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Perspective functions: proximal calculus and applications in high-dimensional statistics

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2018年第2期457卷 1283-1306页

作者： Combettes, Patrick L. Muller, Christian L. North Carolina State Univ Dept Math Raleigh NC 27695 USA Simons Fdn Flatiron Inst New York NY 10010 USA

Perspective functions arise explicitly or implicitly in various forms in applied mathematics and in statistical data analysis. To date, no systematic strategy is available to solve the associated, typically nonsmooth, optimization problems. In this paper, we fill this gap by showing that proximal methods provide an efficient framework to model and solve problems involving perspective functions. We study the construction of the proximity operator of a perspective function under general assumptions and present important instances in which the proximity operator can be computed explicitly or via straightforward numerical operations. These results constitute central building blocks in the design of proximal optimization algorithms. We showcase the versatility of the framework by designing novel proximal algorithms for state-of-the-art regression and variable selection schemes in high-dimensional statistics. (C) 2016 The Authors. Published by Elsevier Inc.

关键词： Convex function Perspective function proximal algorithm Proximity operator Statistics

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A NEW proximal METHOD FOR JOINT IMAGE RESTORATION AND EDGE DETECTION WITH THE MUMFORD-SHAH MODEL

A NEW PROXIMAL METHOD FOR JOINT IMAGE RESTORATION AND EDGE D...

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

作者： Foare, Marion Pustelnik, Nelly Condat, Laurent Univ Lyon 1 Univ Lyon CNRS ENS LyonLab Phys F-69342 Lyon France Univ Grenoble Alpes CNRS Grenoble INP GIPSA Lab F-38000 Grenoble France

ISBN: (纸本)9781538646588

In this paper, we propose an adaptation of the PAM algorithm to the minimization of a nonconvex functional designed for joint image denoising and contour detection. This new functional is based on the Ambrosio-Tortorelli approximation of the well-known Mumford-Shah functional. We motivate the proposed approximation, offering flexibility in the choice of the possibly non-smooth penalization, and we derive closed form expression for the proximal steps involved in the algorithm. We focus our attention on two types of penalization: l(1)-norm and a proposed quadratic-l(1) function. Numerical experiments show that the proposed method is able to detect sharp contours and to reconstruct piecewise smooth approximations with low computational cost and convergence guarantees. We also compare the results with state-of-the-art relaxations of the Mumford-Shah functional and a recent discrete formulation of the Ambrosio-Tortorelli functional.

关键词： Segmentation restoration Ambrosio-Tortorelli non-smooth optimization proximal algorithm

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Monotone operator theory in convex optimization

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MATHEMATICAL PROGRAMMING 2018年第1期170卷 177-206页

作者： Combettes, Patrick L. North Carolina State Univ Dept Math Raleigh NC 27695 USA

Several aspects of the interplay between monotone operator theory and convex optimization are presented. The crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized. We review the properties of subdifferentials as maximally monotone operators and, in tandem, investigate those of proximity operators as resolvents. In particular, we study new transformations which map proximity operators to proximity operators, and establish connections with self-dual classes of firmly nonexpansive operators. In addition, new insights and developments are proposed on the algorithmic front.

关键词： Firmly nonexpansive operator Monotone operator Operator splitting proximal algorithm Proximity operator Proximity-preserving transformation Self-dual class Subdifferential

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GLOBALLY CONVERGENT JACOBI-TYPE algorithmS FOR SIMULTANEOUS ORTHOGONAL SYMMETRIC TENSOR DIAGONALIZATION

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2018年第1期39卷 1-22页

作者： Li, Jianze Usevich, Konstantin Comon, Pierre Tianjin Univ Sch Math Tianjin 300072 Peoples R China Univ Lorraine Campus SciBP 70239 F-54506 Vandoeuvre Les Nancy France CNRS CRAN UMR 7039 Campus SciBP 70239 F-54506 Vandoeuvre Les Nancy France Univ Grenoble Alpes CNRS Grenoble INP GIPSA Lab F-38000 Grenoble France

In this paper, we consider a family of Jacobi-type algorithms for a simultaneous orthogonal diagonalization problem of symmetric tensors. For the Jacobi-based algorithm of [M. Ishteva, P.-A. Absil, and P. Van Dooren, SIAM T. Matrix Anal. Appl., 34 (2013), pp. 651-672], we prove its global convergence for simultaneous orthogonal diagonalization of symmetric matrices and 3rd-order tensors. We also propose a new Jacobi-based algorithm in the general setting and prove its global convergence for sufficiently smooth functions.

