A universal learner achieves best rates for all distributions.
problem Improving learning algorithm rates under various settings.
method Simple extension of Levin's universal search.
result Achieves best-possible rates for all distributions.
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48 results for “distributional universality”
MPE framework proves universal approximation for quantum data distribution.
problem Challenges in generating quantum data from underlying distributions.
method Many-body Projected Ensemble (MPE) framework for quantum state design.
result MPE can approximate any quantum distribution within 1-Wasserstein distance error.
Exact distribution of split conformal prediction coverage found.
problem Determining the reliability of prediction sets in batch mode.
method Analysis of exchangeable data to find universal distribution of empirical coverage.
result Exact distribution of empirical coverage is universal and determined by nominal miscoverage level and calibration sample size.
Universal tester-learner for halfspaces over structured distributions.
problem Learning halfspaces over a wide class of structured distributions.
method Uses a fully polynomial tester-learner based on hypercontractivity and sum-of-squares (SOS) programs.
result Achieves error on any labeled distribution that the tester accepts.
New method for reducing dimensions of distributional data.
problem Nonlinear sufficient dimension reduction for distribution-on-distribution regression.
method Building universal kernels on metric spaces to characterize conditional independence.
result Method outperforms competing methods in synthetic and real data applications.
A note on the distribution of Neumann eigenvalues of the Laplacian on a Euclidean convex domainmath.SP
Universal inequalities for Laplacian eigenvalues on convex domains.
problem Eigenvalue distribution of the Laplacian on convex domains.
method Established two universal inequalities.
result Two new inequalities for Laplacian eigenvalues.
Stochastic gradient descent converges to universal limits in high dimensions.
problem Statistical tasks in high dimensions with specific data projections.
method Stochastic gradient descent applied to mixture distributions, proving universality of limits.
result The ODE limits are universal for mixtures of arbitrary product distributions.
The Breakdown of Gaussian Universality in Classification of High-dimensional Linear Factor Mixturesstat.ML
New findings show Gaussian universality breaks down in high-dimensional linear factor mixtures.
problem The limitations of Gaussian universality in high-dimensional classification.
method Characterization of empirical risk minimization for classification under linear factor mixture models.
result Gaussian universality breaks down under high-dimensional linear factor mixtures.
This paper studies universal rates of ERM for binary classification under agnostic learning.
problem The challenge of achieving universal rates of ERM for binary classification under agnostic learning.
method The paper explores the agnostic universal rates of ERM for binary classification, revealing three possible rates: , , or arbitrarily slow.
result The paper provides a complete characterization of which concept classes fall into each of the three categories of agnostic universal rates.
New framework explains normalizing flows' power and limitations.
problem Understanding the expressive power and limitations of normalizing flows.
method Theoretical framework for well-conditioned coupling-based normalizing flows and volume-preserving flows.
result RealNVP is distributionally universal, but volume-preserving flows are not.
We study the spherical cap packing problem with a probabilistic approach. Such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector. When the set of unit vectors …
Comment on ``Tests of scaling and universality of the distributions of trade size and share volume: Evidence from three distinct markets" by Plerou and Stanley, Phys. Rev. E 76, 046109 (2007)
The problem of universal outlying sequence detection is studied, where the goal is to detect outlying sequences among sequences of samples. A sequence is considered as outlying if the observations therein are generated by a distribution different from those generating the observations in the majority of the sequenc…
The K-sample testing problem involves determining whether K groups of data points are each drawn from the same distribution. Analysis of variance is arguably the most classical method to test mean differences, along with several recent methods to test distributional differences. In this paper, we demonstrate the existe…
Geometric Gaussian approximations capture any distribution.
problem Approximating complex probability distributions.
method Geometric Gaussian approximations through diffeomorphisms or exponential maps.
result Geometric Gaussian approximations are universal, capturing any distribution.
Gaussian Universality in Neural Network Dynamics with Generalized Structured Input Distributionsstat.ML
Deep learning models converge to Gaussian dynamics with mixed structured inputs.
problem Understanding neural network dynamics with complex input distributions.
method Extended hidden manifold model to Gaussian mixtures, analyzed via SGD.
result Learning dynamics with mixed inputs converge to Gaussian behavior.
