Paper argues for using functional theory of randomness for better understanding of data exchangeability and conformal prediction.
problem Understanding relationships between IID data assumptions and data exchangeability.
method Translation of conformal prediction results into the language of functional theory of randomness.
result Every confidence predictor valid for IID data can be transformed to a conformal predictor without losing much predictive efficiency.
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.
We propose a general formalism of iterated random functions with semigroup property, under which exact and approximate Bayesian posterior updates can be viewed as specific instances. A convergence theory for iterated random functions is presented. As an application of the general theory we analyze convergence behaviors…
Random surfaces with long systoles created from graph theory ideas.
problem Finding surfaces with long systoles.
method Two constructions inspired by graph theory.
result Proved a new lower bound on systole length.
Random Function Descent improves optimization in high dimensions.
problem Lack of effective optimization methods in high-dimensional spaces.
method Introducing a 'random function' framework to optimize classical optimization problems.
result Random Function Descent (RFD) is a scalable optimization method that bridges Bayesian and classical optimization.
RFRBoost uses random features to boost deep residual neural networks, improving performance and computational efficiency.
problem Improving performance of deep residual neural networks (RFNNs) while preserving convex optimization benefits.
method Random Feature Representation Boosting (RFRBoost) using boosting theory and random features at each layer.
result RFRBoost significantly outperforms RFNNs and end-to-end trained MLP ResNets in small- to medium-scale tabular datasets.
This paper analyzes how randomizing rewards in MBRL can improve performance without being overly optimistic.
problem The gap between theoretical worst-case regret analysis and empirical performance in MBRL.
method Reward randomization in model-based reinforcement learning (MBRL) with kernelized linear regulator (KNR) model.
result Reward randomization guarantees partial optimism and near-optimal worst-case regret.
RVFL networks can efficiently approximate Lipschitz functions in L∞ norm.
problem Efficiently approximating Lipschitz continuous functions in L∞ norm.
method Random Vector Functional Link (RVFL) network with ReLU activation functions, proving approximation in L∞ norm.
result An RVFL with ReLU activation functions can approximate Lipschitz continuous functions in L∞ norm.
The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important difference is that the population covariance matrices, which are assumed to be non-random…
Study on random matrices in deep neural networks with IID entries.
problem Distribution of singular values in product of random matrices for deep neural networks.
method Random matrix theory with a streamlined approach for non-Gaussian data.
result Generalization of macroscopic universality property to non-Gaussian data.
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave functions on the unit simplex measure the concentration of capital, and their gr…
We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the averaged system disappears, due to the cooperation of the generators. We investi…
For covering spaces and properly discontinuous actions with compatible diffusion processes, we discuss Lyons-Sullivan discretizations of the processes and the associated function theory.
Mean field theory has been successfully used to analyze deep neural networks (DNN) in the infinite size limit. Given the finite size of realistic DNN, we utilize the large deviation theory and path integral analysis to study the deviation of functions represented by DNN from their typical mean field solutions. The para…
We consider a problem of optimal investment with intermediate consumption and random endowment in an incomplete semimartingale model of a financial market. We establish the key assertions of the utility maximization theory assuming that both primal and dual value functions are finite in the interiors of their domains a…
RST improves environmental time series classification accuracy using randomized B-spline trees.
problem Improving accuracy in classifying complex environmental time series.
method Randomized Spline Trees (RST) integrates randomized functional representations into ensemble learning.
result RST variants outperform standard Random Forests and Gradient Boosting on most environmental time series datasets.
We construct a cut-off version of nonpertubative closed Bosonic string field theory in the light-cone gauge with imaginary string coupling constant. We show that the partition function is a continuous function of the string coupling constant, and conjecture a relation between the formal power series expansion of this p…
The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measu…
Proposes a method to estimate functional graphical models from multivariate random functions.
problem Estimating conditional independence structure of multivariate random functions.
method Neighborhood selection approach combining function-on-function regression and graph recovery.
result Statistical consistency of the method in high-dimensional settings.
Random utility theory models an agent's preferences on alternatives by drawing a real-valued score on each alternative (typically independently) from a parameterized distribution, and then ranking the alternatives according to scores. A special case that has received significant attention is the Plackett-Luce model, fo…
Machine learning model predicts DFT total energy to complete basis set limit.
problem Finding a model to extrapolate DFT calculations to complete basis set limit.
method Quantile-random-forest model trained on binary solids data.
result Random-forest model achieves <25% symmetric MAPE for both DFT codes.
Study geodesics on random hyperbolic surfaces, finding variance similar to prime number theory.
problem Distribution of closed geodesics on random hyperbolic surfaces.
method Investigate random variable counting geodesics with norms in short intervals, comparing to prime number theory.
result Establishes variance of geodesic counting function is asymptotic to \(2H \log X\).
Introduces epistemic deep learning for better uncertainty estimation in neural networks.
problem Uncertainty quantification in deep neural networks.
method Random-set convolutional neural networks with belief function-based loss functions.
result Epistemic approach produces better performance in uncertainty estimation.
We propose semi-random features for nonlinear function approximation. The flexibility of semi-random feature lies between the fully adjustable units in deep learning and the random features used in kernel methods. For one hidden layer models with semi-random features, we prove with no unrealistic assumptions that the m…
Study loop corrections in random feature models affecting training and test errors.
problem Analyzing loop corrections in random feature models to understand training and test errors.
method Statistical physics and effective field theory approach to study loop corrections.
result Derived loop corrections to training error, test error, and generalization gap.
