arXiv research

Sharp estimates for heat flow on nonconvex domains.

Researchers create nonconvex, non-soliton ancient flows in various dimensions.

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…

PWGF escapes saddle points in nonconvex optimization.

We establish Evans-Krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial Legendre transformations. The equations under consideration arise in part from the study of the "pluriclosed flow" introduced by the first author and Tian

We show that the pluriclosed flow preserves generalized Kähler structures with the extra condition $[J_+,J_-] = 0$, a condition referred to as "split tangent bundle." Moreover, we show that in this in this case the flow reduces to a nonconvex fully nonlinear parabolic flow of a scalar potential function. We prove a num…

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

We consider the problem of demixing a sequence of source signals from the sum of noisy bilinear measurements. It is a generalized mathematical model for blind demixing with blind deconvolution, which is prevalent across the areas of dictionary learning, image processing, and communications. However, state-of- the-art c…

We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in $\mathbb{R}^{n+1}$ for $n \geq 2$. These provide examples of mean convex yet nonconvex ancient solutions that are not solitons, meaning that they do not evolve by rigid motions or homotheties. …

Ancient solutions to curve shortening flow are constructed and analyzed.

Existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying "true" statistical models. To address this issue, we take a first step towards taming model misspecification by studying the high-dimensional sparse phase retrieval problem with misspecified link…

We consider a compact, star-shaped, mean convex hypersurface $Σ^2\subset \mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which …

Study birth-death dynamics for sampling Gibbs measures with nonconvex potentials.

We study the phase retrieval problem, which solves quadratic system of equations, i.e., recovers a vector $\boldsymbol{x}\in \mathbb{R}^n$ from its magnitude measurements $y_i=|\langle \boldsymbol{a}_i, \boldsymbol{x}\rangle|, i=1,..., m$. We develop a gradient-like algorithm (referred to as RWF representing reshaped W…

In this article, we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these constructions to create pathological examples of flows. We find a sequence of flows that …

Paper proves conjecture about star-shaped curves evolving under GAPF, but not always preserves star shape.

Adaptive sampling for multimodal distributions converges faster than classical methods.

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.

New algorithm for privacy-preserving nonconvex optimization.

Sparse generalized eigenvalue problem (GEP) plays a pivotal role in a large family of high-dimensional statistical models, including sparse Fisher's discriminant analysis, canonical correlation analysis, and sufficient dimension reduction. Sparse GEP involves solving a non-convex optimization problem. Most existing met…

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.

We consider the robust phase retrieval problem of recovering the unknown signal from the magnitude-only measurements, where the measurements can be contaminated by both sparse arbitrary corruption and bounded random noise. We propose a new nonconvex algorithm for robust phase retrieval, namely Robust Wirtinger Flow to …

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

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

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…

Probabilistic optimal power flow (POPF) is an important analytical tool to ensure the secure and economic operation of power systems. POPF needs to solve enormous nonlinear and nonconvex optimization problems. The huge computational burden has become the major bottleneck for the practical application. This paper presen…

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.

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.

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.

Safe reinforcement learning with nonconvex constraints using convex approximations.

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

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