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.
Paper studies equatorial concentration of measure in sphere immersions and submersions.
problem Equatorial concentration of measure in sphere immersions and submersions.
method Analyzes concentration of measure phenomena in the sphere.
result Describes an equatorial concentration of measure for minimal immersions and submersions.
Eigenfunctions on manifolds concentrate near nodal sets.
problem Understanding concentration of eigenfunctions on manifolds.
method Volume measure analysis and Colding-Minicozzi method.
result Exponential concentration of volume measure around nodal sets.
The paper derives concentration inequalities for dynamic risk measures in a Brownian filtration context.
problem Liquidity risk in financial markets.
method Backward stochastic differential equations (BSDEs) and their dual formulation.
result Derives concentration inequalities for time-consistent dynamic risk measures in a Brownian filtration.
Paper proposes a new measure for systemic credit concentration risk.
problem Systemic credit concentration risk not fully captured by single institution measures.
method Network model to describe overlapping portfolios and quantify systemic risk.
result Network metric quantifies systemic risk not reflected in single portfolio measures.
This paper empirically measures intrinsic robustness of image classification models.
problem Understanding the robustness of image classification models under small perturbations.
method Empirical measurement of concentration in concrete datasets, using ℓ∞ and ℓ2 perturbations. result Empirical estimates of intrinsic robustness for various image classification benchmarks.
The paper studies concentration of measure on manifolds with boundary, focusing on 1-Lipschitz functions.
problem Concentration of measure phenomena of non-negative 1-Lipschitz functions on manifolds with Dirichlet boundary condition. method Examined relation between boundary concentration phenomena and large spectral gap phenomena of Dirichlet eigenvalues of Laplacian. Introduced new invariant called the observable inscribed radius.
result Formulated comparison theorems for the observable inscribed radius under lower Ricci curvature and mean curvature bounds for the boundary.
Improved estimation of concentration using half-spaces for adversarial vulnerability.
problem Understanding the concentration of measure phenomenon and its impact on adversarial vulnerability.
method Extending Gaussian Isoperimetric Inequality to non-spherical Gaussian measures and arbitrary ℓ_p-norms, using half-spaces to estimate concentration.
result Proposed method finds tighter intrinsic robustness bounds, providing evidence against concentration as a cause of adversarial vulnerability.
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…
New axioms justify ES without NRC, linking it to mean-ES portfolio selection.
problem Economic axioms for portfolio risk assessment and mean-ES portfolio selection.
method Introducing concentration aversion as an alternative to NRC, establishing axiomatic foundations.
result Concentration aversion uniquely characterizes the family of ES and provides new formulas.
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.
In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures correspondin…
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.
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 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.
Estimates spectral risk measures from i.i.d. samples.
problem Estimating spectral risk measures from limited data.
method Numerical integration method for SRM estimation.
result Estimate concentrates exponentially for bounded support distributions.
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 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.
New local ID estimators based on data separability.
problem Estimating intrinsic dimensionality locally in multi-dimensional data.
method Local estimators based on concentration of measure.
result Empirical comparison with other ID estimators.
Paper uses Wasserstein distance to improve risk measure bounds.
problem Improving concentration bounds for various risk measures.
method Unified approach based on Wasserstein distance to derive bounds for two risk measure classes.
result Bounds match or improve previous results for specific risk measures.
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…
Proves near-tight concentration for polynomial functions of high-temperature Ising models.
problem Understanding interactions in high-dimensional data like social networks.
method Proves concentration of measure for polynomial functions of the Ising model.
result Polynomial functions of high-temperature Ising models exhibit exponential tails with optimal radius.
A new outlier measure CFOF avoids concentration in high dimensions.
problem Outlier detection in high-dimensional data.
method Formalizes concentration of outlier scores, determines closed form distribution, and introduces fast-CFOF algorithm.
result CFOF is immune to concentration in high dimensions and has a clear statistical characterization.
Sample measures of top centile contributions to the total (concentration) are downward biased, unstable estimators, extremely sensitive to sample size and concave in accounting for large deviations. It makes them particularly unfit in domains with power law tails, especially for low values of the exponent. These estima…
Upper bounds on Wasserstein distance for empirical measures in unbounded functional spaces.
problem Analyzing convergence and concentration of empirical measures in unbounded functional spaces.
method Generalized upper bounds using Wasserstein distance, covering large dimensional Euclidean spaces and Gaussian processes.
result Rate-optimal upper bounds for functional data distributions with specific decay rates.
Study shows adversarial attacks are more likely when test instances are concentrated, leading to vulnerabilities in robust classifiers.
problem Adversarial attacks on robust classifiers when test instances are concentrated.
method Theoretical study linking concentration of measure to adversarial vulnerability, focusing on Levy families and product spaces.
result Adversarial attacks can be made effective with minimal perturbations when test instances are concentrated.
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.
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.
The measure concentration property of an mm-space X is roughly described as that any 1-Lipschitz map on X to a metric space Y is almost close to a constant map. The target space Y is called the screen. The case of Y=R is widely studied in many literature (see \cite{gromov}, \cite{ledoux}, \cite{mil2}…
Physics-informed neural networks improve subsurface transport parameter estimation from sparse data.
problem Estimating subsurface transport parameters from sparse measurements.
method Physics-informed neural networks (DNNs) for joint inversion of conductivity, hydraulic head, and concentration fields.
result Physics-informed DNNs yield significantly more accurate parameter estimates than standard DNNs.
