Journal
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
Volume 30, Issue 9, Pages 2825-2839Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TNNLS.2018.2885699
Keywords
Generalized proximal thresholding operators; global convergence analysis; Kurdyka-Lojasiewica (KL) property; nonconvex sparse and low-rank optimization; proximal Jacobian iteration method (PJIM)
Categories
Funding
- National Science Fund of China [U1713208, 61472187, 61502235, 61876083, 61602244]
- Fundamental Research Funds for the Central Universities [30918011321]
- 973 Program [2014CB349303]
- Program for Changjiang Scholars
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The recent studies have found that the nonconvex relaxation functions usually perform better than the convex counterparts in the l(0)-norm and rank function minimization problems. However, due to the absence of convexity in these nonconvex problems, developing efficient algorithms with convergence guarantee becomes very challenging. Inspired by the basic ideas of both the Jacobian alternating direction method of multipliers (JADMMs) for solving linearly constrained problems with separable objectives and the proximal gradient methods (PGMs) for optimizing the unconstrained problems with one variable, this paper focuses on extending the PGMs to the proximal Jacobian iteration methods (PJIMs) for handling with a family of nonconvex composite optimization problems with two splitting variables. To reduce the total computational complexity by decreasing the number of iterations, we devise the accelerated version of PJIMs through the well-known Nesterov's acceleration strategy and further extend both to solve the multivariable cases. Most importantly, we provide a rigorous convergence analysis, in theory, to show that the generated variable sequence globally converges to a critical point by exploiting the Kurdyka-Lojasiewica (KL) property for a broad class of functions. Furthermore, we also establish the linear and sublinear convergence rates of the obtained variable sequence in the objective function. As the specific application to the nonconvex sparse and low-rank recovery problems, several numerical experiments can verify that the newly proposed algorithms not only keep fast convergence speed but also have high precision.
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