In this note, we derive concentration inequalities for random vectors with subGaussian norm (a generalization of both subGaussian random vectors and norm bounded random vectors), which are tight up to logarithmic factors.
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48 results for “Concentrated Random Vectors”
The paper generalizes product inequalities for random vectors and their applications.
problem Understanding concentration of measure for products of random vectors.
method Develops expressions for the concentration of functionals of random vectors based on product norms.
result Provides generalized Hanson-Wright inequalities and applications to random matrices.
The paper studies how norms of random vectors are preserved by random projections.
problem Understanding how random matrix affects norms of random vectors.
method Proved the distribution of the norm of random vector is preserved by random projection.
result Random matrix preserves the distribution of the norm of random vectors with i.i.d. entries.
Deep learning representations of GAN data are like Gaussian mixtures, according to this study.
problem Understanding the statistical nature of deep learning representations of GAN-generated data.
method Using Random Matrix Theory, the study shows that DL representations of GAN data are concentrated random vectors that behave like Gaussian mixtures.
result Deep learning representations of GAN data can be fully described by their first two statistical moments.
Study robust covariance estimation in large data with concentrated vectors.
problem Estimating robust covariance in large data with concentrated vectors.
method Fixed point of a contracting function using stable semi-metric and concentration of measure.
result Existence and uniqueness of robust estimator with evaluated limiting spectral distribution.
A concentration graph associated with a random vector is an undirected graph where each vertex corresponds to one random variable in the vector. The absence of an edge between any pair of vertices (or variables) is equivalent to full conditional independence between these two variables given all the other variables. In…
The paper introduces a new method for tail bounds of random vectors and matrices.
problem Estimating norms of random vectors and matrices under moment assumptions.
method Variational tail bounds for norms of random vectors and matrices.
result Dimension-free concentration inequalities for various norms of random vectors and matrices.
Study spectral properties of sparse random graphs to recover latent vectors.
problem Recovering latent vectors in sparse random geometric graphs.
method Analyzes spectral concentration and uses orthogonal polynomial expansions, decoupling, and matrix concentration.
result Sharpens spectral norm bounds and proves exact recovery for Gaussian mixture models.
Repelling Random Walksstat.ML
Repelling random walks improve graph-based sampling efficiency.
problem Efficient graph-based sampling and statistical estimation.
method Induces correlations between trajectories of an ensemble of walkers on a graph, maintaining unbiasedness.
result Improves concentration of statistical estimators on graphs.
The paper analyzes sparse high-dimensional linear regression with random design and unknown error variance, providing adaptiveness and concentration rates.
problem Sparse high-dimensional linear regression with random design and unknown error variance.
method Analysis of posterior concentration rates, employing techniques to address model misspecification.
result Adaptiveness and concentration rates of the posterior for sparse high-dimensional linear regression.
Improved concentration inequalities for sub-Weibull variables enhance statistical and machine learning applications.
problem Improving concentration inequalities for sub-Weibull random variables.
method Developed new concentration inequalities for sums of independent sub-Weibull random variables, including a new sub-Weibull parameter.
result New concentration inequalities with sharper constants and a mixture of sub-Gaussian and sub-Weibull tails.
New method improves feasibility of fitting Gaussian vectors to an ellipsoid.
problem Feasibility of fitting Gaussian vectors to an ellipsoid boundary.
method Improved concentration of Gram matrices using Bartl & Mendelson (2022) results.
result Feasibility of with high probability when .
Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projectionsstat.ML
A new method approximates the Sliced-Wasserstein distance without random projections.
problem Efficiently approximating the Sliced-Wasserstein distance for machine learning applications.
method Utilizing the concentration of measure phenomenon to develop a deterministic approximation.
result The approximation error goes to zero as the dimension increases, under a weak dependence condition.
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.
Study on inequalities for multinomial variables.
problem Understanding concentration inequalities for multinomial variables.
method Investigation of Dirichlet and Multinomial random variables.
result Results on concentration inequalities for multinomial variables.
Sharp concentration inequalities for sub-Orlicz random variables with phase transition at α=2.
problem Developing concentration inequalities for sub-Orlicz random variables with phase transition.
method New theoretical analysis framework involving variance and min/max functions of Orlicz tails.
result Sharp concentration inequalities with phase transition at α=2 for sub-Orlicz random variables.
