arXiv research

New method efficiently learns positive-definite curvature for neural nets.

The curvature tensor and the scalar curvature are computed in the space of positive definite real matrices endowed by the Kubo-Mori inner product as a Riemannian metric.

Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.

We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly roun…

Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.

Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.

For homogeneous metrics on the spaces of the title it is shown that the Ricci flow can move a metric of stricly positive sectional curvature to one with some negative sectional curvature and one of positive definite Ricci tensor to one with indefinite signature.

The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form can be satisfied only by extrinsic spheres.

Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.

New metrics defined on SPD matrices link to divergences and curvature.

Mathematical foundation for phylogenetic tree uncertainty quantification.

Local positive definite Z2^n-superfunctions can be extended.

Proves inequality for tensor fields on curved spaces.

We establish a one-to-one correspondence between Finsler structures on the $2$-sphere with constant curvature $1$ and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics are closed on the other hand. As a…

The paper defines and studies the geometric mean for tensors and its associated Riemannian geometry.

For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers and a theorem of S. A.…

Gaussian kernels on complex manifolds are never positive definite.

Study examines how wind affects shortest paths on Finsler manifolds.

Investigates concavity of spacetimes, showing conditions for local concavity.

Gaussian kernel fails on circle and related spaces.

Study of $H$-eigenvalues for complex tensors and their applications in differential geometry.

Fix a number $g>1$, let $S$ be a close surface of genus $g$, and $Teich(S)$ be the Teichmüller space of $S$ endowed with the Weil-Petersson metric. In this paper we show that the Riemannian sectional curvature operator of $Teich(S)$ is non-positive definite. As an application we show that any twist harmonic map from ra…

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and v…

We prove that a positive definite smooth four-manifold with $b_2^+ \geq 2$ and having either no 1-handles or no 3-handles cannot admit a symplectic structure.

New Sliced-Wasserstein distances for non-Euclidean data.

The Bergman kernels of holomorphic vector bundles are studied to extend the Fubini-Study map.

Improved Bayesian learning rule handles positive-definite constraints efficiently.

This paper studies geometric properties of Wasserstein metric on SPD(n).

New method optimizes on curved manifolds without curvature dependence.

On a Hermitian manifold we construct a symmetric $(1,1)$- tensor $H$ using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor $H$ for a harmonic $1$-form to be analytic and for an analytic $1$-form to be harm…

Gaussian belief propagation (GaBP) is an iterative algorithm for computing the mean of a multivariate Gaussian distribution, or equivalently, the minimum of a multivariate positive definite quadratic function. Sufficient conditions, such as walk-summability, that guarantee the convergence and correctness of GaBP are kn…

The paper defines functions that induce bounded composition operators on RKHSs with analytic positive definite functions.

Considering a non-constant smooth solution $f$ of the Tanno equation on a closed, connected Kähler manifold $(M,g,J)$ with positively definite metric $g$, Tanno showed that the manifold can be finitely covered by $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS})$, where $g_{FS}$ denotes the Fubini-Study metric of constant hol…

New $k$-means method clusters radar image sequences using SPD matrices.

Paper finds flag curvature of submanifolds in Randers-Minkowski space using Zermelo data.

This work improves understanding of symmetrizing Bregman divergences on positive definite matrices.

We consider a model for linear transient price impact for multiple assets that takes cross-asset impact into account. Our main goal is to single out properties that need to be imposed on the decay kernel so that the model admits well-behaved optimal trade execution strategies. We first show that the existence of such s…

Coercivity condition ensures learning of interacting particle systems.

This paper derives radial fields on manifolds of symmetric positive definite matrices.

Positive definite kernels are an important tool in machine learning that enable efficient solutions to otherwise difficult or intractable problems by implicitly linearizing the problem geometry. In this paper we develop a set-theoretic interpretation of the Earth Mover's Distance (EMD) and propose Earth Mover's Interse…

The Gaussian kernel is never positive-definite on Riemannian symmetric spaces.

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…

Extends Gaussian process theory to Banach spaces.

Study calculates Hessian of Busemann function on Damek-Ricci spaces.

In this article we raise some new questions about positive definite functions on free groups, and explain how these are related to more well-known questions. The article is intended as a survey of known results that also offers some new perspectives and interesting observations; therefore the style is expository.

Paper proves MS convergence for radially symmetric kernels with large bandwidths.

New method classifies manifold-valued data using Riemannian geometry.

In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold $(M,g)$ corresponding to e…