Simplified proof of Gaussian concentration inequality using covariance.
problem Gaussian concentration inequality proof
method Covariance representation based on characteristic functions
result Elementary proof of Gaussian concentration inequality
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48 results for “concentration inequalities”
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.
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.
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.
Develops inequalities for high-dimensional linear processes with dependent innovations.
problem Estimating high-dimensional VAR(p) systems and HAC covariance estimation.
method Concentration inequalities for norm of vector linear processes with sub-Weibull, mixingale innovations.
result Obtained concentration bounds for the maximum entrywise norm of lag- autocovariance matrices.
New concentration inequality for U-statistics of Markov chains.
problem Proving a concentration inequality for U-statistics of order two in uniformly ergodic Markov chains.
method Inductive analysis using martingale techniques, uniform ergodicity, Nummelin splitting, and Bernstein's inequality.
result Recovery of convergence rate for U-statistics of independent random variables and canonical kernels, with improved results for dependent kernels.
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.
We investigate concentration inequalities for Dirichlet and Multinomial random variables.
New method improves missing mass concentration bounds.
problem Missing mass concentration problem
method New method of estimating concentration of heterogenic sums
result Slightly improved state-of-the-art bounds
The paper extends Hoeffding's inequality for Markov chains using a generalized concentrability condition.
problem Applying Hoeffding's inequality to non-ergodic Markov chains.
method Integrates generalized concentrability condition via IPM to extend traditional hypotheses.
result Demonstrates utility in machine learning applications such as empirical risk minimization and bandits.
The paper provides concentration inequalities for Markov chain variance estimators.
problem Estimating the variance of Markov chains with concentration properties.
method Martingale decomposition method for uniformly geometrically ergodic Markov chains.
result Explicit control of the p-th moment of the OBM estimator difference and dependence on p and mixing time.
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.
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.
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.
Study on deep neural networks using concentration inequalities and optimal stopping.
problem Understanding the performance and structure of stochastic deep neural networks.
method Introduced concentration inequalities for SDNN outputs and an EC classifier. Determined the optimal number of layers via an optimal stopping procedure.
result Optimal number of layers for SDNNs determined via an optimal stopping procedure.
Study Finsler metric measure manifolds' concentration properties.
problem Understanding concentration properties in Finsler metric measure manifolds.
method Established relationships with observable diameter, isoperimetric inequalities, and first eigenvalue.
result Derived a Cheng type upper bound estimate for the first closed eigenvalue.
Sharp bounds on uniform generalization errors in binary linear classification.
problem Understanding the uniform generalization errors in binary linear classification.
method Isoperimetric arguments, Poincaré and log-Sobolev inequalities for joint distributions.
result Sharp concentration bounds on uniform generalization errors, almost sure convergence in broad settings.
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.
We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration inequality, proved using the same method. Together these constitute a finite-time versi…
The paper proves LOO CV is reliable under estimator stability.
problem Ensuring the reliability of leave-one-out cross validation.
method Using concentration inequalities based on logarithmic Sobolev inequality.
result LOO CV is a valid procedure under estimator stability.
We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.
The paper proves concentration inequalities for diffusion processes.
problem Proving concentration inequalities for diffusion processes.
method Analysis via the Poisson equation for a broad class of subexponentially ergodic processes.
result Demonstrates power of concentration inequalities in validating conditions for Lasso estimation and sampling algorithms.
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.
It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a…
The study characterizes heat flow and concentration on directed graphs with a lower Ricci curvature bound.
problem Understanding heat flow and concentration on directed graphs with a specific curvature bound.
method Characterization via gradient estimate and transportation inequality for the heat semigroup.
result Concentration of measure inequality for directed graphs with positive Ricci curvature.
The study proves inequalities and curvature properties for Markov chains.
problem Isoperimetric and concentration inequalities for Markov chains.
method Laplacian separation principle for eikonal equation; modified log-Sobolev constant; Ollivier curvature.
result Affirmative answers to open questions and new inequalities.
