Shiqian Ma

Riemannian Stochastic Proximal Gradient Methods for Nonsmooth Optimization over the Stiefel Manifold

  • Bokun Wang
  • Shiqian Ma
  • Lingzhou Xue
    • Categories Nonlinear Optimization, Statistics

    Riemannian optimization has drawn a lot of attention due to its wide applications in practice. Riemannian stochastic first-order algorithms have been studied in the literature to solve large-scale machine learning problems over Riemannian manifolds. However, most of the existing Riemannian stochastic algorithms require the objective function to be differentiable, and they do not apply to … Read more

    Efficiently Escaping Saddle Points in Bilevel Optimization

  • Minhui Huang
  • Kaiyi Ji
  • Lifeng Lai
  • Shiqian Ma
    • Categories Nonlinear Optimization Tags , , ,

    Bilevel optimization is one of the fundamental problems in machine learning and optimization. Recent theoretical developments in bilevel optimization focus on finding the first-order stationary points for nonconvex-strongly-convex cases. In this paper, we analyze algorithms that can escape saddle points in nonconvex-strongly-convex bilevel optimization. Specifically, we show that the perturbed approximate implicit differentiation (AID) with … Read more

    On the Convergence of Projected Alternating Maximization for Equitable and Optimal Transport

  • Minhui Huang
  • Shiqian Ma
  • Lifeng Lai
    • Categories Convex Optimization Tags , , , , , ,

    This paper studies the equitable and optimal transport (EOT) problem, which has many applications such as fair division problems and optimal transport with multiple agents etc. In the discrete distributions case, the EOT problem can be formulated as a linear program (LP). Since this LP is prohibitively large for general LP solvers, Scetbon \etal \cite{scetbon2021equitable} … Read more

    A Riemannian Block Coordinate Descent Method for Computing the Projection Robust Wasserstein Distance

  • Minhui Huang
  • Shiqian Ma
  • Lifeng Lai
    • Categories Nonlinear Optimization Tags , , ,

    The Wasserstein distance has become increasingly important in machine learning and deep learning. Despite its popularity, the Wasserstein distance is hard to approximate because of the curse of dimensionality. A recently proposed approach to alleviate the curse of dimensionality is to project the sampled data from the high dimensional probability distribution onto a lower-dimensional subspace, … Read more

    Projection Robust Wasserstein Barycenters

  • Minhui Huang
  • Shiqian Ma
  • Lifeng Lai
    • Categories Nonlinear Optimization Tags , ,

    Collecting and aggregating information from several probability measures or histograms is a fundamental task in machine learning. One of the popular solution methods for this task is to compute the barycenter of the probability measures under the Wasserstein metric. However, approximating the Wasserstein barycenter is numerically challenging because of the curse of dimensionality. This paper … Read more

    A Riemannian smoothing steepest descent method for non-Lipschitz optimization on submanifolds

  • Xiaojun Chen
  • Shiqian Ma
  • Chao Zhang
    • Categories Nonlinear Optimization Tags , , , ,

    In this paper, we propose a Riemannian smoothing steepest descent method to minimize a nonconvex and non-Lipschitz function on submanifolds. The generalized subdifferentials on Riemannian manifold and the Riemannian gradient sub-consistency are defined and discussed. We prove that any accumulation point of the sequence generated by the Riemannian smoothing steepest descent method is a stationary … Read more

    Zeroth-Order Algorithms for Nonconvex Minimax Problems with Improved Complexities

  • Krishnakumar Balasubramanian
  • Shiqian Ma
  • Meisam Razaviyayn
  • Zhongruo Wang
    • Categories Nonlinear Optimization

    In this paper, we study zeroth-order algorithms for minimax optimization problems that are nonconvex in one variable and strongly-concave in the other variable. Such minimax optimization problems have attracted significant attention lately due to their applications in modern machine learning tasks. We first design and analyze the Zeroth-Order Gradient Descent Ascent (ZO-GDA) algorithm, and provide … Read more

    Manifold Proximal Point Algorithms for Dual Principal Component Pursuit and Orthogonal Dictionary Learning

  • Shixiang Chen
  • Shiqian Ma
  • Anthony Man-Cho So
  • Zengde Deng
    • Categories Nonlinear Optimization

    We consider the problem of maximizing the $\ell_1$ norm of a linear map over the sphere, which arises in various machine learning applications such as orthogonal dictionary learning (ODL) and robust subspace recovery (RSR). The problem is numerically challenging due to its nonsmooth objective and nonconvex constraint, and its algorithmic aspects have not been well … Read more

    Riemannian Stochastic Proximal Gradient Methods for Nonsmooth Optimization over the Stiefel Manifold

  • Shiqian Ma
  • Bokun Wang
  • Lingzhou Xue
    • Categories Nonlinear Optimization

    Riemannian optimization has drawn a lot of attention due to its wide applications in practice. Riemannian stochastic first-order algorithms have been studied in the literature to solve large-scale machine learning problems over Riemannian manifolds. However, most of the existing Riemannian stochastic algorithms require the objective function to be differentiable, and they do not apply to … Read more

    A Manifold Proximal Linear Method for Sparse Spectral Clustering with Application to Single-Cell RNA Sequencing Data Analysis

  • Shixiang Chen
  • Shiqian Ma
  • Bingyuan Liu
  • Zhongruo Wang
  • Lingzhou Xue
  • Hongyu Zhao
    • Categories Nonlinear Optimization

    Spectral clustering is one of the fundamental unsupervised learning methods widely used in data analysis. Sparse spectral clustering (SSC) imposes sparsity to the spectral clustering and it improves the interpretability of the model. This paper considers a widely adopted model for SSC, which can be formulated as an optimization problem over the Stiefel manifold with … Read more