 
+The disadvantage of this paper is that the recommendation algorithm is relatively 
+single, and the latent semantic model is used for both the shadow recommender and 
+the target recommender. The attack performance of the attack model when the shad-
+ow data user and the target recommender use different recommendation algorithms 
+has not been verified. Therefore, the attack model lacks certain generalization ability. 
+In addition, due to the time relationship, this paper does not test defenses against 
+member speculation attacks, such as defense strategies such as reducing the number 
+of categories, publishing data using differential privacy protection, popular random-
+ized recommendation algorithms, etc., which have been obtained in previous work. 
+Effective tests [19][20]. 
+5.2 
+Outlook for future work 
+This paper studies the member speculation attack of recommendation system based on 
+latent semantic model. Due to factors such as time and energy, the shadow recom-
+mender and target recommender in this paper are the same recommendation algo-
+rithm. In future research, the attack performance of different recommendation algo-
+rithms can be tested. , such as item-based recommendation, neural network collabora-
+tive filtering recommendation algorithm. For example, the shadow recommender uses 
+a neural network collaborative filtering algorithm, while the target recommender uses 
+a latent semantic recommendation algorithm to test the generalization ability of the 
+attack model based on different collocations. Furthermore, the impact of hyperparam-
+eters on attack performance can be analyzed. Testing the attack performance from 
+different aspects can effectively improve the attack generalization ability of the mod-
+el, and also help us to fully understand the attack mechanism of the members to pro-
+pose a better defense mechanism. 
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+

LdE2T4oBgHgl3EQfAwbl/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf,len=356
+page_content='Memership Inference Attacks Against Latent Factor  Model  Dazhi Hu  College of Information Science and Technology, Donghua University, Shanghai China  2222053@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='dhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='cn  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  The advent of the information age has led to the problems of information over-  load and unclear demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=" As an information filtering system, personalized  recommendation systems predict users' behavior and preference for items and  improves users' information acquisition efficiency." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' However, recommendation  systems usually use highly sensitive user data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' In this paper, we use  the latent factor model as the recommender to get the list of recommended  items, and we representing users from relevant items Compared with the tradi-  tional member inference against machine learning classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' We construct a  multilayer perceptron model with two hidden layers as the attack model to  complete the member inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Moreover,a shadow recommender is estab-  lished to derive the labeled training data for the attack model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The attack model  is trained on the dataset generated by the shadow recommender and tested on  the dataset generated by the target recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The experimental data show  that the AUC index of our attack model can reach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='857 on the real dataset  MovieLens, which shows that the attack model has good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  Keywords: membership inference, recommender systems, collaborative filter-  ing, LFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content="  1  Introduction  A recommender system is an information screening system that can predict a user's  evaluation or preference for an item." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' With the advent of the Internet age, human be-  ings have moved from "information shortage" to "information power".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Consumers  hope to find content they are interested in in the massive information, and it is diffi-  cult for information producers to differentiate their products and attract people\'s atten-  tion, which leads to the problem of "information overload".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' As a tool to link users  with information, recommender systems can better deal with the problems of "infor-  mation overload" and "long tail", so as to provide users with personalized services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  There are three common recommendation methods: 1) Content-based recommenda-  tion algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' 2) Collaborative filtering recommendation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' 3) Hybrid rec-  ommendation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content="  However, the success of recommender systems is based on large-scale user data,  which is very likely to leak users' privacy, such as users' facial information, social  2  information, location information, medical information, etc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=" For example, as one of  the biological features of the human body, face information, like fingerprints and iris,  has become a means of proving one's identity." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Face recognition technology, which  performs identity verification based on human facial features, has been widely used,  such as mobile phone unlocking, registration verification of various APPs, lending  and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' According to security industry sources, using tools such as PS, you can  create a face image with a background, and then through dynamic video software, the  face photo can realize actions such as blinking, nodding, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=', combined with the now  very mature AI face changing Technology, with some audio simulation technology  and personal privacy information, will be used by criminals for video chat fraud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' This  very confusing fraud method is not easy for ordinary people to identify, so it brings  serious information security risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' It is true that the wide application of machine  learning has brought people a lot of convenience, but it has also brought about the  problem of user privacy protection [1,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  Therefore, this paper attempts to study the member speculation attack mecha-  nism in the implicit semantic recommendation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=" The attacker's goal [20]is to  determine whether the target recommender uses the user's data for training, and to  evaluate the performance of the attack model through indicators such as AUC, so as  to gain a more comprehensive understanding of the recommendation system." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' There-  fore, defense measures such as popular randomization are proposed to achieve the  goal of privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  In this paper, a latent semantic recommendation system based on matrix decom-  position is used to generate a user-item matrix, and a stochastic gradient descent algo-  rithm is used to minimize the loss function and improve the recommendation perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=" In the past, most of the member speculation attacks of machine learning clas-  sifiers [30]were at the sample level, while this paper focuses on the user-level mem-  ber speculation attacks, that is, to determine whether the target recommender uses the  user's data for training, based on the user's interaction with the item and the recom-  mendation system." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=" For user's recommendation, we extract the user's feature vector as  the input of the attack model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Compared with the previous work of most members  speculating on the attack, the attacker can only observe the recommended items from  the recommender system, not as the posterior probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Recommended results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  User-level member speculation attack has wider application fields, and it can also  make us better understand the mechanism of member speculation attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  2  Recommender system related background knowledge  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='1  Recommendation system definition and value  People are looking for what they need in the massive information, and at the same  time, the massive information is also looking for the right person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Two important  preconditions for the emergence of the system arise, namely information overload and  unclear requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Recommendation systems are widely used now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=" The most  common example is Taobao's recommendation algorithm, which can recommend  3  things that you may be interested in based on your basic information, such as gender,  region, browsing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The recommender system essentially solves the problem of match-  ing users, items and environments, and helps to establish connections between users  and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The recommendation system is a branch application of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  The recommendation system uses a lot of machine learning technology, and uses  various algorithms to build recommendation models to improve the accuracy, sur-  prise, coverage, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=" of recommendations, and even try to feedback changes in users'  interests, such as Today’s Toutiao APP pulls down to display new news, and feedback  changes in users’ interests in real time." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The recommender system well meets the  needs of the "target" provider, the platform, and the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Taking Taobao shopping as  an example, the provider of the "subject matter" is Taobao\'s thousands of store own-  ers, the platform is Taobao, and the user is the natural person or enterprise shopping  on Taobao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Through the recommendation system, products can be better exposed to  users who need to buy, and the allocation efficiency of social resources can be im-  proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='2  Existing Recommender System Algorithms  The recommendation algorithm is the core of the recommendation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' With the  continuous development of the recommendation system, the types of recommendation  algorithms are also increasing, and more and more attention has been paid by the  academic and industrial circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Existing recommendation algorithms [21]can be di-  vided into content-based recommendation, collaborative filtering recommendation  and hybrid recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Collaborative filtering recommendation can be divided  into neighborhood-based recommendation and model-based recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' In  neighborhood-based recommendation, user-based recommendation and item-based  recommendation are its two categories, and in model-based recommendation, latent  semantics, latent semantic model and graph model are the main research hotspot  models [16,40-43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='3  Latent Semantic Model Recommendation System Algorithm  The recommendation algorithm adopted in this work is the implicit semantic model  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The latent semantic model algorithm was first proposed in the text field to  find the hidden semantics of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' In 2006, it was used for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Its  core idea is to link user interests and items through implicit features, find potential  topics and categories based on user behavior, and then automatically cluster items into  different categories or theme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  4  3  Member Speculation Attack Design in Recommender Systems  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='1  Member Speculation Attack Definition  At present, most machine learning systems are designed with only a weak threat mod-  el in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' When faced with natural input, the system performance can be better re-  flected, but when encountering malicious users, the performance of these systems  may be greatly threatened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content="  A common definition of a membership inference attack is that, given access to a  data record and a target model, an attacker needs to determine whether that record is  in the model's training dataset." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=" Specifically, it will directly reveal their private infor-  mation if it is known that the user's data is part of the recommender system's training  data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The essence of the member speculation attack is a binary classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The perfor-  mance of machine learning models on their training data and the test set data they  encounter for the first time is often different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=" The attacker's goal is to build an attack  model that can distinguish the behavioral differences of the target model, and use this  attack model to identify the target model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' members and non-members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  After synthesizing the shadow dataset 𝐷′, the attacker starts to simulate the tar-  get model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' If the target model is trained on a machine learning service platform, the  attacker can use the same service to train a simulated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The purpose of this step  is to train a simulated model with the same or similar predictive ability as the target  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' In this way, the attacker can clearly grasp the information of the training data  and test data of the simulated model, thereby providing training data for the attacking  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Based on the obtained attack model data, the attacker uses machine learning  models such as Naive Bayes, decision tree, and neural network to train the attack  model  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content="2  Member speculates attacking adversary information  enemy attack target  The adversary target means the degree of damage caused by the attacker's target,  which can be divided into: (1) Confidentiality attacks." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Attackers try to steal structural  parameters about the target model, as well as confidential information such as training  data that contains sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' (2) Integrity attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=" Attackers try to induce  model output, affect the integrity of the training process, or affect the output of the  model's prediction phase." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' (3) Availability attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Attackers attempt to affect model  performance or quality of service, hinder or interrupt normal user requests for the  model, and make it untrustworthy in the target environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  Adversary Background Information  The knowledge of the adversary inferred by the members of the attack is divided  into three categories: black-box knowledge, gray-box knowledge, and white-box  knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Black-box knowledge: When the attacker does not have any expertise  5  about the training data, the attacker has black-box knowledge at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Grey-box  knowledge: When the attacker knows part of the expertise about the training data, the  attacker has grey-box knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' White-box knowledge: An attacker with white-box  knowledge can obtain a certain version of the real data of the training data, and can  train the corresponding mirror model according to this version, or can use all the  knowledge of the network structure and parameters to attack the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  4  Evaluation of Member Speculation Attack Performance in  Recommender Systems  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='1  Experimental setup  data set  The data used in this experiment is the MovieLens dataset provided by  GroupLens, which is a public dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The MovieLens dataset has been widely used as  an effective dataset that can be widely used and validated for algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' This dataset  provides a good source for recommender system scientists and researchers, and many  recommendation algorithms have been developed and validated with this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The  dataset has been denoised and necessary processing and can be used directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The  dataset used in this paper contains 100,836 ratings data for 9,742 movies by 610 users  during the period from 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='29 to 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Users are randomly selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' All se-  lected users have rated at least 20 movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Demographics are not included, each user  is represented by an id, and no other information is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Rating scores ranging  from 1 to 5 indicate how much the user likes the movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The dataset is randomly di-  vided into two disjoint subsets, namely the shadow dataset and the target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  Recommended method  Recommendation algorithms output recommended items based on the infor-  mation learned from the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' In this paper, the latent semantic algorithm is adopted,  the stochastic gradient descent algorithm is used to update the parameters with a  learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='01, and the model is constructed with a regularization coefficient of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='01 to enhance the generalization ability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The adversary can observe the  list of recommended items from the recommender system and employ a matrix factor-  ization method to project user movies into a shared latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content="2  Experimental results  To evaluate the attack performance, it is assumed that the adversary has a shadow  dataset from the same distribution as the target recommender's training data and  knows the target recommender's algorithm, for this paper, both the target recommend-  er and the shadow recommender are Use latent semantic models." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Experimental results  6  show that our attack model has good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Plotting the ROC curve, we get the  following image:  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' MLP model experiment ROC curve  Next, we try to explore the influence of the length k of the vector on the attack per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' We evaluate the attack model performance from 10 to 100 different values,  as shown in Figure 4-2, the abscissa is the vector length, and the ordinate is the value  of AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' According to the experimental results, we get the following conclusions:  when the vector length is less than 50, the attack performance improves with the in-  crease of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' When the value of k exceeds 50, the attack performance will theoretically  be improved, because a larger length vector can provide more dimensional perspec-  tives, however, when the vector length is large enough, the performance of the attack  model tends to be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  ROC  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='2  Val AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='857  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='6  80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='0  False Positive Rate7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Influence curve of vector length on attack performance  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Model attack performance under different learning rates  5  Conclusion and Outlook  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='1  Conclusion of the work of this paper  Personalized recommender systems have received more and more attention with the  deepening of research and the emergence of the needs of the Internet industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Not  only traditional electronic websites need the functions and services of recommender  systems, but some emerging fields also introduce recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Music, mov-  ies, news are all recommended objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' This paper introduces several widely used  recommendation algorithms and the related work of member speculation attack, in-  troduces its basic principles, and proposes the recommendation algorithm and mem-  ber speculation attack model used in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' In this paper, a personalized  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='9  k=10 k=20 k=30 k=40 k=50 k=60 k=70 k=80 k=90 k=100  AUC  vector length  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content="9  1E-5  1E-4  1E-3  1E-2  1E-1  AUC  learning rate  8  recommendation implicit semantic model is used to generate a recommendation list,  and then the adversary extracts the user's feature vector, that is, the difference be-  tween the interaction and the center vector of the recommendation set, as the user's  feature vector." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The user feature vector and labels are then used as input to the attack  model, which is trained on the shadow dataset and tested on the target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  The disadvantage of this paper is that the recommendation algorithm is relatively  single, and the latent semantic model is used for both the shadow recommender and  the target recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' The attack performance of the attack model when the shad-  ow data user and the target recommender use different recommendation algorithms  has not been verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Therefore, the attack model lacks certain generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  In addition, due to the time relationship, this paper does not test defenses against  member speculation attacks, such as defense strategies such as reducing the number  of categories, publishing data using differential privacy protection, popular random-  ized recommendation algorithms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=', which have been obtained in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  Effective tests [19][20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content='2  Outlook for future work  This paper studies the member speculation attack of recommendation system based on  latent semantic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Due to factors such as time and energy, the shadow recom-  mender and target recommender in this paper are the same recommendation algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' In future research, the attack performance of different recommendation algo-  rithms can be tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' , such as item-based recommendation, neural network collabora-  tive filtering recommendation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' For example, the shadow recommender uses  a neural network collaborative filtering algorithm, while the target recommender uses  a latent semantic recommendation algorithm to test the generalization ability of the  attack model based on different collocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Furthermore, the impact of hyperparam-  eters on attack performance can be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
+page_content=' Testing the attack performance from  different aspects can effectively improve the attack generalization ability of the mod-  el, and also help us to fully understand the attack mechanism of the members to pro-  pose a better defense mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfAwbl/content/2301.03596v1.pdf'}
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+Incorporating prior information into distributed
+lag nonlinear models with zero-inflated
+monotone regression trees
+Daniel Mork∗ and Ander Wilson†
+Abstract.
+In environmental health research there is often interest in the effect of an expo-
+sure on a health outcome assessed on the same day and several subsequent days or
+lags. Distributed lag nonlinear models (DLNM) are a well-established statistical
+framework for estimating an exposure-lag-response function. We propose methods
+to allow for prior information to be incorporated into DLNMs. First, we impose
+a monotonicity constraint in the exposure-response at lagged time periods which
+matches with knowledge on how biological mechanisms respond to increased levels
+of exposures. Second, we introduce variable selection into the DLNM to identify
+lagged periods of susceptibility with respect to the outcome of interest. The vari-
+able selection approach allows for direct application of informative priors on which
+lags have nonzero association with the outcome. We propose a tree-of-trees model
+that uses two layers of trees: one for splitting the exposure time frame and one
+for fitting exposure-response functions over different time periods. We introduce a
+zero-inflated alternative to the tree splitting prior in Bayesian additive regression
+trees to allow for lag selection and the addition of informative priors. We develop
+a computational approach for efficient posterior sampling and perform a compre-
+hensive simulation study to compare our method to existing DLNM approaches.
+We apply our method to estimate time-lagged extreme temperature relationships
+with mortality during summer or winter in Chicago, IL.
+MSC2020 subject classifications: .
+Keywords: Bayesian additive regression trees, monotone, distributed lag, variable
+selection, nonparametric estimation.
+1
+Introduction
+In many applications, there is interest in estimating the relationship between a predictor
+and an outcome when there is substantial prior information about the exposure-response
+relationship. Including prior information into a statistical analysis can increase the pre-
+cision of an estimator. In this paper, we consider estimation of the relationship between
+temperature exposure and mortality due to exposure on the same day and 20 subse-
+quent days. In the environmental epidemiology literature this is commonly referred to
+as lagged effects. There is substantial prior research that confirms high summer temper-
+atures and low winter temperatures are both associated with increased risk of mortality,
+∗Harvard T.H. Chan School of Public Health [email protected]
+†Colorado State University [email protected]
+arXiv:2301.12937v1  [stat.ME]  30 Jan 2023
+
+2
+Monotone exposure-lag-response function
+the relationship between high or low temperatures, and mortality are each monotonic
+and exposures with the largest impact on mortality occur on the same day and the pre-
+vious 3 to 10 days (Baccini et al., 2008; Yu et al., 2012; Guo et al., 2014; Ragettli et al.,
+2017). However, there is a lack of appropriate methods to estimate lagged health effects
+with shape constraints or informative priors on the length of the lagged association.
+When estimating the association between an exposure and an outcome on each of
+the days following exposure, henceforth lags, the most common statistical approach is
+a distributed lag model (DLM). DLMs are commonly used in time-series studies to es-
+timate the lagged relationships between an exposure and a health outcome (Schwartz,
+2000) and in the analysis of perinatal cohort studies to identify time periods during
+pregnancy when exposures are related to changes in birth or children’s health outcomes
+(Hsu et al., 2015). In a DLM, an outcome yt at time t is regressed on repeated mea-
+surements of past exposures, xt−l, for lagged times l = 0, . . . , L. Such models assume
+a linear exposure-response relationship at each lag with lag-specific slope. Exposures
+measured at fine temporal resolution tend to be highly correlated. To reduce the effect
+of multicolinearity, DLMs are typically constrained so that the lagged relationships vary
+smoothly in time. Methods to constrain DLMs include splines (Zanobetti et al., 2000),
+principal components (Wilson et al., 2017a), Gaussian processes (Warren et al., 2012),
+and regression trees (Mork and Wilson, 2022a). To allow for nonlinear associations be-
+tween a lagged predictor and an outcome, Gasparrini et al. (2010) proposed distributed
+lag nonlinear models (DLNMs) that allow for a smooth, nonlinear exposure-lag-response
+function at each time point using a bi-dimensional spline basis. Gasparrini et al. (2017)
+later extended DLNM to penalized regression splines that reduce sensitivity of model
+selection. Mork and Wilson (2022b) proposed a regression tree DLNM (TDLNM) ap-
+proach as an equally powered and more precise alternative. Both DLM and DLNM are
+now standard tools in the environmental epidemiology literature.