Sharp concentration results for sums of heavy-tailed random variables.
problem Analyzing sums of independent heavy-tailed random variables.
method Using concentration inequalities and large deviation principles for distributions satisfying specific tail bounds.
result Sharp concentration inequalities and large deviation results for sums of heavy-tailed random variables.
Lecture notes on kernel functions and Random Fourier Features.
problem Understanding and approximating kernel functions in machine learning.
method Mathematical background and proofs of concentration results.
result Estimation of error in Random Fourier Features approximation.
We improve bounds for stochastic processes, especially those with heavy tails.
problem Bounding the concentration of sub- processes with heavy tails.
method Variational approach to concentration, focusing on sub-Gaussian and other tail conditions.
result First dimension-free self-normalized empirical Bernstein inequality.
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.
The paper offers a framework to analyze machine learning problems using concentration of measure.
problem Analyzing machine learning algorithms defined by implicit equations.
method Develops a concentration of measure framework to solve convex problems and implicit formulations.
result Provides precise estimations for the first moments of the solution, describing the behavior and performance of machine learning classifiers.
Proves new concentration inequalities for sub-gaussian and sub-exponential variables.
problem Understanding functions of independent random variables better.
method Sub-gaussian and sub-exponential conditions, Rademacher complexities, Lipschitz function classes.
result Extension of Rademacher complexities to unbounded sub-exponential distributions.
A new matrix concentration inequality for random products of matrices.
problem Understanding the behavior of random matrix products under bounded independent positive semidefinite matrices.
method Developed a non-asymptotic concentration inequality for the product of matrices.
result The inequality provides a bound on the deviation of the matrix product from its expected value.
Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.
problem Limited understanding of self-normalized concentration for vector-valued processes outside sub-Gaussian frameworks.
method Developed concentration inequalities for self-normalized processes with light tails (e.g., Bennett, Bernstein bounds) for vector-valued data.
result Provided new insights and bounds for self-normalized processes with non-sub-Gaussian distributions.
The paper provides concentration bounds for embeddings of generative models.
problem Establishing accurate statistical analysis of generative models.
method Data Kernel Perspective Space embedding method.
result Required number of sample responses for accurate approximation.
This work gives a simultaneous analysis of both the ordinary least squares estimator and the ridge regression estimator in the random design setting under mild assumptions on the covariate/response distributions. In particular, the analysis provides sharp results on the ``out-of-sample'' prediction error, as opposed to…
Paper simplifies concentration inequalities for easier probabilistic analysis.
problem Complexity in probabilistic analysis of random variables.
method Compact notations for concentration inequalities.
result Simplified expressions for typical sizes and tails of random variables.
The paper establishes concentration bounds for embeddings of generative models.
problem Analyzing statistical properties of generative models.
method High probability concentration bounds on sample vector embeddings using Data Kernel Perspective Space.
result Determines the number of samples needed for accurate approximation of generative model embeddings.
Devoted to multi-task learning and structured output learning, operator-valued kernels provide a flexible tool to build vector-valued functions in the context of Reproducing Kernel Hilbert Spaces. To scale up these methods, we extend the celebrated Random Fourier Feature methodology to get an approximation of operator-…
The paper develops concentration inequalities for structured random data, extending beyond independent terms.
problem Developing concentration inequalities for structured weighted sums of random data, including tensors and matrix-valued data.
method The paper develops Hoeffding and Bernstein bounds for structured weighted sums under exchangeability, extending beyond the classical framework of independent terms.
result The paper develops a sharper concentration bound for combinatorial sums of matrix arrays.
Random feature matrices' singular values concentrate near their full expectation in high dimensions.
problem Characterizing the spectra of random feature matrices for regression problems.
method Analyzing two settings of input variables (random or well-separated) with conditions on dimension, complexity ratio, and sampling variance.
result The singular values of random feature matrices concentrate near their full expectation and near one with high probability.
G-Net constructs binary neural networks with high accuracy using randomized binary embeddings.
problem Creating high-accuracy binary neural networks with theoretical guarantees.
method Proposes a novel floating-point G-Net family with randomized binary embeddings and theoretical accuracy guarantees.
result Empirically, G-Net achieves almost 30% higher accuracy on CIFAR-10 compared to prior HDC models.
