This work uses statistical mechanics to explain AI learning.
problem Understanding the statistical principles behind AI learning.
method Starting from sample concentration behaviors, the study applies statistical mechanics principles to AI and machine learning.
result Exponential families and statistical quantities are key in AI and machine learning.
This paper introduces a novel measure-theoretic theory for machine learning that does not require statistical assumptions. Based on this theory, a new regularization method in deep learning is derived and shown to outperform previous methods in CIFAR-10, CIFAR-100, and SVHN. Moreover, the proposed theory provides a the…
Statistical learning theory provides bounds of the generalization gap, using in particular the Vapnik-Chervonenkis dimension and the Rademacher complexity. An alternative approach, mainly studied in the statistical physics literature, is the study of generalization in simple synthetic-data models. Here we discuss the c…
Survey on statistical learning theory for control, focusing on linear systems.
problem Applying machine learning techniques to control systems, especially linear ones.
method Adapting tools from modern high-dimensional statistics and learning theory.
result Recent advances in statistical learning theory for control, particularly for linear systems.
Extreme value theory enhances statistical learning extrapolation for rare events.
problem Challenges in traditional machine learning methods for extreme data.
method Asymptotic theory and statistical tools for tail behavior.
result Effective extrapolation methods for extreme quantiles and anomalies.
Paper discusses the Fisher metric and differentiability in statistical models.
problem Understanding the relationship between Fisher metric and differentiability in statistical models.
method Comparison of different concepts and models in Information Geometry, mathematical statistics, and measure theory.
result Discussion of various models and their differentiability properties.
Statistical field theory aids in understanding deep learning complexities.
problem Complexity and lack of theoretical understanding in deep learning.
method Statistical field theory as a theoretical framework.
result Field theory provides insights into generalization, bias, and feature learning.
Paper reviews algebraic research in machine learning theory.
problem Understanding phase transitions in machine learning models.
method Algebraic approaches in statistical mechanics.
result Algebraic methods are essential for analyzing machine learning models with singularities.
New algebraic geometry and statistical manifold connections proven.
problem Understanding the structure of statistical manifolds and their algebraic properties.
method Developed relations between algebraic geometry, information theory, and Topological Field Theory.
result Statistical pre-Frobenius manifolds form algebraic varieties and have hexagonal, isoclinic webs.
Develops theory of homogeneous statistical manifolds and classifies Lie groups.
problem Understanding statistical manifolds and Lie groups.
method Constructs examples and classifies Lie groups using information geometry.
result Explicit examples of homogeneous statistical manifolds of low dimension constructed.
Statistical learning theory provides the theoretical basis for many of today's machine learning algorithms. In this article we attempt to give a gentle, non-technical overview over the key ideas and insights of statistical learning theory. We target at a broad audience, not necessarily machine learning researchers. Thi…
Novel mutual information bound improves statistical inference rates.
problem Improving statistical inference rates in Bayesian nonparametrics.
method Introduces a novel mutual information bound.
result Improved contraction rates for fractional posteriors.
Sharp statistical theory for conditional diffusion models.
problem Lack of theoretical foundation for conditional diffusion models.
method Sharp statistical theory with approximation of conditional score function.
result Sample complexity bound that adapts to data distribution smoothness.
Develops a new asymptotic efficiency theory for non-Euclidean parameter spaces.
problem Lack of a unified efficiency theory for non-Euclidean parameter spaces.
method Introduces a new theory for Riemannian manifolds with regularity conditions.
result Establishes efficiency bounds for non-Euclidean parameter spaces.
Paper uses stats to predict treatment choice based on illness probability.
problem Improving treatment decision-making in personalized medicine.
method Statistical decision theory with maximum regret evaluation.
result Estimates illness probability for better treatment choice.
This work provides statistical guarantees for VAEs using PAC-Bayesian theory.
problem Theoretical properties of VAEs remain open questions.
method PAC-Bayesian theory to derive statistical guarantees.
result Upper bounds on Wasserstein distance between input and generative model.