+Imposing monotonicity in a distributed lag nonlinear model is appealing because it
+forces the resulting estimate to correspond with the underlying biological belief that an
+increase in exposure will not result in an improved health outcome. Yet, there are no ex-
+isting methods to estimate an exposure-lag-response function subject to a monotonicity
+constraint. Some previous work has estimated monotone exposure-response functions
+for exposure to air pollution or weather and a health outcome assessed on the same
+day (Powell et al., 2012; Wilson et al., 2014), and there is a rich statistical literature
+on shape constrained regression for a general regression function. Approaches for shape
+constrained regression in a general regression setting include piecewise linear functions
+(Hildreth, 1954; Brunk, 1955), kernel smoothers Mammen (1991), a large number of
+spine based approaches (Ramsay, 1988; Neelon and Dunson, 2004; Meyer, 2008; Wang
+and Li, 2008; Meyer et al., 2011; Meyer, 2012; Powell et al., 2012) and Bernstein poly-
+nomial methods (Chang et al., 2005, 2007; Curtis and Ghosh, 2011; Wilson et al., 2014;
+Ding and Zhang, 2016; Wilson et al., 2020). Chipman et al. (2021) proposed a mono-
+tone regression model based on the popular nonparametric Bayesian additive regression
+trees (BART) framework (Chipman et al., 2010) that constrains a general regression
+function to be monotone with respect to some or all predictors. None of these methods
+are applicable to repeated measures of exposure.
+
+D. Mork, A. Wilson
+3
+A second appealing area to apply prior information is on which lags have a nonzero
+exposure effect. Incorporating prior information on lag selection can improve precision
+in the identified lags during which exposure is associated with an outcome, which are
+critical to targeting public health interventions. These time periods are of particular
+interest when estimating the association between maternal exposure to air pollution
+during pregnancy and birth outcomes where time periods of interest are referred to as
+critical windows of susceptibility and hypothesized to correspond to sensitive stages of
+fetal development (Wright, 2017). There is limited work on including prior information
+on which lags represent susceptible periods. To incorporate biological information into
+the analysis of maternal exposure to air pollution and childhood asthma, Hazlehurst
+et al. (2021) used average exposure over clinically defined developmental periods and
+assessed which periods demonstrated the greatest association between exposure and
+asthma risk. Using average exposure over predefined windows can cause bias (Wilson
+et al., 2017b) and there are no existing methods to add prior information on which
+lags have nonzero association with the outcome in the distributed lag framework. For
+linear DLMs, previous work focuses on adding prior knowledge about the smoothness of
+lag-to-lag variation in the magnitude of the effect in linear DLMs. This can be achieve
+through Bayesian priors (Heaton and Peng, 2012) or a priori knot selection in spline
+models (Gasparrini, 2016). In a time series study it is typically to allow more flexibility
+on the shape of the distributed lag function for short lags during which the majority of
+the exposure effect is posited to occur (Gasparrini, 2016). Yet, these previous methods
+do not specify a prior on whether there is an effect at specific lags or not, with the
+exception of smoothly going to zero as the lag increases in some cases (Heaton and
+Peng, 2012).
+Adding prior information on which lags are associated with the outcome is difficult
+with most existing models because the probability of a nonzero effect at each lag is
+not directly parameterized in most distributed-lag-type models. This not only hinders
+assigning prior probability of a nonzero effect but also reduces interpretability and
+inference on lag selection. Warren et al. (2020) proposed a Bayesian variable selection
+approach to identify time periods where there is a nonzero linear relationship. This
+approach directly parameterizes inclusion or exclusion of time periods, but does not
+apply prior information to the inclusion probabilities. For a nonlinear DLNM, time
+periods are typically identified using confidence or credible intervals to compare expected
+outcomes compared to a reference outcome, such as zero or median exposure. This
+both results in a multiple testing issue and is unsatisfying because comparing to a
+single reference exposure value may miss associations that are only present over certain
+exposure ranges such as very high exposures. Hence, there is a need for DLNM models
+that can allow for both assigning prior information to which lags have nonzero effect
+and for direct inference on time periods when there is a true exposure-response function.
+In this paper, we propose a monotone model using BART-style models that is specif-
+ically tailored to repeated measures of exposure using the DLNM framework. Our ap-
+proach, which we call monotone-TDLNM throughout this paper, utilizes a nested tree
+BART framework (Chipman et al., 2010; Mork et al., 2022). Specifically, we employ an
+ensemble of regression trees that subdivide the lagged time periods of exposure. Within
+a single time-tree, each terminal node corresponds to a mutually exclusive time period
+
+4
+Monotone exposure-lag-response function
+of exposure which is affiliated with a nested regression tree that specifies a monotone
+exposure-response function for the exposure observed in the given time period. Hence,
+the exposure-trees are nested in the time-tree to make a tree of trees. To ensure mono-
+tonicity we implement a constraint on terminal node parameters of the exposure-trees
+using a transformation that allows for Gibbs sampling in a hybrid Markov chain Monte
+Carlo (MCMC) approach to posterior sampling. We introduce variable selection into the
+DLNM through a zero-inflated alternative to the standard BART tree splitting prior.
+By combining the zero-inflated splitting prior in an ensemble regression trees setting
+we are able assign prior probabilities of a nonzero effect at each lag and to make direct
+inference on the time periods of susceptibility to exposure in our monotone model. The
+elegance of the nested tree approach is that the exposure-response relationship at each
+time point is specified by an ensemble of univariate regression trees–nested trees that
+only splits on exposure concentration. By using univariate trees we can efficiently add
+monotonicity constraints and selection at each time point.
+We demonstrate the advantage of our monotone-TDLNM compared to unconstrained
+TDLNM and penalized spline DLNMs through a simulation study. Specifically, we show
+the monotonicity constraint results in more precise estimates of both the exposure-lag-
+response function and of time periods when there is a nonzero association. We apply
+monotone-TDLNMs to a reanalysis of temperature exposure and mortality in a time-
+series study from Chicago, Illinois, USA. Previous analyses of these data have looked
+broadly at temperature, which tends to have an “inverted-J” shape with both high sum-
+mer temperatures and low winter temperatures being associated with increased mortal-
+ity. We consider separate analysis of summer and winter temperatures. We separately
+estimate the monotonic relationship between high summer temperatures and mortality
+and between low winter temperatures and mortality. Software is made available in the R
+package dlmtree (github.com/danielmork/dlmtree). Code to replicate our simulation
+and data analysis are available at https://github.com/danielmork/monotone_dlnm.
+2
+Distributed lag nonlinear models
+We begin by introducing the DLNM in the context of a time-series study on tempera-
+ture related mortality. Let yt represent the observed mortality count at time t. We are
+interested in how extreme temperature during preceding days is related to changes in
+yt. Denote xt, xt−1, . . . , xt−L to be the temperatures during the same day as the out-
+come and the previous L days. The time-lagged relationship between temperature and
+mortality is described through the equation
+g [E(yt)] =
+L
+�
+l=0
+w(xt−l, l) + h(t; ζ)
+(1)
+where g(·) is a link function; h(t; ζ) is a function of time parameterized by ζ and
+w(xt−l, l) is a nonlinear exposure-lag-response function for characterizing the effect of
+temperature xt−l on mortality l days after exposure. We assume h(t; ζ) contains an
+intercept and in some cases may contain other covariates such as day of week.
+
+D. Mork, A. Wilson
+5
+Two assumptions are often imposed when estimating w. First, it is assumed that
+w varies smoothly across the range of xt−l at each lag time l. This assumption follows
+from biological plausibility, that an exposure-response will exhibit a smooth trend across
+the range of exposure concentration. The second assumption is that w(·, l) is similar for
+proximal lags. This assumption is both biologically motivated and statistically practical.
+From a biological perspective it is assumed that exposure on proximal days will have
+a similar effect on the outcome. Statistically, there typically exists high autocorrelation
+between exposure measurements taken close in time and some form of regularization is
+needed to reduce the effects of multicolinearity in the predictors and ensure biologically
+plausible estimates of the exposure-lag-response function. Regularization in the lag di-
+mension can take the form of a smoothness constraint or piecewise smooth or constant
+parameters over a small number of time segments. Combined, these assumptions result
+in a smoothly or piecewise smoothly varying exposure-lag-response function.
+3
+Nested tree framework for DLNM
+Our approach to estimating a DLNM with monotone exposure-response and variable
+selection uses a nested regression tree framework (Mork et al., 2022). Figure 1 visualizes
+the nested tree framework. Like most tree methods, we us an ensemble approach. In our
+case, we use an ensemble of A nested tree units indexed by a. Let Ta denote a binary tree
+with dichotomous splits on the lagged time periods of exposure l = 0, . . . , L into one
+or more mutually exclusive segments. The regression tree Ta consists of internal nodes
+with splits on the available times and terminal nodes denoted {ηab}Ba
+b=1 that identify the
+tree endpoints. In contrast to the previously proposed, unconstrained TDLNM, internal
+nodes of Ta split only on time and do not split on values of exposure concentration.
