๐ฏ Top Personalized Recommendations
AI Summary
- When IQ(X) is large, a relatively large sample size or privacy budget is needed to release reliable results. [3]
- Small sample sizes and extreme observations can result in large local sensitivity, potentially pushing it close to worst-case levels. [3]
- Smooth sensitivity: A measure of how sensitive a function is to changes in its input, used to determine the amount of noise needed for differential privacy. [3]
- One possible strategy is to allocate part of the privacy budget to privately estimate some upper quantile q of the values in X, say the 99th percentile. [2]
- The differentially private Gini index algorithm can perform well when the normalized range IQ(X) of X is not too large. [1]
Abstract
The Gini index is a widely reported measure of income inequality. In some settings, the underlying data used to compute the Gini index are confidential. The organization charged with reporting the Gini index may be concerned that its release could leak information about the underlying data. We present an approach for bounding this information leakage by releasing a differentially private version of the Gini index. In doing so, we analyze how adding, deleting, or altering a single observation in any specific dataset can affect the computation of the Gini index; this is known as the local sensitivity. We then derive a smooth upper bound on the local sensitivity. Using this bound, we define a mechanism that adds noise to the Gini index, thereby satisfying differential privacy. Using simulated and genuine income data, we show that the mechanism can reduce the errors from noise injection substantially relative to differentially private algorithms that rely on the global sensitivity, that is, the maximum of the local sensitivities over all possible datasets. We characterize settings where using smooth sensitivity can provide highly accurate estimates, as well as settings where the noise variance is simply too large to provide reliably useful results. We also present a Bayesian post-processing step that provides interval estimates about the value of the Gini index computed with the confidential data.
The Gini index is a widely reported measure of income inequality. In some settings, the underlying data used to compute the Gini index are confidential. The organization charged with reporting the Gini index may be concerned that its release could leak information about the underlying data. We present an approach for bounding this information leakage by releasing a differentially private version of the Gini index. In doing so, we analyze how adding, deleting, or altering a single observation in any specific dataset can affect the computation of the Gini index; this is known as the local sensitivity. We then derive a smooth upper bound on the local sensitivity. Using this bound, we define a mechanism that adds noise to the Gini index, thereby satisfying differential privacy. Using simulated and genuine income data, we show that the mechanism can reduce the errors from noise injection substantially relative to differentially private algorithms that rely on the global sensitivity, that is, the maximum of the local sensitivities over all possible datasets. We characterize settings where using smooth sensitivity can provide highly accurate estimates, as well as settings where the noise variance is simply too large to provide reliably useful results. We also present a Bayesian post-processing step that provides interval estimates about the value of the Gini index computed with the confidential data.
Why we think this paper is great for you:
This paper directly addresses the measurement of income inequality, a core area of your interest, by focusing on the Gini index. It also considers the privacy implications of handling sensitive data related to these crucial economic metrics.
This paper directly addresses the measurement of income inequality, a core area of your interest, by focusing on the Gini index. It also considers the privacy implications of handling sensitive data related to these crucial economic metrics.
AI Summary
- The rise of digital ghosts and deadbots forces us to confront fundamental questions about how we remember our dead. [3]
- Digital ghosts can become the blind spot between memory and trickery, a prolonged mourning disguised as dialogue. [3]
- Deadbots: AI replicas that simulate the presence of deceased individuals. [3]
- Digital ghosts: AI-generated representations of deceased people that can interact with the living. [3]
- The era of AI 'afterlives' is here, and it falls upon us to ensure this technology is used in a way that supports memory without becoming an imposture, and helps heal without betraying the dignity of those we love and lose. [3]
- The article cites various studies and papers on the topic of digital ghosts and deadbots, including works by authors such as Jed R. [3]
- From an ethical standpoint, arguably intent and transparency make a substantial difference in creating or engaging with a simulacrum of a loved one. [2]
- Brubaker and John Danaher. [1]
Abstract
Advances in artificial intelligence now make it possible to simulate the dead through chatbots, voice clones, and video avatars trained on a person's digital traces. These "digital ghosts" are moving from fiction to commercial reality, reshaping how people mourn and remember. This paper offers a conceptual and ethical analysis of AI-mediated digital afterlives. We define what counts as a digital ghost, trace their rise across personal, commercial, and institutional contexts, and identify core ethical tensions around grief and well-being, truthfulness and deception, consent and posthumous privacy, dignity and misrepresentation, and the commercialization of mourning. To analyze these challenges, we propose a nine-dimensional taxonomy of digital afterlife technologies and, building on it, outline the features of an ethically acceptable digital ghost: premortem intent, mutual consent, transparent and limited data use, clear disclosure, restricted purposes and access, family or estate stewardship, and minimal behavioral agency. We argue for targeted regulation and professional guidelines to ensure that digital ghosts can aid remembrance without slipping into forms of deception.
Advances in artificial intelligence now make it possible to simulate the dead through chatbots, voice clones, and video avatars trained on a person's digital traces. These "digital ghosts" are moving from fiction to commercial reality, reshaping how people mourn and remember. This paper offers a conceptual and ethical analysis of AI-mediated digital afterlives. We define what counts as a digital ghost, trace their rise across personal, commercial, and institutional contexts, and identify core ethical tensions around grief and well-being, truthfulness and deception, consent and posthumous privacy, dignity and misrepresentation, and the commercialization of mourning. To analyze these challenges, we propose a nine-dimensional taxonomy of digital afterlife technologies and, building on it, outline the features of an ethically acceptable digital ghost: premortem intent, mutual consent, transparent and limited data use, clear disclosure, restricted purposes and access, family or estate stewardship, and minimal behavioral agency. We argue for targeted regulation and professional guidelines to ensure that digital ghosts can aid remembrance without slipping into forms of deception.
Why we think this paper is great for you:
This paper explores ethical considerations in AI, a field that often intersects with issues of fairness and societal impact. Understanding ethical frameworks for AI is important for ensuring equitable technological development.
This paper explores ethical considerations in AI, a field that often intersects with issues of fairness and societal impact. Understanding ethical frameworks for AI is important for ensuring equitable technological development.
AI Summary
- Social AI is a type of artificial intelligence that can interact with humans in a way that simulates human-like conversation and behavior. [3]
- The development of Social AI requires the integration of multiple disciplines, including computer science, psychology, sociology, and philosophy. [3]
- Social AI has the potential to revolutionize various industries, such as healthcare, education, and customer service, by providing personalized support and improving user experience. [3]
- Social AI: A type of artificial intelligence that can interact with humans in a way that simulates human-like conversation and behavior. [3]
- Human-AI interaction: The process by which humans interact with artificial intelligence systems, such as chatbots or virtual assistants. [3]
- The development of Social AI has the potential to transform various industries and improve user experience. [3]
- However, the development of Social AI also raises several challenges and concerns, including ensuring transparency, accountability, and ethics in AI decision-making. [2]
Abstract
As artificial intelligence systems become increasingly integrated into human social contexts, Artificial Social Intelligence (ASI) has emerged as a critical capability that enables AI to perceive, understand, and engage meaningfully in complex human social interactions. This chapter introduces a comprehensive framework for Human-Centered Artificial Social Intelligence (HC-ASI), built upon the Technology-Human Factors-Ethics (THE) Triangle, which systematically addresses both technical foundations and human-centered design principles necessary for developing socially intelligent AI systems. This chapter provides a comprehensive overview of current ASI research. This chapter begins by establishing the theoretical foundations of ASI, tracing its evolution from classical psychological theories of human social intelligence to contemporary computational models, then examines the mechanisms underlying human-AI social interaction with particular emphasis on establishing shared social understanding and appropriate role positioning. The chapter further explores ASI's practical implications for individuals and groups through comprehensive evaluation frameworks that combine technical benchmarks with human-centered experiential assessments, demonstrating real-world applications through detailed case studies spanning healthcare, companionship, education, and customer service domains. Building on the overview and the framework of HC -ASI, this chapter articulates core HC-ASI design principles and translates them into actionable methodologies and implementation guidelines that provide practical guidance for researchers and practitioners. This chapter concludes with a critical discussion of current challenges and promising directions for developing comprehensive HC-ASI ecosystems.