Sharp Gaussian isoperimetry proven along Ricci flow.
problem Proving sharp Gaussian isoperimetric inequality for Ricci flow.
method Using monotonicity formula to prove inequality.
result Exact Gaussian enlargement theorem and concentration estimates.
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.
Expanding on techniques of concentration of measure, we develop a quantitative framework for modeling liquidity risk using convex risk measures. The fundamental objects of study are curves of the form , where is a convex risk measure and a random variable, and we call such a curve a \emph{liqu…
Study proves uniqueness of Yang-Mills field tangent cones in arbitrary dimensions.
problem Proving uniqueness of Yang-Mills field tangent cones.
method Log-epiperimetric inequality, Luckhaus type lemma, and curvature concentration exclusion.
result Uniqueness of tangent cones for Yang-Mills fields in arbitrary dimensions.
Paper shows how online betting algorithms' regret can be used to create tight confidence sequences.
problem Estimating the expectation of random variables from samples and creating time-uniform confidence sequences.
method Converts the regret guarantee of universal portfolio algorithms into time-uniform concentration inequalities and confidence sequences.
result Numerically obtained confidence sequences are never vacuous and satisfy the law of iterated logarithm.
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.
Estimates barycenter in geodesic spaces with finite sample bounds.
problem Estimating the barycenter of a distribution in geodesic spaces.
method Finite sample error bounds, Hoeffding- and Bernstein-type concentration inequalities, efficient algorithms.
result Statistical guarantees for efficient barycenter computation.
Motivated by liquidity risk in mathematical finance, D. Lacker introduced concentration inequalities for risk measures, i.e. upper bounds on the \emph{liquidity risk profile} of a financial loss. We derive these inequalities in the case of time-consistent dynamic risk measures when the filtration is assumed to carry a …
The paper develops new inequalities for Markov chain sums, linking them to mixing time.
problem Establishing concentration inequalities for Markov chain sums.
method Developed novel concentration inequalities for geometrically ergodic Markov chains, linking bounds to mixing time constants.
result Explicit bounds for additive functionals of Markov chains, linked to Rosenthal inequality constants and mixing properties.
New inequality for ternary variables improves on existing measures.
problem Analyzing excess losses and weighted majority votes with ternary random variables.
method Developed a split-kl inequality and its PAC-Bayes extension.
result Outperforms existing inequalities in certain regimes.
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…
This paper presents compact notations for concentration inequalities and convenient results to streamline probabilistic analysis. The new expressions describe the typical sizes and tails of random variables, allowing for simple operations without heavy use of inessential constants. They bridge classical asymptotic nota…
The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Generalized Lions' concentration compactness and rigidity results of Sobolev inequalities on singular spaces.
result Almost extremal functions are close to extremal functions on the round sphere and Euclidean Sobolev inequality.
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.
We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequa…
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.
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.
Concentration Inequalities for Two-Sample Rank Processes with Application to Bipartite Rankingmath.ST
The paper proves concentration inequalities for two-sample rank processes and applies them to ranking performance criteria.
problem Measuring the performance of ranking statistics between two populations.
method Proves concentration inequalities for two-sample rank processes indexed by VC classes of scoring functions.
result Generalization capacity of empirical maximizers of ranking performance criteria is investigated.
We study the isoperimetric, functional and concentration properties of -dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension is negative, and more generally, is in the range , extending the scope from the traditional range $N \i…
A crucial assumption in most statistical learning theory is that samples are independently and identically distributed (i.i.d.). However, for many real applications, the i.i.d. assumption does not hold. We consider learning problems in which examples are dependent and their dependency relation is characterized by a gra…
New inequalities for matrix supermartingales converge under various conditions.
problem Convergence and maximal inequalities of supermartingales in positive semidefinite matrices.
method Developed new concentration inequalities for matrix supermartingales.
result New inequalities for matrix supermartingales under different tail conditions.
A universal inequality that bounds the charge of a body by its size is presented, and is proven as a consequence of the Einstein equations in the context of initial data sets which satisfy an appropriate energy condition. We also present a general sufficient condition for the formation of black holes due to concentrati…