An Alternating Direction Method of Multipliers for constrained joint diagonalization by congruence

In this paper, we address the problem of joint diagonalization by congruence (i.e. the canonical polyadic decomposition of semi-symmetric 3rd order tensors) subject to arbitrary convex constraints. Sufficient conditions for the existence of a solution are given. An efficient algorithm based on the Alternating Direction Method of Multipliers (ADMM) is then designed. ADMM provides an elegant approach for handling the additional constraint terms, while taking advantage of the structure of the objective function. Numerical tests on simulated matrices show the benefits of the proposed method for low signal to noise ratios. Simulations in the context of nuclear magnetic resonance spectroscopy are also provided.

[1]  Patrick L. Combettes,et al.  Stochastic Quasi-Fejér Block-Coordinate Fixed Point Iterations with Random Sweeping , 2014 .

[2]  Gabriele Steidl,et al.  Deblurring Poissonian images by split Bregman techniques , 2010, J. Vis. Commun. Image Represent..

[3]  Bingsheng He,et al.  The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent , 2014, Mathematical Programming.

[4]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[5]  Lu Wang,et al.  Nonnegative Joint Diagonalization by Congruence Based on LU Matrix Factorization , 2013, IEEE Signal Processing Letters.

[6]  Nikos D. Sidiropoulos,et al.  Parallel Algorithms for Constrained Tensor Factorization via Alternating Direction Method of Multipliers , 2014, IEEE Transactions on Signal Processing.

[7]  Hyunsoo Kim,et al.  Nonnegative Matrix Factorization Based on Alternating Nonnegativity Constrained Least Squares and Active Set Method , 2008, SIAM J. Matrix Anal. Appl..

[8]  J.-C. Pesquet,et al.  A Douglas–Rachford Splitting Approach to Nonsmooth Convex Variational Signal Recovery , 2007, IEEE Journal of Selected Topics in Signal Processing.

[9]  Laurent Albera,et al.  Line search and trust region strategies for canonical decomposition of semi-nonnegative semi-symmetric 3rd order tensors , 2014 .

[10]  Arie Yeredor,et al.  Joint Matrices Decompositions and Blind Source Separation: A survey of methods, identification, and applications , 2014, IEEE Signal Processing Magazine.

[11]  Lotfi Senhadji,et al.  Semi-nonnegative joint diagonalization by congruence and semi-nonnegative ICA , 2014, Signal Process..

[12]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[13]  José M. Bioucas-Dias,et al.  Restoration of Poissonian Images Using Alternating Direction Optimization , 2010, IEEE Transactions on Image Processing.

[14]  Lu Wang,et al.  Canonical polyadic decomposition of third-order semi-nonnegative semi-symmetric tensors using LU and QR matrix factorizations , 2014, EURASIP J. Adv. Signal Process..

[15]  Arie Yeredor,et al.  Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation , 2002, IEEE Trans. Signal Process..

[16]  Harvey J. Everett.  Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources , 1963 .

[17]  J. Pesquet,et al.  A Parallel Inertial Proximal Optimization Method , 2012 .

[18]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[19]  M. J. D. Powell,et al.  A method for nonlinear constraints in minimization problems , 1969 .