low rank matrix

An extrapolated and provably convergent algorithm for nonlinear matrix decomposition with the ReLU function

  • Nicolas Gillis
  • Margherita Porcelli
  • Giovanni Seraghiti
    • Categories Data Science Algorithms, Optimization in Data Science, Other Topics Tags , ,

    Nonlinear matrix decomposition (NMD) with the ReLU function, denoted ReLU-NMD, is the following problem: given a sparse, nonnegative matrix \(X\) and a factorization rank \(r\), identify a rank-\(r\) matrix \(\Theta\) such that \(X\approx \max(0,\Theta)\). This decomposition finds application in data compression, matrix completion with entries missing not at random, and manifold learning. The standard ReLU-NMD … Read more

    A new perspective on low-rank optimization

  • Dimitris Bertsimas
  • Ryan Cory-Wright
  • Jean Pauphilet
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Semi-definite Programming Tags , , ,

    A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable convex relaxations. We invoke the matrix perspective function — the matrix analog of the perspective function — and characterize explicitly … Read more

    Augmented L1 and Nuclear-Norm Models with a Globally Linearly Convergent Algorithm

  • Ming-Jun Lai
  • Wotao Yin
    • Categories Nonsmooth Optimization Tags , , , ,

    This paper studies the long-existing idea of adding a nice smooth function to “smooth” a non-differentiable objective function in the context of sparse optimization, in particular, the minimization of $||x||_1+1/(2\alpha)||x||_2^2$, where $x$ is a vector, as well as those of the minimization of $||X||_*+1/(2\alpha)||X||_F^2$, where $X$ is a matrix and $||X||_*$ and $||X||_F$ are the … Read more