The paper outlines future work in random sets theory.
problem Developing a theory of statistical reasoning with random sets.
method Generalizing logistic regression, probability laws, and geometric uncertainty.
result A new geometric approach to uncertainty with general random sets.
Develops a statistical framework for coherent risk estimation.
problem Constructing coherent risk estimators with sound financial and statistical properties.
method Inspired by axiomatic risk measure theory, defines coherent risk estimators through robust representations linked to L-estimators. result Demonstrates that coherence of a risk measure does not necessarily carry over to its estimators and shows alternative weight structures can lead to different outcomes.
The study provides statistical theory for WGANs in time series forecasting.
problem Statistical analysis of WGANs for time series forecasting.
method Statistical theory and upper bounds for excess Bayes risk, weak convergence, and confidence intervals.
result Developed confidence intervals for time series forecasting using WGANs.
Paper uses SLT to improve model selection for SHM.
problem Model selection for SHM using data-based systems.
method Utilizes Statistical Learning Theory to rigorously estimate generalisation.
result Incorporating domain knowledge improves model generalisation.
Global EQG sums boundary states over manifold diffeomorphism classes.
problem Summing boundary states over manifold diffeomorphism classes.
method Formulated as classical statistical physics, weights determined by general principles.
result Hartle-Hawking state as a probability measure.
Simplified neural network EFTs reveal a single critical condition.
problem Understanding neuron statistics in neural networks at initialization.
method Diagrammatic approach to effective field theories (EFTs).
result A single condition governs criticality of all neuron preactivations.
The main aim of this paper is to extend Bochner's technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical structures. It deals, in particular, with Hodge's theory, Bochner-Weitzenbock and Simon's type formulas. Moreover, a few global and local theorem…
The study examines how quantum resources enhance the complexity of quantum circuits.
problem Quantum resource enhancement on circuit complexity.
method Utilizing quantum resource theories, the study analyzes statistical complexities of quantum circuits with limited quantum resources.
result Bounds for statistical complexities of quantum circuits are derived and applied to specific cases.
New theory challenges traditional machine learning assumptions.
problem Traditional machine learning theories are critiqued.
method A new theory is proposed and discussed.
result Learning true probabilities is not equivalent to other learning goals.
New learnability criteria for non-iid processes equivalent to online learning.
problem Statistical learning under non-iid stochastic processes is underdeveloped.
method Defined two learnability notions and showed their equivalence to online learning.
result Learnability criteria for non-iid processes are equivalent to online learning.
High-dimensional statistics advances in complex data domains.
problem Complex, rich datasets challenge traditional methods.
method Evolved to address sophisticated estimation and inference problems.
result Deepened connections with optimization, concentration, and information theory.
Study classifies submanifolds in probability simplex.
problem Classifying submanifolds in the probability simplex.
method Complete classification through geometric analysis.
result Doubly totally-umbilical submanifolds identified and classified.
Counting the number of clusters, when these clusters overlap significantly is a challenging problem in machine learning. We argue that a purely mathematical quantum theory, formulated using the path integral technique, when applied to non-physics modeling leads to non-physics quantum theories that are statistical in na…
The main goal of statistical learning theory is to provide a fundamental framework for the problem of decision making and model construction based on sets of data. Here, we present a brief introduction to the fundamentals of statistical learning theory, in particular the difference between empirical and structural risk…
Investigates financial and economic systems using statistical mechanics and information theory.
problem Complexity, asymmetry, stochasticity, and non-linearity in financial and economic systems.
method Model-based and empirical analyses using statistical mechanics and information theory.
result Derives probability distribution functions for better understanding of financial and economic dynamics.
Lectures on deep learning from a learning theory perspective.
problem Understanding how deep learning architectures lead to inductive bias.
method Statistical learning theory and stochastic optimization.
result Gradient descent on linear diagonal networks can lead to various forms of implicit bias.
