arXiv research

A new GAN loss function based on cumulant generating functions improves stability and robustness.

This paper introduces a variational approximation framework using direct optimization of what is known as the {\it scale invariant Alpha-Beta divergence} (sAB divergence). This new objective encompasses most variational objectives that use the Kullback-Leibler, the R{é}nyi or the gamma divergences. It also gives access…

Study of limits of Einstein-Bogomol'nyi metrics on P^1 in two regimes.

In this paper we construct new solutions of the Kahler-Yang-Mills equations, by applying dimensional reduction methods to the product of the complex projective line with a compact Riemann surface. The resulting equations, that we call gravitating vortex equations, describe Abelian vortices on the Riemann surface with b…

The paper finds shape modes for vortices in a specific sigma model.

In this work we consider the gravitating vortex equations. These equations couple a metric over a compact Riemann surface with a hermitian metric over a holomorphic line bundle equipped with a fixed global section --- the Higgs field ---, and have a symplectic interpretation as moment-map equations. As a particular cas…

We obtain all possible solutions of a 1/4 Bogomol'nyi-Prasad-Sommerfield equation exactly, containing configurations made of walls, vortices and monopoles in the Higgs phase. We use supersymmetric U(N_C) gauge theories with eight supercharges with N_F fundamental hypermultiplets in the strong coupling limit. The moduli…

Gradient descent converges with arbitrary stepsize for separable data under Fenchel-Young losses.

Existence and uniqueness of gravitating vortices on Riemann surfaces with specific properties.

Solves existence of gravitating vortices with positive curvature.

Paper introduces Lambda EVaR, a new risk measure.

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the $L_{r}$ operator associated to immersed hypersurfaces with locally bounded $(r+1)$-th mean curvature $H_{r+1}$ of the space forms …

We consider a general 4n-dimensional quaternionic Kahler geometry with a free action of the torus T^(n+1). The toric action lifts onto the Swann bundle of the quaternionic Kahler space to a tri-holomorphic action that commutes with the standard H* action on the bundle. By matching Pedersen and Poon's generalized Gibbon…

We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\…

We present a systematic method to construct exactly all Bogomol'nyi-Prasad-Sommerfield (BPS) multi-wall solutions in supersymmetric (SUSY) U(N_C) gauge theories in five dimensions with N_F hypermultiplets in the fundamental representation for infinite gauge coupling. The moduli space of these non-Abelian walls is found…

A gas of $N$ Bogomol'nyi vortices in the Abelian Higgs model is studied on a compact Riemann surface of genus $g$ and area $A$. The volume of the moduli space is computed and found to depend on $N, g$ and $A$, but not on other details of the shape of the surface. The volume is then used to find the thermodynamic partit…

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

New KL-divergence for Gaussian distributions based on Wasserstein geometry.

Develops a contraction framework for MCMC mixing rates.

The paper introduces a new divergence measure for variational autoencoders to improve reconstruction and generation.

We establish area bounds for two-dimensional immersions in R^3 and R^n. Namely, for μ-stable immersions in R^3 (R^n), for graphs in $\mathbb R^3$ which solve quasilinear equations in divergence form, and for graphs which are critical for Fermat-type variational problems in R^n.

This handbook translates lead time analysis into R code.

We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any $n$-dimensional ($n\geq 4$) gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite q…

New proof of Willmore inequality using geometric divergence inequality.

PLA improves sampling from distributions under isoperimetry with faster KL divergence convergence.

Let $\mathcal{R}$ be an open Riemann surface. In this paper we prove that every continuous function $M \to \mathbb{R}^n$, $n\ge 3$, defined on a divergent Jordan arc $M \subset \mathcal{R}$ can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theor…

Paper connects rejection learning to Bhattacharyya divergence.

New variational formula for Rényi divergences improves neural network estimation in high dimensions.

New method minimizes robust density power-based divergences for general parametric densities.

A study on $\alpha$-GANs proving convergence and estimation guarantees.

Unified framework for network model assessment using maximum entropy.

New solutions found for a complex equation, diverging from a cone.

The Bogomol'nyi-Prasad-Sommerfield (BPS) multi-wall solutions are constructed in supersymmetric U(N_C) gauge theories in five dimensions with N_F(>N_C) hypermultiplets in the fundamental representation. Exact solutions are obtained with full generic moduli for infinite gauge coupling and with partial moduli for finite …

A new method speeds up computation of Sinkhorn divergences to linear time.

The study explains YouTube commenters' behavior using rational inattention models.

This paper raises an implicit manifold learning perspective in Generative Adversarial Networks (GANs), by studying how the support of the learned distribution, modelled as a submanifold $\mathcal{M}_θ$, perfectly match with $\mathcal{M}_{r}$, the support of the real data distribution. We show that optimizing Jensen-Sha…

Graph Laplacians converge under symmetric divergence conditions.

Let G be the real points of a semisimple algebraic Q-group, let H be an arithmetic subgroup of G and let T be the real points of an R-split torus in G. We prove that if there is a divergent T-orbit in G/H, and Q-rank(G) > 1, then the dimension of T is not larger than Q-rank(G). This provides a partial answer to a quest…

Our main result is that if a generic convex domain in $\R^n$ collapses to a domain in $\R^{n-1}$, then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the domain need not be smooth, merely Lipschitz continuous. To motivate th…

In this letter, we generalize the convolutional NMF by taking the $β$-divergence as the contrast function and present the correct multiplicative updates for its factors in closed form. The new updates unify the $β$-NMF and the convolutional NMF. We state why almost all of the existing updates are inexact and approximat…

Study optimal transport costs with zero MTW tensor, finding new families of costs and divergence functions.

We prove a Filling Theorem for the Heisenberg Groups $H^{2n+1}$: For a given $k$-cycle $a$ we construct a $(k+1)$-chain $b$ (the filling) with boundary $\partial b=a$ and controlled volume. For this filling $b$ we prove a uniform bound on the distance of points in $b$ to its boundary $a$. Using this we compute the high…

We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than 1/r. The divergence theorem then yields a pair of integral equations for the norm…

This study provides an explicit expansion of KL divergence's gradient flow in Fisher-Rao geometry.

We give a generalization of the Nambu mechanics based on vector Hamiltonians theory. It is shown that any divergence-free phase flow in $\mathbb{R}^n$ can be represented as a generalized Nambu mechanics with $n-1$ integral invariants. For the case when the phase flow in $\mathbb{R}^n$ has $n-3$ or less first integrals,…

Unified framework for data-free sampling using Wasserstein gradient flows.

In this paper, we consider an infinite dimensional exponential family, $\mathcal{P}$ of probability densities, which are parametrized by functions in a reproducing kernel Hilbert space, $H$ and show it to be quite rich in the sense that a broad class of densities on $\mathbb{R}^d$ can be approximated arbitrarily well i…

The coefficient of determination, known as $R^2$, is commonly used as a goodness-of-fit criterion for fitting linear models. $R^2$ is somewhat controversial when fitting nonlinear models, although it may be generalised on a case-by-case basis to deal with specific models such as the logistic model. Assume we are fittin…