Sharp inequalities for matrix means with unknown variance.
problem Estimating matrix means with unknown variance.
method Empirical Bernstein inequalities for symmetric random matrices.
result Adapts to unknown variance with tight deviation bounds.
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48 results for “matrix Bernstein inequality”
This note gives a simple analysis of a randomized approximation scheme for matrix multiplication proposed by Sarlos (2006) based on a random rotation followed by uniform column sampling. The result follows from a matrix version of Bernstein's inequality and a tail inequality for quadratic forms in subgaussian random ve…
We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the dimens…
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.
Improved streaming PCA algorithm matches matrix Bernstein guarantees.
problem Efficiently estimating the top eigenvector of a covariance matrix in streaming data.
method Oja's algorithm with a suitable choice of step size.
result Streaming algorithm nearly matches batch method's accuracy and reduces sample complexity.
Concentration for matrix martingales in continuous time and microscopic activity of social networksmath.PR
This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermarti…
Improved analysis of UCRL2 with empirical Bernstein inequality reduces exploration-exploitation regret.
problem Exploration-exploitation in communicating Markov Decision Processes.
method Analysis of UCRL2 with Empirical Bernstein inequalities (UCRL2B).
result Regret bound of for UCRL2B.
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.
Study introduces new Bernstein inequalities for dependent data in Hilbert spaces.
problem Learning from non-independent and non-identically distributed data.
method Data-dependent Bernstein inequalities tailored for vector-valued processes in Hilbert space.
result Achieved novel risk bounds for covariance operator estimation and operator learning.
The paper explores nonconvex penalties using Bernstein functions for sparse estimation.
problem Sparse estimation in high-dimensional problems.
method Nonconvex penalties based on Bernstein functions, with coordinate descent and proximal alternating linearized minimization methods.
result The Bernstein penalty leads to effective sparse estimation and classification.
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.
New PAC-Bayesian bound improves generalization performance.
problem Improving generalization bounds for learning algorithms.
method Developed a new PAC-Bayesian generalization bound using a Bernstein-like inequality.
result The new bound consistently outperforms existing bounds and converges faster under certain conditions.
Develops a method for identifying structured dynamical systems from data.
problem Identifying structured dynamical systems from undersampled and noisy data.
method Sparse least-squares fitting via optimization with the alternating direction method of multipliers.
result The method is stable and successful under certain conditions, as shown by theoretical guarantees and computational results.
We prove new concentration inequalities for random variables.
problem Concentration of random variables in nonlinear functions.
method Efron-Stein inequalities and PAC-Bayesian approach.
result User-friendly concentration bounds for various applications.
A Bernstein-type inequality for stochastic processes of quadratic forms of Gaussian variablesmath.ST
We introduce a Bernstein-type inequality which serves to uniformly control quadratic forms of gaussian variables. The latter can for example be used to derive sharp model selection criteria for linear estimation in linear regression and linear inverse problems via penalization, and we do not exclude that its scope of a…
Minimal surface equation results in constant solutions on RCD spaces.
problem Analyzing minimal surfaces on RCD spaces.
method Using properties of RCD spaces and the minimal surface equation.
result Positive solutions to the minimal surface equation are constant on RCD spaces.
We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as $\l…
Law of iterated logarithm derived from betting strategy.
problem Deriving the law of the iterated logarithm.
method Using a betting strategy's wealth guarantee.
result Law of the iterated logarithm established.
New method improves RL in continuous spaces with kernel smoothing.
problem Sample efficiency and structural assumptions in classical RL.
method Kernel smoothing model-based approach with Bernstein-style exploration bonus.
result Achieves improved regret bound in finite-horizon settings.
Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.
problem Cohomology of BGG complexes on bounded Lipschitz domains.
method Computes cohomology of conformal deformation and Hessian complexes in Sobolev spaces, allowing multiple input complexes.
result Establishes conformal Korn inequality and proposes generalizations of continuum models with microstructures.
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.
We are concerned with obtaining novel concentration inequalities for the missing mass, i.e. the total probability mass of the outcomes not observed in the sample. We not only derive - for the first time - distribution-free Bernstein-like deviation bounds with sublinear exponents in deviation size for missing mass, but …
Paper refines PAC-Bayes bounds for bandit problems.
problem Improving probabilistic bounds for off-policy learning.
method Optimizes PAC-Bayesian bounds using a new parameter optimization approach.
result Provides two parameter-free PAC-Bayes bounds that nearly match optimal rates.
Optimizes sub-Gaussian matrices for preserving data distances.
problem Improving the performance of sub-Gaussian matrices in preserving data distances.
method Analyzes sub-Gaussian matrices and their dependence on the sub-Gaussian norm, presenting optimal bounds.
result Optimal dependence on the sub-Gaussian norm for sub-Gaussian matrices as near isometries on sets.
UCRL-V algorithm achieves optimal regret in unknown MDPs.
problem Optimizing reinforcement learning in unknown finite Markov decision processes.
method Empirical Bernstein inequalities for variance-based confidence intervals.
result Achieves optimal regret up to logarithmic factors.
The paper analyzes learning schemes for various stationary stochastic processes.
problem Analyzing learning schemes with different stationary stochastic processes.
method Unified treatment of various mixing processes using generalized Bernstein-type inequality.
result Sharp oracle inequalities and convergence rates for learning schemes.
