arXiv research

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

Proves $C^{1,1}$ regularity for multiple membrane solutions.

ERM with $f$-divergence regularization yields unique solution.

Convex solutions to a specific equation are smooth when the phase is smooth enough.

We compute approximate solutions to L0 regularized linear regression using L1 regularization, also known as the Lasso, as an initialization step. Our algorithm, the Lass-0 ("Lass-zero"), uses a computationally efficient stepwise search to determine a locally optimal L0 solution given any L1 regularization solution. We …

Study proves existence of expanding solutions for multiphase surfaces with regular junctions.

New regularization method reduces support of empirical risk minimization solutions.

Non-convex regularizers usually improve the performance of sparse estimation in practice. To prove this fact, we study the conditions of sparse estimations for the sharp concave regularizers which are a general family of non-convex regularizers including many existing regularizers. For the global solutions of the regul…

We study the regularity properties for solutions of a class of Schrödinger equations $(Δ+ V) u = 0$ on a stratified space $M$ endowed with an iterated edge metric. The focus is on obtaining optimal Hölder regularity of these solutions assuming fairly minimal conditions on the underlying metric and potential.

We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler--Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under…

Paper proves smoothness of solutions to a complex geometric problem.

We establish interior regularity for convex viscosity solutions of the special Lagrangian equation. Our result states that all such solutions are real analytic in the interior of the domain.

The study shows how certain ODEs and integrals are regular under Borel summation.

New proof for convex solutions of Monge-Ampère equation.

Classifies regularity for Lagrangian mean curvature type equations.

Smooth solutions found for a specific type of Yamabe problem.

Investigates regularity of solutions to complex Hessian equation.

Improved iterative hard thresholding for faster, sparser solutions.

New algorithmic view of ℓ2 regularization using ODEs and path-following methods.

We establish higher-order weighted Sobolev and Holder regularity for solutions to variational equations defined by the elliptic Heston operator, a linear second-order degenerate-elliptic operator arising in mathematical finance. Furthermore, given $C^\infty$-smooth data, we prove $C^\infty$-regularity of solutions up t…

The regularity of weak solutions of a two-dimensional nonlinear sigma model with coarse gravitino is shown. Here the gravitino is only assumed to be in $L^p$ for some $p>4$. The precise regularity results depend on the value of $p$.

We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreove…

The paper studies how different entropic regularizations affect GAN solutions.

Regularizes 3D inverse scattering with tangent-point energy for better solutions.

This research smooths out fluid equations to avoid sudden shocks.

Solutions to a quadratic matrix equation are linked to strongly regular graphs and multiplicative characters.

Study on the smoothness of solutions to a specific type of stochastic differential equation.

Flexible per-class regularization improves binary classifiers.

Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.

We recast the Calabi flow in DeGiorgi's language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions, and that sufficiently regular minimizing movements are smooth soluti…

Unique solution found for financial system risk.

New black hole solutions with lens space horizons in 5D Kaluza-Klein theory.

Paper explores weak solutions' regularity in critical dimensions without conservation law.

For a variety of regularized optimization problems in machine learning, algorithms computing the entire solution path have been developed recently. Most of these methods are quadratic programs that are parameterized by a single parameter, as for example the Support Vector Machine (SVM). Solution path algorithms do not …

Proves higher regularity for anisotropic inverse mean curvature flow.

Careful tuning of a regularization parameter is indispensable in many machine learning tasks because it has a significant impact on generalization performances. Nevertheless, current practice of regularization parameter tuning is more of an art than a science, e.g., it is hard to tell how many grid-points would be need…

In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on Kähler manifolds. We first define a notion of weak solution of CSCK for an $L^\infty$ Kähler metric. The main result is to show that such a weak solution (with uniform $L^\infty$ bound…

Smooth solutions found for Hamiltonian stationary equations in low dimensions.

Concave regularization methods provide natural procedures for sparse recovery. However, they are difficult to analyze in the high dimensional setting. Only recently a few sparse recovery results have been established for some specific local solutions obtained via specialized numerical procedures. Still, the fundamental…

The paper confirms the existence of 5D regular static vacuum solutions with multiple black holes and Kasner asymptotics.

This work is focused on the solvability of initial-boundary value problems for degenerate parabolic partial differential equations that arise in the pricing of Asian options, and on the investigation of differential and certain qualitative properties of solutions of such equations. The generalized solvability for such …

We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle interrelation with solutions of the geodesic equations. Then we turn to the initial value …

We use convex relaxation techniques to provide a sequence of solutions to the matrix completion problem. Using the nuclear norm as a regularizer, we provide simple and very efficient algorithms for minimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm iteratively replaces the missing…

Entropy asymmetry affects regularization in ERM, leading to biased solutions.

The purpose of this work is to develop and study a distributed strategy for Pareto optimization of an aggregate cost consisting of regularized risks. Each risk is modeled as the expectation of some loss function with unknown probability distribution while the regularizers are assumed deterministic, but are not required…

We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerat…

Paper solves a complex equation for unbounded convex sets.

We provide theoretical analysis of the statistical and computational properties of penalized $M$-estimators that can be formulated as the solution to a possibly nonconvex optimization problem. Many important estimators fall in this category, including least squares regression with nonconvex regularization, generalized …