Salar Fattahi

A Parametric Approach for Solving Convex Quadratic Optimization with Indicators Over Trees

  • Aaresh Bhathena
  • Salar Fattahi
  • Andrés Gómez
  • Simge Küçükyavuz
    • Categories (Mixed) Integer Nonlinear Programming, Dynamic Programming, Quadratic Programming Tags , , , , ,

    This paper investigates convex quadratic optimization problems involving $n$ indicator variables, each associated with a continuous variable, particularly focusing on scenarios where the matrix $Q$ defining the quadratic term is positive definite and its sparsity pattern corresponds to the adjacency matrix of a tree graph. We introduce a graph-based dynamic programming algorithm that solves this … Read more

    Solution Path of Time-varying Markov Random Fields with Discrete Regularization

  • Salar Fattahi
  • Andrés Gómez
    • Categories Dynamic Programming, Integer Programming, Optimization in Data Science

    We study the problem of inferring sparse time-varying Markov random fields (MRFs) with different discrete and temporal regularizations on the parameters. Due to the intractability of discrete regularization, most approaches for solving this problem rely on the so-called maximum-likelihood estimation (MLE) with relaxed regularization, which neither results in ideal statistical properties nor scale to the … Read more

    On the Optimization Landscape of Burer-Monteiro Factorization: When do Global Solutions Correspond to Ground Truth?

  • Jianhao Ma
  • Salar Fattahi
    • Categories Global Optimization, Nonlinear Optimization, Optimization in Data Science Tags , ,

    In low-rank matrix recovery, the goal is to recover a low-rank matrix, given a limited number of linear and possibly noisy measurements. Low-rank matrix recovery is typically solved via a nonconvex method called Burer-Monteiro factorization (BM). If the rank of the ground truth is known, BM is free of sub-optimal local solutions, and its true solutions … Read more

    Simple Alternating Minimization Provably Solves Complete Dictionary Learning

  • lianggy
  • Gavin Zhang
  • Salar Fattahi
  • Richard Y. Zhang
    • Categories Optimization in Data Science

    This paper focuses on complete dictionary learning problem, where the goal is to reparametrize a set of given signals as linear combinations of atoms from a learned dictionary. There are two main challenges faced by theoretical and practical studies of dictionary learning: the lack of theoretical guarantees for practically-used heuristic algorithms, and their poor scalability … Read more

    Behind the Scenes of Gradient Descent: A Trajectory Analysis via Basis Function Decomposition

  • Jianhao Ma
  • Lingjun Guo
  • Salar Fattahi
    • Categories Optimization in Data Science

    This work analyzes the solution trajectory of gradient-based algorithms via a novel basis function decomposition. We show that, although solution trajectories of gradient-based algorithms may vary depending on the learning task, they behave almost monotonically when projected onto an appropriate orthonormal function basis. Such projection gives rise to a basis function decomposition of the solution … Read more

    Blessing of Nonconvexity in Deep Linear Models: Depth Flattens the Optimization Landscape Around the True Solution

  • Jianhao Ma
  • Salar Fattahi
    • Categories Nonlinear Optimization, Statistics Tags , ,

    This work characterizes the effect of depth on the optimization landscape of linear regression, showing that, despite their nonconvexity, deeper models have more desirable optimization landscape. We consider a robust and over-parameterized setting, where a subset of measurements are grossly corrupted with noise and the true linear model is captured via an $N$-layer linear neural … Read more

    Preconditioned Gradient Descent for Overparameterized Nonconvex Burer–Monteiro Factorization with Global Optimality Certification

  • Gavin Zhang
  • Salar Fattahi
  • Richard Y. Zhang
    • Categories Nonlinear Optimization Tags , ,

    We consider using gradient descent to minimize the nonconvex function $f(X)=\phi(XX^{T})$ over an $n\times r$ factor matrix $X$, in which $\phi$ is an underlying smooth convex cost function defined over $n\times n$ matrices. While only a second-order stationary point $X$ can be provably found in reasonable time, if $X$ is additionally \emph{rank deficient}, then its … Read more

    Global Convergence of Sub-gradient Method for Robust Matrix Recovery: Small Initialization, Noisy Measurements, and Over-parameterization

  • Jianhao Ma
  • Salar Fattahi
    • Categories Statistics, Unconstrained Optimization Tags ,

    In this work, we study the performance of sub-gradient method (SubGM) on a natural nonconvex and nonsmooth formulation of low-rank matrix recovery with $\ell_1$-loss, where the goal is to recover a low-rank matrix from a limited number of measurements, a subset of which may be grossly corrupted with noise. We study a scenario where the … Read more

    A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators

  • Peijing Liu
  • Salar Fattahi
  • Andrés Gómez
  • Simge Küçükyavuz
    • Categories (Mixed) Integer Nonlinear Programming, Statistics Tags , , , , , ,

    In this paper, we consider convex quadratic optimization problems with indicator variables when the matrix Q defining the quadratic term in the objective is sparse. We use a graphical representation of the support of Q, and show that if this graph is a path, then we can solve the associated problem in polynomial time. This … Read more

    Implicit Regularization of Sub-Gradient Method in Robust Matrix Recovery: Don’t be Afraid of Outliers

  • Salar Fattahi
  • Jianhao Ma
    • Categories Nonlinear Optimization, Nonsmooth Optimization, Statistics Tags , ,

    It is well-known that simple short-sighted algorithms, such as gradient descent, generalize well in the over-parameterized learning tasks, due to their implicit regularization. However, it is unknown whether the implicit regularization of these algorithms can be extended to robust learning tasks, where a subset of samples may be grossly corrupted with noise. In this work, … Read more