Low-rank matrix factorization (MF) is an important technique in data science. The key idea of MF is that there exists latent structures in the data, by uncovering which we could obtain a compressed representation of the data. By factorizing an original matrix to low-rank matrices, MF provides a unified method for dimension reduction, clustering, and matrix completion. In this article we review several important variants of MF, including: Basic MF, Non-negative MF, Orthogonal non-negative MF. As can be told from their names, non-negative MF and orthogonal non-negative MF are variants of basic MF with non-negativity and/or orthogonality constraints. Such constraints are useful in specific senarios. In the first part of this article, we introduce, for each of these models, the application scenarios, the distinctive properties, and the optimizing method. By properly adapting MF, we can go beyond the problem of clustering and matrix completion. In the second part of this article, we will extend MF to sparse matrix compeletion, enhance matrix compeletion using various regularization methods, and make use of MF for (semi-)supervised learning by introducing latent space reinforcement and transformation. We will see that MF is not only a useful model but also as a flexible framework that is applicable for various prediction problems.
University of California（加利福尼亚大学）；University of Washington（华盛顿大学）
Matrix factorization is a well-studied task in machine learning for compactly representing large, noisy data. In our approach, instead of using the traditional concept of matrix rank, we define a new notion of link-rank based on a non-linear link function used within factorization. In particular, by applying the round function on a factorization to obtain ordinal-valued matrices, we introduce generalized round-rank (GRR). We show that not only are there many full-rank matrices that are low GRR, but further, that these matrices cannot be approximated well by low-rank linear factorization. We pr…
University of Electronic Science and Technology of China（电子科技大学）；Southwest Jiaotong University（西南交通大学）
As we all known, the nonnegative matrix factorization (NMF) is a dimension reduction method that has been widely used in image processing, text compressing and signal processing etc. In this paper, an algorithm for nonnegative matrix approximation is proposed. This method mainly bases on the active set and the quasi-Newton type algorithm, by using the symmetric rank-one and negative curvature direction technologies to approximate the Hessian matrix. Our method improves the recent results of those methods in [Pattern Recognition, 45(2012)3557-3565; SIAM J. Sci. Comput., 33(6)(2011)3261-3281; Ne…
Institute of Artificial Intelligence, Medical University of Vienna（维也纳医科大学人工智能研究所）
Learning good quality neural graph embeddings has long been achieved by minimizing the point-wise mutual information (PMI) for co-occurring nodes in simulated random walks. This design choice has been mostly popularized by the direct application of the highly-successful word embedding algorithm word2vec to predicting the formation of new links in social, co-citation, and biological networks. However, such a skeuomorphic design of graph embedding methods entails a truncation of information coming from pairs of nodes with low PMI. To circumvent this issue, we propose an improved approach to lear…
What learning algorithms can be run directly on compressively-sensed data? In this work, we consider the question of accurately and efficiently computing low-rank matrix or tensor factorizations given data compressed via random projections. We examine the approach of first performing factorization in the compressed domain, and then reconstructing the original high-dimensional factors from the recovered (compressed) factors. In both the matrix and tensor settings, we establish conditions under which this natural approach will provably recover the original factors. While it is well-known that ra…
Rank minimization is of interest in machine learning applications such as recommender systems and robust principal component analysis. Minimizing the convex relaxation to the rank minimization problem, the nuclear norm, is an effective technique to solve the problem with strong performance guarantees. However, nonconvex relaxations have less estimation bias than the nuclear norm and can more accurately reduce the effect of noise on the measurements. We develop efficient algorithms based on iteratively reweighted nuclear norm schemes, while also utilizing the low rank factorization for semidefi…
Improved SVD-based Initialization for Nonnegative Matrix Factorization using Low-Rank Correction
Graduate School of Science and Engineering（研究生院）；PAF-Karachi Institute of Economics and Technology（卡拉奇经济和技术学院）；Department of Mathematics and Operational Research（数学与运筹学系）
University of Liège（列日大学）；Mines ParisTech（巴黎高等矿业学院）；INRIA-Lille（里尔法国国家信息与自动化研究所）
Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fix…
Lossless Model Compression via Joint Low-Rank Factorization Optimization
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Peng Cheng Laboratory（鹏城实验室）；the School of Microelectronics, South China University of Technology（华南理工大学微电子学院）
Nowadays, the availability of large-scale data in disparate application domains urges the deployment of sophisticated tools for extracting valuable knowledge out of this huge bulk of information. In that vein, low-rank representations (LRRs) which seek low-dimensional embeddings of data have naturally appeared. In an effort to reduce computational complexity and improve estimation performance, LRR has been viewed via a matrix factorization (MF) perspective. Recently, low-rank MF (LRMF) approaches have been proposed for tackling the inherent weakness of MF i.e., the unawareness of the dimension…
CANDECOMP/PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank. In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian trea…
The goal of tensor completion is to recover a tensor from a subset of its entries, often by exploiting its low-rank property. Among several useful definitions of tensor rank, the low-tubal-rank was shown to give a valuable characterization of the inherent low-rank structure of a tensor. While some low-tubal-rank tensor completion algorithms with favorable performance have been recently proposed, these algorithms utilize second-order statistics to measure the error residual, which may not work well when the observed entries contain large outliers. In this paper, we propose a new objective funct…
We develop a new collaborative filtering (CF) method that combines both previously known users' preferences, i.e. standard CF, as well as product/user attributes, i.e. classical function approximation, to predict a given user's interest in a particular product. Our method is a generalized low rank matrix completion problem, where we learn a function whose inputs are pairs of vectors -- the standard low rank matrix completion problem being a special case where the inputs to the function are the row and column indices of the matrix. We solve this generalized matrix completion problem using tenso…
