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Extended alternating structure-adapted proximal gradient algorithm for nonconvex nonsmooth problems

Ying Gao, Chunfeng Cui, Wenxing Zhang, Deren Han

Abstract

Alternating structure-adapted proximal (ASAP) gradient algorithm (M. Nikolova and P. Tan, SIAM J Optim, 29:2053-2078, 2019) has drawn much attention due to its efficiency in solving nonconvex nonsmooth optimization problems. However, the multiblock nonseparable structure confines the performance of ASAP to far-reaching practical problems, e.g., coupled tensor decomposition. In this paper, we propose an extended ASAP (eASAP) algorithm for nonconvex nonsmooth optimization whose objective is the sum of two nonseperable functions and a coupling one. By exploiting the blockwise restricted prox-regularity, eASAP is capable of minimizing the objective whose coupling function is multiblock nonseparable. Moreover, we analyze the global convergence of eASAP by virtue of the Aubin property on partial subdifferential mapping and the Kurdyka-Łojasiewicz property on the objective. Furthermore, the sublinear convergence rate of eASAP is built upon the proximal point algorithmic framework under some mild conditions. Numerical simulations on multimodal data fusion demonstrate the compelling performance of the proposed method.

Extended alternating structure-adapted proximal gradient algorithm for nonconvex nonsmooth problems

Abstract

Alternating structure-adapted proximal (ASAP) gradient algorithm (M. Nikolova and P. Tan, SIAM J Optim, 29:2053-2078, 2019) has drawn much attention due to its efficiency in solving nonconvex nonsmooth optimization problems. However, the multiblock nonseparable structure confines the performance of ASAP to far-reaching practical problems, e.g., coupled tensor decomposition. In this paper, we propose an extended ASAP (eASAP) algorithm for nonconvex nonsmooth optimization whose objective is the sum of two nonseperable functions and a coupling one. By exploiting the blockwise restricted prox-regularity, eASAP is capable of minimizing the objective whose coupling function is multiblock nonseparable. Moreover, we analyze the global convergence of eASAP by virtue of the Aubin property on partial subdifferential mapping and the Kurdyka-Łojasiewicz property on the objective. Furthermore, the sublinear convergence rate of eASAP is built upon the proximal point algorithmic framework under some mild conditions. Numerical simulations on multimodal data fusion demonstrate the compelling performance of the proposed method.

Paper Structure

This paper contains 16 sections, 15 theorems, 100 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subdifferential and partial subdifferential
  4. Restricted prox-regularity
  5. KŁ property
  6. Aubin property
  7. Extended ASAP algorithm
  8. Convergence analysis
  9. Descent property of Algorithm \ref{['algo1']}
  10. Subsequetial convergence
  11. Convergence of the iterative sequence
  12. Convergence rate of the iterative sequence
  13. Numerical experiments
  14. Synthetic data
  15. Real image data
  16. ...and 1 more sections

Key Result

Lemma 2.5

Let $f\in C_{L_f}^{1,1}(\mathbb{R}^n)$. Then

Figures (7)

  • Figure 1: The objective function values with respect to iterations (left) and CPU time (right) for solving problem \ref{['Lap']}.
  • Figure 2: The relative error (Relerr) and factor match score (FMS) with respect to iterations for solving problem \ref{['Lap']}.
  • Figure 3: Illustration of the hyperspectral super-resolution task kanatsoulis2018hyperspectral.
  • Figure 4: Testing hyperspectral images. (a) $80\times 80\times 204$ subscene of Salinas datasets. (b) $144\times 144\times 220$ subscene of Indian Pines dataset. (c) $300\times 300\times 102$ subscene of Pavia Centre dataset.
  • Figure 5: Results of Salinas reconstructions by solving \ref{['hyper']}. The first row: the 32-th band of recovered SRIs. The second row: the 32-th band of corresponding residual images. The last row: the SAM maps.
  • ...and 2 more figures

Theorems & Definitions (39)

  • Definition 2.1: Varia-Ana
  • Definition 2.2: nikolova2019alternating
  • Definition 2.3: wang2019global
  • Definition 2.4: bolte2010characterizations
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 29 more