arXiv research

Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.

Upper bound for max-sliced 2-Wasserstein distance between measures.

Paper introduces symmetric divergence link models for probability distributions.

New neural network models learn symmetric functions of varying input sizes.

The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.

Study parametrized Kähler class for cocycles on Hermitian symmetric spaces.

We consider a finitely generated torsion free Kleinian group $H$ and a random walk on $H$ with respect to a symmetric nondegenerate probability measure $μ$ with finite support. When $H$ is geometrically infinite without parabolics or when $H$ is Gromov hyperbolic with parabolics, we prove that the Patterson-Sullivan me…

The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdré). Then we give a quan…

Let $Γ$ be a relatively hyperbolic group and let $μ$ be an admissible symmetric finitely supported probability measure on $Γ$. We extend Floyd-Ancona type inequalities up to the spectral radius of $μ$. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk o…

We associate certain probability measures on $\R$ to geodesics in the space $\H_L$ of positively curved metrics on a line bundle $L$, and to geodesics in the finite dimensional symmetric space of hermitian norms on $H^0(X, kL)$. We prove that the measures associated to the finite dimensional spaces converge weakly to t…

New dimension concept for groups based on percolation probability.

We consider fundamental questions of arbitrage pricing arising when the uncertainty model is given by a set of possible mutually singular probability measures. With a single probability model, essential equivalence between the absence of arbitrage and the existence of an equivalent martingale measure is a folk theorem,…

This article introduces a Bayesian nonparametric method for quantifying the relative evidence in a dataset in favour of the dependence or independence of two variables conditional on a third. The approach uses Polya tree priors on spaces of conditional probability densities, accounting for uncertainty in the form of th…

Due to Čencov's theorem, there exists a unique family of invariant symmetric $(0,2)$-tensor fields on the space of positive probability measures on a set of $n$-points indexed by $n\in \mathbb{N}$ under Markov embeddings. We deform Markov embeddings keeping sufficiency, and prove existence and uniqueness of invariant f…

We introduce an universum of the Polish (=complete separable metric) space - the convex cone of distance matrices and study its geometry. It happened that the generic Polish spaces in this sense of this universum is so called Urysohn spaces defined by P.S.Urysohn in 20-th, and generic metric triple (= metric space with…

Given an initial (resp., terminal) probability measure $μ$ (resp., $ν$) on $\mathbb{R}^d$, we characterize those optimal stopping times $τ$ that maximize or minimize the functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Brownian motion with initial law $B_0\sim μ$ and with final distribution --once stop…

Unified framework for information-theoretic bounds on learning algorithms.

New measures generalize existing ones, linking information and risk.

Nonlinear SGD achieves high-probability rates in non-convex optimization with heavy-tailed noise.

WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance …

Graph Laplacians converge under symmetric divergence conditions.

New findings on null measurability in symmetrization interface of VC learning.

We consider cocycles of isometries on spaces of nonpositive curvature $H$. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there…

Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.

Estimates the probability of a random symmetric tensor being close to rank-one.

Study optimal transport on simplex boundary, proving transport map and potential regularity.

This work accelerates constrained sampling using large deviation principles.

Solves Christoffel-Minkowski problem for axially symmetric bodies.

Bayesian approach to robust risk measures under model uncertainty.

The paper proves an inequality for symmetric polynomials under a fixed point measure.

Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. For instance, they were essential in the groundbreaking work of Preiss on the rectifiability of Radon measures. However, relatively little is understood about the structure of general uniform meas…

We investigate the probability distributions of the recurrence intervals $τ$ between consecutive 1-min returns above a positive threshold $q>0$ or below a negative threshold $q<0$ of two indices and 20 individual stocks in China's stock market. The distributions of recurrence intervals for positive and negative thresho…

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a …

For incomplete sub-Riemannian manifolds, and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditi…

The paper explores geometry of probability measures and barycenter maps.

The framework of this paper is that of risk measuring under uncertainty, which is when no reference probability measure is given. To every regular convex risk measure on ${\cal C}_b(Ω)$, we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless non positive elements of $…

We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with "lifting" and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techn…

The isotropy action on certain symmetric spaces is shown to be equivariantly formal.

In a market of deterministic cash flows, given as an additive, symmetric relation of exchangeability on the finite signed Borel measures on the non-negative real time axis, it is shown that the only arbitrage-free price functional that fulfills some additional mild requirements is the integral of the unit zero-coupon b…

We connect shift-invariant characteristic kernels to infinitely divisible distributions on $\mathbb{R}^{d}$. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures. The contribution of this paper is two-fold. First, we show, usi…

The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.

Paper introduces a new distance measure for Gaussian Mixture Models.

A novel method compares 3D point clouds using information geometry.

The paper reviews historical and modern approaches to asset pricing probability measures.

A new method quantizes conditional probability measures using deep learning.

A scalable approach to learning from probability measures using quantization.

A new approach to sensitivity analysis without the Sobol decomposition.

The paper constructs a Dirichlet form and proves functional inequalities for a specific measure.