+Instead, for each terminal node {ηab}Ba
+b=1 in tree Ta, we define a binary nested tree
+Eab. The internal nodes of each Eab split on values of exposure concentration and the
+terminal nodes are denoted by {λabc}Cab
+c=1. To complete the nested tree model we define a
+scalar parameter δabc corresponding to each λabc. Each δabc characterizes the exposure-
+response for a given exposure-concentration and time combination defined by Ta and
+Eab.
+From the nested tree model, the exposure-lag-response function, w, is calculated
+w(xt−l, l) =
+A
+�
+a=1
+Ba
+�
+b=1
+Cab
+�
+c=1
+δabcψ(xt−l, l; ηab, λabc, σx).
+(2)
+Here, the sums are over, from left to right: a) nested tree unit in the ensemble, b)
+terminal node of the time tree Ta, and c) terminal nodes of the nested exposure-splitting
+tree Eab. The summand contains both the terminal node parameter δabc and an exposure-
+specific weight function denoted by ψ that depends on exposure timing, the terminal
+nodes and a hyperparameter σx. The weight function ψ can take many forms, two
+explored by Mork and Wilson (2022b) include a step function and a smooth weight
+function to incorporate smoothness in the exposure-concentration dimension. As most
+biological applications assume a smooth effect of exposure, we follow the latter approach
+
+6
+Monotone exposure-lag-response function
+𝑙 < 𝑙!
+𝑙 > 𝑙!
+𝑙 < 𝑙"
+𝑙 > 𝑙"
+𝜂#!
+𝜂#"
+Lag
+Exposure-concentration
+𝑙!
+𝑙"
+𝑥"
+𝑥!
+𝛿!""
+𝛿!#"
+𝛿!##
+𝛿!$"
+𝑥$
+𝛿!"$
+𝛿!"#
+𝜂#$
+ℰ!"
+𝑥 < 𝑥!
+𝑥 > 𝑥!
+𝑥 < 𝑥"
+𝑥 > 𝑥"
+𝜆!""
+𝜆!"#
+𝜆!"$
+ℰ!#
+𝑥 < 𝑥#
+𝑥 > 𝑥#
+𝜆!##
+𝜆!#"
+ℰ!$
+𝜆!$"
+𝒯#
+Figure 1: Diagram of the nested tree DLNM framework. On the left is a single binary
+tree Ta from the ensemble of trees, a = 1, . . . , A. Tree Ta partitions the lag dimension via
+terminal nodes ηab, b = 1, 2, 3, that partition the time dimension. The nested trees, Eab,
+and corresponding terminal nodes, λabc, partition the exposure-concentration dimension
+within each time period. For each λabc there is a corresponding scalar parameter, δabc,
+shown in the resulting exposure-lag-response surface at right. To impose monotonicity
+we require δabc ≤ δabc′ for c < c′. For identifiability and variable selection we set δab1 = 0.
+and define
+ψ(xt−l, l; ηab, λabc, σx) =
+�
+Φ
+�⌈xabc⌉ − xt−l
+σx
+�
+− Φ
+�⌊xabc⌋ − xt−l
+σx
+��
+· I(l ∈ ηab),
+(3)
+where ⌈xabc⌉ and ⌊xabc⌋ are the upper and lower exposure-concentration limits of node
+λabc, respectively, Φ is the normal cumulative density function, I(l ∈ ηab) is an indicator
+function that equals 1 when lag time l is in node ηab and 0 otherwise, and σx is a tuning
+parameter that is fixed for all exposure-time-response functions. Larger σx will increase
+the smoothness of the exposure-response curves while smaller σx allows for sharper
+changes in the exposure-response relationship. As σx → 0 we arrive at the step weight.
+We treat σx as fixed due to the computational expense required to estimate it in our
+model and follow Mork and Wilson (2022b) by setting σx equal to half the standard
+deviation of the exposure data.
+4
+Monotone treed exposure-lag-response function
+The exposure-response relationship is monotone if w(xt−l, l) ≤ w(x∗
+t−l, l) for any two
+exposures such that xt−l < x∗
+t−l at the same time t − l. We do not assume strict mono-
+tonicity as we wish to allow for a null exposure-response relationship. The monotonicity
+constraint in the exposure-response at each lag time l is satisfied if the terminal node
+parameters {δabc}Cab
+c=1 are non-decreasing in the exposure-concentrations spanned by
+terminal nodes {λabc}Cab
+c=1 for each nested tree Eab. Without loss of generality, we order
+the terminal nodes on each nested tree according to their exposure ranges such that
+
+D. Mork, A. Wilson
+7
+⌊xab1⌋ < · · · < ⌊xabCab⌋, where ⌊xabc⌋ is the minimum exposure concentration value in
+terminal node λabc. We then constrain the terminal node parameters δab1 ≤ · · · ≤ δabCab.
+We set δab1 = 0, which allows for identifiability of the exposure-lag-response function.
+We implement the constraint on the δ’s through a reparameterization of the node-
+specific parameters for each nested tree following the approach used by (Wang and Li,
+2008) for monotone regression with Bernstein polynomials. Let θab = (θab1, . . . , θabCab)′
+and δab = (δab1, . . . , δabCab)′. We consider first-order difference transformation matrix
+Dab such that Dabδab = θab. Specifically, we define the transformation δabc −δab(c−1) =
+θabc for c ≥ 2 and θab1 = δab1 = 0. Using this transformation, the parameter space for
+each θabc is fixed to be θabc ≥ 0. This facilitates more efficient sampling via MCMC. In
+contrast, the parameter space of each δabc is constrained such that δabc−1 ≤ δabc ≤ δabc+1
+and is challenging to estimate with MCMC. The matrix Dab follows a block style with
+adjacent 1 and −1 entries at a unique location in each row and zeros elsewhere. For
+example, with Cab = 4, the transformation matrix is
+Dab =
+�
+���
+1
+0
+0
+0
+−1
+1
+0
+0
+0
+−1
+1
+0
+0
+0
+−1
+1
+�
+��� .
+(4)
+A block-style transformation can be extended to the entire nested tree, such that
+Daδa = θa where δa = (δ′
+a1, . . . , δ′
+aBa)′, θa = (θ′
+a1, . . . , θ′
+aBa)′, and
+Da =
+�
+������
+Da1
+0
+· · ·
+0
+0
+0
+Da2
+· · ·
+0
+0
+...
+...
+...
+...
+...
+0
+0
+· · ·
+Da(Ba−1)
+0
+0
+0
+· · ·
+0
+DaBa
+�
+������
+(5)
+where Dab make up the diagonal block matrices with zeros elsewhere. We assign inde-
+pendent priors to θabc, where each follows a truncated normal distribution with range
+[0, ∞), mean 0, and variance σ2ν2. In the variance, σ2 is the residual variance from a
+Gaussian model and is fixed to a value of 1 for binomial models. This transformation
+and prior specification allows us to implement efficient multivariate truncated normal
+sampling procedures.
+5
+Zero-inflated regression trees for lag selection
+The standard BART model uses a splitting probability that decays quickly as the depth
+of the split increases but assigns a large probability of splitting to the initial node
+(Chipman et al., 2010). Specifically, BART defines the probability of a split at node
+λ by psplit(λ) = α(1 + dλ)−β, α ∈ (0, 1), β > 0, where dλ is the depth of the node
+beginning at zero. In addition, a probability on the dichotomous splitting rule (i.e.,
+variable and location) is defined, most commonly by a uniform probability across vari-
+ables and locations. In our situation, we have two sets of trees, the time trees Ta’s and
+the exposure-response trees Eab’s, that both have different purposes and require separate
+consideration for an appropriate prior structure.
+
+8
+Monotone exposure-lag-response function
+5.1
+Prior on time-splitting trees Ta
+We retain the original BART splitting prior for trees Ta. Imposing default hyperpriors
+αT = 0.95 and βT = 2, results in a prior distribution for the number of terminal
+nodes in the time dimension with P(Ba > 1) = 0.95 with E(Ba) = 2.51. Having a
+high prior probability of multiple terminal nodes in the time dimension allows us to
+identify critical windows and have flexibility across lags. A high prior probability on Ba
+being relatively small helps to regularize the model by retaining the assumption that
+the lagged exposure-response functions are similar across nearby times.
+For a model with l = 0, . . . , L lags there are L possible splitting points. To determine
+the time splitting points, we incorporate work by Linero (2018) where the probability
+of a rule splitting between lags l and l + 1 is equal to Pl/l+1 with P0/1, . . . , PL−1/L ∼
+Dirichlet{κL−1, . . . , κL−1} and κ(κ + L)−1 ∼ β(1, 1). This allows for a data-informed
+approach to identifying change points in the exposure-lag-response function by sharing
+information about split points across trees in the ensemble.
+5.2
+Prior on exposure-response trees Eab
+For the nested trees, Eab, the default BART splitting prior is less desirable because at
+least two terminal nodes in exposure concentration implies a nonzero effect of exposure.
+The prior probability of no splits across an ensemble of A trees is (1 − α)A. Changing α
+to reflect prior belief that there is a non-zero effect requires an extremely low α value for
+even a moderate ensemble size A. A negative consequence of setting α to be low is that
+it will also decrease the probability of splits at all subsequent levels of the nested tree.