This paper shows DPPs can outperform random coresets in machine learning tasks.
problem Building efficient coresets for machine learning models.
method Using determinantal point processes (DPPs) to construct coresets with provable improvements over random sampling.
result DPPs can provably outperform independently drawn coresets in terms of approximation of total loss.
The paper reviews and improves concentration inequalities for statistical inference.
problem Analyzing statistical inference in various settings with high-dimensional data.
method Review and improvement of concentration inequalities for different types of random variables and statistical measures.
result Fresh new results and improved bounds with sharper constants.
Random features and KRR generalize similarly when N is large enough.
problem Understanding the generalization error of random features and KRR methods.
method Analyzing spectral conditions and hypercontractivity on kernel eigenfunctions.
result The test error of random features is larger than KRR when N is small, but they achieve the same error when N is large.
New approach to concentration inequalities for unbounded state space dynamical systems.
problem Concentration inequalities for unbounded state space dynamical systems.
method Functional analytic framework, transport-entropy inequality.
result Exponential concentration inequalities for sampling from stationary distribution.
Study tail risk aggregation under dependence uncertainty.
problem Risk aggregation under dependence uncertainty and hidden dependence.
method Introduce hidden dependence, show compatibility with small perturbations, quantify portfolio risk.
result Small deviations in dependence structure can lead to significant risk underestimation.
Paper develops a new inequality for non-causal machine learning.
problem Current concentration inequalities cannot be applied to non-causal machine learning.
method Develops a framework for non-causal random fields and proves a Hoeffding-type inequality.
result Obtains a Hoeffding-type concentration inequality for non-causal random fields.
Sharp concentration bounds for i.i.d. variables.
problem Controlling the tail probabilities of independent variables.
method Extension of Sanov's theorem using large deviations and information theory.
result Matching concentration and anti-concentration bounds for i.i.d. samples of any size.
In this paper, we study randomized reduction methods, which reduce high-dimensional features into low-dimensional space by randomized methods (e.g., random projection, random hashing), for large-scale high-dimensional classification. Previous theoretical results on randomized reduction methods hinge on strong assumptio…
We study the concentration of random kernel matrices around their mean. We derive nonasymptotic exponential concentration inequalities for Lipschitz kernels assuming that the data points are independent draws from a class of multivariate distributions on , including the strongly log-concave distributions u…
We present a new paradigm for speeding up randomized computations of several frequently used functions in machine learning. In particular, our paradigm can be applied for improving computations of kernels based on random embeddings. Above that, the presented framework covers multivariate randomized functions. As a bypr…
Theoretical study of random forests for nonlinear time series.
problem Theoretical justification for using random forests in time series modeling.
method Uniform concentration inequality for regression trees and random forests consistency proof.
result Consistency of random forests for nonlinear autoregressive processes.
This paper studies node embeddings of networks, revealing their geometric properties.
problem Understanding the geometric properties of node embeddings in random networks.
method Characterization of ergodic limits, generalization, and convex relaxations of random walk node embedding objectives.
result The optimal node embedding Grammians have rank 1 for a nuclear norm relaxation of the non-randomized objective.
Improved bounds for Monte Carlo Rademacher Averages using self-bounding functions.
problem Proving sharper concentration bounds for MCERA.
method Deriving new bounds through self-bounding functions and concentration of measure.
result Novel bounds depend on data-dependent quantities, improving over standard methods.
We prove semi-empirical concentration inequalities for random variables which are given as possibly nonlinear functions of independent random variables. These inequalities describe concentration of random variable in terms of the data/distribution-dependent Efron-Stein (ES) estimate of its variance and they do not requ…
We propose fast approximations for the generalized sliced-Wasserstein distance.
problem Efficient approximation of the generalized sliced-Wasserstein distance in high dimensions.
method Deterministic approximations using random projections and concentration of measure results.
result One-dimensional projections of high-dimensional random vectors are approximately Gaussian.
Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors and a constraint family , find a set that maximizes the squared volume of the simplex spanned by the vectors in . A motivatin…