The Diagonalized Newton Algorithm for Nonnegative Matrix Factorization

Non-negative matrix factorization (NMF) has become a popular machine learning approach to many problems in text mining, speech and image processing, bio-informatics and seismic data analysis to name a few. In NMF, a matrix of non-negative data is approximated by the low-rank product of two matrices with non-negative entries. In this paper, the approximation quality is measured by the Kullback-Leibler divergence between the data and its low-rank reconstruction. The existence of the simple multiplicative update (MU) algorithm for computing the matrix factors has contributed to the success of NMF. Despite the availability of algorithms showing faster convergence, MU remains popular due to its simplicity. In this paper, a diagonalized Newton algorithm (DNA) is proposed showing faster convergence while the implementation remains simple and suitable for high-rank problems. The DNA algorithm is applied to various publicly available data sets, showing a substantial speed-up on modern hardware.Comment: 8 pages + references; International Conference on Learning Representations, 201

Convergence of Gradient Descent for Low-Rank Matrix Approximation

This paper provides a proof of global convergence of gradient search for low-rank matrix approximation. Such approximations have recently been of interest for large-scale problems, as well as for dictionary learning for sparse signal representations and matrix completion. The proof is based on the interpretation of the problem as an optimization on the Grassmann manifold and Fubiny-Study distance on this space

Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective

The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of problems, e.g., nonnegativity or sparsity-constrained factorization, we take a {\it top-down} approach: we start with general optimization theory (e.g., inexact and accelerated block coordinate descent, stochastic optimization, and Gauss-Newton methods) that covers a wide range of factorization problems with diverse constraints and regularization terms of engineering interest. Then, we go `under the hood' to showcase specific algorithm design under these introduced principles. We pay a particular attention to recent algorithmic developments in structured tensor and matrix factorization (e.g., random sketching and adaptive step size based stochastic optimization and structure-exploiting second-order algorithms), which are the state of the art---yet much less touched upon in the literature compared to {\it block coordinate descent} (BCD)-based methods. We expect that the article to have an educational values in the field of structured factorization and hope to stimulate more research in this important and exciting direction.Comment: Final Version; to appear in IEEE Signal Processing Magazine; title revised to comply with the journal's rul

Accurate and Efficient Expression Evaluation and Linear Algebra

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: Most of our results will use the so-called Traditional Model (TM). We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high accuracy algorithm or a proof that none exists. When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such as $x+y+z$, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case. Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl

Regularized Compression of A Noisy Blurred Image

Both regularization and compression are important issues in image processing and have been widely approached in the literature. The usual procedure to obtain the compression of an image given through a noisy blur requires two steps: first a deblurring step of the image and then a factorization step of the regularized image to get an approximation in terms of low rank nonnegative factors. We examine here the possibility of swapping the two steps by deblurring directly the noisy factors or partially denoised factors. The experimentation shows that in this way images with comparable regularized compression can be obtained with a lower computational cost

Expansions and factorizations of matrices and their applications

Abstract. Linear algebra is a foundation to decompositions and algorithms for extracting simple structures from complex data. In this thesis, we investigate and apply modern techniques from linear algebra to solve problems arising in signal processing and computer science. In particular, we focus on data that takes the shape of a matrix and we explore how to represent it as products of circulant and diagonal matrices. To this end, we study matrix decompositions, approximations, and structured matrix expansions whose elements are products of circulant and diagonal matrices. Computationally, we develop a matrix expansion with DCD matrices for approximating a given matrix. Remarkably, DCD matrices, i.e., a product of diagonal matrix, circulant matrix, and another diagonal matrix, give an natural extension to rank-one matrices. Inspired from the singular value decomposition, we introduce a notion of a matrix rank closely related to the expansion and compute the rank of some specific structured matrices. Specifically, Toeplitz matrix is a sum of two DCD matrices. Here, we present a greedy algorithmic framework to devise the expansion numerically. Finally, we show that the practical uses of the DCD expansion can be complemented by the proposed framework and perform two experiments with a view towards applications