A grand challenge of the 21st century cosmology is to accurately estimate the cosmological parameters of our Universe. A major approach to estimating the cosmological parameters is to use the large-scale matter distribution of the Universe. Galaxy surveys provide the means to map out cosmic large-scale structure in thr…
Estimating symmetric properties of a distribution, e.g. support size, coverage, entropy, distance to uniformity, are among the most fundamental problems in algorithmic statistics. While each of these properties have been studied extensively and separate optimal estimators are known for each, in striking recent work, Ac…
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance …
Unified plug-in approach for estimating symmetric properties of distributions efficiently.
problem Estimating symmetric properties of distributions with high accuracy and efficiency.
method Profile-maximum-likelihood (PML) based estimator.
result Achieves theoretical limit for universal symmetric property estimation.
We present a relatively detailed analysis of the persistence probability distributions in financial dynamics. Compared with the auto-correlation function, the persistence probability distributions describe dynamic correlations non-local in time. Universal and non-universal behaviors of the German DAX and Shanghai Index…
We analyze the constituents stocks of the Dow Jones Industrial Average (DJIA30) and the Standard & Poor's 100 index (S&P100) of the NYSE stock exchange market. Surprisingly, we discover the data collapse of the histograms of the DJIA30 price fluctuations and of the S&P100 price fluctuations to the universal non-paramet…
Paper establishes universal lower bounds and optimal rates for clustering sub-exponential mixture models.
problem Achieving optimal error rates in clustering sub-exponential mixture models.
method Establishes universal lower bounds and demonstrates iterative algorithms' optimality in sub-exponential mixture models.
result Iterative algorithms achieve the universal lower bound in sub-exponential mixture models.
Signatures of universality are detected by comparing individual eigenvalue distributions and level spacings from financial covariance matrices to random matrix predictions. A chopping procedure is devised in order to produce a statistical ensemble of asset-price covariances from a single instance of financial data sets…
New flows model distributions on Riemannian manifolds without domain knowledge.
problem Limited modeling of distributions on Riemannian manifolds.
method Riemannian convex potential maps using optimal transport.
result These flows can model standard distributions on spheres and tori.
The goal of this paper is to characterize function distributions that deep learning can or cannot learn in poly-time. A universality result is proved for SGD-based deep learning and a non-universality result is proved for GD-based deep learning; this also gives a separation between SGD-based deep learning and statistic…
Transformer pretraining yields strong EB performance without explicit adaptation.
problem Empirical Bayes problems with unknown test distributions.
method Indirect analysis of pretrained transformer's performance under universal priors.
result Near-optimal regret bound of for arbitrary test distributions.
CF-INNs can approximate any invertible function, resolving a long-standing problem.
problem Whether CF-INNs can approximate any invertible function.
method Demonstrated CF-INNs are universal approximators for invertible functions by showing a convenient criterion.
result CF-INNs are universal approximators for invertible functions.
Theory extends optimal learning rates without realizability assumption.
problem Agnostic binary classification without realizability assumption.
method Identifies tetrachotomy of optimal rates and combinatorial structures.
result Optimal universal rates for binary classification in agnostic setting.
We show that deep narrow Boltzmann machines are universal approximators of probability distributions on the activities of their visible units, provided they have sufficiently many hidden layers, each containing the same number of units as the visible layer. We show that, within certain parameter domains, deep Boltzmann…
In this work we show that, using the eigen-decomposition of the adjacency matrix, we can consistently estimate feature maps for latent position graphs with positive definite link function , provided that the latent positions are i.i.d. from some distribution F. We then consider the exploitation task of vertex classi…
Record statistics for random walk bridgescond-mat.stat-mech
We investigate the statistics of records in a random sequence of time steps. The sequence 's represents the position at step of a random walk `bridge' of steps that starts and ends at the origin. At each step, the increment of the position is a random ju…
LLMs learn peaked distributions slowly due to power-law losses.
problem Slow convergence of loss in training large language models.
method Systematic analysis of toy models and empirical evaluation of LLMs.
result Power-law time scaling with an exponent of 1/3 for learning peaked distributions.