As artificial intelligence systems become increasingly integrated into human social contexts, Artificial Social Intelligence (ASI) has emerged as a critical capability that enables AI to perceive, understand, and engage meaningfully in complex human social interactions. This chapter introduces a comprehensive framework for Human-Centered Artificial Social Intelligence (HC-ASI), built upon the Technology-Human Factors-Ethics (THE) Triangle, which systematically addresses both technical foundations and human-centered design principles necessary for developing socially intelligent AI systems. This chapter provides a comprehensive overview of current ASI research. This chapter begins by establishing the theoretical foundations of ASI, tracing its evolution from classical psychological theories of human social intelligence to contemporary computational models, then examines the mechanisms underlying human-AI social interaction with particular emphasis on establishing shared social understanding and appropriate role positioning. The chapter further explores ASI's practical implications for individuals and groups through comprehensive evaluation frameworks that combine technical benchmarks with human-centered experiential assessments, demonstrating real-world applications through detailed case studies spanning healthcare, companionship, education, and customer service domains. Building on the overview and the framework of HC -ASI, this chapter articulates core HC-ASI design principles and translates them into actionable methodologies and implementation guidelines that provide practical guidance for researchers and practitioners. This chapter concludes with a critical discussion of current challenges and promising directions for developing comprehensive HC-ASI ecosystems.
Why we think this paper is great for you:
This work delves into Artificial Social Intelligence, which aims to help AI understand and engage in human social contexts. Insights from this field could contribute to a deeper understanding of complex social dynamics.
This work delves into Artificial Social Intelligence, which aims to help AI understand and engage in human social contexts. Insights from this field could contribute to a deeper understanding of complex social dynamics.
AI Summary
- The authors highlight the potential benefits of using AI in research, including increased productivity and well-being for mathematicians. [3]
- They also note that AI can excel at routine but lengthy calculations, freeing up time for more creative work. [3]
- They also note that human-AI collaboration can lead to new insights and solutions. [3]
- LLM: Large Language Model AI: Artificial Intelligence The use of AI in research has the potential to increase productivity and well-being for mathematicians. [3]
- Human-AI collaboration can lead to new insights and solutions. [3]
- The paper discusses the use of a large language model (LLM) in solving a research problem in mathematical statistics. [2]
Abstract
Over the last few months, AI models including large language models have improved greatly. There are now several documented examples where they have helped professional mathematical scientists prove new results, sometimes even helping resolve known open problems. In this short note, we add another example to the list, by documenting how we were able to solve a previously unsolved research problem in robust mathematical statistics with crucial help from GPT-5. Our problem concerns robust density estimation, where the observations are perturbed by Wasserstein-bounded contaminations.In a previous preprint (Chao and Dobriban, 2023, arxiv:2308.01853v2), we have obtained upper and lower bounds on the minimax optimal estimation error; which were, however, not sharp. Starting in October 2025, making significant use of GPT-5 Pro, we were able to derive the minimax optimal error rate (reported in version 3 of the above arxiv preprint). GPT-5 provided crucial help along the way, including by suggesting calculations that we did not think of, and techniques that were not familiar to us, such as the dynamic Benamou-Brenier formulation, for key steps in the analysis. Working with GPT-5 took a few weeks of effort, and we estimate that it could have taken several months to get the same results otherwise. At the same time, there are still areas where working with GPT-5 was challenging: it sometimes provided incorrect references, and glossed over details that sometimes took days of work to fill in. We outline our workflow and steps taken to mitigate issues. Overall, our work can serve as additional documentation for a new age of human-AI collaborative work in mathematical science.
Over the last few months, AI models including large language models have improved greatly. There are now several documented examples where they have helped professional mathematical scientists prove new results, sometimes even helping resolve known open problems. In this short note, we add another example to the list, by documenting how we were able to solve a previously unsolved research problem in robust mathematical statistics with crucial help from GPT-5. Our problem concerns robust density estimation, where the observations are perturbed by Wasserstein-bounded contaminations.In a previous preprint (Chao and Dobriban, 2023, arxiv:2308.01853v2), we have obtained upper and lower bounds on the minimax optimal estimation error; which were, however, not sharp. Starting in October 2025, making significant use of GPT-5 Pro, we were able to derive the minimax optimal error rate (reported in version 3 of the above arxiv preprint). GPT-5 provided crucial help along the way, including by suggesting calculations that we did not think of, and techniques that were not familiar to us, such as the dynamic Benamou-Brenier formulation, for key steps in the analysis. Working with GPT-5 took a few weeks of effort, and we estimate that it could have taken several months to get the same results otherwise. At the same time, there are still areas where working with GPT-5 was challenging: it sometimes provided incorrect references, and glossed over details that sometimes took days of work to fill in. We outline our workflow and steps taken to mitigate issues. Overall, our work can serve as additional documentation for a new age of human-AI collaborative work in mathematical science.
Why we think this paper is great for you:
This paper highlights how AI can assist in solving problems in mathematical statistics. Advancements in statistical methods, aided by AI, are invaluable for rigorous analysis in social and economic research.
This paper highlights how AI can assist in solving problems in mathematical statistics. Advancements in statistical methods, aided by AI, are invaluable for rigorous analysis in social and economic research.
