A Flexible and Efficient Algorithmic Framework for Constrained Matrix and Tensor Factorization

We propose a general algorithmic framework for constrained matrix and tensor factorization, which is widely used in signal processing and machine learning. The new framework is a hybrid between alternating optimization (AO) and the alternating direction method of multipliers (ADMM): each matrix factor is updated in turn, using ADMM, hence the name AO-ADMM. This combination can naturally accommodate a great variety of constraints on the factor matrices, and almost all possible loss measures for the fitting. Computation caching and warm start strategies are used to ensure that each update is evaluated efficiently, while the outer AO framework exploits recent developments in block coordinate descent (BCD)-type methods which help ensure that every limit point is a stationary point, as well as faster and more robust convergence in practice. Three special cases are studied in detail: non-negative matrix/tensor factorization, constrained matrix/tensor completion, and dictionary learning. Extensive simulations and experiments with real data are used to showcase the effectiveness and broad applicability of the proposed framework

ADMM for Multiaffine Constrained Optimization

We expand the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the Kurdyka-\L{}ojasiewicz (K-\L{}) property holds, this is strengthened to convergence to a single constrained stationary point. Our analysis applies under assumptions that we have endeavored to make as weak as possible. It applies to problems that involve nonconvex and/or nonsmooth objective terms, in addition to the multiaffine constraints that can involve multiple (three or more) blocks of variables. To illustrate the applicability of our results, we describe examples including nonnegative matrix factorization, sparse learning, risk parity portfolio selection, nonconvex formulations of convex problems, and neural network training. In each case, our ADMM approach encounters only subproblems that have closed-form solutions.Comment: v3: 37 pages, 7 figures v2: 32 pages, 0 figures. v1: 26 pages, 0 figure

Global hard thresholding algorithms for joint sparse image representation and denoising

Sparse coding of images is traditionally done by cutting them into small patches and representing each patch individually over some dictionary given a pre-determined number of nonzero coefficients to use for each patch. In lack of a way to effectively distribute a total number (or global budget) of nonzero coefficients across all patches, current sparse recovery algorithms distribute the global budget equally across all patches despite the wide range of differences in structural complexity among them. In this work we propose a new framework for joint sparse representation and recovery of all image patches simultaneously. We also present two novel global hard thresholding algorithms, based on the notion of variable splitting, for solving the joint sparse model. Experimentation using both synthetic and real data shows effectiveness of the proposed framework for sparse image representation and denoising tasks. Additionally, time complexity analysis of the proposed algorithms indicate high scalability of both algorithms, making them favorable to use on large megapixel images

Linearized ADMM for Non-convex Non-smooth Optimization with Convergence Analysis

Linearized alternating direction method of multipliers (ADMM) as an extension of ADMM has been widely used to solve linearly constrained problems in signal processing, machine leaning, communications, and many other fields. Despite its broad applications in nonconvex optimization, for a great number of nonconvex and nonsmooth objective functions, its theoretical convergence guarantee is still an open problem. In this paper, we propose a two-block linearized ADMM and a multi-block parallel linearized ADMM for problems with nonconvex and nonsmooth objectives. Mathematically, we present that the algorithms can converge for a broader class of objective functions under less strict assumptions compared with previous works. Furthermore, our proposed algorithm can update coupled variables in parallel and work for less restrictive nonconvex problems, where the traditional ADMM may have difficulties in solving subproblems.Comment: 29 pages, 2 tables, 2 figure

Fast Convolutional Sparse Coding in the Dual Domain

Convolutional sparse coding (CSC) is an important building block of many computer vision applications ranging from image and video compression to deep learning. We present two contributions to the state of the art in CSC. First, we significantly speed up the computation by proposing a new optimization framework that tackles the problem in the dual domain. Second, we extend the original formulation to higher dimensions in order to process a wider range of inputs, such as RGB images and videos. Our results show up to 20 times speedup compared to current state-of-the-art CSC solvers

A Fast and Efficient Algorithm for Reconstructing MR images From Partial Fourier Samples

In this paper, the problem of Magnetic Resonance (MR) image reconstruction from partial Fourier samples has been considered. To this aim, we leverage the evidence that MR images are sparser than their zero-filled reconstructed ones from incomplete Fourier samples. This information can be used to define an optimization problem which searches for the sparsest possible image conforming with the available Fourier samples. We solve the resulting problem using the well-known Alternating Direction Method of Multipliers (ADMM). Unlike most existing methods that work with small over-lapping image patches, the proposed algorithm considers the whole image without dividing it into small blocks. Experimental results prominently confirm its promising performance and advantages over the existing methods