Unified theorem for deep and shallow joint-equivariant machines.
problem Universal approximation of joint-equivariant machines.
method Constructive universal approximation theorem based on ridgelet transform.
result Unified approximation of deep and shallow networks.
OptFormer learns universal HPO from diverse datasets.
problem Learning HPO from experiments with different hyperparameters.
method Text-based Transformer framework for joint policy and function prediction.
result OptFormer can imitate multiple HPO algorithms and improve predictions.
Using detailed statistical analyses of the size distribution of a universe of equity exchange-traded funds (ETFs), we discover a discrete hierarchy of sizes, which imprints a log-periodic structure on the probability distribution of ETF sizes that dominates the details of the asymptotic tail. This allows us to propose …
The durations of recession and prosperity: does their distribution follow a power or an exponential law?cond-mat.stat-mech
Following findings by Ormerod and Mounfield, Wright rises the problem whether a power or an exponential law describes the distribution of occurrences of economic recession periods. In order to clarify the controversy a different set of GDP data is hereby examined. The conclusion about a power law distribution of recess…
We consider the group of sense-preserving diffeomorphisms $\Diff S^1$ of the unit circle and its central extension, the Virasoro-Bott group, with their respective horizontal distributions chosen to be Ehresmann connections with respect to a projection to the smooth universal Teichmüller space and the universal Teichmül…
New model suggests universe emerges from single particle quantum mechanics.
problem Exploring how the universe might arise from a single particle in quantum mechanics.
method Novel spontaneous symmetry breaking acting on probability distributions of Hamiltonians.
result Evidence supports the hypothesis that nature seeks tensor decompositions.
Data augmentation affects estimates' uncertainty and distribution in complex ways.
problem Understanding how data augmentation impacts the variance and limiting distribution of estimates.
method Developed an adaptation of Lindeberg's technique for block dependence.
result Data augmentation can increase rather than decrease uncertainty, and it may shift the double-descent peak of an empirical risk.
Unified method for CNNs to approximate equivariant maps across various groups.
problem Limited universal approximation theorems for CNNs with specific groups and settings.
method Unified approach to derive universal approximation theorems for equivariant maps by CNNs in diverse settings.
result Ability to handle non-linear equivariant maps between infinite-dimensional spaces for non-compact groups.
Study characterizes learning from heavy-tailed data in high dimensions using superstatistical methods.
problem Characterizing learning from heavy-tailed data in high-dimensional settings.
method Empirical risk minimization with double-stochastic processes and superstatistical analysis.
result Analytical characterization of separability transition and generalization performance.
A plethora of natural, artificial and social systems exist which do not belong to the Boltzmann-Gibbs (BG) statistical-mechanical world, based on the standard additive entropy and its associated exponential BG factor. Frequent behaviors in such complex systems have been shown to be closely related to -stati…
Physics of Personal Incomecond-mat.stat-mech
We report empirical studies on the personal income distribution, and clarify that the distribution pattern of the lognormal with power law tail is the universal structure. We analyze the temporal change of Pareto index and Gibrat index to investigate the change of the inequality of the income distribution. In addition …
Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo…
Universal Risk Budgetingq-fin.PM
A new risk budgeting scheme derived from universal portfolio theory.
problem Risk allocation in portfolio management.
method Integrates Cover's universal portfolio selection with modern risk allocation models.
result Proves mathematical equivalence to a novel universal portfolio scheme.
SURF simplifies distribution estimation with simple, robust, and fast algorithms.
problem Efficient and accurate distribution estimation in statistics and machine learning.
method Piecewise polynomial approximation using empirical probability interpolation and divide-and-conquer merging.
result Surpassing state-of-the-art algorithms in efficiency and accuracy, SURF estimates distributions robustly and quickly.
New model analyzes dynamic correlations in stock returns.
problem Analyzing time-varying correlations in high-dimensional data.
method Dynamic factor correlation model with novel parametrization.
result Model accurately captures heterogeneous heavy-tailed distributions and dependent shocks.