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The dual IRLS scheme for (hyper-)graph $p$-Laplacians and $\ell^p$ regression with large exponents

Johannes Storn

Abstract

We introduce an iterative scheme for discrete convex minimization problems of $p$-Laplace type such as variational graph $p$-Laplace problems and $\ell^p$ regression. In each iteration, the scheme solves only a weighted least-squares problem. We verify linear convergence for suitably regularized problems and derive convergence to any prescribed tolerance.

The dual IRLS scheme for (hyper-)graph $p$-Laplacians and $\ell^p$ regression with large exponents

Abstract

We introduce an iterative scheme for discrete convex minimization problems of -Laplace type such as variational graph -Laplace problems and regression. In each iteration, the scheme solves only a weighted least-squares problem. We verify linear convergence for suitably regularized problems and derive convergence to any prescribed tolerance.

Paper Structure

This paper contains 14 sections, 11 theorems, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The (Hyper-)Graph $p$-Laplacian
  3. Minimization problem
  4. Applications
  5. Graph-based semi-supervised learning
  6. Hypergraph models
  7. Linear regression in $\ell^p$
  8. Dual IRLS scheme
  9. Dual problem
  10. Dual IRLS
  11. Regularization
  12. Applications
  13. Linear regression in $\ell^p$
  14. Graph $p$-Laplacian

Key Result

proposition 1

Let $\phi\colon \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ be an N-function and define the right-continuous inverse Then we have the following.

Figures (3)

  • Figure 1: Plot of derivatives $\phi'$ and $(\phi^*)'$. The area marked dark gray equals $\phi(s)$, the area marked light gray equals $\phi^*(r)$. The area surrounded by the dashed line equals $rs$.
  • Figure 2: Distance $\lVert Au^n - b \rVert_{\ell^p} - \lVert Au - b \rVert_{\ell^p}$ plotted against the number of iterations $n$ for the dIRLS scheme \ref{['line1']}, Newton's method \ref{['line2']}, and the $p$-IRLS scheme \ref{['line3']}.
  • Figure 3: The left-hand side displays the accuracy in predicting the $70\, 000$ digits/fashion items. The right-hand side displays the energy errors for the MNIST (thick line) and the Fashion MNIST (thin line).

Theorems & Definitions (24)

  • definition 1: N-function
  • proposition 1: Conjugate of an N-function
  • theorem 1: Duality
  • proof
  • remark 1: Built-in error control
  • definition 2: Uniform convexity
  • lemma 1: Alternative characterization
  • proof
  • theorem 2: Convergence of the dIRLS scheme
  • lemma 2: Equivalent assumptions
  • ...and 14 more