LSALSA: Accelerated Source Separation via Learned Sparse Coding

We propose an efficient algorithm for the generalized sparse coding (SC) inference problem. The proposed framework applies to both the single dictionary setting, where each data point is represented as a sparse combination of the columns of one dictionary matrix, as well as the multiple dictionary setting as given in morphological component analysis (MCA), where the goal is to separate a signal into additive parts such that each part has distinct sparse representation within a corresponding dictionary. Both the SC task and its generalization via MCA have been cast as $\ell_1$-regularized least-squares optimization problems. To accelerate traditional acquisition of sparse codes, we propose a deep learning architecture that constitutes a trainable time-unfolded version of the Split Augmented Lagrangian Shrinkage Algorithm (SALSA), a special case of the Alternating Direction Method of Multipliers (ADMM). We empirically validate both variants of the algorithm, that we refer to as LSALSA (learned-SALSA), on image vision tasks and demonstrate that at inference our networks achieve vast improvements in terms of the running time, the quality of estimated sparse codes, and visual clarity on both classic SC and MCA problems. Finally, we present a theoretical framework for analyzing LSALSA network: we show that the proposed approach exactly implements a truncated ADMM applied to a new, learned cost function with curvature modified by one of the learned parameterized matrices. We extend a very recent Stochastic Alternating Optimization analysis framework to show that a gradient descent step along this learned loss landscape is equivalent to a modified gradient descent step along the original loss landscape. In this framework, the acceleration achieved by LSALSA could potentially be explained by the network's ability to learn a correction to the gradient direction of steeper descent.Comment: ECML-PKDD 2019 via journal track; Special Issue Mach Learn (2019

Tomographic Image Reconstruction using Training images

We describe and examine an algorithm for tomographic image reconstruction where prior knowledge about the solution is available in the form of training images. We first construct a nonnegative dictionary based on prototype elements from the training images; this problem is formulated as a regularized non-negative matrix factorization. Incorporating the dictionary as a prior in a convex reconstruction problem, we then find an approximate solution with a sparse representation in the dictionary. The dictionary is applied to non-overlapping patches of the image, which reduces the computational complexity compared to other algorithms. Computational experiments clarify the choice and interplay of the model parameters and the regularization parameters, and we show that in few-projection low-dose settings our algorithm is competitive with total variation regularization and tends to include more texture and more correct edges.Comment: 25 pages, 12 figure

Penalty Dual Decomposition Method For Nonsmooth Nonconvex Optimization

Many contemporary signal processing, machine learning and wireless communication applications can be formulated as nonconvex nonsmooth optimization problems. Often there is a lack of efficient algorithms for these problems, especially when the optimization variables are nonlinearly coupled in some nonconvex constraints. In this work, we propose an algorithm named penalty dual decomposition (PDD) for these difficult problems and discuss its various applications. The PDD is a double-loop iterative algorithm. Its inner iterations is used to inexactly solve a nonconvex nonsmooth augmented Lagrangian problem via block-coordinate-descenttype methods, while its outer iteration updates the dual variables and/or a penalty parameter. In Part I of this work, we describe the PDD algorithm and rigorously establish its convergence to KKT solutions. In Part II we evaluate the performance of PDD by customizing it to three applications arising from signal processing and wireless communications.Comment: Two part paper, 27 figure

Noisy Matrix Completion under Sparse Factor Models

This paper examines a general class of noisy matrix completion tasks where the goal is to estimate a matrix from observations obtained at a subset of its entries, each of which is subject to random noise or corruption. Our specific focus is on settings where the matrix to be estimated is well-approximated by a product of two (a priori unknown) matrices, one of which is sparse. Such structural models - referred to here as "sparse factor models" - have been widely used, for example, in subspace clustering applications, as well as in contemporary sparse modeling and dictionary learning tasks. Our main theoretical contributions are estimation error bounds for sparsity-regularized maximum likelihood estimators for problems of this form, which are applicable to a number of different observation noise or corruption models. Several specific implications are examined, including scenarios where observations are corrupted by additive Gaussian noise or additive heavier-tailed (Laplace) noise, Poisson-distributed observations, and highly-quantized (e.g., one-bit) observations. We also propose a simple algorithmic approach based on the alternating direction method of multipliers for these tasks, and provide experimental evidence to support our error analyses.Comment: 42 Pages, 7 Figures, Submitted to IEEE Transactions on Information Theor