arXiv research

Derives Lagrangian for minimal surfaces, proving tangential variations vanish.

In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.

We provide sharp empirical estimates of expectation, variance and normal approximation for a class of statistics whose variation in any argument does not change too much when another argument is modified. Examples of such weak interactions are furnished by U- and V-statistics, Lipschitz L-statistics and various error f…

Study solves complex equation on specific types of manifolds.

New method proves hypersurface symmetry under weaker conditions.

This paper investigates the use of methods from partial differential equations and the Calculus of variations to study learning problems that are regularized using graph Laplacians. Graph Laplacians are a powerful, flexible method for capturing local and global geometry in many classes of learning problems, and the tec…

Sharp inequalities and eigenvalue problems on Finsler manifolds with nonnegative Ricci curvature.

We advocate an optimization-centric view on and introduce a novel generalization of Bayesian inference. Our inspiration is the representation of Bayes' rule as infinite-dimensional optimization problem (Csiszar, 1975; Donsker and Varadhan; 1975, Zellner; 1988). First, we use it to prove an optimality result of standard…

Simplified proof for Frank and Lieb's inequality on Heisenberg group.

Generalizes Nakano-positivity to Hilbert space fields.

Study optimal consumption with drawdown limits over a fixed time frame.

The aim of this paper is to provide new theoretical and computational understanding on two loss regularizations employed in deep learning, known as local entropy and heat regularization. For both regularized losses we introduce variational characterizations that naturally suggest a two-step scheme for their optimizatio…

Through the direct study of the analysis estimator we derive oracle inequalities with fast and slow rates by adapting the arguments involving projections by Dalalyan, Hebiri and Lederer (2017). We then extend the theory to the square root analysis estimator. Finally, we focus on (square root) total variation regularize…

Geometric structures help in understanding thermodynamics.

This work interprets SFA through variational inference, relaxing linearity constraints.

he celebrated formula of Schlafli relates the variation of the dihedral angles of a smooth family of polyhedra in a space form and the variation of volume. We give a smooth analogue of this classical formula -- our result relates the variation of the volume bounded by a hypersurface moving in a general Einstein manifol…

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the correspondi…

We characterize convex isoperimetric sets in the Heisenberg group endowed with horizontal perimeter. We first prove Sobolev regularity for a certain class of vector fields in the plane with bounded variation, related to the curvature equations. Then, by an approximation-reparameterization argument, we show that the bou…

In the paper "On Truncated Variation of Brownian Motion with Drift" (Bull. Pol. Acad. Sci. Math. 56 (2008), no.4, 267 - 281) we defined truncated variation of Brownian motion with drift, $W_t = B_t + μt, t\geq 0,$ where $(B_t)$ is a standard Brownian motion. Truncated variation differs from regular variation by neglect…

In this paper we are investigating variational homogeneous second order differential equations by considering the questions of how many different variational principles exist for a given spray. We focus our attention on h(2)-variationality; that is, the regular Lagrange function is homogeneous of degree two in the dire…

We study Hamiltonian stationary Lagrangian surfaces in C^2, i.e. Lagrangian surfaces in C^2 which are stationary points of the area functional under smooth Hamiltonian variations. Using loop groups, we propose a formulation of the equation as a completely integrable system. We construct a Weierstrass type representatio…

New deep learning model robust to adversarial attacks using stochastic LWTA units.

We combine conditional variational autoencoders (VAE) with adversarial censoring in order to learn invariant representations that are disentangled from nuisance/sensitive variations. In this method, an adversarial network attempts to recover the nuisance variable from the representation, which the VAE is trained to pre…

It is shown that $m$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with minimum Gaussian surface area must be $(m-1)$-dimensional. This follows from a second variation argument using infinitesimal translations. The special case $m=3$ proves the Double Bubble problem for the Gaussian measure,…

The paper connects neural network ensembles to Bayesian inference using variational methods.

A Closer Look at Disentangling in β-VAE shows non-monotonic inference performance.

Study finds existence of $Q$-curvature metrics on even-dimensional manifolds with conical singularities.

The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.

In this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime. The proof is based on both variational and geometric arguments involving the causal structure of the spacetime, the completeness of suitable Finsler metrics associated to it an…

In this paper, we study the prescribed $Q$-curvature problem on closed four-dimensional Riemannian manifolds when the total integral of the $Q$-curvature is a positive integer multiple of the one of the four-dimensional round sphere. This problem has a variational structure with a lack of compactness. Using some topolo…

This paper shows how to learn variational inequalities fast with strong monotonicity.

This paper investigates the dependence of functional portfolio generation, introduced by Fernholz (1999), on an extra finite variation process. The framework of Karatzas and Ruf (2017) is used to formulate conditions on trading strategies to be strong arbitrage relative to the market over sufficiently large time horizo…

DAGSurv uses deep neural networks to analyze survival data based on causal graphs.

We solve explicitly a two-dimensional singular control problem of finite fuel type for infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative and transient price impact. Liquidity is stochastic in that the volume effect process, which determi…

In the first part of the paper we investigate some geometric features of Moser-Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopo…

When used as a surrogate objective for maximum likelihood estimation in latent variable models, the evidence lower bound (ELBO) produces state-of-the-art results. Inspired by this, we consider the extension of the ELBO to a family of lower bounds defined by a particle filter's estimator of the marginal likelihood, the …

We prove that Morrison and Nieh's categorification of the su(3) quantum knot invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified su(2) theory, which was not functorial as originally defined. We use methods of Bar-Natan to construct explicit chain maps for each variation of…

Generating versatile and appropriate synthetic speech requires control over the output expression separate from the spoken text. Important non-textual speech variation is seldom annotated, in which case output control must be learned in an unsupervised fashion. In this paper, we perform an in-depth study of methods for…

In this work we address the problem of argument search. The purpose of argument search is the distillation of pro and contra arguments for requested topics from large text corpora. In previous works, the usual approach is to use a standard search engine to extract text parts which are relevant to the given topic and su…

We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form $H(X)=1+{A}{|X|^{-γ}}$ for $|X|$ large, when $A<0$ and $γ\in(0,2)$. Such surfaces are close to sections of unduloids with small necksize, fold…

A new geometric flow $K$-flow on 3-manifolds shrinks or preserves homogeneous spheres.

Several models of stock trading [P. Bak et al, Physica A {\bf 246}, 430 (1997)] are analyzed in analogy with one-dimensional, two-species reaction-diffusion-branching processes. Using heuristic and scaling arguments, we show that the short-time market price variation is subdiffusive with a Hurst exponent $H=1/4$. Biase…

Study on holomorphic curves in 6-sphere with boundary conditions.

Study optimal consumption for loss-averse agents considering past spending peaks.

The paper solves curvature prescription on a disk with negative Gaussian curvature.

The classical approach to inverse problems is based on the optimization of a misfit function. Despite its computational appeal, such an approach suffers from many shortcomings, e.g., non-uniqueness of solutions, modeling prior knowledge, etc. The Bayesian formalism to inverse problems avoids most of the difficulties en…

We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carathéodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preservin…

Stochastic LWTA networks resist adversarial attacks while maintaining accuracy.