+This is undesirable as it is likely to induce strong shrinkage on the exposure-response
+relationship. Our objective is, therefore, to allow user-specification of the probability of
+the first split without impacting the prior probability of subsequent splits conditional
+on there being a first split.
+We enlist two strategies to help accomplish variable selection in our model. First we
+define nested trees without splits (i.e. Cab = 1) to be zero effect (as described by setting
+δab1 = 0 in Section 4), ensuring that specific time periods will be excluded from the
+exposure-lag-response function (e.g., these times are not selected). Second, we propose
+an alternative splitting probability on trees that allows for the time periods with or
+without effects to also be learned from the data.
+For nested tree Eab we define the zero-inflated tree splitting prior as
+psplitZI(λ|ηab, γ, α, β) = π0(γ, ηab)I(dλ = 0) + αE(1 + dλ)−βEI(dλ > 0)
+(6)
+where γ = [γ0, . . . , γL] is a vector of parameters corresponding to each lag time point;
+αE = 0.95 and βE = 2 are the standard BART parameters set to typical default values;
+dλ is the depth of node λ; and
+logit{π0(γ, ηab)} =
+�
+l∈ηab
+γl.
+(7)
+
+D. Mork, A. Wilson
+9
+By defining π0 with the logistic function, we allow for continuous parameters γl ∈ R and
+the probability of a split to be based on the time periods in a given terminal node, ηab.
+If ηab contains many time points with higher valued γl there will be a high probability
+of an effect within Eab. If ηab instead contains times with many lower valued γl, nested
+tree Eab will have a much lower probability of splitting and therefore no effect.
+For the zero-inflated splitting parameters, γl, we define prior γ ∼ MVN(γ0, Σρ),
+where γ0 = (γ01, . . . , γ0L)′ is a prior mean and Σρ is a covariance matrix with parameter
+ρ. We specify an exponential covariance matrix, Σρ(l, l′) = exp{−(l−l′)/ρ}. It is natural
+to set γ0l = 0 for all l = 0, . . . , L implying a prior probability of effect at time l equal to
+0.5. However, prior information on which weeks have a nonzero association can be easily
+incorporated by assigning lag-specific values to each γ0l. We assign a discrete uniform
+prior probability to the candidate values of ρ with possible values corresponding to the
+lag-1 covariance (i.e. when |l − l′| = 1) of 0.05, 0.1, 0.15, . . . , 0.95. We scale Σρ such that
+95% of the time, π0(γ0l) falls between 0.05 and 0.95 for each l. For the splitting rules on
+specific exposure concentration values, we define a discrete uniform prior distribution.
+5.3
+Variable-selection based inference on lags
+In the spline-based DLNM or TDLNM, periods of susceptibility (i.e., lags with nonzero
+exposure-response relationships) are not well defined. They are typically identified as lag
+times that have credible intervals not containing zero for some user-specified exposure
+contrast of interest. In the proposed method, we can directly infer the probability of a
+lag-specific susceptibility through a posterior analysis of terminal nodes at a given lag
+time. Specifically, for posterior samples r = 1, . . . , R define E(r)
+l
+= 1 if any nested tree
+Eab with l ∈ ηab has 2 or more terminal nodes, otherwise E(r)
+l
+= 0. Then,
+ˆP(susceptibility at lag l) = R−1
+R
+�
+r=1
+E(r)
+l
+.
+(8)
+By specifying a reasonable level of confidence (e.g., probability ≥ 0.95) we can make a
+conclusion about which lag times show susceptibility to the exposure. It is noted that at
+ˆP(susceptibility at lag l) = 0.95, the corresponding central credible interval will equal
+zero as the lower bound is at the 2.5 percentile. If this result is not desired, alternative
+solutions are to use an upper 95% credible interval or a 90% credible interval alongside
+the probability of effect.
+6
+Prior specification and posterior computation
+6.1
+Prior specification
+For simplicity, we focus on a Gaussian model in this section. The model is
+yt =
+L
+�
+l=0
+w(xt−l, l) + h(t; ζ) + εt,
+(9)
+
+10
+Monotone exposure-lag-response function
+where εt ∼ N(0, σ2) and assumed independent. We discuss binomial outcomes in Sec-
+tion 6.3.
+We complete the Bayesian specification of the model by assigning priors to the
+remaining parameters. We specify half-Cauchy priors for the variance parameters, with
+σ, ν ∼ C+(0, 1). Because σ2 is the variance parameter for the residuals in the fixed
+effect model, it can be interpreted as a scaling factor on residual variance in the tree
+ensemble, ν2, and is equivalent to scaled BART variance prior described by Linero and
+Yang (2018). We specify a non-informative prior on the regression parameters for the
+time trend and covariate model, ζ ∼ MVN(0, cσ2I), where c is fixed to a large value.
+6.2
+MCMC approach for Gaussian model
+We estimate the model parameters using MCMC with a hybrid Gibbs-Metropolis-
+Hastings algorithm. Specifically, the terminal node parameters can be efficiently sampled
+via a Gibbs sampler as can the variance components and regression parameters for the
+covariates. The tree structures are updated with the Metropolis-Hastings (MH) algo-
+rithm using the grow, prune, and change proposals steps. When updating the exposure-
+splitting trees we also consider the zero-inflation probability and update that probability.
+Updating terminal node parameters: θa
+Define the total exposure effect as f(xt) = �L
+l=0 w(xt−l, l). Let utabc = �L
+l=0 I(xt−l ∈
+λabc, l ∈ ηab) be the count of lagged exposure measurements prior to t within the
+exposure concentration range of terminal node λabc and the time range of terminal
+node ηab. Define uta = (uta11, . . . , uta1Ca1, uta21, . . . , utaBaCaBa )′ and Ua be the matrix
+containing rows u′
+ta. Applying the transformation from Section 4, u′
+taδa = u′
+taD−1
+a θa
+and the total exposure effect is f(xt) = �A
+a=1 u′
+taD−1
+a θa.
+To estimate the exposure-lag-response function we integrate out ζ from the full
+likelihood. Let y = (y1, . . . , yn)′ and f = [f(x1), . . . , f(xn)]′ be vectors of length n.
+Then,
+y|σ2 ∼ MVN(f, σ2VZ)
+VZ = (I − Z′VζZ)−1
+Vζ = (Z′Z + c−1I)−1,
+where Z is a matrix with rows z′
+t that are vector functions of the covariates or spline
+basis expansion for time. Using a Bayesian backfitting approach (Hastie and Tib-
+shirani, 2000) we consider the partial residuals Ra = [R1a, . . . , Rna]′ where Rta =
+yt − �
+a′̸=a uta′D−1
+a′ θa′ resulting in
+Ra|Ta, Ea1, . . . , EaBa, θa, σ2 ∼ MVN(UaD−1
+a θa, σ2VZ).
+The transformed terminal node parameters θa for tree Ta and nested trees {Eab}Ba
+b=1 are
+simulated as a block from their full conditional
+θa|− ∼ T N [0,∞)[Vθa(UaD−1
+a )′V−1
+Z Ra, σ2Vθa]
+
+D. Mork, A. Wilson
+11
+Vθa =
+�
+(UaD−1
+a )′V−1
+Z UaD−1
+a
++ ν−2I
+�−1 .
+Draws for θa are done via efficient sampling methods proposed by Li and Ghosh (2015).
+Updating Eab
+When updating the nested tree, Eab we must consider the additional zero-inflated split-
+ting probability. For alterations to the tree involving more than two terminal nodes, the
+acceptance ratio is identical to previous BART implementations. However, the accep-
+tance ratio is different when moving Eab between 1 and 2 terminal nodes. The probability
+of a nested tree Eab is written
+p(Eab) =
+�
+λ internal
+psplitZI(λ|ηab, γ, α, β)prule(λ)
+�
+λ terminal
+[1 − psplitZI(λ|ηab, γ, α, β)] .
+The MH algorithm acceptance probability for growing tree Eab with 1 terminal node to
+proposed tree E∗
+ab with 2 terminal nodes equals
+r
+=
+min
+�p(Ra|Ta, {Eab}∗, ν, σ2)p(E∗
+ab)p(Eab|E∗
+ab)
+p(Ra|Ta, {Eab}, ν, σ2)p(Eab)p(E∗
+ab|Eab) , 1
+�
+=
+min
+�
+p(Ra|Ta, {Eab}∗, ν, σ2)π0(γ, ηab)
+�
+1 − α · 2−β�2
+p(Ra|Ta, {Eab}, ν, σ2)[1 − π0(γ, ηab)]
+, 1
+�
+,
+where {Eab} = Ea1, . . . , EaBa is the set of nested trees; {Eab}∗ = Ea1, . . . , E∗
+ab, . . . , EaBa is
+the set of nested trees with a tree proposal; and p(Ra|Ta, {Eab}, ν, σ2) is the likelihood
+for the partial residuals integrated over the terminal node parameters, θa, to account
+for the changing dimensionality of the parameter space. We calculate
+p(Ra|Ta, {Eab}, ν, σ2) =
+�
+p(Ra|Ta, {Eab}, θa, σ2)p(θa|σ, ν)dθa
+= (2πσ2)−n/2(22σ2ν2)−p/2|VZ|−1/2|Vθa|1/2
+× exp
+�
+−R′
+aV−1
+Z Ra
+2σ2
++ R′
+aV−1
+Z UaD−1
+a Vθa(UaD−1
+a )′V−1
+Z Ra
+2σ2
+�
+×
+��
+[0,∞)
+p(θa|Ra, −)dθa
+�
+,
+where the last integral is the normalizing constant for the full conditional distribution
+of θa, which can be numerically approximated through algorithms described by Genz
+and Bretz (2012). The acceptance probability for pruning from 2 to 1 terminal nodes
+is the multiplicative inverse of grow; a change proposal reduces to the quotient of the
+integrated likelihoods.
+Updating Ta
+In our nested tree framework an additional consideration must be made when updating
+tree Ta. When we grow Ta we must replace a nested tree, Eab with two new nested trees,
+
+12
+Monotone exposure-lag-response function
+E∗
+ab1 and E∗
+ab2. Therefore, proposed tree T ∗
+a has a different number of terminal nodes,
+B∗
+a, each with a corresponding nested tree. When we apply a grow proposal T ∗
+a , we draw
+each new E∗
+abi, i ∈ {1, 2}, using the zero-inflated tree prior described in Section 5. The
+other nested trees remain the same. To calculate the MH acceptance ratio for proposal
+Ta we follow similar steps to updating Eab by first calculating p(Ra|T ∗
+a , {Eab}∗, ν, σ2)
+and p(Ra|Ta, {Eab}, ν, σ2). The resulting acceptance ratio equals
+r = min
+�p(Ra|T ∗
+a , {Eab}∗, ν, σ2)p(T ∗
+a )p(Ta|T ∗
+a )
+p(Ra|Ta, {Eab}, ν, σ2)p(Ta)p(T ∗
+a |Ta) , 1
+�
+.
+(10)
+A prune proposal for Ta replaces two nested trees corresponding to the pruned nodes
+with a new nested tree while a change proposal is the same as changing an internal node
+in standard BART. In a change proposal for Ta we retain the structure of all nested
+trees.
+Updating remaining parameters: γl, ζ, σ, and ν
+Each terminal node ηab contributes one binary ‘observation’ (effect or no effect) to a
+logistic model relating the probability of an effect at lag time l and parameters γl. We
+employ a Poly´a-gamma augmentation approach to updating γl (Polson et al., 2013).
+The remaining parameters are updated with Gibbs steps using standard full condi-
+tionals. The variance components σ and ν are updated using the approach of Makalic
+and Schmidt (2016). The parameters for the time trend and covariate model, γ and ζ,
+have multivariate normal full conditionals.
+6.3
+Implementation for binomial response
+For a binomial response, yt, we define the logistic model,
+yt|xt, t ∼ Binomial{nt, 1/(1 + e−ψt)}
+(11)
+where ψt = �L
+l=0 w(xt−l, l) + h(t; ζ). To estimate the exposure-lag-response function,
+w(xt−l, l), we follow a Poly´a-gamma augmented variable approach. Briefly, for Poly´a-
+gamma random variable, ωt ∼ PG(nt, ψt), the data-likelihood is proportional to exp{(yt−
+nt/2)ψt}E{exp(−ωtψ2
+t /2)}, where the expectation is with respect to a PG(nt, 0) ran-
+dom variable (see e.g. Polson et al. (2013)). The inclusion of ωt creates a conditionally
+Gaussian likelihood and allows for Gibbs updates of θa and ζ parameters.
+6.4
+Strategies for incorporating additional prior information
+Two goals that often exist simultaneously in estimating exposure-lag-response functions
+are identifying periods of susceptibility or nonzero effects in the exposure response and
+change points in the lag response dimension. The specification of monotone-TDLNM
+allows for the inclusion of prior information in each dimension.
+
+D. Mork, A. Wilson
+13
+To increase or decrease the prior probability of a nonzero exposure relationship
+we can alter the priors in the zero-inflated regression tree for specific time periods,
+specifically γ0 and Σp. For example, we can define γ0l and Σp(l, l) such that 1/{1+e−γ0l}
+falls within a given range 95% of the time. In practice, high probability of effect in some
+lags should be balanced by lower probability of effect in other lags to maintain roughly
+zero mean across γ0 to prevent false positives when many lag periods are included in
+the same terminal node.
+Informative priors for lag change points are easily introduced via two alternate
+approaches. First, we can fix the time split probability rules, Pl/l+1, to increase the
+probability of a tree splitting at a given lag and increasing the likelihood that differ-
+ent exposure-response relationships will exist before or after the split. In some anal-
+yses where there is a well-defined change point in the exposure time-series, this ap-
+proach makes the most sense. For example, air pollution exposures during fetal devel-
+opment and early childhood may assume a change in the exposure-lag-response at the
+time of birth–here it makes sense to apply a large fixed probability of split at that
+time point. The second strategy to incorporating prior information into the time split
+probabilities is through a modification to the Dirichlet prior. Here, we introduce prior
+P0/1, . . . , PL−1/L ∼ Dirichlet(d0/1κ, . . . , dL−1/Lκ) where � dl/l+1 = 1, and dl/l+1 > 0
+is the prior probability of a split point between lag l and l + 1 in a given tree. The
+hyper parameter κ may be set for a specific variance in the probabilities or assigned
+a prior distribution as in Section 5.1. Using a fixed κ in the Dirichlet prior can allow
+for additional posterior inference on the change points through Bayes Factors or similar
+methods.
+An alternative method to introduce prior information into the regression tree model
+is using fixed time trees corresponding to previously estimated periods of susceptibility.
+For example, if two other studies identify periods of susceptibility during specific lag
+periods, we may fix two trees in our model with change points corresponding to these
+past studies. The model and data will determine whether the exposure-lag-response
+function in the nested trees is nonzero; however, we remove the need to select splitting
+times in these trees. Other trees in the model can continue to explore other splitting
+times in case the relationships in the data do not agree with previously estimated
+lag periods of susceptibility. Furthermore, the idea of fixed time trees could allow for
+information transfer from larger studies, where the time periods can be more effectively
+estimated, to small studies. The idea of using one sample to estimate the periods of
+susceptibility and another sample to estimate the exposure response is closely tied to
+the idea of honest inference (Wager and Athey, 2018; Athey and Imbens, 2016).
+7
+Simulation study
+We developed a simulation study based on our data analysis to compare our proposed
+method alongside existing methods TDLNM (Mork and Wilson, 2022b) and the penal-
+ized spline DLNM (GAM) (Gasparrini et al., 2017). The goals of our simulation study
+were to first, show that the monotonicity assumptions in monotone-TDLNM improve
+estimation of the exposure-lag-response function while providing valid inference in terms
+
+14
+Monotone exposure-lag-response function
+of confidence intervals and high precision for identifying periods of susceptibility and
+second, that the inclusion of informative priors can additionally supplement ability of
+DLNM methods to estimate exposure-lag-response functions.
+For each simulation replicate we sampled n = 1000 days from the summer tempera-
+ture time series in our data analysis and used L = 20 days of lagged temperature expo-
+sures. We created 9 simulation scenarios based on combinations of 3 exposure-response
+relationships given by
+fx(x) = I(x > 25)
+�
+�
+�
+�
+�
+0.1 · (x − 25)
+linear
+0.2 · log(x − 25)
+sublinear
+0.2 · [exp{0.25 · (x − 25)} − 1]
+exponential
+(12)
+and 3 lag-response relationships,
+fl(l) =
+�
+��
+�
+�
+�
+20 · I(l < 4)
+piecewise
+max{0, 6 · (6 − l)}
+linear
+max{0, 0.2 · (l + 1) · (l − 8)2}
+quadratic
+(13)
+where the exposure-lag-response function is calculated as w(xt−l, l) = fx(xt−l) · fl(l).
+Figure 2 shows plots of fx and fl. The outcome was sampled by yt = �20
+l=0 w(xt−l, l)+εt
+where εt was drawn independently from a normal distribution with mean zero and
+standard deviation 2, 4, and 8 times the standard deviation of the exposure-lag-response.
+We repeated each combination of exposure-response, lag-response, and error for 50
+simulation replicates.
+Figure 2: The exposure-lag-response function in our simulations is defined by
+w(xt−l, l) = fx(xt−l) · fl(l), based on combinations of fx and fl.
