Paper refines Talagrand inequality on Euclidean spaces.
problem Improving Talagrand inequality for Euclidean spaces.
method Symmetrization and alternative proof methods.
result Several refined functional inequalities derived.
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48 results for “Talagrand inequality”
Paper proves generalized Talagrand inequality for Sinkhorn distance.
problem Proving a generalized Talagrand inequality for Sinkhorn distance.
method Using entropy power inequality and infinitesimal displacement convexity of optimal transport map.
result Extends previous results of Gaussian Talagrand inequality for Sinkhorn distance to strongly log-concave case.
Paper introduces information-constrained optimal transport, generalizing Talagrand's inequality.
problem Optimal transport problem with information constraints.
method Information constrained variation of optimal transport, using Marton's approach.
result Recovery of concentration of measure results and solution to Cover's open problem.
We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a …
Proves error bounds for PGD, extending log-Sobolev and Talagrand inequalities.
problem Maximum likelihood estimation of large latent variable models.
method Extending log-Sobolev and Talagrand inequalities to models with strongly concave log-likelihoods.
result Non-asymptotic error bounds for PGD in models satisfying LSI and PŁI.
This research analyzes the error convergence rate of GAN models.
problem Understanding the error convergence rate of GAN models.
method Applying Talagrand inequality and Borel-Cantelli lemma to establish a tight convergence rate.
result Established a tight convergence rate for the error of GAN models.
Paper improves risk bound for MTL with graph-dependent data.
problem Sub-optimal risk bound in multi-task learning with graph-dependent data.
method Proposes a new Bennett-type inequality and develops new Talagrand-type inequality and local fractional Rademacher complexity.
result Derives a sharper risk bound of .
We derive weighted log-Sobolev inequalities from a class of super Poincaré inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\d…
Study generalizes matrix completion with side info in low noise settings.
problem Matrix completion with side information in low noise conditions.
method Inductive matrix completion with i.i.d. subgaussian noise, uniform sampling, and side information.
result Generalization bounds with noise scaling, convergence to zero, and logarithmic dependence on matrix size.
Study non-Gaussian measures' concentration properties in metric spaces.
problem Concentration properties for non-linear Gaussian functionals with non-Gaussian tails.
method Prove generalised Transportation-Cost Inequalities (TCIs) for specific functionals.
result Extended TCIs for rough volatility and Parabolic Anderson Model.
Unified framework for information-theoretic bounds on learning algorithms.
problem Deriving generalization bounds for learning algorithms.
method Probabilistic decorrelation lemma, symmetrization, couplings, chaining, Young's inequality.
result New upper bounds on generalization error in expectation and high probability.
We investigate the -relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement -convexity of the -relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the -convexity of the weig…
We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci cur…
Prove non-asymptotic bounds for minimal risk in statistical learning
problem Estimating minimal risk in statistical learning
method Using concentration inequalities
result Non-asymptotic bounds for minimal risk
Learning and Concentration for High Dimensional Linear Gaussians: an Invariant Subspace Approachstat.ML
Study non-asymptotic bounds on correlation in high-dimensional linear systems, revealing invariant subspaces and bottlenecks.
problem Understanding correlation and mixing in high-dimensional linear systems with Gaussian noise.
method Sampling from sub-trajectories, using Talagrand's inequality, and analyzing invariant subspaces.
result Large discrepancy between algebraic and geometric multiplicity leads to bottlenecks between invariant subspaces.
This manuscript presents some new impossibility results on adversarial robustness in machine learning, a very important yet largely open problem. We show that if conditioned on a class label the data distribution satisfies the Talagrand transportation-cost inequality (for example, this condition is satisfied if t…
Kernel-based L2-boosting with structure constraints improves regression efficiency.
problem Developing efficient kernel methods for regression.
method Kernel-based re-scaled boosting with truncation (KReBooT).
result KReBooT achieves near overfitting resistance and sparse estimates.
Proves subgaussian distributions are SoS-certifiably subgaussian, enabling efficient algorithms for various statistical tasks.
problem Efficiently learning from subgaussian distributions in high dimensions.
method Universal constant and polynomial sum of squares (SoS) approach.
result Proves subgaussian distributions are SoS-certifiably subgaussian.
New methods improve stability of Sinkhorn algorithm in machine learning.
problem Stability of Sinkhorn semigroups in high-dimensional settings.
method Semigroup analysis based on contraction coefficients and Lyapunov-type operator-theoretic techniques.
result Unified and simplified arguments in Sinkhorn algorithm stability.
New bounds link generalization to stochastic optimizer's lower tail exponents.
problem Understanding the impact of stochastic optimization algorithms on generalization in non-convex settings.
method Proves novel bounds linking generalization to the lower tail exponent of the transition kernel of stochastic optimizers, both discrete- and continuous-time.
result Empirical results show correlations between generalization error and lower tail exponents.
The paper develops a new probabilistic framework for denoising diffusion models using free entropy and stochastic analysis.
problem Developing a mathematical framework for denoising diffusion models in noncommutative settings.
method Formulating diffusion and reverse processes governed by operator-valued stochastic dynamics, using tools from free stochastic analysis.
result Establishing an information-geometric link between entropy production, transport, and deconvolution.
RHMC accelerates sampling from log-concave distributions.
problem Sampling from log-concave probability distributions efficiently.
method RHMC uses simulated Hamiltonian dynamics with random integration times.
result RHMC converges exponentially fast in KL divergence for log-concave distributions.