AI Summary
- Explainable Artificial Intelligence (XAI) is rapidly evolving, moving beyond post-hoc methods towards a more comprehensive mechanistic understanding of AI model behavior and decision-making process. [3]
- Embedding causal reasoning into interpretability pipelines through counterfactuals or mechanistic priors provides the necessary structure to distinguish genuine drivers from spurious correlations. [3]
- Explainable Artificial Intelligence (XAI): AI that can explain its own decision-making process and provide insights into how it arrived at a particular conclusion. [3]
- Causality: The relationship between cause and effect, essential for understanding the underlying mechanisms of complex systems. [3]
- Interpretability: The ability to understand and interpret the results of an algorithm or model. [3]
- XAI has the potential to contribute to trustworthy and discovery-driven science by providing insights into AI decision-making processes. [3]
- Causality plays a decisive role in XAI, without causal grounding even stable and reproducible explanations risk being superficially plausible while scientifically misleading. [2]
- Interpretability results that are accurate but misaligned with domain concepts may fail to resonate with experts, rendering them ineffective in real scientific workflows. [1]
Abstract
Artificial intelligence and machine learning are reshaping how we approach scientific discovery, not by replacing established methods but by extending what researchers can probe, predict, and design. In this roadmap we provide a forward-looking view of AI-enabled science across biology, chemistry, climate science, mathematics, materials science, physics, self-driving laboratories and unconventional computing. Several shared themes emerge: the need for diverse and trustworthy data, transferable electronic-structure and interatomic models, AI systems integrated into end-to-end scientific workflows that connect simulations to experiments and generative systems grounded in synthesisability rather than purely idealised phases. Across domains, we highlight how large foundation models, active learning and self-driving laboratories can close loops between prediction and validation while maintaining reproducibility and physical interpretability. Taken together, these perspectives outline where AI-enabled science stands today, identify bottlenecks in data, methods and infrastructure, and chart concrete directions for building AI systems that are not only more powerful but also more transparent and capable of accelerating discovery in complex real-world environments.
Artificial intelligence and machine learning are reshaping how we approach scientific discovery, not by replacing established methods but by extending what researchers can probe, predict, and design. In this roadmap we provide a forward-looking view of AI-enabled science across biology, chemistry, climate science, mathematics, materials science, physics, self-driving laboratories and unconventional computing. Several shared themes emerge: the need for diverse and trustworthy data, transferable electronic-structure and interatomic models, AI systems integrated into end-to-end scientific workflows that connect simulations to experiments and generative systems grounded in synthesisability rather than purely idealised phases. Across domains, we highlight how large foundation models, active learning and self-driving laboratories can close loops between prediction and validation while maintaining reproducibility and physical interpretability. Taken together, these perspectives outline where AI-enabled science stands today, identify bottlenecks in data, methods and infrastructure, and chart concrete directions for building AI systems that are not only more powerful but also more transparent and capable of accelerating discovery in complex real-world environments.
Why we think this paper is great for you:
This roadmap outlines the future directions of AI for scientific pursuit across various fields. The broader application of AI in science could provide new tools for analyzing and addressing societal challenges.
This roadmap outlines the future directions of AI for scientific pursuit across various fields. The broader application of AI in science could provide new tools for analyzing and addressing societal challenges.
AI Summary
- The article discusses the Model Context Protocol (MCP) and its performance evaluation in serving model cards, comparing it to REST protocol. [2]
- FAIR Signposting Profile: Implementation guidelines for exposing machine-actionable navigation links using standardized HTTP headers and HTML link elements. [1]
Abstract
AI/ML model cards can contain a benchmarked evaluation of an AI/ML model against intended use but a one time assessment during model training does not get at how and where a model is actually used over its lifetime. Through Patra Model Cards embedded in the ICICLE AI Institute software ecosystem we study model cards as dynamic objects. The study reported here assesses the benefits and tradeoffs of adopting the Model Context Protocol (MCP) as an interface to the Patra Model Card server. Quantitative assessment shows the overhead of MCP as compared to a REST interface. The core question however is of active sessions enabled by MCP; this is a qualitative question of fit and use in the context of dynamic model cards that we address as well.
AI/ML model cards can contain a benchmarked evaluation of an AI/ML model against intended use but a one time assessment during model training does not get at how and where a model is actually used over its lifetime. Through Patra Model Cards embedded in the ICICLE AI Institute software ecosystem we study model cards as dynamic objects. The study reported here assesses the benefits and tradeoffs of adopting the Model Context Protocol (MCP) as an interface to the Patra Model Card server. Quantitative assessment shows the overhead of MCP as compared to a REST interface. The core question however is of active sessions enabled by MCP; this is a qualitative question of fit and use in the context of dynamic model cards that we address as well.
Why we think this paper is great for you:
This paper discusses AI/ML model cards for evaluating AI models. Robust evaluation and transparency are crucial for developing AI systems that are fair and unbiased, which can indirectly support equitable outcomes.
This paper discusses AI/ML model cards for evaluating AI models. Robust evaluation and transparency are crucial for developing AI systems that are fair and unbiased, which can indirectly support equitable outcomes.
Abstract
Edge General Intelligence (EGI) represents a paradigm shift in mobile edge computing, where intelligent agents operate autonomously in dynamic, resource-constrained environments. However, the deployment of advanced agentic AI models on mobile and edge devices faces significant challenges due to limited computation, energy, and storage resources. To address these constraints, this survey investigates the integration of Knowledge Distillation (KD) into EGI, positioning KD as a key enabler for efficient, communication-aware, and scalable intelligence at the wireless edge. In particular, we emphasize KD techniques specifically designed for wireless communication and mobile networking, such as channel-aware self-distillation, cross-model Channel State Information (CSI) feedback distillation, and robust modulation/classification distillation. Furthermore, we review novel architectures natively suited for KD and edge deployment, such as Mamba, RWKV (Receptance, Weight, Key, Value) and Cross-Architecture distillation, which enhance generalization capabilities. Subsequently, we examine diverse applications in which KD-driven architectures enable EGI across vision, speech, and multimodal tasks. Finally, we highlight the key challenges and future directions for KD in EGI. This survey aims to provide a comprehensive reference for researchers exploring KD-driven frameworks for mobile agentic AI in the era of EGI.
Edge General Intelligence (EGI) represents a paradigm shift in mobile edge computing, where intelligent agents operate autonomously in dynamic, resource-constrained environments. However, the deployment of advanced agentic AI models on mobile and edge devices faces significant challenges due to limited computation, energy, and storage resources. To address these constraints, this survey investigates the integration of Knowledge Distillation (KD) into EGI, positioning KD as a key enabler for efficient, communication-aware, and scalable intelligence at the wireless edge. In particular, we emphasize KD techniques specifically designed for wireless communication and mobile networking, such as channel-aware self-distillation, cross-model Channel State Information (CSI) feedback distillation, and robust modulation/classification distillation. Furthermore, we review novel architectures natively suited for KD and edge deployment, such as Mamba, RWKV (Receptance, Weight, Key, Value) and Cross-Architecture distillation, which enhance generalization capabilities. Subsequently, we examine diverse applications in which KD-driven architectures enable EGI across vision, speech, and multimodal tasks. Finally, we highlight the key challenges and future directions for KD in EGI. This survey aims to provide a comprehensive reference for researchers exploring KD-driven frameworks for mobile agentic AI in the era of EGI.
Why we think this paper is great for you:
This research focuses on deploying advanced AI on mobile and edge devices. Expanding the accessibility and reach of intelligent agents could have broad societal implications, potentially influencing resource distribution and access.
This research focuses on deploying advanced AI on mobile and edge devices. Expanding the accessibility and reach of intelligent agents could have broad societal implications, potentially influencing resource distribution and access.
Economic Inequality
Abstract
Equality saturation is a technique for program optimization based on non-destructive rewriting and a form of program analysis called e-class analysis. The current form of e-class analysis is pessimistic and therefore ineffective at analyzing cyclic programs, such as those in SSA form. We propose an abstract interpretation algorithm that can precisely analyze cycles during equality saturation. This results in a unified algorithm for optimistic analysis and non-destructive rewriting. We instantiate this approach on a prototype abstract interpreter for SSA programs using a new semantics of SSA. Our prototype can analyze simple example programs more precisely than clang and gcc.
Equality saturation is a technique for program optimization based on non-destructive rewriting and a form of program analysis called e-class analysis. The current form of e-class analysis is pessimistic and therefore ineffective at analyzing cyclic programs, such as those in SSA form. We propose an abstract interpretation algorithm that can precisely analyze cycles during equality saturation. This results in a unified algorithm for optimistic analysis and non-destructive rewriting. We instantiate this approach on a prototype abstract interpreter for SSA programs using a new semantics of SSA. Our prototype can analyze simple example programs more precisely than clang and gcc.
AI Summary
- Abstract interpretation and equality saturation can be combined using a fixpoint algorithm that alternates between phases of abstract interpretation and equality saturation. [2]
- Analysis results from abstract interpretation can help improve the precision of equality saturation by providing flow-insensitive abstractions that can be used in rewrite rules. [1]
AI and Society
Abstract
Algorithms have been estimated to increase AI training FLOP efficiency by a factor of 22,000 between 2012 and 2023 [Ho et al., 2024]. Running small-scale ablation experiments on key innovations from this time period, we are able to account for less than 10x of these gains. Surveying the broader literature, we estimate that additional innovations not included in our ablations account for less than 10x, yielding a total under 100x. This leads us to conduct scaling experiments, which reveal that much of this efficiency gap can be explained by algorithms with scale-dependent efficiency improvements. In particular, we conduct scaling experiments between LSTMs and Transformers, finding exponent differences in their compute-optimal scaling law while finding little scaling difference for many other innovations. These experiments demonstrate that - contrary to standard assumptions - an algorithm's efficiency gains are tied to compute scale. Using experimental extrapolation and literature estimates, we account for 6,930x efficiency gains over the same time period, with the scale-dependent LSTM-to-Transformer transition accounting for the majority of gains. Our results indicate that algorithmic progress for small models has been far slower than previously assumed, and that measures of algorithmic efficiency are strongly reference-dependent.
Algorithms have been estimated to increase AI training FLOP efficiency by a factor of 22,000 between 2012 and 2023 [Ho et al., 2024]. Running small-scale ablation experiments on key innovations from this time period, we are able to account for less than 10x of these gains. Surveying the broader literature, we estimate that additional innovations not included in our ablations account for less than 10x, yielding a total under 100x. This leads us to conduct scaling experiments, which reveal that much of this efficiency gap can be explained by algorithms with scale-dependent efficiency improvements. In particular, we conduct scaling experiments between LSTMs and Transformers, finding exponent differences in their compute-optimal scaling law while finding little scaling difference for many other innovations. These experiments demonstrate that - contrary to standard assumptions - an algorithm's efficiency gains are tied to compute scale. Using experimental extrapolation and literature estimates, we account for 6,930x efficiency gains over the same time period, with the scale-dependent LSTM-to-Transformer transition accounting for the majority of gains. Our results indicate that algorithmic progress for small models has been far slower than previously assumed, and that measures of algorithmic efficiency are strongly reference-dependent.
AI Summary
- Algorithmic progress in language models exhibits fundamentally different behavior across compute scales. [3]
- Algorithmic progress: The improvement in training efficiency and capabilities of language models over time. [3]
- Scale-dependent innovations: Innovations whose impact on efficiency gains varies depending on the compute scale. [3]
- Algorithmic progress is not a single number, but rather depends on both the reference algorithm and target compute scale. [3]
- Scale-dependent innovations are critical to understanding algorithmic progress and its implications for the future of AI. [3]
- The study's experiments are conducted at small scales compared to more recent scaling studies. [3]
- Scale-dependent innovations, such as the LSTM-to-Transformer transition and Chinchilla rebalancing, account for most of the efficiency gains at frontier scales. [2]
- The concentration of progress in architectural transitions suggests that future progress may depend on discovering fundamentally new architectures rather than incremental refinements of existing ones. [1]
Deep Learning
How to Use Deep Learning to Identify Sufficient Conditions: A Case Study on Stanley's $e$-Positivity
Abstract
In a study, published in \emph{Nature}, researchers from DeepMind and mathematicians demonstrated a general framework using machine learning to make conjectures in pure mathematics. Their work uses neural networks and attribution techniques to guide human intuition towards making provable conjectures. Here, we build upon this framework to develop a method for identifying sufficient conditions that imply a given mathematical statement. Our approach trains neural networks with a custom loss function that prioritizes high precision. Then uses attribution techniques and exploratory data analysis to make conjectures. As a demonstration, we apply this process to Stanley's problem of $e$-positivity of graphs--a problem that has been at the center of algebraic combinatorics for the past three decades. Guided by AI, we rediscover that one sufficient condition for a graph to be $e$-positive is that it is co-triangle-free, and that the number of claws is the most important factor for $e$-positivity. Based on the most important factors in Saliency Map analysis of neural networks, we suggest that the classification of $e$-positive graphs is more related to continuous graph invariants rather than the discrete ones. Furthermore, using neural networks and exploratory data analysis, we show that the claw-free and claw-contractible-free graphs with $10$ and $11$ vertices are $e$-positive, resolving a conjecture by Dahlberg, Foley, and van Willigenburg.
In a study, published in \emph{Nature}, researchers from DeepMind and mathematicians demonstrated a general framework using machine learning to make conjectures in pure mathematics. Their work uses neural networks and attribution techniques to guide human intuition towards making provable conjectures. Here, we build upon this framework to develop a method for identifying sufficient conditions that imply a given mathematical statement. Our approach trains neural networks with a custom loss function that prioritizes high precision. Then uses attribution techniques and exploratory data analysis to make conjectures. As a demonstration, we apply this process to Stanley's problem of $e$-positivity of graphs--a problem that has been at the center of algebraic combinatorics for the past three decades. Guided by AI, we rediscover that one sufficient condition for a graph to be $e$-positive is that it is co-triangle-free, and that the number of claws is the most important factor for $e$-positivity. Based on the most important factors in Saliency Map analysis of neural networks, we suggest that the classification of $e$-positive graphs is more related to continuous graph invariants rather than the discrete ones. Furthermore, using neural networks and exploratory data analysis, we show that the claw-free and claw-contractible-free graphs with $10$ and $11$ vertices are $e$-positive, resolving a conjecture by Dahlberg, Foley, and van Willigenburg.
AI Summary
- The authors used a precision-optimized model to identify the top four features that impact e-positivity in graphs. [3]
- The model achieved 100% precision on the test set, indicating high reliability for its positive predictions. [3]
- The study demonstrates how AI can be used to guide human intuition and advance mathematics by identifying underlying patterns associated with e-positivity. [3]
- The authors' approach can be applied to other areas of mathematics where pattern recognition is crucial. [3]
- E-positivity: a property of graphs that refers to the existence of certain combinatorial structures, such as cycles or paths. [3]
- Chromatic symmetric function: a polynomial invariant of graphs that encodes information about their coloring properties. [3]
- The study demonstrates the potential of AI in advancing mathematics by identifying underlying patterns associated with e-positivity. [3]
- The precision-optimized model achieved high reliability for its positive predictions, indicating that it can be trusted with high confidence when classifying graphs as e-positive. [3]
- The approach used in this study can be applied to other areas of mathematics where pattern recognition is crucial. [3]
- Saliency Map analysis: a technique used to identify the most important features or variables in a dataset by computing the average gradient of the model's output with respect to its input features. [2]
Abstract
This paper argues that DNNs implement a computational Occam's razor -- finding the `simplest' algorithm that fits the data -- and that this could explain their incredible and wide-ranging success over more traditional statistical methods. We start with the discovery that the set of real-valued function $f$ that can be $ฮต$-approximated with a binary circuit of size at most $cฮต^{-ฮณ}$ becomes convex in the `Harder than Monte Carlo' (HTMC) regime, when $ฮณ>2$, allowing for the definition of a HTMC norm on functions. In parallel one can define a complexity measure on the parameters of a ResNets (a weighted $\ell_1$ norm of the parameters), which induce a `ResNet norm' on functions. The HTMC and ResNet norms can then be related by an almost matching sandwich bound. Thus minimizing this ResNet norm is equivalent to finding a circuit that fits the data with an almost minimal number of nodes (within a power of 2 of being optimal). ResNets thus appear as an alternative model for computation of real functions, better adapted to the HTMC regime and its convexity.
This paper argues that DNNs implement a computational Occam's razor -- finding the `simplest' algorithm that fits the data -- and that this could explain their incredible and wide-ranging success over more traditional statistical methods. We start with the discovery that the set of real-valued function $f$ that can be $ฮต$-approximated with a binary circuit of size at most $cฮต^{-ฮณ}$ becomes convex in the `Harder than Monte Carlo' (HTMC) regime, when $ฮณ>2$, allowing for the definition of a HTMC norm on functions. In parallel one can define a complexity measure on the parameters of a ResNets (a weighted $\ell_1$ norm of the parameters), which induce a `ResNet norm' on functions. The HTMC and ResNet norms can then be related by an almost matching sandwich bound. Thus minimizing this ResNet norm is equivalent to finding a circuit that fits the data with an almost minimal number of nodes (within a power of 2 of being optimal). ResNets thus appear as an alternative model for computation of real functions, better adapted to the HTMC regime and its convexity.
AI Summary
- The HTMC norm is a measure of the complexity of a function, and it has several useful properties, including compositionality and convexity. [3]
- The construction of the ResNet involves two main parts: first, the input is mapped to the weighted binary representations of its surrounding vertices; second, a sorting algorithm is used to recover the simplex that contains the input. [3]
- The Lipschitz constant of this network is bounded by cp doutb|C|p=2/3. [3]
- The ResNet representation of Tetrakis functions has several useful properties, including compositionality and convexity. [3]
- HTMC norm: a measure of the complexity of a function. [3]
- Hยจ older continuous: a property of a function that implies it can be represented as a sum of Tetrakis functions. [3]
- Tetrakis function: a type of function that is both HTMC computable and Hยจ older continuous. [3]
- ResNet: a type of neural network that can represent functions that are both HTMC computable and Hยจ older continuous. [3]
- ResNets can be used to represent functions that are both HTMC computable and Hยจ older continuous. [2]
We did not find tons of content matching your interests we've included some additional topics that are popular. Also be aware that if the topics is not present in arxiv we wont be able to recommend it.
AI Agents
Abstract
AI/ML model cards can contain a benchmarked evaluation of an AI/ML model against intended use but a one time assessment during model training does not get at how and where a model is actually used over its lifetime. Through Patra Model Cards embedded in the ICICLE AI Institute software ecosystem we study model cards as dynamic objects. The study reported here assesses the benefits and tradeoffs of adopting the Model Context Protocol (MCP) as an interface to the Patra Model Card server. Quantitative assessment shows the overhead of MCP as compared to a REST interface. The core question however is of active sessions enabled by MCP; this is a qualitative question of fit and use in the context of dynamic model cards that we address as well.
AI/ML model cards can contain a benchmarked evaluation of an AI/ML model against intended use but a one time assessment during model training does not get at how and where a model is actually used over its lifetime. Through Patra Model Cards embedded in the ICICLE AI Institute software ecosystem we study model cards as dynamic objects. The study reported here assesses the benefits and tradeoffs of adopting the Model Context Protocol (MCP) as an interface to the Patra Model Card server. Quantitative assessment shows the overhead of MCP as compared to a REST interface. The core question however is of active sessions enabled by MCP; this is a qualitative question of fit and use in the context of dynamic model cards that we address as well.
AI Summary
- The article discusses the Model Context Protocol (MCP) and its performance evaluation in serving model cards, comparing it to REST protocol. [2]
- FAIR Signposting Profile: Implementation guidelines for exposing machine-actionable navigation links using standardized HTTP headers and HTML link elements. [1]
Abstract
Edge General Intelligence (EGI) represents a paradigm shift in mobile edge computing, where intelligent agents operate autonomously in dynamic, resource-constrained environments. However, the deployment of advanced agentic AI models on mobile and edge devices faces significant challenges due to limited computation, energy, and storage resources. To address these constraints, this survey investigates the integration of Knowledge Distillation (KD) into EGI, positioning KD as a key enabler for efficient, communication-aware, and scalable intelligence at the wireless edge. In particular, we emphasize KD techniques specifically designed for wireless communication and mobile networking, such as channel-aware self-distillation, cross-model Channel State Information (CSI) feedback distillation, and robust modulation/classification distillation. Furthermore, we review novel architectures natively suited for KD and edge deployment, such as Mamba, RWKV (Receptance, Weight, Key, Value) and Cross-Architecture distillation, which enhance generalization capabilities. Subsequently, we examine diverse applications in which KD-driven architectures enable EGI across vision, speech, and multimodal tasks. Finally, we highlight the key challenges and future directions for KD in EGI. This survey aims to provide a comprehensive reference for researchers exploring KD-driven frameworks for mobile agentic AI in the era of EGI.
Edge General Intelligence (EGI) represents a paradigm shift in mobile edge computing, where intelligent agents operate autonomously in dynamic, resource-constrained environments. However, the deployment of advanced agentic AI models on mobile and edge devices faces significant challenges due to limited computation, energy, and storage resources. To address these constraints, this survey investigates the integration of Knowledge Distillation (KD) into EGI, positioning KD as a key enabler for efficient, communication-aware, and scalable intelligence at the wireless edge. In particular, we emphasize KD techniques specifically designed for wireless communication and mobile networking, such as channel-aware self-distillation, cross-model Channel State Information (CSI) feedback distillation, and robust modulation/classification distillation. Furthermore, we review novel architectures natively suited for KD and edge deployment, such as Mamba, RWKV (Receptance, Weight, Key, Value) and Cross-Architecture distillation, which enhance generalization capabilities. Subsequently, we examine diverse applications in which KD-driven architectures enable EGI across vision, speech, and multimodal tasks. Finally, we highlight the key challenges and future directions for KD in EGI. This survey aims to provide a comprehensive reference for researchers exploring KD-driven frameworks for mobile agentic AI in the era of EGI.