+For each simulation scenario, we compare three models: GAM, TDLNM, and monotone-
+TDLNM. We also compare with the same models using informative priors. For GAM,
+we use a penalized B-spline crossbasis with 10 degrees of freedom in both exposure and
+lag dimensions. To incorporate informative priors, we add varying ridge penalization to
+the latter 6 lag degrees of freedom which will increase the shrinkage of estimates in later
+
+0.5 
+0.4 -
+0.3 -
+fX
+0.2
+0.1 -
+0.0 -
+15
+20
+25
+30
+X30 -
+20
+10
+0
+0
+5
+10
+15
+20
+LagD. Mork, A. Wilson
+15
+lag periods toward zero (see e.g., Gasparrini et al. (2017) for more details). We apply
+TDLNM and monotone-TDLNM with default priors as describe in Mork and Wilson
+(2022b) and this paper. To add informative prior information to TDLNM we define
+the probability of a split in time to be 10 times higher during the lag periods of true
+effect compared to other lag periods. For monotone-TDLNM, we add informative priors
+in two places as described in Section 6.4: first we set mean and variance of the prior
+normal distribution for γl such that there is a 95% probability π0(γl) is between 0.8
+and 0.99 during lag periods of effect and between 0.01 and 0.8 during other lag periods;
+second we set splitting probabilities such that Pl/l+1 divided by Pl′/l′+1 is equal to
+10 for l and l′ being periods of effect or no effect, respectively, and we fix κ = 2. For
+monotone-TDLNM we fix the smoothing parameter σx equal to half the standard de-
+viation of the temperature data. Because a smooth exposure-response function at each
+lag enforces strict monotonicity, we add a small quantity of 0.05 to each side of the
+estimated credible intervals to capture places where the exposure-lag-response function
+remains at zero across a range of exposure-concentration values (e.g., less than 25 for
+periods of nonzero effect). For the tree models, we run each model for 2,000 iterations
+thinned by 10 following 2,000 burn-in iterations; we restart the MCMC procedure 5
+times and combine all results.
+For each simulation scenario, we estimate the exposure-lag-response function across a
+grid of values spanning the temperature and lag data: x = 3, 4, 5 . . . , 30 and l = 0, . . . , 20.
+We calculate the root mean square error (RMSE), first averaged at each point on the
+grid then averaged across the entire exposure-lag-response function. We determine the
+average coverage based on how often 95% credible intervals cover the true exposure-lag-
+response value at each point on the grid. We also calculate the average credible interval
+width by taking the average difference between the upper and lower interval limits.
+Finally, we calculate precision as the proportion of correctly identified time periods of
+nonzero effect (true positive) relative to the total identified time periods of nonzero
+effect (true positive plus false positive). In TDLNM and GAM we use credible intervals
+to determine precision, in monotone-TDLNM we consider ˆP(susceptibility at time l) ≥
+0.95 as the criteria for a nonzero effect.
+Simulation results are presented in Figure 3. Monotone-TDLNM and TDLNM con-
+sistently have the lowest RMSE across all simulation scenario and error combinations.
+Informative priors further decrease the RMSE of monotone-TDLNM in the largest er-
+ror setting. Targeted penalization also decreases the RMSE for the GAM approach,
+but this method consistently has the highest RMSE. Both TDLNM and monotone-
+TDLNM are able to shrink estimates towards zero in places of no effect while GAM
+often overgeneralizes periods of effect and retains additional wiggliness across the es-
+timated exposure-lag-response function. Additionally, the spline based GAM displays
+more extreme behavior at the boundaries, a characteristic not shared by the tree-based
+methods.
+Monotone-TDLNM generally maintains nominal coverage of the true exposure-lag-
+response function by 95% credible intervals and has the smallest credible interval width
+across the range of simulation scenarios and error settings. Adding informative prior in-
+formation further decreases the credible interval width in the highest error setting. We
+
+16
+Monotone exposure-lag-response function
+Figure 3: Simulation results (row panels, y-axis) comparing all scenarios (x-axis: fx/fl
+combination) and error combinations (column panels). Each model is represented by a
+different shape and color, models with informative priors are filled shapes.
+
+Error = 2
+4
+8
+1.5
+口
+RMSE
+1.0
+口
+口
+口
+0.5 
+口
+6
+0
+CI Width
+4 -
+口
+2
+1.00
+L
+L
+4
+4
+4
+4
+0.75
+Coverage
+0.50 
+0.25
+0.00
+1.00
+(
+6
+6
+4
+2
+！
+0.75
+Precision
+0.50
+0.25
+0.00
+1.00
+0.75
+True Positive
+0
+0.50
+0.25
+0.00
+4
+1.00
+0.75
+False Positive 
+0.50
+0.25
+0.00
+口
+GAM
+TDLNM
+Monotone-TDLNM
+GAM IP
+TDLNM IP
+Monotone-TDLNM IPD. Mork, A. Wilson
+17
+note that the strict monotonicity in monotone-TDLNM, due to the smoothing weight,
+makes exact coverage of a flat exposure-response impossible (e.g., below 25C during
+periods of nonzero effect) resulting in non-coverage at lower exposure-concentrations
+despite a roughly flat exposure response curve. TDLNM also maintains nominal cover-
+age of the true effects but the credible interval width is generally larger than monotone-
+TDLNM. GAM models have adaquate coverage by 95% confidence intervals, but the
+largest interval width. The wiggliness of splines in GAM contributes to additional un-
+certainty across the exposure-lag-response function, especially near boundaries where
+spline-based methods have the largest uncertainty. Addition targeted penalization to
+GAM decreases average width substantially, making it similar to TDLNM.
+Our proposed method has the highest precision compared to TDLNM and GAM
+across all scenarios and error settings. We note that in the highest error setting with
+quadratic lag-response, monotone-TDLNM has slightly increase false positives, decreas-
+ing the precision metric. Compared to GAM, TDLNM also has higher precision on
+average. The inclusion of targeting shrinkage in GAM increases precision slightly.
+With respect to true positives, or the probability of detecting a correct period of
+nonzero effect, GAM consistently outperforms the other methods with the trade off of
+the largest false positive rate (probability of identifying an effect where non exists). The
+addition of informative prior information improves the true positive rate for monotone-
+TDLNM with the biggest difference occurring the highest error setting. Changing the
+splitting probabilities for TDLNM does not have a large effect on the model outcomes.
+8
+Temperature related mortality
+We illustrate our method by analyzing the association between extreme heat or cold and
+daily mortality. Let dt represent the number of deaths on day t while temperature during
+the past 20 days is given by xt, xt−1, . . . , xt−20. We use mortality and temperature data
+from the National Morbidity, Mortality and Air Pollution Study, in the city of Chicago
+between 1987 to 2000 (Samet et al., 2000). The data is publicly available in the R
+package dlnm. Because extreme hot and cold temperatures may each increase mortality
+and break the monotonicity assumption in our model, we separately analyze summer
+(May-September) and winter (October-April) time periods. Days were removed from
+the analysis if lagged temperature was missing, resulting in 2,142 summer days and
+2,952 winter days. Separately for summer and winter, we fit the model
+E[log(dt)] = γmy(t) + δdow(t) +
+20
+�
+l=0
+w(xt−l, l)
+(14)
+where γmy(t) is an intercept for month and year at time t, δdow(t) is an intercept for
+the day of the week. We assume a constant variance for the log-rate death outcome
+conditional on the time and temperature, i.e., Var[log(dt)|t, xt] = σ2. We reversed the
+sign of temperature in the winter model so that w(xt−l, l) is monotone increasing for
+monotone-TDLNM.
+
+18
+Monotone exposure-lag-response function
+For the summer analyses we applied informative priors reflecting the stronger body
+of evidence relating lagged relationships with heat-related mortality. Specifically, in
+monotone-TDLNM we set a normally distributed prior on γl for l = 0, . . . , 6 such
+that π0(γl) had a 95% prior probability between 0.8 and 0.99, greatly increasing the
+probability of a split in the nested tree during these lags. We also set the corresponding
+time splitting probabilities so they were ten times large during the first six lags than
+later time lags and we fix κ = 2. For the winter temperature and mortality analysis we
+retained default vague priors to reflect the added uncertainty around cold-temperature
+related mortality.
+We compare results to TDLNM (no monotonicity assumption) and penalized spline
+DLNM (GAM) (Mork and Wilson, 2022b; Gasparrini et al., 2017). The treed mod-
+els were run for 5,000 MCMC iterations thinned by 10 after 2,000 burn-in iterations.
+We combined posterior samples from 10 independent Markov chains. For GAM, we
+specify third degree B-splines with 10 degrees of freedom in both lag and temperature
+dimensions and second order difference penalties. In the GAM summer model we added
+varying ridge penalization to the last seven degrees of freedom in the lag dimension as an
+informative prior (see e.g., Gasparrini et al. (2017) for more details). In both summer
+and winter mortality data, we estimate temperature-mortality relationships from the
+DLNM relative to 20 degrees Celsius and back transform results to estimated percent
+change in mortality at a given lag-temperature combination.
+Monotone-TDLNM allows us to compute a posterior probability of non-zero temper-
+ature related mortality during a given lag period. The posterior probabilities of effect are
+shown in Figure 4. We use posterior probability above 0.95 to indicate a nonzero effect.
+The results from monotone-TDLNM indicate a relationship for heat-related (summer)
+mortality during lags 0 − 3 days prior and cold-related (winter) mortality during lags
+1 − 9 days prior. Due to the monotonicity assumption in monotone-TDLNM we can
+specifically say these relationships are due to heat or cold. TDLNM and GAM do not
+share a similar variable selection method and we identify possible periods of suscep-
+tibility by identifying where 95% credible intervals do not contain zero. Because the
+exposure-response functions in TDLNM and GAM are not monotone, the identified pe-
+riods of susceptibility may be due to increased or decreased temperatures and requires
+inspection of the posterior distribution of the exposure-time-response surface. Based on
+the credible intervals, TDLNM indicates nonzero relationships between mortality and
+temperature in summer at lags 0 − 11 days prior and for winter temperatures at lags
+0−10 days prior; GAM finds nonzero relationships between mortality and summer tem-
+peratures during lags 0 − 6, 8 − 11 and 13 days prior, and due to winter temperatures
+0 − 5 and 7 − 9 days prior.
