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.
Paper addresses concentration of distances for fractional quasi p-norms, identifying conditions for concentration and anti-concentration.
problem Understanding concentration of distances for fractional quasi p-norms in high dimensions.
method Analyzes conditions for concentration and anti-concentration of distances for fractional quasi p-norms.
result Identifies conditions for concentration and anti-concentration of fractional quasi p-norms, ruling out some approaches and specifying conditions for control.
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.
Unified framework for measuring concentration in weighted networks considering both weight distributions and network structure.
problem Traditional indices neglect the topology of relationships among network elements.
method Develops a family of topology-aware concentration indices that jointly account for weight distributions and network structure.
result The proposed indices preserve key properties and allow concentration to be evaluated across different dimensions of dependence.
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.
We analyze a plug-in estimator for a large class of integral functionals of one or more continuous probability densities. This class includes important families of entropy, divergence, mutual information, and their conditional versions. For densities on the d-dimensional unit cube [0,1]d that lie in a β-Hölder s…
The paper studies Dirac operators and their solutions concentrating near singular sets.
problem Understanding concentration properties of solutions to Dirac equations.
method Analyzes Dirac operators of the form Dε=D+ε−1A and their solutions. result Solutions concentrate exponentially near the locus where the rank of ker(A) jumps. We obtain sharp bounds on the performance of Empirical Risk Minimization performed in a convex class and with respect to the squared loss, without assuming that class members and the target are bounded functions or have rapidly decaying tails. Rather than resorting to a concentration-based argument, the method used her…
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.
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 (ρ(λX))λ≥0, where ρ is a convex risk measure and X a random variable, and we call such a curve a \emph{liqu…
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.
This work achieves exponential concentration in heavy-tailed data over CAT(κ) spaces using the Fréchet median.
problem Achieving robust estimation in heavy-tailed data distributions.
method Developing a concentration bound for the Fréchet median in CAT(κ) spaces.
result Exponential concentration of the Fréchet median in CAT(κ) spaces over heavy-tailed data.
This paper presents a unified approach based on Wasserstein distance to derive concentration bounds for empirical estimates for two broad classes of risk measures defined in the paper. The classes of risk measures introduced include as special cases well known risk measures from the finance literature such as condition…
Improved analysis for extreme multi-class CRL with better sample complexity.
problem Theoretical sample complexity of CRL in extreme multi-class settings is poorly understood.
method Improved U-Statistics estimator to capture class concentration, proving O(k) sample complexity. result Sample complexity is O(k) for extreme multi-class learning, independent of class distribution. Greedy algorithm achieves sublinear regret for various distributions.
problem Efficient performance of greedy algorithms in linear contextual bandit problems.
method Introduced Local Anti-Concentration (LAC) condition to ensure sublinear regret.
result Greedy algorithm achieves O(polylogT) cumulative expected regret. 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…
Paper tackles offline RL with weak assumptions on both function classes and data coverage.
problem Achieve sample-efficient offline RL with weak assumptions on both factors.
method Simple algorithm based on primal-dual formulation of MDPs, with density-ratio function modeling dual variables.
result Polynomial sample complexity achieved under realizability and single-policy concentrability.
We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point with…
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 Rd, including the strongly log-concave distributions u…
We study prediction and estimation problems using empirical risk minimization, relative to a general convex loss function. We obtain sharp error rates even when concentration is false or is very restricted, for example, in heavy-tailed scenarios. Our results show that the error rate depends on two parameters: one captu…
We consider closed immersed hypersurfaces in R3 and R4 evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time fo…
New method certifies anti-concentration for various non-Gaussian distributions.
problem Efficiently certifying anti-concentration for non-Gaussian distributions.
method Sum-of-Squares relaxation of integer program for anti-concentration.
result Quasi-polynomial time certificates for non-Gaussian distributions.
This paper contains a feasibility study of deep neural networks for the classification of Euro banknotes with respect to requirements of central banks on the ATM and high speed sorting industry. Instead of concentrating on the accuracy for a large number of classes as in the famous ImageNet Challenge we focus thus on c…
New methods correct spectral distortions using known analyte concentrations.
problem Distorted spectral shapes from absorbing and scattering contributions.
method Modified penalized baseline correction methods that incorporate known analyte concentrations.
result Improved prediction performance on near infra-red data sets.
We improve bounds for stochastic processes, especially those with heavy tails.
problem Bounding the concentration of sub-ψ 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.
Study convergence of Yamabe flow on singular spaces with positive constant.
problem Analyzing convergence of Yamabe flow on singular spaces.
method Normalized Yamabe flow with positive Yamabe constant on pseudo-manifolds, including stratified spaces.
result Established convergence under low energy condition and investigated alternatives.
Study on Dirichlet process mixtures for clustering consistency.
problem Consistency of clustering with Dirichlet process mixtures.
method Analysis of posterior distribution as sample size increases, focusing on consistency for the number of clusters.
result Consistency for the number of clusters can be achieved with a properly adapted concentration parameter in a Bayesian setting.
New inequalities for unbounded functions improve denoising score matching.
problem Statistical error bounds for denoising score matching with unbounded objective functions.
method Derive new concentration inequalities using McDiarmid's inequality and Rademacher complexity bounds.
result Improved statistical error bounds for denoising score matching.
Theory for algebraic data on categories via concentration structures.
problem Defining algebraic structures on categories.
method Introducing concentration structures and concentration monoids.
result Every group can be represented as a concentration monoid of a trivial category.
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.
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.
Many modern machine learning classifiers are shown to be vulnerable to adversarial perturbations of the instances. Despite a massive amount of work focusing on making classifiers robust, the task seems quite challenging. In this work, through a theoretical study, we investigate the adversarial risk and robustness of cl…
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 study examines the generalization of Macro-AUC in multi-label learning, identifying label imbalance as a critical factor.
problem Theoretical understanding of Macro-AUC in multi-label learning is lacking.
method Characterization of generalization properties of learning algorithms based on surrogate losses w.r.t. Macro-AUC, identification of label imbalance as a critical factor.
result The widely-used univariate loss-based algorithm is more sensitive to label imbalance than pairwise and reweighted loss-based ones, implying worse performance.
New concept of partial comonotonicity connects riskmetrics and dependence.
problem Understanding and quantifying risk metrics under partial comonotonicity.
method Developed a new notion of partial comonotonicity and established its connection to distortion riskmetrics.
result Partial comonotonicity uniquely characterizes a class of distortion riskmetrics through additivity.
Surfaces in 3-manifolds concentrate at curvature critical points.
problem Understanding concentration of surfaces in 3-manifolds.
method Proving surfaces concentrate at critical points of scalar curvature.
result Simply connected H-surfaces concentrate at curvature critical points.
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 survey recent results related to the concentration of eigenfunctions. We also prove some new results concerning ball-concentration, as well as showing that eigenfunctions saturating lower bounds for L1-norms must also, in a measure theoretical sense, have extreme concentration near a geodesic.
Paper develops sparse learning for heavy-tailed time series with locally stationary dynamics.
problem Sparse learning for high-dimensional heavy-tailed locally stationary time series.
method Additive modeling with kernel smoothing, sparsity-inducing penalized estimation.
result Prediction-error bounds and convergence rates for different sparsity structures.
A new model CDTM improves text classification by concentrating document topics.
problem Unsupervised text classification with diverse topic distributions.
method Imposes an exponential entropy penalty on document topic distribution to encourage concentration.
result More coherent topics and concentrated, sparse document-topic distributions.
New insights on offline RL with state aggregation and trajectory data.
problem Understanding sample complexity in offline policy evaluation.
method Analyzing concentrability coefficient in aggregated Markov Transition Model.
result Sample complexity depends on concentrability coefficient in aggregated model.
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
We consider the following singularly perturbed Neumann problem \begin{eqnarray*} \ve^2 Δu -u +u^p = 0 \, \quad u>0 \quad {\mbox {in}} \quad Ω, \quad {\partial u \over \partial ν}=0 \quad {\mbox {on}} \quad \partial Ω, \end{eqnarray*} where p>2 and Ω is a smooth and bounded domain in R2. We construct a new class…
Let B1 be a ball of radius r1 in $S^n(\Hy^n)$, and let B0 be a smaller ball of radius r0 such that B0ˉ⊂B1. For Sn we consider r1<π. Let u be a solution of the problem $-\La u =1$ in $\Om := B_1\setminus \bar{B_0}$ vanishing on the boundary. It is shown that the associated functional…
Study provides bounds for estimating intrinsic dimension using Gaussian kernels.
problem Estimating intrinsic dimension from data.
method Finite-sample concentration and anti-concentration bounds for Gaussian kernel sums.
result Explicit dependence on sample size, bandwidth, and geometric parameters.
Developed concentrated liquidity in n-dimensional AMM with polar coordinates in Rust.
problem Risk of stacking too many stablecoin pools.
method Building concentrated liquidity positions with ticks in polar coordinates in Rust.
result Hedging risk of stacking stablecoin pools.
In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative 1-Lipschitz functions with Dirichlet boundary condition around zero, which is called boundary concentration phenomena. We first examine relation between boundary concen…
Study on volume of tubes and concentration in Riemannian geometry.
problem Understanding concentration loci in Riemannian manifolds and their relation to tube volumes.
method Provided a general formula for tube volumes, specialized to totally geodesic submanifolds, and investigated concentration loci.
result Explicitly proved concentration for codimension one cases and explored characterizations in Wasserstein and Box distances.