AI and Society
Abstract
Algorithms have been estimated to increase AI training FLOP efficiency by a factor of 22,000 between 2012 and 2023 [Ho et al., 2024]. Running small-scale ablation experiments on key innovations from this time period, we are able to account for less than 10x of these gains. Surveying the broader literature, we estimate that additional innovations not included in our ablations account for less than 10x, yielding a total under 100x. This leads us to conduct scaling experiments, which reveal that much of this efficiency gap can be explained by algorithms with scale-dependent efficiency improvements. In particular, we conduct scaling experiments between LSTMs and Transformers, finding exponent differences in their compute-optimal scaling law while finding little scaling difference for many other innovations. These experiments demonstrate that - contrary to standard assumptions - an algorithm's efficiency gains are tied to compute scale. Using experimental extrapolation and literature estimates, we account for 6,930x efficiency gains over the same time period, with the scale-dependent LSTM-to-Transformer transition accounting for the majority of gains. Our results indicate that algorithmic progress for small models has been far slower than previously assumed, and that measures of algorithmic efficiency are strongly reference-dependent.
Algorithms have been estimated to increase AI training FLOP efficiency by a factor of 22,000 between 2012 and 2023 [Ho et al., 2024]. Running small-scale ablation experiments on key innovations from this time period, we are able to account for less than 10x of these gains. Surveying the broader literature, we estimate that additional innovations not included in our ablations account for less than 10x, yielding a total under 100x. This leads us to conduct scaling experiments, which reveal that much of this efficiency gap can be explained by algorithms with scale-dependent efficiency improvements. In particular, we conduct scaling experiments between LSTMs and Transformers, finding exponent differences in their compute-optimal scaling law while finding little scaling difference for many other innovations. These experiments demonstrate that - contrary to standard assumptions - an algorithm's efficiency gains are tied to compute scale. Using experimental extrapolation and literature estimates, we account for 6,930x efficiency gains over the same time period, with the scale-dependent LSTM-to-Transformer transition accounting for the majority of gains. Our results indicate that algorithmic progress for small models has been far slower than previously assumed, and that measures of algorithmic efficiency are strongly reference-dependent.
AI Summary
- Algorithmic progress in language models exhibits fundamentally different behavior across compute scales. [3]
- Algorithmic progress: The improvement in training efficiency and capabilities of language models over time. [3]
- Scale-dependent innovations: Innovations whose impact on efficiency gains varies depending on the compute scale. [3]
- Algorithmic progress is not a single number, but rather depends on both the reference algorithm and target compute scale. [3]
- Scale-dependent innovations are critical to understanding algorithmic progress and its implications for the future of AI. [3]
- The study's experiments are conducted at small scales compared to more recent scaling studies. [3]
- Scale-dependent innovations, such as the LSTM-to-Transformer transition and Chinchilla rebalancing, account for most of the efficiency gains at frontier scales. [2]
- The concentration of progress in architectural transitions suggests that future progress may depend on discovering fundamentally new architectures rather than incremental refinements of existing ones. [1]
Abstract
Advances in artificial intelligence now make it possible to simulate the dead through chatbots, voice clones, and video avatars trained on a person's digital traces. These "digital ghosts" are moving from fiction to commercial reality, reshaping how people mourn and remember. This paper offers a conceptual and ethical analysis of AI-mediated digital afterlives. We define what counts as a digital ghost, trace their rise across personal, commercial, and institutional contexts, and identify core ethical tensions around grief and well-being, truthfulness and deception, consent and posthumous privacy, dignity and misrepresentation, and the commercialization of mourning. To analyze these challenges, we propose a nine-dimensional taxonomy of digital afterlife technologies and, building on it, outline the features of an ethically acceptable digital ghost: premortem intent, mutual consent, transparent and limited data use, clear disclosure, restricted purposes and access, family or estate stewardship, and minimal behavioral agency. We argue for targeted regulation and professional guidelines to ensure that digital ghosts can aid remembrance without slipping into forms of deception.
Advances in artificial intelligence now make it possible to simulate the dead through chatbots, voice clones, and video avatars trained on a person's digital traces. These "digital ghosts" are moving from fiction to commercial reality, reshaping how people mourn and remember. This paper offers a conceptual and ethical analysis of AI-mediated digital afterlives. We define what counts as a digital ghost, trace their rise across personal, commercial, and institutional contexts, and identify core ethical tensions around grief and well-being, truthfulness and deception, consent and posthumous privacy, dignity and misrepresentation, and the commercialization of mourning. To analyze these challenges, we propose a nine-dimensional taxonomy of digital afterlife technologies and, building on it, outline the features of an ethically acceptable digital ghost: premortem intent, mutual consent, transparent and limited data use, clear disclosure, restricted purposes and access, family or estate stewardship, and minimal behavioral agency. We argue for targeted regulation and professional guidelines to ensure that digital ghosts can aid remembrance without slipping into forms of deception.
AI Summary
- The rise of digital ghosts and deadbots forces us to confront fundamental questions about how we remember our dead. [3]
- Digital ghosts can become the blind spot between memory and trickery, a prolonged mourning disguised as dialogue. [3]
- Deadbots: AI replicas that simulate the presence of deceased individuals. [3]
- Digital ghosts: AI-generated representations of deceased people that can interact with the living. [3]
- The era of AI 'afterlives' is here, and it falls upon us to ensure this technology is used in a way that supports memory without becoming an imposture, and helps heal without betraying the dignity of those we love and lose. [3]
- The article cites various studies and papers on the topic of digital ghosts and deadbots, including works by authors such as Jed R. [3]
- From an ethical standpoint, arguably intent and transparency make a substantial difference in creating or engaging with a simulacrum of a loved one. [2]
- Brubaker and John Danaher. [1]
Research Automation with AI
Abstract
Over the last few months, AI models including large language models have improved greatly. There are now several documented examples where they have helped professional mathematical scientists prove new results, sometimes even helping resolve known open problems. In this short note, we add another example to the list, by documenting how we were able to solve a previously unsolved research problem in robust mathematical statistics with crucial help from GPT-5. Our problem concerns robust density estimation, where the observations are perturbed by Wasserstein-bounded contaminations.In a previous preprint (Chao and Dobriban, 2023, arxiv:2308.01853v2), we have obtained upper and lower bounds on the minimax optimal estimation error; which were, however, not sharp. Starting in October 2025, making significant use of GPT-5 Pro, we were able to derive the minimax optimal error rate (reported in version 3 of the above arxiv preprint). GPT-5 provided crucial help along the way, including by suggesting calculations that we did not think of, and techniques that were not familiar to us, such as the dynamic Benamou-Brenier formulation, for key steps in the analysis. Working with GPT-5 took a few weeks of effort, and we estimate that it could have taken several months to get the same results otherwise. At the same time, there are still areas where working with GPT-5 was challenging: it sometimes provided incorrect references, and glossed over details that sometimes took days of work to fill in. We outline our workflow and steps taken to mitigate issues. Overall, our work can serve as additional documentation for a new age of human-AI collaborative work in mathematical science.
Over the last few months, AI models including large language models have improved greatly. There are now several documented examples where they have helped professional mathematical scientists prove new results, sometimes even helping resolve known open problems. In this short note, we add another example to the list, by documenting how we were able to solve a previously unsolved research problem in robust mathematical statistics with crucial help from GPT-5. Our problem concerns robust density estimation, where the observations are perturbed by Wasserstein-bounded contaminations.In a previous preprint (Chao and Dobriban, 2023, arxiv:2308.01853v2), we have obtained upper and lower bounds on the minimax optimal estimation error; which were, however, not sharp. Starting in October 2025, making significant use of GPT-5 Pro, we were able to derive the minimax optimal error rate (reported in version 3 of the above arxiv preprint). GPT-5 provided crucial help along the way, including by suggesting calculations that we did not think of, and techniques that were not familiar to us, such as the dynamic Benamou-Brenier formulation, for key steps in the analysis. Working with GPT-5 took a few weeks of effort, and we estimate that it could have taken several months to get the same results otherwise. At the same time, there are still areas where working with GPT-5 was challenging: it sometimes provided incorrect references, and glossed over details that sometimes took days of work to fill in. We outline our workflow and steps taken to mitigate issues. Overall, our work can serve as additional documentation for a new age of human-AI collaborative work in mathematical science.
AI Summary
- The authors highlight the potential benefits of using AI in research, including increased productivity and well-being for mathematicians. [3]
- They also note that AI can excel at routine but lengthy calculations, freeing up time for more creative work. [3]
- They also note that human-AI collaboration can lead to new insights and solutions. [3]
- LLM: Large Language Model AI: Artificial Intelligence The use of AI in research has the potential to increase productivity and well-being for mathematicians. [3]
- Human-AI collaboration can lead to new insights and solutions. [3]
- The paper discusses the use of a large language model (LLM) in solving a research problem in mathematical statistics. [2]
Abstract
Artificial intelligence and machine learning are reshaping how we approach scientific discovery, not by replacing established methods but by extending what researchers can probe, predict, and design. In this roadmap we provide a forward-looking view of AI-enabled science across biology, chemistry, climate science, mathematics, materials science, physics, self-driving laboratories and unconventional computing. Several shared themes emerge: the need for diverse and trustworthy data, transferable electronic-structure and interatomic models, AI systems integrated into end-to-end scientific workflows that connect simulations to experiments and generative systems grounded in synthesisability rather than purely idealised phases. Across domains, we highlight how large foundation models, active learning and self-driving laboratories can close loops between prediction and validation while maintaining reproducibility and physical interpretability. Taken together, these perspectives outline where AI-enabled science stands today, identify bottlenecks in data, methods and infrastructure, and chart concrete directions for building AI systems that are not only more powerful but also more transparent and capable of accelerating discovery in complex real-world environments.
Artificial intelligence and machine learning are reshaping how we approach scientific discovery, not by replacing established methods but by extending what researchers can probe, predict, and design. In this roadmap we provide a forward-looking view of AI-enabled science across biology, chemistry, climate science, mathematics, materials science, physics, self-driving laboratories and unconventional computing. Several shared themes emerge: the need for diverse and trustworthy data, transferable electronic-structure and interatomic models, AI systems integrated into end-to-end scientific workflows that connect simulations to experiments and generative systems grounded in synthesisability rather than purely idealised phases. Across domains, we highlight how large foundation models, active learning and self-driving laboratories can close loops between prediction and validation while maintaining reproducibility and physical interpretability. Taken together, these perspectives outline where AI-enabled science stands today, identify bottlenecks in data, methods and infrastructure, and chart concrete directions for building AI systems that are not only more powerful but also more transparent and capable of accelerating discovery in complex real-world environments.
AI Summary
- Explainable Artificial Intelligence (XAI) is rapidly evolving, moving beyond post-hoc methods towards a more comprehensive mechanistic understanding of AI model behavior and decision-making process. [3]
- Embedding causal reasoning into interpretability pipelines through counterfactuals or mechanistic priors provides the necessary structure to distinguish genuine drivers from spurious correlations. [3]
- Explainable Artificial Intelligence (XAI): AI that can explain its own decision-making process and provide insights into how it arrived at a particular conclusion. [3]
- Causality: The relationship between cause and effect, essential for understanding the underlying mechanisms of complex systems. [3]
- Interpretability: The ability to understand and interpret the results of an algorithm or model. [3]
- XAI has the potential to contribute to trustworthy and discovery-driven science by providing insights into AI decision-making processes. [3]
- Causality plays a decisive role in XAI, without causal grounding even stable and reproducible explanations risk being superficially plausible while scientifically misleading. [2]
- Interpretability results that are accurate but misaligned with domain concepts may fail to resonate with experts, rendering them ineffective in real scientific workflows. [1]
AGI: Artificial General Intelligence
Abstract
As artificial intelligence systems become increasingly integrated into human social contexts, Artificial Social Intelligence (ASI) has emerged as a critical capability that enables AI to perceive, understand, and engage meaningfully in complex human social interactions. This chapter introduces a comprehensive framework for Human-Centered Artificial Social Intelligence (HC-ASI), built upon the Technology-Human Factors-Ethics (THE) Triangle, which systematically addresses both technical foundations and human-centered design principles necessary for developing socially intelligent AI systems. This chapter provides a comprehensive overview of current ASI research. This chapter begins by establishing the theoretical foundations of ASI, tracing its evolution from classical psychological theories of human social intelligence to contemporary computational models, then examines the mechanisms underlying human-AI social interaction with particular emphasis on establishing shared social understanding and appropriate role positioning. The chapter further explores ASI's practical implications for individuals and groups through comprehensive evaluation frameworks that combine technical benchmarks with human-centered experiential assessments, demonstrating real-world applications through detailed case studies spanning healthcare, companionship, education, and customer service domains. Building on the overview and the framework of HC -ASI, this chapter articulates core HC-ASI design principles and translates them into actionable methodologies and implementation guidelines that provide practical guidance for researchers and practitioners. This chapter concludes with a critical discussion of current challenges and promising directions for developing comprehensive HC-ASI ecosystems.
As artificial intelligence systems become increasingly integrated into human social contexts, Artificial Social Intelligence (ASI) has emerged as a critical capability that enables AI to perceive, understand, and engage meaningfully in complex human social interactions. This chapter introduces a comprehensive framework for Human-Centered Artificial Social Intelligence (HC-ASI), built upon the Technology-Human Factors-Ethics (THE) Triangle, which systematically addresses both technical foundations and human-centered design principles necessary for developing socially intelligent AI systems. This chapter provides a comprehensive overview of current ASI research. This chapter begins by establishing the theoretical foundations of ASI, tracing its evolution from classical psychological theories of human social intelligence to contemporary computational models, then examines the mechanisms underlying human-AI social interaction with particular emphasis on establishing shared social understanding and appropriate role positioning. The chapter further explores ASI's practical implications for individuals and groups through comprehensive evaluation frameworks that combine technical benchmarks with human-centered experiential assessments, demonstrating real-world applications through detailed case studies spanning healthcare, companionship, education, and customer service domains. Building on the overview and the framework of HC -ASI, this chapter articulates core HC-ASI design principles and translates them into actionable methodologies and implementation guidelines that provide practical guidance for researchers and practitioners. This chapter concludes with a critical discussion of current challenges and promising directions for developing comprehensive HC-ASI ecosystems.
AI Summary
- Social AI is a type of artificial intelligence that can interact with humans in a way that simulates human-like conversation and behavior. [3]
- The development of Social AI requires the integration of multiple disciplines, including computer science, psychology, sociology, and philosophy. [3]
- Social AI has the potential to revolutionize various industries, such as healthcare, education, and customer service, by providing personalized support and improving user experience. [3]
- Social AI: A type of artificial intelligence that can interact with humans in a way that simulates human-like conversation and behavior. [3]
- Human-AI interaction: The process by which humans interact with artificial intelligence systems, such as chatbots or virtual assistants. [3]
- The development of Social AI has the potential to transform various industries and improve user experience. [3]
- However, the development of Social AI also raises several challenges and concerns, including ensuring transparency, accountability, and ethics in AI decision-making. [2]
Deep Learning
How to Use Deep Learning to Identify Sufficient Conditions: A Case Study on Stanley's $e$-Positivity
Abstract
In a study, published in \emph{Nature}, researchers from DeepMind and mathematicians demonstrated a general framework using machine learning to make conjectures in pure mathematics. Their work uses neural networks and attribution techniques to guide human intuition towards making provable conjectures. Here, we build upon this framework to develop a method for identifying sufficient conditions that imply a given mathematical statement. Our approach trains neural networks with a custom loss function that prioritizes high precision. Then uses attribution techniques and exploratory data analysis to make conjectures. As a demonstration, we apply this process to Stanley's problem of $e$-positivity of graphs--a problem that has been at the center of algebraic combinatorics for the past three decades. Guided by AI, we rediscover that one sufficient condition for a graph to be $e$-positive is that it is co-triangle-free, and that the number of claws is the most important factor for $e$-positivity. Based on the most important factors in Saliency Map analysis of neural networks, we suggest that the classification of $e$-positive graphs is more related to continuous graph invariants rather than the discrete ones. Furthermore, using neural networks and exploratory data analysis, we show that the claw-free and claw-contractible-free graphs with $10$ and $11$ vertices are $e$-positive, resolving a conjecture by Dahlberg, Foley, and van Willigenburg.
In a study, published in \emph{Nature}, researchers from DeepMind and mathematicians demonstrated a general framework using machine learning to make conjectures in pure mathematics. Their work uses neural networks and attribution techniques to guide human intuition towards making provable conjectures. Here, we build upon this framework to develop a method for identifying sufficient conditions that imply a given mathematical statement. Our approach trains neural networks with a custom loss function that prioritizes high precision. Then uses attribution techniques and exploratory data analysis to make conjectures. As a demonstration, we apply this process to Stanley's problem of $e$-positivity of graphs--a problem that has been at the center of algebraic combinatorics for the past three decades. Guided by AI, we rediscover that one sufficient condition for a graph to be $e$-positive is that it is co-triangle-free, and that the number of claws is the most important factor for $e$-positivity. Based on the most important factors in Saliency Map analysis of neural networks, we suggest that the classification of $e$-positive graphs is more related to continuous graph invariants rather than the discrete ones. Furthermore, using neural networks and exploratory data analysis, we show that the claw-free and claw-contractible-free graphs with $10$ and $11$ vertices are $e$-positive, resolving a conjecture by Dahlberg, Foley, and van Willigenburg.
AI Summary
- The authors used a precision-optimized model to identify the top four features that impact e-positivity in graphs. [3]
- The model achieved 100% precision on the test set, indicating high reliability for its positive predictions. [3]
- The study demonstrates how AI can be used to guide human intuition and advance mathematics by identifying underlying patterns associated with e-positivity. [3]
- The authors' approach can be applied to other areas of mathematics where pattern recognition is crucial. [3]
- E-positivity: a property of graphs that refers to the existence of certain combinatorial structures, such as cycles or paths. [3]
- Chromatic symmetric function: a polynomial invariant of graphs that encodes information about their coloring properties. [3]
- The study demonstrates the potential of AI in advancing mathematics by identifying underlying patterns associated with e-positivity. [3]
- The precision-optimized model achieved high reliability for its positive predictions, indicating that it can be trusted with high confidence when classifying graphs as e-positive. [3]
- The approach used in this study can be applied to other areas of mathematics where pattern recognition is crucial. [3]
- Saliency Map analysis: a technique used to identify the most important features or variables in a dataset by computing the average gradient of the model's output with respect to its input features. [2]
Abstract
This paper argues that DNNs implement a computational Occam's razor -- finding the `simplest' algorithm that fits the data -- and that this could explain their incredible and wide-ranging success over more traditional statistical methods. We start with the discovery that the set of real-valued function $f$ that can be $ฮต$-approximated with a binary circuit of size at most $cฮต^{-ฮณ}$ becomes convex in the `Harder than Monte Carlo' (HTMC) regime, when $ฮณ>2$, allowing for the definition of a HTMC norm on functions. In parallel one can define a complexity measure on the parameters of a ResNets (a weighted $\ell_1$ norm of the parameters), which induce a `ResNet norm' on functions. The HTMC and ResNet norms can then be related by an almost matching sandwich bound. Thus minimizing this ResNet norm is equivalent to finding a circuit that fits the data with an almost minimal number of nodes (within a power of 2 of being optimal). ResNets thus appear as an alternative model for computation of real functions, better adapted to the HTMC regime and its convexity.
This paper argues that DNNs implement a computational Occam's razor -- finding the `simplest' algorithm that fits the data -- and that this could explain their incredible and wide-ranging success over more traditional statistical methods. We start with the discovery that the set of real-valued function $f$ that can be $ฮต$-approximated with a binary circuit of size at most $cฮต^{-ฮณ}$ becomes convex in the `Harder than Monte Carlo' (HTMC) regime, when $ฮณ>2$, allowing for the definition of a HTMC norm on functions. In parallel one can define a complexity measure on the parameters of a ResNets (a weighted $\ell_1$ norm of the parameters), which induce a `ResNet norm' on functions. The HTMC and ResNet norms can then be related by an almost matching sandwich bound. Thus minimizing this ResNet norm is equivalent to finding a circuit that fits the data with an almost minimal number of nodes (within a power of 2 of being optimal). ResNets thus appear as an alternative model for computation of real functions, better adapted to the HTMC regime and its convexity.
AI Summary
- The HTMC norm is a measure of the complexity of a function, and it has several useful properties, including compositionality and convexity. [3]
- The construction of the ResNet involves two main parts: first, the input is mapped to the weighted binary representations of its surrounding vertices; second, a sorting algorithm is used to recover the simplex that contains the input. [3]
- The Lipschitz constant of this network is bounded by cp doutb|C|p=2/3. [3]
- The ResNet representation of Tetrakis functions has several useful properties, including compositionality and convexity. [3]
- HTMC norm: a measure of the complexity of a function. [3]
- Hยจ older continuous: a property of a function that implies it can be represented as a sum of Tetrakis functions. [3]
- Tetrakis function: a type of function that is both HTMC computable and Hยจ older continuous. [3]
- ResNet: a type of neural network that can represent functions that are both HTMC computable and Hยจ older continuous. [3]
- ResNets can be used to represent functions that are both HTMC computable and Hยจ older continuous. [2]
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