arXiv research

Variationality of conformal geodesics fails in higher dimensions.

The paper studies the dimension of limit sets using variational principles and stationary measures.

In 3D, conformal geodesics are variational.

Recent work used importance sampling ideas for better variational bounds on likelihoods. We clarify the applicability of these ideas to pure probabilistic inference, by showing the resulting Importance Weighted Variational Inference (IWVI) technique is an instance of augmented variational inference, thus identifying th…

There are a number of examples of variations of Hodge structure of maximum dimension. However, to our knowledge, those that are global on the level of the period domain are totally geodesic subspaces that arise from an orbit of a subgroup of the group of the period domain. That is, they are defined by Lie theory rather…

The paper calculates variations of Einstein-Hilbert action on CR manifolds.

A new framework for Einstein-Hilbert action with topological variations.

The paper shows that almost every path structure is not variational.

We analyze the structure of the boundary terms in the conformal anomaly integrated over a manifold with boundaries. We suggest that the anomalies of type B, polynomial in the Weyl tensor, are accompanied with the respective boundary terms of the Gibbons-Hawking type. Their form is dictated by the requirement that they …

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

Over the years data has become increasingly higher dimensional, which has prompted an increased need for dimension reduction techniques. This is perhaps especially true for clustering (unsupervised classification) as well as semi-supervised and supervised classification. Although dimension reduction in the area of clus…

Study fourth-order geometric flow of shape operator for co-dimension one immersions.

Improves bandit convex optimization with gradient variations.

BBVI converges nearly dimensionally independent for log-concave targets.

tvGP-VAE models tensor-valued latent variables with Gaussian processes for better data structure representation.

We use a method, inspired by Pohozeav's work, to study asymptotic behaviors of non-variational elliptic systems in dimension n greater than two. The results apply to changing sign solutions.

Paper optimizes classification of distributions using Wasserstein metric.

A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a ge…

The paper studies the smoothness of critical points of variational integrals on Hessian spaces.

Proves continuity and singular set dimension for 2D maps with Q values.

Anomaly detection using dimensionality reduction has been an essential technique for monitoring multidimensional data. Although deep learning-based methods have been well studied for their remarkable detection performance, their interpretability is still a problem. In this paper, we propose a novel algorithm for estima…

Bayesian neural networks improve uncertainty calibration without sacrificing accuracy.

The space of Sobolev connections, as it has been introduced for studying the variation of Yang-Mills Lagrangian in the critical dimension $4$, happens not to be weakly sequentially complete in dimension larger than $4$. This is a major obstruction for studying the variations of this important Lagrangian in high dimensi…

Develops a fast variational approximation for high-dimensional empirical Bayes posteriors.

The study provides a sample complexity estimate for multi-category classifiers with bounded variation.

VFlow enhances generative flows by augmenting data dimensions for better expressiveness.

The main result of this paper is, that for convex billiards in higher dimensions, in contrast with 2D case, for every point on the boundary and for every $n$ there always exist billiard trajectories developing conjugate points at the $n$-th collision with the boundary. We shall explain that this is a consequence of the…

This paper is a sequel to \cite{Choi} in Math. Ann. In that paper we studied the subharmonicity of Kähler-Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to $3$. In this paper, we study the variations Kähler-Einstein metrics on bounded strongly pseudoconvex domains of dimension $2$.…

In this short note we prove that, in dimension three, flat metrics are the only complete metrics with non-negative scalar curvature which are critical for the $σ_{2}$-curvature functional.

VAE global minima can learn correct manifold dimensions, even with conditioning variables.

In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study $(κ,N)$-convex functions on metric spaces where $κ$ is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that …

Extends Dirac structures to infinite dimensions for mechanical systems.

We present a new class of solutions for the inverse problem in the calculus of variations in arbitrary dimension $n$. This is the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. We also provide a number of new theorems concerning the in…

A conformally invariant generalization of the Willmore energy for compact immersed submanifolds of even dimension in a Riemannian manifold is derived and studied. The energy arises as the coefficient of the log term in the renormalized area expansion of a minimal submanifold in a Poincare-Einstein space with prescribed…

It is proved that the set of geodesic circles in two dimensions may be given a variational description and the explicit form of it is presented. In the limit case of the Euclidean geometry a certain claim of uniqueness of such description is proved. A formal notion of 'spin' force is discovered as a by-product of the v…

Proposes VAE-KRnet for density estimation and variational Bayes.

This study analyzes VAEs using ID and II, revealing a transition in behaviour and distinct training phases.

We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes in particular all doubling spaces, metric trees, …

Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Emery's curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distribut…

Black box variational inference allows researchers to easily prototype and evaluate an array of models. Recent advances allow such algorithms to scale to high dimensions. However, a central question remains: How to specify an expressive variational distribution that maintains efficient computation? To address this, we …

Improved robustness for high-dimensional Kalman filtering.

This work analyzes Gibbs samplers for Bayesian hierarchical models without dimensionality constraints.

Study efficient neural operator learning using variation spaces.

Non-trivial obstructions found for topological solitons in Yang-Mills-Chern-Simons theories.

uHMC achieves fast mixing in high dimensions with gradient evaluations.

In recent years, data have become increasingly higher dimensional and, therefore, an increased need has arisen for dimension reduction techniques for clustering. Although such techniques are firmly established in the literature for multivariate data, there is a relative paucity in the area of matrix variate, or three-w…

We present a Donaldson-Witten type field theory in eight dimensions on manifolds with $Spin(7)$ holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi…

GD-VAEs learn dynamics from observations using geometric and topological information.