arXiv research

We study in a uniform manner the properties of biconservative surfaces in arbitrary Riemannian manifolds. Biconservative surfaces being characterized by the vanishing of the divergence of a symmetric tensor field $S_2$ of type $(1,1)$, their properties will follow from general properties of a symmetric tensor field of …

The study proves inequalities for complex operators on curved spaces.

This paper studies generic properties of connections on vector bundles, solving cohomological equations and proving opaque connections.

New loss functions based on f-divergences improve language model performance.

We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more general than similar classes defined by other authors. We revisit Spakula's uniform K-…

The study establishes inequalities for functions on manifolds using Green function estimates.

The paper finds inequalities for eigenvalues of operators on immersed manifolds.

The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the "Bregman function"). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, o…

New sampler improves uniform sampling over convex bodies with fewer queries.

Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.

Paper analyzes Annealed Langevin Dynamics for multimodal sampling stability.

Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be the Hausdorff dimension of the Gromov boundary $\partial X$. We define the critical exponent $δ(μ)$ of any discrete invariant random subgroup $μ$ of the locally compa…

The paper proves inequalities for twisted differential forms on manifolds.

We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to $r$-stability as well as to almost-Einstein hypersurfaces.

Study odd generalized Einstein metrics on 3D Lie groups.

This paper tackles post-trade allocation inefficiencies and presents a uniform return allocation method.

Study Euler characteristic of manifolds with almost nonnegative curvature operator, showing nonnegativity under certain conditions.

Upper bounds for eigenvalues on submanifolds in weighted manifolds.

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eige…

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constru…

This work presents a parametrized family of divergences, namely Alpha-Beta Log- Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infi…

The paper proves rigidity results for manifolds with special holonomy.

Improved error estimate for SGLD sampling algorithm.

We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based upon the Weyl's asymptotic formula. As an application, we find a lower bound for th…

The paper is devoted to differential geometry of singular distributions (i.e., of varying dimension) on a Riemannian manifold. Such distributions are defined as images of the tangent bundle under smooth endomorphisms. We prove the novel divergence theorem with the divergence type operator and deduce the Codazzi equatio…

New algorithms sample convex bodies using Markov chains and restricted Gaussian oracles.

We compute the expected value of the Kullback-Leibler divergence to various fundamental statistical models with respect to canonical priors on the probability simplex. We obtain closed formulas for the expected model approximation errors, depending on the dimension of the models and the cardinalities of their sample sp…

Establishes uniform Hörmander estimates for flat line bundles on Kähler manifolds.

We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K-homology, and that this class only depends on the principal symbol of the operator.

Continuous-time PCD for MLE with explicit error bounds.

We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Ampère equation of divergent type and, using the Gualtieri $\bar{\partial}$ operator, we characterize the conservation laws and the generating function of such equation as generalized holomorphic objects.

We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. The generalization will follow from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the uniform estimates …

Study flute surfaces and Loch Ness monster, proving their parabolicity and uniformization.

Uniform K-homology theory applied to elliptic operators on manifolds with boundary.

We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. This generalization will follow as a corollary from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the un…

Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.

Online learning of linear operators between infinite-dimensional spaces is possible but with limitations.

The paper explores how information geometry impacts classical CR inequalities.

We first present the natural definitions of the horizontal differential, the divergence (as an adjoint operator), and a $p$-harmonic form on a Finsler manifold. Next, we prove a Hodge-type theorem for a Finsler manifold in the sense that a horizontal $p$-form is harmonic if and only if the horizontal Laplacian vanishes…

We prove the Reilly formula for a class of elliptic divergence differential operator $L_Au=div(A\nabla u)$, where $A$ is a (1,1)-Codazzi tensor field. Then we get some estimates for the first positive eigenvalue of the operator.

We study the distribution of hard-, soft-, and adaptive soft-thresholding estimators within a linear regression model where the number of parameters k can depend on sample size n and may diverge with n. In addition to the case of known error-variance, we define and study versions of the estimators when the error-varian…

Defines T-duality and generalised Ricci flow relations using Courant algebroid relations.

New method detects communities in complex hypergraphs, matching theoretical limits.

This study approximates distances between Gaussian processes and covariance operators using RKHS.

Proposes a new divergence measure for probability distributions.

We define a very general "parametric connect sum" construction which can be used to eliminate isolated conical singularities of Riemannian manifolds. We then show that various important analytic and elliptic estimates, formulated in terms of weighted Sobolev spaces, can be obtained independently of the parameters used …

A new data-adaptive prior stabilizes kernel learning in operators.

Solves nonlinear problems on metric structures through eigenvalue counting.