A theory of deep learning is emerging, focusing on training dynamics and statistics.
problem Develop a scientific theory to understand deep learning.
method Synthesize research into five areas: idealized settings, tractable limits, mathematical laws, hyperparameters, and universal behaviors.
result The emerging theory is a mechanics of the learning process, named learning mechanics.
Statistical test rejects market efficiency using entropy from price returns.
problem Determining market efficiency using information theory.
method Symbolic representation of price returns, Shannon entropy, and statistical test.
result Rejects market efficiency hypothesis for various datasets.
We analyze the relationships between game theory and quantum mechanics and the extensions to statistical physics and information theory. We use certain quantization relationships to assign quantum states to the strategies of a player. These quantum states are contained in a density operator which describes the new quan…
Develops a dynamic mean field theory for reinforcement learning.
problem Finite state and action Bayesian reinforcement learning in large state spaces.
method Analogies with statistical physics, interpreting probabilities as couplings and values as spins, solving mean field equations.
result State-action values are statistically independent in the asymptotic state space limit, with exact or approximate equations for computation.
We study lightlike submanifolds of indefinite statistical manifolds. Contrary to the classical theory of submanifolds of statistical manifolds, lightlike submanifolds of indefinite statistical manifolds need not to be statistical submanifold. Therefore we obtain some conditions for a lightlike submanifold of indefinite…
Modern statistical inference tasks often require iterative optimization methods to compute the solution. Convergence analysis from an optimization viewpoint only informs us how well the solution is approximated numerically but overlooks the sampling nature of the data. In contrast, recognizing the randomness in the dat…
Overview of high-dimensional time series regression methods.
problem Estimation and inference with high-dimensional time series data.
method Limit theory for high-dimensional dependent data, asymptotic theory for time series regression, statistical learning methods.
result Main limit theory results and asymptotic theory for high-dimensional time series regression.
These notes gather recent results on robust statistical learning theory. The goal is to stress the main principles underlying the construction and theoretical analysis of these estimators rather than provide an exhaustive account on this rapidly growing field. The notes are the basis of lectures given at the conference…
We apply the geometric-topology surgery theory on spacetime manifolds to study the constraints of quantum statistics data in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable creating anyon excitations of particles and strings, well-defined in gapped st…
New statistical theory explains contrastive learning effectiveness.
problem Understanding why contrastive learning works well for representation extraction.
method Developed a new theoretical framework based on approximate sufficient statistics.
result Near-sufficient encoders derived from contrastive learning can be adapted for downstream tasks.
In this paper, we explore various statistical techniques for anomaly detection in conjunction with the popular Long Short-Term Memory (LSTM) deep learning model for transportation networks. We obtain the prediction errors from an LSTM model, and then apply three statistical models based on (i) the Gaussian distribution…
Non-linear image reconstruction and signal analysis deal with complex inverse problems. To tackle such problems in a systematic way, I present information field theory (IFT) as a means of Bayesian, data based inference on spatially distributed signal fields. IFT is a statistical field theory, which permits the construc…
The paper explores how information theory aids in statistical learning models.
problem Characterizing fundamental performance limits in statistical learning models.
method Introduces divergence measures and evidence lower bound (ELBO) in model training.
result Provides a systematic derivation for generative diffusion models.
Renormalization in neural networks linked to quantum field theory.
problem Implementing renormalization in neural networks.
method Mapping neural networks to quantum field theory, applying renormalization techniques.
result Changing weight standard deviation corresponds to a renormalization flow.
Learning in restricted Boltzmann machine is typically hard due to the computation of gradients of log-likelihood function. To describe the network state statistics of the restricted Boltzmann machine, we develop an advanced mean field theory based on the Bethe approximation. Our theory provides an efficient message pas…
The study reveals how attention paths in Transformers influence learning outcomes.
problem Understanding the theoretical basis of Transformers' performance.
method Developed a statistical mechanics theory for a simplified attention network.
result The predictor statistics are influenced by the combination of attention paths.