关键词： orthogonal tensor diagonalization Jacobi rotation global convergence Lojasiewicz gradient inequality proximal algorithm

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Efficient Learning with a Family of Nonconvex Regularizers by Redistributing Nonconvexity

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JOURNAL OF MACHINE LEARNING RESEARCH 2018年第154期18卷 1-52页

作者： Yao, Quanming Kwok, James T. Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China

The use of convex regularizers allows for easy optimization, though they often produce biased estimation and inferior prediction performance. Recently, nonconvex regularizers have attracted a lot of attention and outperformed convex ones. However, the resultant optimization problem is much harder. In this paper, a popular subclass of l(1) -based nonconvex sparsity-inducing and low-rank regularizers is considered. This includes nonconvex variants of lasso, sparse group lasso, tree-structured lasso, nuclear norm and total variation regularizers. We propose to move the nonconvexity from the regularizer to the loss. The nonconvex regularizer is then transformed to a familiar convex one, while the resultant loss function can still be guaranteed to be smooth. Learning with the convexified regularizer can be performed by existing efficient algorithms originally designed for convex regularizers (such as the proximal algorithm, Frank-Wolfe algorithm, alternating direction method of multipliers and stochastic gradient descent). This is further extended to consider cases where the convexified regularizer does not have a closed-form proximal step, and when the loss function is nonconvex nonsmooth. Extensive experiments on a variety of machine learning application scenarios show that optimizing the transformed problem is much faster than running the state-of-the-art on the original problem.

关键词： Nonconvex optimization Nonconvex regularization proximal algorithm Frank-Wolfe algorithm Matrix completion

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A Variational Method for Dejittering Large Fluorescence Line Scanner Images

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IEEE TRANSACTIONS ON COMPUTATIONAL IMAGING 2018年第2期4卷 241-256页

作者： Hoai-Nam Nguyen Paveau, Vincent Cauchois, Cyril Kervrann, Charles INRIA Ctr Rennes Bretagne Atlantique F-35042 Rennes France Innopsys Inc F-31390 Carbonne France

We propose a variational method dedicated to jitter correction of large fluorescence scanner images. Our method consists in minimizing a global energy functional to estimate a dense displacement field representing the spatially-varying jitter. The computational approach is based on a half-quadratic splitting of the energy functional, which decouples the realignment data term and the dedicated differential-based regularizer. The resulting problem amounts to alternatively solving two convex and nonconvex optimization subproblems with appropriate algorithms. Experimental results on artificial and large real fluorescence images demonstrate that our method is not only capable to handle large displacements but is also efficient in terms of subpixel precision without inducing additional intensity artifacts.

关键词： Dejittering fluorescence scanner variational method regularization optimization quadratic relaxation proximal algorithm

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Interior proximal algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2012年第1期154卷 217-234页

作者： Brito, Arnaldo S. da Cruz Neto, J. X. Lopes, Jurandir O. Roberto Oliveira, P. Univ Fed Rio de Janeiro Rio De Janeiro Brazil COPPE Sistemas Univ Fed Rio de Janeiro Rio De Janeiro Brazil

In this paper, we propose two interior proximal algorithms inspired by the logarithmic-quadratic proximal method. The first method we propose is for general linearly constrained quasiconvex minimization problems. For this method, we prove global convergence when the regularization parameters go to zero. The latter assumption can be dropped when the function is assumed to be pseudoconvex. We also obtain convergence results for quasimonotone variational inequalities, which are more general than monotone ones.

关键词： Quasiconvex function proximal algorithm Quasiconvex linearly constrained problems Variational inequalities Quasimonotone operator

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Integrating spatial-anatomical regularization and structure sparsity into SVM: Improving interpretation of Alzheimer's disease classification

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NEUROIMAGE 2018年 178卷 445-460页

作者： Sun, Zhuo Qiao, Yuchuan Lelieveldt, Boudewijn P. F. Staring, Marius Leiden Univ Dept Radiol Div Image Proc Med Ctr NL-2300 RC Leiden Netherlands Delft Univ Technol Fac EEMCS Intelligent Syst Grp NL-2600 GA Delft Netherlands