This study assesses risk concentration in MDB portfolios using Monte Carlo simulations.
problem Risk concentration in MDB portfolios of a few borrowers.
method Realistic MDB portfolio simulations and Monte Carlo analysis.
result Current risk adjustments may be overly conservative.
Study non-asymptotic behavior of Coulomb gas on compact manifolds.
problem Understand the behavior of Coulomb gas on compact manifolds.
method Use Kantorovich-Wasserstein distance, empirical measure, and heat kernel.
result Prove concentration inequality in Kantorovich-Wasserstein distance.
Quantum neural networks converge to Gaussian processes as they grow.
problem Understanding the convergence of quantum neural networks to Gaussian processes.
method Analyzing Haar random unitary and orthogonal deep QNNs, considering input states, measurement observables, and non-independence of unitary matrix entries.
result Quantum neural networks outputs converge to Gaussian processes in the limit of large Hilbert space dimension.
Max-sliced Wasserstein metric reduces high-dimensional data to 1D for better estimation.
problem Curse of dimensionality in optimal transport.
method Introduces max-sliced Wasserstein metric to reduce high-dimensional problems to 1D.
result Uniform ratio bounds of empirical measures on RKHS concentrate uniformly fast at parametric rates.
New centrality measures for uncertain graphs using graphon theory.
problem Uncertainty in graph-based datasets hinders traditional centrality measures.
method Introducing centrality measures for graphons, a statistical approach based on graphon theory.
result Graphon centrality functions are robust to stochastic variations and provide uncertainty bounds.
Deep learning method improves risk assessment for small loan portfolios.
problem Measuring name concentration risk in small loan portfolios.
method Deep learning approach using Monte Carlo simulations with importance sampling.
result New method outperforms existing analytical methods for small portfolios.
This work improves information concentration for exp-concave distributions, making it dimension-independent.
problem Challenges in information concentration for log-concave distributions with dimension dependence.
method Proves exp-concavity leads to dimension-independent information concentration using a novel variance Brascamp-Lieb inequality.
result Information concentration depends only on the exp-concavity parameter, not the dimension.
In this paper, I shall demonstrate that sufficiently high-dimensional closed positively-curved Riemannian manifolds are either diffeomorphic to a spherical space form, or isometric to a locally compact rank one symmetric space. This surprising classification of positively-curved Riemannian manifolds results from combin…
Study shows how to reduce variational inference bias by concentrating likelihood ratio distribution.
problem Bias and variance issues in variational inference.
method Upper bound variational gap using dispersion measure of likelihood ratio, suggesting methods to reduce bias.
result Reducing bias in variational inference can be achieved by making likelihood ratio distribution more concentrated.
Study the singular limit of a boundary reaction equation, showing energy concentration and varifold support.
problem Analyzing the singular limit of a boundary reaction equation.
method Investigates the critical points of the boundary reaction equation \((-Δ)^{\frac{1}{2}}u = \frac{1}{\varepsilon}(u-u^3)\) in \(U \subset \mathbb{R}^n\).
result Shows existence of an (n−1)-rectifiable energy concentration set and associates limit energy measures to a stationary varifold. Study bounds expansion coefficient from observable diameter in metric measure spaces.
problem Bound the expansion coefficient from below in terms of the observable diameter.
method Considered concentration of measure phenomenon, connected observable diameter and expansion coefficient, derived upper bound, combined with lower bound to obtain upper bound for observable diameter.
result Obtained upper bound for observable diameter in terms of expansion coefficient.
Quantum kernel methods can lead to trivial models due to exponential concentration of kernel values.
problem Exponential concentration of quantum kernel values can lead to trivial models in QML.
method Analyzing the resources needed to accurately estimate quantum kernel values and identifying four sources of concentration.
result Quantum kernel values can be exponentially concentrated, leading to trivial models.
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.
Study improves convergence rates for GVI under prior misspecification.
problem Improving convergence rates for GVI under prior misspecification.
method Proves rates of convergence and robustness to prior misspecification in GVI framework.
result Establishes sufficient conditions for existence and uniqueness of GVI posteriors.
Machine learning models estimate nutrient concentrations from water quality surrogates.
problem Estimating high frequency nutrient concentrations from limited in-situ measurements.
method Used machine learning (Random Forests) to estimate nutrient concentrations using surrogate measures.
result Reduced RMSE by up to 60.1% compared to linear models, with additional sensors not providing significant benefits.
New wavelets use Monte Carlo for efficient discretization.
problem Efficiently discretizing continuous wavelets on general domains.
method Defined continuous wavelets via spectral calculus and proposed a Monte Carlo discretization.
result Convergence of Monte Carlo wavelets under natural regularity assumptions.
Study on stock portfolio concentration among Finnish households and investors.
problem Understanding the concentration of stock portfolios owned by Finnish households and investors.
method Analysis of stock portfolios using Herfindahl-Hirschman index over 20 years.
result High portfolio concentration observed in Finnish retail investors, similar to institutional investors.
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.