arXiv research

Study Finsler metric measure manifolds' concentration properties.

Paper studies equatorial concentration of measure in sphere immersions and submersions.

Improved estimation of concentration using half-spaces for adversarial vulnerability.

Many recent works have shown that adversarial examples that fool classifiers can be found by minimally perturbing a normal input. Recent theoretical results, starting with Gilmer et al. (2018b), show that if the inputs are drawn from a concentrated metric probability space, then adversarial examples with small perturba…

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.

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 $(ρ(λX))_{λ\ge 0}$, where $ρ$ is a convex risk measure and $X$ a random variable, and we call such a curve a \emph{liqu…

Measurement and management of credit concentration risk is critical for banks and relevant for micro-prudential requirements. While several methods exist for measuring credit concentration risk within institutions, the systemic effect of different institutions' exposures to the same counterparties has been less explore…

New axioms justify ES without NRC, linking it to mean-ES portfolio selection.

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 $L^1$-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.

The paper offers a framework to analyze machine learning problems using concentration of measure.

The study characterizes heat flow and concentration on directed graphs with a lower Ricci curvature bound.

The paper generalizes product inequalities for random vectors and their applications.

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 …

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…

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…

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…

Study on volume of tubes and concentration in Riemannian geometry.

Data assimilation for parameter and state estimation in subsurface transport problems remains a significant challenge due to the sparsity of measurements, the heterogeneity of porous media, and the high computational cost of forward numerical models. We present a physics-informed deep neural networks (DNNs) machine lea…

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…

Study robust covariance estimation in large data with concentrated vectors.

Intrinsic dimensionality (ID) is one of the most fundamental characteristics of multi-dimensional data point clouds. Knowing ID is crucial to choose the appropriate machine learning approach as well as to understand its behavior and validate it. ID can be computed globally for the whole data point distribution, or comp…

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 $Y=\mathbb{R}$ is widely studied in many literature (see \cite{gromov}, \cite{ledoux}, \cite{mil2}…

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…

This study assesses risk concentration in MDB portfolios using Monte Carlo simulations.

Quantum neural networks converge to Gaussian processes as they grow.

As relational datasets modeled as graphs keep increasing in size and their data-acquisition is permeated by uncertainty, graph-based analysis techniques can become computationally and conceptually challenging. In particular, node centrality measures rely on the assumption that the graph is perfectly known -- a premise …

We present a novel notion of outlier, called the Concentration Free Outlier Factor, or CFOF. As a main contribution, we formalize the notion of concentration of outlier scores and theoretically prove that CFOF does not concentrate in the Euclidean space for any arbitrary large dimensionality. To the best of our knowled…

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…

We consider the problem of estimating a spectral risk measure (SRM) from i.i.d. samples, and propose a novel method that is based on numerical integration. We show that our SRM estimate concentrates exponentially, when the underlying distribution has bounded support. Further, we also consider the case when the underlyi…

Max-sliced Wasserstein metric reduces high-dimensional data to 1D for better estimation.

Deep learning method improves risk assessment for small loan portfolios.

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 the singular limit of a boundary reaction equation, showing energy concentration and varifold support.

Information concentration of probability measures have important implications in learning theory. Recently, it is discovered that the information content of a log-concave distribution concentrates around their differential entropy, albeit with an unpleasant dependence on the ambient dimension. In this work, we prove th…

Concerns about interpretability, computational resources, and principled inductive priors have motivated efforts to engineer sparse neural models for NLP tasks. If sparsity is important for NLP, might well-trained neural models naturally become roughly sparse? Using the Taxi-Euclidean norm to measure sparsity, we find …

Quantum kernel methods can lead to trivial models due to exponential concentration of kernel values.

We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence…

Study non-Gaussian measures' concentration properties in metric spaces.

Study improves convergence rates for GVI under prior misspecification.

Study on stock portfolio concentration among Finnish households and investors.

The paper proves concentration inequalities for two-sample rank processes and applies them to ranking performance criteria.

The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established -- along with its equality conditions.

We prove near-tight concentration of measure for polynomial functions of the Ising model under high temperature. For any degree $d$, we show that a degree-$d$ polynomial of a $n$-spin Ising model exhibits exponential tails that scale as $\exp(-r^{2/d})$ at radius $r=\tildeΩ_d(n^{d/2})$. Our concentration radius is opti…

NUTS mixing time scales as d^(1/4) for Gaussian distributions.

The paper analyzes sparse high-dimensional linear regression with random design and unknown error variance, providing adaptiveness and concentration rates.

We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the $\mathrm{U}$-functional of a convex body. For both results we provide stronger versions in the sense of stability i…