New theory approximates functions between metric spaces using random probability measures.
problem Building universal functions approximators between arbitrary metric spaces.
method Using elementary functions between Euclidean spaces, randomization to output discrete probability measures over target space.
result Very general qualitative guarantees and quantitative guarantees for Hölder-like maps.
1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our measure will depend on the topology of the space, but also on how nicely the spa…
This work's purpose is to understand the dynamics of some social systems whose properties can be captured by certain iterated function systems. To achieve this intension, we start from the theory of iterated function systems, and then we study two specific economic models on random utility function and optimal stochast…
Develops RF-GLS for binary geospatial data.
problem Challenges in extending RF to binary geospatial data.
method Proposes RF-GLS for binary data, embedding it in generalized mixed effects models.
result Establishes consistency of RF-GP for mean function and covariate effect estimation.
Random Matrix Theory explains loss surface Hessians in neural networks.
problem Understanding the loss surfaces of neural networks.
method Investigation of local spectral statistics of neural network Hessians.
result Excellent agreement with Gaussian Orthogonal Ensemble statistics.
We investigate the random dynamics of polynomial maps on the Riemann sphere and the dynamics of semigroups of polynomial maps on the Riemann sphere. In particular, the dynamics of a semigroup G of polynomials whose planar postcritical set is bounded and the associated random dynamics are studied. In general, the Juli…
Recently, based on the idea of randomizing space theory, random convex analysis has been being developed in order to deal with the corresponding problems in random environments such as analysis of conditional convex risk measures and the related variational problems and optimization problems. Random convex analysis is …
New concentration inequalities for tensors with heavy-tailed coefficients.
problem Developing bounds for Euclidean functions of tensors with sub-Weibull distributions.
method Extending concentration inequalities to sub-Weibull random tensors, using new inequalities for heavy-tailed random variables and martingale analysis.
result Established a phase transition between sub-gaussian and heavy-tailed regimes for Euclidean functions of tensors.
Random matrix theory explains how neural networks adapt to data.
problem Understanding how neural networks learn and generalize from data.
method Random matrix analysis of two-layer neural networks.
result Sharp characterization of feature spectrum and generalization error.
The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.
problem Analyzing Riemannian Gaussian distributions on symmetric spaces.
method Analytical computation of marginals using orthogonal and skew orthogonal polynomials, and diffusion kernels.
result Riemannian Gaussian distributions are random matrix types, and their probability density functions can be computed analytically.
Multivariate splines linked to infinitely-wide neural networks with improved numerical performance.
problem Understanding the relationship between multivariate splines and neural networks.
method Showed multivariate splines can be represented as random features in infinitely-wide neural networks with a homogeneous activation function.
result The function space of multivariate splines is a Sobolev space on a Euclidean ball with explicit norm bounds on derivatives.
New method calculates Shapley values for uncertain functions.
problem Uncertain value functions in explainable machine learning.
method Definition of Shapley values using probability theory.
result Shapley values can be applied to uncertain functions.
This work uses sampling theory to analyze smoothness and error bounds of finite neural networks.
problem Analyzing the function space of finite neural networks and providing error bounds.
method Applying sampling theory to finite neural networks with non-expansive activation functions, considering both deterministic and random sampling.
result Novel error bounds for univariate neural networks under band-limited input assumption, highlighting the advantage of deterministic uniform sampling.
Paper analyzes error bounds for learning with vector-valued RF, improving existing analyses.
problem Learning with vector-valued random features in infinite-dimensional settings.
method Direct analysis of risk functional, avoiding random matrix theory.
result Strong consistency and minimax optimal convergence rates established.
Global approximation for piecewise linear paths via signatures.
problem Global approximation theorems for piecewise linear paths.
method Using signatures of piecewise linear paths and their density in Lp-norms. result Linear functionals of signatures are dense in Lp-norms under an integrability condition. Randomized SINDy learns dynamic data structures using probabilistic methods.
problem Learning time-dependent data structures in dynamic systems.
method Sequential machine learning with a probabilistic approach, incorporating feature augmentation and Tikhonov regularization.
result Demonstrated effectiveness in regression and binary classification using real-world data.
This paper optimizes sampling for least-squares approximation.
problem Optimizing sampling for least-squares approximation in arbitrary linear spaces.
method Introducing the Christoffel function to construct near-optimal random sampling strategies.
result The number of samples scales log-linearly in the dimension of the approximation space.
Kernel methods are an extremely popular set of techniques used for many important machine learning and data analysis applications. In addition to having good practical performances, these methods are supported by a well-developed theory. Kernel methods use an implicit mapping of the input data into a high dimensional f…
Analyzes bias-variance in overparameterized linear models using random features.
problem Understanding bias-variance trade-off in overparameterized models.
method Zero-temperature cavity method and random matrix theory.
result Three phase transitions in the linear random features model.
Paper develops a method for estimating PFLM with minimized rates in high dimensions.
problem Estimating PFLM with minimized rates in high dimensions.
method Least square approach with mixed regularizations of function-norm and ℓ1-norm.
result Established optimal minimax rates of estimation for PFLM.
Randomly biased data makes complex models as easy to learn as simple ones.
problem Learning complex models like multi-index and sparse Boolean functions.
method Introducing a small random shift in the first moment of the data distribution.
result Randomly biased data makes Gaussian single index models and sparse Boolean functions as easy to learn as linear functions.
We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed s…
Study on geodesics on random hyperbolic surfaces, showing variance asymptotic to X log X.
problem Distribution of closed geodesics on random hyperbolic surfaces.
method Viewing surfaces as random points in moduli space, studying weighted counting function.
result Variance in large genus limit is asymptotic to X log X, with exceptions.