In recent years, machine learning approaches have been successfully applied to the field of neuroimaging for classification and regression tasks. However, many approaches do not give an intuitive relation between the raw features and the diagnosis. Therefore, they are difficult for clinicians to interpret. Moreover, most approaches treat the features extracted from the brain (for example, voxelwise gray matter concentration maps from brain MRI) as independent variables and ignore their spatial and anatomical relations. In this paper, we present a new Support Vector Machine (SVM)-based learning method for the classification of Alzheimer's disease (AD), which integrates spatial-anatomical information. In this way, spatial-neighbor features in the same anatomical region are encouraged to have similar weights in the SVM model. Secondly, we introduce a group lasso penalty to induce structure sparsity, which may help clinicians to assess the key regions involved in the disease. For solving this learning problem, we use an accelerated proximal gradient descent approach. We tested our method on the subset of ADNI data selected by Cuingnet et al. (2011) for Alzheimer's disease classification, as well as on an independent larger dataset from ADNI. Good classification performance is obtained for distinguishing cognitive normals (CN) vs. AD, as well as on distinguishing between various sub-types (e.g. CN vs. Mild Cognitive Impairment). The model trained on Cuignet's dataset for AD vs. CN classification was directly used without re-training to the independent larger dataset. Good performance was achieved, demonstrating the generalizability of the proposed methods. For all experiments, the classification results are comparable or better than the state-of-the-art, while the weight map more clearly indicates the key regions related to Alzheimer's disease.

关键词： Alzheimer's disease Support vector machine (SVM) Spatial-anatomical regularization Structure sparsity proximal algorithm

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A New proximal Method for Joint Image Restoration and Edge Detection with the Mumford-Shah Model

A New Proximal Method for Joint Image Restoration and Edge D...

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IEEE International Conference on Acoustics, Speech and Signal Processing

作者： Marion Foare Nelly Pustelnik Laurent Condat ENS de Lyon Univ. Lyon 1 CNRS Laboratoire de Physique Univ. Lyon Lyon F-69342 France CNRS Grenoble INP GIPSA-lab Univ. Grenoble Alpes Grenoble 38000 France

ISBN: (纸本)9781538646595

In this paper, we propose an adaptation of the PAM algorithm to the minimization of a nonconvex functional designed for joint image denoising and contour detection. This new functional is based on the Ambrosio-Tortorelli approximation of the well-known Mumford-Shah functional. We motivate the proposed approximation, offering flexibility in the choice of the possibly non-smooth penalization, and we derive closed form expression for the proximal steps involved in the algorithm. We focus our attention on two types of penalization: l_1-norm and a proposed quadratic-f. function. Numerical experiments show that the proposed method is able to detect sharp contours and to reconstruct piecewise smooth approximations with low computational cost and convergence guarantees. We also compare the results with state-of-the-art relaxations of the Mumford-Shah functional and a recent discrete formulation of the Ambrosio-Tortorelli functional.

关键词： Segmentation restoration Ambrosio-Tortorelli non-smooth optimization proximal algorithm

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A proximal Approach for Nonnegative Tensor Decomposition 13th

A Proximal Approach for Nonnegative Tensor Decomposition

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13th International Conference on Latent Variable Analysis and Signal Separation (LVA/ICA)

作者： Vu, Xuan Chaux, Caroline Thirion-Moreau, Nadege Maire, Sylvain Aix Marseille Univ CNRS ENSAM LSISUMR 7296 F-13397 Marseille France Univ Toulon & Var CNRS LSIS UMR 7296 F-83957 La Garde France Aix Marseille Univ CNRS Cent Marseille I2M Marseille France

ISBN: (纸本)9783319535463;9783319535470

This communication deals with N-th order tensor decompositions. More precisely, we are interested in the (Canonical) Polyadic Decomposition. In our case, this problem is formulated under a variational approach where the considered criterion to be minimized is composed of several terms: one accounting for the fidelity to data and others that can represent not only regularization (such as sparsity prior) but also hard constraints (such as nonnegativity). The resulting optimization problem is solved by using the Block-Coordinate Variable Metric Forward-Backward (BC-VMFB) algorithm. The robustness and efficiency of the suggested approach is illustrated on realistic synthetic data such as those encountered in the context of environmental data analysis and fluorescence spectroscopy. Our simulations are performed on 4-th order tensors.

关键词： Constrained optimization proximal algorithm Block alternating minimization Nonnegative tensor factorization (NTF)

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