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.
Spacelike surfaces in Generalized Robertson-Walker spacetimes whose mean curvature function satisfies a natural nonlinear inequality are analyzed. Several uniqueness and nonexistence results for such compact spacelike surfaces are proved. In the nonparametric case, new Calabi-Bernstein type problems are solved as a con…
Paper estimates curvature of minimal surfaces in a specific geometric space.
problem Estimating curvature of minimal hypersurfaces in Heisenberg groups.
method Extending Simons formula and Kato inequality to sub-Riemannian setting, applying to stable hypersurfaces.
result Integral curvature estimates for stable hypersurfaces in Heisenberg groups.
New PAC-Bayes bounds for martingale mixtures simplify prior work.
problem Bounding mixtures of martingales uniformly over distributions and times.
method PAC-Bayes approach, extending Bernstein inequalities.
result Tight concentration bounds for martingale mixtures.
Paper analyzes Nyström regularization for time series forecasting with sequential sub-sampling.
problem Learning rate analysis of Nyström regularization for -mixing time series.
method Banach-valued Bernstein inequality and integral operator approach for -mixing sequences.
result Almost optimal learning rates for Nyström regularization with sequential sub-sampling.
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…
Paper develops new inequalities for high-dimensional statistics under sub-Weibull tail assumptions.
problem High-dimensional statistical methods under sub-Weibull tail assumptions.
method Develops new concentration inequalities for sums of independent random variables under sub-Weibull tail assumptions.
result Concentration inequalities match asymptotics of central limit theorem and match sub-Gaussian tail behavior.
Deep neural networks are optimal for dependent data using PAC-Bayes bounds.
problem Optimizing deep neural networks for dependent data.
method PAC-Bayes oracle inequalities and Bernstein inequality.
result Upper and lower bounds match, proving minimax optimality.
Using simple facts from harmonic analysis, namely Bernstein inequality and Plansherel isometry, we prove that the pseudodifferential equation improves the Sobolev regularity of solutions provided the potential is integrable with the critical power .
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.
The betting CI outperforms classical methods in constructing confidence intervals for bounded means.
problem Constructing nonasymptotic confidence intervals for bounded means.
method A betting-based approach to define and time-uniform variants of confidence intervals (CSs).
result The betting CI matches the fundamental limits, outperforming existing empirical Bernstein CIs.
When I first encountered PAC-Bayesian concentration inequalities they seemed to me to be rather disconnected from good old-fashioned results like Hoeffding's and Bernstein's inequalities. But, at least for one flavour of the PAC-Bayesian bounds, there is actually a very close relation, and the main innovation is a cont…
Extends inequalities to nonstationary Markov chains for better risk bounds.
problem Proving inequalities for nonstationary, non-i.i.d. Markov chains.
method Extends Dedecker and Fan's (2015) inequalities to nonstationary Markov chains.
result Provides a Bernstein-type inequality for periodic autoregressive processes.
New algorithm learns policies without uniform overlap assumption.
problem Learning optimal policies from non-uniformly collected data.
method Pessimistic Policy Learning (PPL) using lower confidence bounds.
result Efficient policy learning for adaptively collected data.
Almost sharp local Bernstein estimates for Laplace eigenfunctions on compact Riemannian manifoldsmath.AP
The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.
problem Understanding local growth properties of Laplace eigenfunctions on compact Riemannian manifolds.
method Refined Donnelly-Fefferman method based on --Carleman estimates, combined with elliptic regularity and patching of local Carleman estimates.
result Almost sharp local --Bernstein inequalities for .
Machine learning selects the best prediction rules from noisy data.
problem Selection under uncertainty in machine learning.
method Statistical tools and inequalities to control noise in empirical estimates.
result Theoretical guarantees on selection outcomes under uncertainty.
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.
Develops a deep learning framework for various data types.
problem Handling nonparametric regression and classification across different data types.
method Introduces a general framework with two estimators: NPDNN and SPDNN, based on data satisfying generalized Bernstein-type inequalities.
result Both NPDNN and SPDNN estimators are minimax optimal in many classical settings.
We explore a connection between the Finslerian area functional and well-investigated Cartan functionals to prove new Bernstein theorems, uniqueness and removability results for Finsler-minimal graphs, as well as enclosure theorems and isoperimetric inequalities for minimal immersions in Finsler spaces. In addition, we …
Random Fourier Features adapted for operator-valued kernels to scale multi-task and structured output learning.
problem Scaling operator-valued kernels for multi-task and structured output learning.
method Adapted Random Fourier Features for operator-valued kernels, using a generalization of Bochner's theorem.
result Uniform convergence of kernel approximation for operator-valued Random Fourier Features.
Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.
problem Finding compact surfaces in Euclidean 3-space with specific curvature properties.
method Representation of solutions to linear elliptic equations with discontinuous coefficients.
result Compact surfaces of genus zero with specific curvature properties are round spheres.
We develop a coherent framework for integrative simultaneous analysis of the exploration-exploitation and model order selection trade-offs. We improve over our preceding results on the same subject (Seldin et al., 2011) by combining PAC-Bayesian analysis with Bernstein-type inequality for martingales. Such a combinatio…