+Figure 5a shows slices of the estimated DLNM at three different temperatures (25,
+28, and 32 degrees C). This is the estimated percent difference in mean mortality for
+each temperature value compared to 20 degrees C by lag. Figure 5b shows slices at three
+different lag periods (1, 2, and 5 days prior) for each of the three compared methods.
+This is the exposure-response function between temperature and mortality on 1, 2,
+and 5 days post exposure. Only monotone-TDLNM shows longer lagged effects at lower
+temperatures (28 degrees C from 0−3 days prior), the other methods show an immediate
+
+D. Mork, A. Wilson
+19
+Figure 4: Posterior probability of susceptibility at a given lag, for summer (solid line)
+and winter (dashed line) models.
+effect (same day of exposure) as well as intermittent nonzero relationships at longer lags
+(e.g., GAM shows increased mortality for temperatures below 20C during lags 8−11 as
+well as decreased mortality above 20C at lag 13). For higher temperatures (32 degrees
+C), GAM shows effects extending back 6 days while the lagged relationships for TDLNM
+and monotone-TDLNM are similar but only extend to 4 and 3 days prior, respectively.
+When looking at slices in the lag dimension, all models indicate an exponential-like
+relationship between temperature and mortality. At 2 days prior, monotone-TDLNM
+indicates the relationship to increased mortality extends to lower temperatures than
+the other models. We also note that the credible interval widths for monotone-TDLNM
+are similar or smaller than the competing methods, especially at low temperatures and
+later lags where the effects are shrunk towards zero.
+Figure 6a shows the lagged relationship to mortality at temperatures -15, 0, and
+10 degrees C. TDLNM and GAM indicate decreased mortality at lag 0 followed by in-
+creased mortality related to low temperatures during previous days. Monotone-TDLNM
+and TDLNM show potentially longer lagged relationships between lower temperatures
+and mortality compared to GAM. The wiggliness of GAM also indicates a no relation-
+ship between mortality at cold temperatures at 6 days prior which is likely a spline-
+induced artifact. Figure 6b shows the exposure-response relationship between mortality
+and temperature (degrees below zero) at 1, 2, and 5 days prior. We see a more linear
+relationship between lower temperatures and mortality in all models where GAM ex-
+tends up to temperatures near 20C while the tree-based methods only indicate increased
+mortality at lower temperatures.
+
+1.00
+0.75
+P(susceptibility at lag
+0.50
+0.25
+0.00-
+0
+5
+10
+15
+20
+Lag (days)
+Summer
+Winter20
+Monotone exposure-lag-response function
+9
+Discussion
+DLNMs have become a standard tool in environmental epidemiology. Most commonly
+researchers use spline-based DLNMs to estimate the association between an exposure
+and a lagged health outcome, such a temperature and mortality on the same day and
+following 20 days as we consider in this paper. In such analyses, it is rare to consider
+prior information on the shape of the exposure-lag-response function. This is despite
+ample prior research that shows that the exposure-response function is monotone for
+many exposure-response pairs or sheds light on which lags are likely to be associated
+with the outcome. The decision to not include information on monotonicity is in large
+part due to a lack of available statistical methods to estimate monotone DLNMs.
+We propose a regression tree based DLNM that is constrained to have a monotone
+exposure-response relationship at each lag time. Our work builds of the popular spline-
+based DLNM (Gasparrini et al., 2017) and existing tree-based TDLNM (Mork and
+Wilson, 2022b), both of which impose no constraints on the shape of the exposure-
+response relationship. We propose a nested-tree approach that uses one tree to partition
+of lag period into discrete time segments and a set of nested trees that capture the
+exposure-response relationship during each time segment. The approach can be thought
+of as a Bayesian tree analog to the crossbasis DLNM that uses two basis expansion, one
+in the lag direction and a second in the exposure-concentrations direction, to estimate
+a DLNM (Gasparrini et al., 2010).
+Our proposed monotone-TDLNM allows for easy inclusion of prior information and
+inference on lags for which there there is a nonzero exposure-response relationship
+through Bayesian variable selection methods incorporated into a regression tree set-
+ting. Identifying lags with nonzero exposure-response relationships is challenging with
+previous implementations of DLNM. Similar time selection methods have been proposed
+for linear DLMs (Warren et al., 2020), but no such methods are available for nonlinear
+DLNMs. No previous methods have considered inclusion of prior information on which
+lags are associated with an outcome.
+Through simulation, we show that the monotone-TDLNM resulted in more precise
+estimation of the exposure-lag-response function and lag selection compared to a pe-
+nalized spline DLNM and unconstrained TDLNM. The monotone-TDLNM resulted in
+smaller credible intervals that still maintained the nominal coverage level. In an analysis
+of summer heat exposure and winter cold exposure and mortality in a Chicago, Illinois,
+USA time-series study we found that both summer heat and winter cold were associated
+with increased mortality. Importantly, the unconstrained models were consistent with
+a monotone relationship and such a relationship has been found in other analyses of
+temperature and mortality. However, the constrained model that includes prior informa-
+tion on monotonicity resulted in more precise estimates the the exposure-lag-response
+function. This highlights one of the advantages of the proposed model. In addition, the
+easy inference on the lags with nonzero effect highlights a second major advantage over
+previous methods.
+A clear limitation of the proposed approach, or any constrained regression method,
+is that violation of the monotonicity assumption would be a major misapplication and
+
+D. Mork, A. Wilson
+21
+result in bias and incorrect inference. In the case of temperature and mortality this
+monotonicity by season is well established. However, in other situations it is impor-
+tant to check these assumptions. For example, a researcher may consider also fitting
+unconstrained models and looking for evidence of departures from monotonicity as we
+demonstrated in our data application.
+Our work adds to a recent literature on incorporating prior information in environ-
+mental epidemiology analyses. Thomas et al. (2007) provides a compelling argument
+for incorporating biological prior information in Bayesian analyses of the health effects
+of environmental exposures. Recent work has promoted the use of informative priors in
+several model types (Reich et al., 2020; McGee et al., 2022), including previous models
+that impose monotonicity in the exposure-response relationship (Powell et al., 2012;
+Wilson et al., 2014). As was so well stated by (Thomas et al., 2007), “by directly in-
+corporating into our analyses information from other studies or allied fields—we can
+improve our ability to distinguish true causes of disease from noise and bias.”
+
+22
+Monotone exposure-lag-response function
+(a) Lagged effects of heat-related mortality at three different temperature
+levels (rows) as estimated by three DLNM methods (columns). The x-axis
+indicates the lag time in days while the y-axis is unique to each row and
+describes the estimated percent change in mortality. The dark black lines
+show the estimated mean relationship while the gray area describes the 95%
+credible interval.
+(b) Temperature-specific effects of heat-related mortality at three different
+lag times (rows) as estimated by three DLNM methods (columns). The x-
+axis indicates the temperature in degrees Celsius while the y-axis is unique
+to each row and describes the estimated percent change in mortality. The
+dark black lines show the estimated mean relationship while the gray area
+describes the 95% credible interval.
+Figure 5
+
+GAM
+TDLNM
+Monotone-TDLNM
+30
+20
+25 deg C 
+10
+Difference in Mortality
+0
+30
+20
+28 deg C
+10
+0
+%
+30
+20
+32 deg 
+10
+c
+0
+0
+5
+10
+15
+200
+5
+10
+15
+200
+5
+10
+15
+20
+Lag (days)GAM
+TDLNM
+Monotone-TDLNM
+30
+1 day prior
+20
+10
+0
+30
+ days prior
+2
+20
+10
+0
+%
+30
+ 5 days prior
+20
+10
+0
+10
+15
+20
+25
+30
+10
+15
+20
+25
+30
+010
+15
+20
+25
+30
+TemperatureD. Mork, A. Wilson
+23
+(a) Lagged effects of cold-related mortality at three different temperature
+levels (rows) as estimated by three DLNM methods (columns). The x-axis
+indicates the lag time in days while the y-axis is unique to each row and
+describes the estimated percent change in mortality. The dark black lines
+show the estimated mean relationship while the gray area describes the 95%
+credible interval.
+(b) Temperature-specific effects of cold-related mortality at three different
+lag times (rows) as estimated by three DLNM methods (columns). The
+x-axis indicates the temperature in degrees Celsius below zero while the
+y-axis is unique to each row and describes the estimated percent change in
+mortality. The dark black lines show the estimated mean relationship while
+the gray area describes the 95% credible interval.
+Figure 6
+
+GAM
+TDLNM
+Monotone-TDLNM
+5 
+-15 deg C 
+0
+-5
+% Difference in Mortality
+O deg C
+0
+5
+5
+10 deg C 
+5
+0
+5
+10
+15
+200
+5
+10
+15
+20
+5
+10
+15
+20
+Lag (days)GAM
+TDLNM
+Monotone-TDLNM
+8 -
+-9
+1 day prior
+4 -
+e in Mortality
+0
+8
+6
+ 2 days prior
+Difference
+4
+2
+0
+%
+8
+6 -
+ 5 days prior
+4 -
+2 -
+0
+-10
+T
+T
+0
+10
+20
+-10
+0
+10
+20
+-10
+0
+10
+20
+Degreesbelowzero24
+Monotone exposure-lag-response function
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+R01AG066793.
+

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