arXiv research

Sharp estimates for heat flow on nonconvex domains.

We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point. Our theorem is the analog of the result of Grayson for curvature flow of clo…

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $Ω\subset \mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $\partial C \cap Ω$ consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show…

Deep learning struggles with out-of-distribution data, so this paper tackles domain generalization.

A new model for dynamic covariance recovery in neuroimaging data.

This work uses Lasry-Lions envelopes to solve nonconvex optimization problems.

Two new methods improve block-sparse signal recovery from noisy data.

Paper improves robust PCA for noisy, outlier, and missing data.

Unified framework for constructing nonconvex sparse recovery methods.

Improved SGD methods converge faster for nonconvex optimization.

Schedule-free SGD is optimal for nonconvex optimization problems.

Stochastic gradient methods can converge in expectation under heavy-tailed noise.

We consider compressed sensing formulated as a minimization problem of nonconvex sparse penalties, Smoothly Clipped Absolute deviation (SCAD) and Minimax Concave Penalty (MCP). The nonconvexity of these penalties is controlled by nonconvexity parameters, and L1 penalty is contained as a limit with respect to these para…

PPGD solves nonconvex nonsmooth optimization problems without KL property.

New framework explains why nonconvex methods work well in low-rank matrix estimation.

Develops shuffling gradient-based methods for nonconvex-concave minimax optimization.

While many solutions for privacy-preserving convex empirical risk minimization (ERM) have been developed, privacy-preserving nonconvex ERM remains a challenge. We study nonconvex ERM, which takes the form of minimizing a finite-sum of nonconvex loss functions over a training set. We propose a new differentially private…

In the paper, we study the stochastic alternating direction method of multipliers (ADMM) for the nonconvex optimizations, and propose three classes of the nonconvex stochastic ADMM with variance reduction, based on different reduced variance stochastic gradients. Specifically, the first class called the nonconvex stoch…

This paper analyzes OGDA and EG methods for nonconvex minimax problems.

Simple DP algorithms find approximate solutions for nonconvex ERM.

Unified parametric assumption improves convergence guarantees for nonconvex optimization.

Support vector machines (SVMs) with sparsity-inducing nonconvex penalties have received considerable attentions for the characteristics of automatic classification and variable selection. However, it is quite challenging to solve the nonconvex penalized SVMs due to their nondifferentiability, nonsmoothness and nonconve…

Symmetric nonnegative matrix factorization has found abundant applications in various domains by providing a symmetric low-rank decomposition of nonnegative matrices. In this paper we propose a Frank-Wolfe (FW) solver to optimize the symmetric nonnegative matrix factorization problem under a simplicial constraint, whic…

In this paper, we study and analyze the mini-batch version of StochAstic Recursive grAdient algoritHm (SARAH), a method employing the stochastic recursive gradient, for solving empirical loss minimization for the case of nonconvex losses. We provide a sublinear convergence rate (to stationary points) for general noncon…

We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonconvex part is smooth and the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gra…

The use of convex regularizers allows for easy optimization, though they often produce biased estimation and inferior prediction performance. Recently, nonconvex regularizers have attracted a lot of attention and outperformed convex ones. However, the resultant optimization problem is much harder. In this paper, for a …

This work addresses the issue of large covariance matrix estimation in high-dimensional statistical analysis. Recently, improved iterative algorithms with positive-definite guarantee have been developed. However, these algorithms cannot be directly extended to use a nonconvex penalty for sparsity inducing. Generally, a…

New algorithm tackles nonconvex machine learning problems with adaptive normalization and independent sampling.

Fuzzy clustering with similarity queries improves efficiency and accuracy.

As surrogate functions of $L_0$-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex l…

With the large rising of complex data, the nonconvex models such as nonconvex loss function and nonconvex regularizer are widely used in machine learning and pattern recognition. In this paper, we propose a class of mini-batch stochastic ADMMs (alternating direction method of multipliers) for solving large-scale noncon…

Paper proposes a new method for training nonconvex models.

Two algorithms solve nonconvex minimax problems with linear constraints, achieving complexity guarantees.

Paper proposes an algorithm to solve complex minimax problems efficiently.

PAGE optimizes nonconvex problems with optimal convergence rates.

New algorithm solves nonconvex-convex minimax problems efficiently.

Develops efficient method for nonconvex problems using Regula Falsi.

New algorithms solve nonconvex-concave minimax problems without parameter knowledge.

AGDA and variance-reduced methods solve nonconvex-nonconcave minimax problems globally and faster.

Two new algorithms solve nonconvex-strongly concave problems efficiently.

New nonconvex penalty smooths at origin for deep learning.

Unified framework for nonconvex matrix completion with linearly parameterized factors.

Paper analyzes nonconvex bandit problems with improved adaptive methods.

Study uncovers statistical optimality of nonconvex tensor completion methods.

The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and gradually nonconvex composite functions have been adopted to obtain more desirable prop…

Innovative method solves nonconvex optimization on manifolds.

This work studies low-rank approximation of a positive semidefinite matrix from partial entries via nonconvex optimization. We characterized how well local-minimum based low-rank factorization approximates a fixed positive semidefinite matrix without any assumptions on the rank-matching, the condition number or eigensp…

We study Frank-Wolfe methods for nonconvex stochastic and finite-sum optimization problems. Frank-Wolfe methods (in the convex case) have gained tremendous recent interest in machine learning and optimization communities due to their projection-free property and their ability to exploit structured constraints. However,…