Two SVGD variants achieve fast convergence with provable guarantees.
problem Understanding and improving SVGD's performance with finite particles.
method Introducing virtual particles and novel stochastic approximations.
result Provable fast convergence rates for finite-particle SVGD variants.
Paper reinterprets majorizing measure theorem in terms of coding theory.
problem Understanding boundedness of random processes.
method Information-theoretic perspective using variable-length codes.
result Boundedness of random processes linked to efficient coding.
A note proves the binary perceptron's capacity is less than 0.847.
problem Determining the capacity of the binary perceptron.
method Conditional first moment method combined with known results on the spherical perceptron.
result Proves the binary perceptron's capacity is less than 0.847.
Improved generalization bounds for CNNs using Rademacher complexity.
problem Establishing non-vacuous generalization bounds for deep learning models.
method Rademacher complexity framework with novel contraction lemmas for high-dimensional mappings.
result Enhanced generalization bounds for a broader class of activation functions.
New findings show Rademacher complexities are not crucial for learning complexities.
problem Understanding the sample complexity of learning with squared loss in convex classes.
method Novel learning procedure combining mean estimation and Talagrand's generic chaining method.
result Sample complexity is determined by the limiting Gaussian process, not Rademacher complexities.
We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the non-spiked one is asymptotically normal below a certai…
The paper provides a new uniform tail bound for empirical processes.
problem Developing a uniform tail bound for empirical processes indexed by a class of functions.
method Introducing a deflation step to the standard generic chaining argument, and using a natural seminorm based on Cramér functions.
result Established a new uniform tail bound for empirical processes.
Stochastic Trace Optimization of Parameter Dependent Matrices Based on Statistical Learning Theorystat.ML
Estimates matrix trace optimization with statistical learning theory.
problem Optimizing trace of parameter-dependent matrices.
method Monte Carlo estimator with bounds derived from epsilon nets and generic chaining.
result Predicts small sampling amount for matrices with small off-diagonal mass.
We study the problem of detecting the presence of a single unknown spike in a rectangular data matrix, in a high-dimensional regime where the spike has fixed strength and the aspect ratio of the matrix converges to a finite limit. This setup includes Johnstone's spiked covariance model. We analyze the likelihood ratio …
We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss can control the test error under all Moreau envelopes of the loss . We use our finite-sample bound to directly recover…
The isoperimetric inequalitymath.DG
The isoperimetric inequality and related inequalities are explored.
problem Proving the isoperimetric inequality and related inequalities.
method Discussing classical and recent proofs.
result Various proofs of the isoperimetric inequality and Sobolev inequality.
New proof of Willmore inequality using geometric divergence inequality.
problem Proving the Willmore inequality for bounded domains.
method Using a parametric geometric inequality derived from a divergence form geometric differential inequality.
result New proofs of quantitative Willmore-type and weighted Minkowski inequalities.
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.
problem Deriving new inequalities on manifolds and convex hypersurfaces.
method Using Fourier theory and geometric implications of Poincare-type inequalities.
result Sharp Minkowski-type inequalities, including stability and Alexandrov-Fenchel inequalities.
The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.
problem Proving -Sobolev and Nash inequalities on Finsler metric measure manifolds.
method Global -Poincaré inequality, -Sobolev inequality, Nash inequality derivation.
result Established global optimal -Sobolev inequality with a sharp constant.
New inequality on sphere generalizes circle inequality.
problem Generalizing circle inequality to sphere.
method Develops a new inequality on the sphere that incorporates mass center deviation.
result Improves Aubin's inequality and Onofri's inequality.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Explains geometric inequalities for minimal hypersurfaces.
problem Geometric inequalities for minimal hypersurfaces.
method Expository discussion of known inequalities.
result Discussion of classical inequalities for minimal hypersurfaces.
The paper finds new inequalities for convex polygons.
problem Finding precise inequalities for convex polygons.
method Analytic isoperimetric inequalities based on Schur convex functions, followed by Bonnesen-style and inverse Bonnesen-style inequalities.
result Sharp discrete isoperimetric inequalities for planar convex polygons.
The study improves Bochner inequality on Finsler manifolds to derive important inequalities.
problem Improving Bochner inequality on Finsler manifolds to derive new inequalities.
method Using improved Bochner inequality and its integrated form, the study derives a sharp Poincaré-Lichnerowicz inequality, a new proof for logarithmic Sobolev inequality, and an estimate of geodesic ball volumes.
result Derivation of new inequalities and estimates on Finsler manifolds.
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
Sharp inequalities for star bodies in 2D space.
problem Understanding star bodies in 2D space.
method Sharp inequalities for star bodies in .
result New inequalities and proofs for star bodies.
The paper develops inequalities for log-concave functions and related surface areas.
problem Understanding log-concave functions and their inequalities.
method Establishing new inequalities through f-divergences and functional affine surface areas.
result New inequalities on functional affine surface area and bounds for Kullback-Leibler divergence.
Study on functional inequalities on simple edge spaces.
problem Whether classical functional inequalities hold in simple edge spaces.
method Analyzing Sobolev and Poincaré inequalities, proving optimality of Sobolev constant.
result Optimality result concerning the B-constant of the Sobolev inequality.
Proves inequalities on curved spaces with positive curvature.
problem Proving inequalities on manifolds with nonnegative Ricci curvature.
method Analyzes manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Proves Heisenberg-Pauli-Weyl, Hardy-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities.