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Linearly-Involved Moreau-Enhanced-Over-Subspace Model: Debiased Sparse Modeling and Stable Outlier-Robust Regression

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2023年 71卷 1232-1247页

作者： Yukawa, Masahiro Kaneko, Hiroyuki Suzuki, Kyohei Yamada, Isao Keio Univ Dept Elect & Elect Engn Yokohama Kanagawa 2238522 Japan Tokyo Inst Technol Dept Informat & Commun Engn Meguro Ku Tokyo 1528550 Japan

We present an efficient mathematical framework to derive promising methods that enjoy "enhanced" desirable properties. The popular minimax concave penalty for sparse modeling subtracts, from the l(1) norm, its Moreau envelope, inducing nearly unbiased estimates and thus yielding considerable performance enhancements. To extend it to underdetermined linear systems, we propose the projective minimax concave penalty, which leads to "enhanced" sparseness over the input subspace. We also present a promising regression method which has an "enhanced" robustness and substantial stability by distinguishing outlier and noise explicitly. The proposed framework, named the linearly-involved Moreau -enhanced-over-subspace (LiMES) model, encompasses those two specific examples as well as two others: stable principal component pursuit and robust classification. The LiMES function involved in the model is an "additively nonseparable" weakly convex function, while the inner' objective function to define the Moreau envelope is "separable". This mixed nature of separability and nonseparability allows an application of the LiMES model to the underdetermined case with an efficient algorithmic implementation. Two linear/affine operators play key roles in the model: one corresponds to the projection mentioned above and the other takes care of robust regression/classification. A necessary and sufficient condition for convexity of the smooth part of the objective function is studied. Numerical examples show the efficacy of LiMES in applications to sparse modeling and robust regression.

关键词： convex optimization weakly convex function proximity operator Moreau envelope

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An Adaptive Alternating Direction Method of Multipliers

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2022年第3期195卷 1019-1055页

作者： Bartz, Sedi Campoy, Ruben Phan, Hung M. Univ Massachusetts Lowell Kennedy Coll Sci Dept Math Sci Lowell MA USA Univ Valencia Dept Stat & Operat Res Valencia Spain

The alternating direction method of multipliers (ADMM) is a powerful splitting algorithm for linearly constrained convex optimization problems. In view of its popularity and applicability, a growing attention is drawn toward the ADMM in nonconvex settings. Recent studies of minimization problems for nonconvex functions include various combinations of assumptions on the objective function including, in particular, a Lipschitz gradient assumption. We consider the case where the objective is the sum of a strongly convex function and a weakly convex function. To this end, we present and study an adaptive version of the ADMM which incorporates generalized notions of convexity and penalty parameters adapted to the convexity constants of the functions. We prove convergence of the scheme under natural assumptions. To this end, we employ the recent adaptive Douglas-Rachford algorithm by revisiting the well-known duality relation between the classical ADMM and the Douglas-Rachford splitting algorithm, generalizing this connection to our setting. We illustrate our approach by relating and comparing to alternatives, and by numerical experiments on a signal denoising problem.

关键词： Alternating direction method of multipliers Douglas-Rachford algorithm weakly convex function Comonotonicity Signal denoising Firm thresholding

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A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2019年第1期2019卷 1-21页

作者： Zhang, Tao Shen, Zhengwei Univ Sci & Technol Beijing Sch Math & Phys Beijing Peoples R China

The convergence of the alternating direction method of multipliers (ADMMs) algorithm to convex/nonconvex combinational optimization has been well established in the literature. Due to the extensive applications of a weakly convex function in signal processing and machine learning, in this paper, we investigate the convergence of an ADMM algorithm to the strongly and weakly convex combinational optimization (SWCCO) problem. Specifically, we firstly show the convergence of the iterative sequences of the SWCCO-ADMM under a mild regularity condition;then we establish the o(1/k) sublinear convergence rate of the SWCCO-ADMM algorithm using the same conditions and the linear convergence rate by imposing the gradient Lipschitz continuity condition on the objective function. The techniques used for the convergence analysis in this paper are fundamental, and we accomplish the global convergence without using the Kurdyka-ojasiewicz (KL) inequality, which is common but complex in the proof of nonconvex ADMM.

关键词： convex optimization weakly convex function Ublinear and linear convergence ADMM

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Growth Conditions on a function and the Error Bound Condition

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MATHEMATICAL NOTES 2021年第3-4期109卷 638-643页

作者： Balashov, M. V. Russian Acad Sci Trapeznikov Inst Control Sci Moscow 117997 Russia

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Stable Robust Regression under Sparse Outlier and Gaussian Noise 30

Stable Robust Regression under Sparse Outlier and Gaussian N...

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30th European Signal Processing Conference (EUSIPCO)

作者： Yukawa, Masahiro Suzuki, Kyohei Yamada, Isao Keio Univ Dept Elect & Elect Engn Tokyo Japan Tokyo Inst Technol Dept Informat & Commun Engn Tokyo Japan

ISBN: (纸本)9789082797091

We propose an efficient regression method which is highly robust against outliers and stable even in the severely noisy situations. The robustness here comes from the adoption of the minimax concave loss, while the stability comes from separate treatments of the outlier and noise by an introduction of an auxiliary vector modeling the Gaussian noise. We present a necessary and sufficient condition for convexity of the smooth part of the entire cost under a certain assumption, where a general model is used with its potential use for other applications envisioned. We show that the proposed formulation can be solved via reformulation by the forward-backward-based primal-dual method under the convexity condition. The numerical examples show the remarkable robustness of the proposed estimator under highly noisy situations.

关键词： weakly convex function minimax concave loss Moreau envelope robust regression

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EXTERNAL DIVISION OF TWO PROXIMITY OPERATORS: AN APPLICATION TO SIGNAL RECOVERY WITH STRUCTURED SPARSITY 49

EXTERNAL DIVISION OF TWO PROXIMITY OPERATORS: AN APPLICATION...

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49th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： Suzuki, Kyohei Yukawa, Masahiro Keio Univ Dept Elect & Elect Engn Minato City Tokyo Japan

ISBN: (纸本)9798350344868;9798350344851

This paper studies the external division operator, an external division (an affine combination with positive and negative weights) of two proximity operators. We show that the external division operator is cocoercive under some condition, and it can be expressed as the proximity operator of a certain weakly convex function. We then consider using the external division operator as an alternative to the proximity operator in the proximal gradient algorithm, which we show converges to a minimizer of the cost function penalized by the weakly convex function under some conditions. Our analysis covers the case when the fidelity function is convex but not strongly convex. In simulations, we employ the octagonal shrinkage and clustering algorithm for regression in the external division operator, and we show that significant improvements are attained in signal recovery with structured sparsity in both overdetermined and underdetermined cases.

关键词： Cocoercive operator proximity operator weakly convex function proximal gradient algorithm OSCAR

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A VARIABLE SMOOTHING FOR NONconvexLY CONSTRAINED NONSMOOTH OPTIMIZATION WITH APPLICATION TO SPARSE SPECTRAL CLUSTERING 49

A VARIABLE SMOOTHING FOR NONCONVEXLY CONSTRAINED NONSMOOTH O...

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49th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： Kume, Keita Yamada, Isao Tokyo Inst Technol Dept Informat & Commun Engn Tokyo Japan

ISBN: (纸本)9798350344868;9798350344851

We propose a variable smoothing algorithm for solving nonconvexly constrained nonsmooth optimization problems. The target problem has two issues that need to be addressed: (i) the nonconvex constraint and (ii) the nonsmooth term. To handle the nonconvex constraint, we translate the target problem into an unconstrained problem by parameterizing the nonconvex constraint in terms of a Euclidean space. We show that under a certain condition, these problems are equivalent in view of finding a stationary point. To find a stationary point of the parameterized problem, the proposed algorithm performs the gradient descent update for the smoothed version of the parameterized problem with replacement of the nonsmooth function by the Moreau envelope, inspired by a variable smoothing algorithm [Bohm-Wright, J. Optim. Theory Appl., 2021] specialized for unconstrained nonsmooth optimization. We also present a convergence analysis of the proposed algorithm as well as its application to a nonconvex reformulation of the sparse spectral clustering.

关键词： nonconvex constraint nonsmooth optimization weakly convex function Moreau envelope sparse spectral clustering

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