Toronto Metropolitan University（多伦多都会大学）；University of Toronto（多伦多大学）
The selection of most informative and discriminative features from high-dimensional data has been noticed as an important topic in machine learning and data engineering. Using matrix factorization-based techniques such as nonnegative matrix factorization for feature selection has emerged as a hot topic in feature selection. The main goal of feature selection using matrix factorization is to extract a subspace which approximates the original space but in a lower dimension. In this study, rank revealing QR (RRQR) factorization, which is computationally cheaper than singular value decomposition (…
Matrix factorization is a simple and natural test-bed to investigate the implicit regularization of gradient descent. Gunasekar et al. (2017) conjectured that Gradient Flow with infinitesimal initialization converges to the solution that minimizes the nuclear norm, but a series of recent papers argued that the language of norm minimization is not sufficient to give a full characterization for the implicit regularization. In this work, we provide theoretical and empirical evidence that for depth-2 matrix factorization, gradient flow with infinitesimal initialization is mathematically equivalent…
University of Science and Technology（科学技术大学）；Electronics and Telecommunications Research Institute（电子通信研究院）
Convolutional Neural Networks (CNNs) is one of successful method in many areas such as image classification tasks. However, the amount of memory and computational cost needed for CNNs inference obstructs them to run efficiently in mobile devices because of memory and computational ability limitation. One of the method to compress CNNs is compressing the layers iteratively, i.e. by layer-by-layer compression and fine-tuning, with CP-decomposition in convolutional layers. To compress with CP-decomposition, rank selection is important. In the previous approach rank selection that is based on sens…
There has recently been considerable interest in completing a low-rank matrix or tensor given only a small fraction (or few linear combinations) of its entries. Related approaches have found considerable success in the area of recommender systems, under machine learning. From a statistical estimation point of view, the gold standard is to have access to the joint probability distribution of all pertinent random variables, from which any desired optimal estimator can be readily derived. In practice high-dimensional joint distributions are very hard to estimate, and only estimates of low-dimensi…
We study the convergence of a variant of distributed gradient descent (DGD) on a distributed low-rank matrix approximation problem wherein some optimization variables are used for consensus (as in classical DGD) and some optimization variables appear only locally at a single node in the network. We term the resulting algorithm DGD+LOCAL. Using algorithmic connections to gradient descent and geometric connections to the well-behaved landscape of the centralized low-rank matrix approximation problem, we identify sufficient conditions where DGD+LOCAL is guaranteed to converge with exact consensus…
Stability of Low-Rank Implicit Regularization in Perturbed Deep Matrix Factorization
Department of Informatics and Networked Systems, University of Pittsburgh（信息学与网络系统系，匹兹堡大学）；Department of Mathematics, University of Pittsburgh（数学系，匹兹堡大学）
Istituto Italiano di Tecnologia（意大利技术研究院）；Johns Hopkins University（约翰斯·霍普金斯大学）
Regularization for matrix factorization (MF) and approximation problems has been carried out in many different ways. Due to its popularity in deep learning, dropout has been applied also for this class of problems. Despite its solid empirical performance, the theoretical properties of dropout as a regularizer remain quite elusive for this class of problems. In this paper, we present a theoretical analysis of dropout for MF, where Bernoulli random variables are used to drop columns of the factors. We demonstrate the equivalence between dropout and a fully deterministic model for MF in which the…
Department of Mathematics, UC Santa Barbara（加州大学圣巴巴拉分校数学系）；Department of ECE, UC Santa Barbara（加州大学圣巴巴拉分校电气工程与计算机科学系）
Streaming tensor factorization is a powerful tool for processing high-volume and multi-way temporal data in Internet networks, recommender systems and image/video data analysis. Existing streaming tensor factorization algorithms rely on least-squares data fitting and they do not possess a mechanism for tensor rank determination. This leaves them susceptible to outliers and vulnerable to over-fitting. This paper presents a Bayesian robust streaming tensor factorization model to identify sparse outliers, automatically determine the underlying tensor rank and accurately fit low-rank structure. We…
Ferdowsi University of Mashhad（马什哈德菲尔多西大学）；University of Tehran（德黑兰大学）
In this paper, we tackle two important problems in low-rank learning, which are partial singular value decomposition and numerical rank estimation of huge matrices. By using the concepts of Krylov subspaces such as Golub-Kahan bidiagonalization (GK-bidiagonalization) as well as Ritz vectors, we propose two methods for solving these problems in a fast and accurate way. Our experiments show the advantages of the proposed methods compared to the traditional and randomized singular value decomposition methods. The proposed methods are appropriate for applications involving huge matrices where the …
The Chinese University of Hong Kong（香港中文大学）；Arizona State University（亚利桑那州立大学）
Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in machine learning, statistics, bioinformatics, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint/mixed relaxations (i.e., l_{1}-norm and trace norm) under certain conditions. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a scalable, provable structured low-rank matrix factorization meth…
We consider the problem of recovering low-rank matrices from random rank-one measurements, which spans numerous applications including covariance sketching, phase retrieval, quantum state tomography, and learning shallow polynomial neural networks, among others. Our approach is to directly estimate the low-rank factor by minimizing a nonconvex quadratic loss function via vanilla gradient descent, following a tailored spectral initialization. When the true rank is small, this algorithm is guaranteed to converge to the ground truth (up to global ambiguity) with near-optimal sample complexity and…
Institute for Interdisciplinary Information Sciences（交叉信息研究院）；Tsinghua University（清华大学）；Paul G. Allen School of Computer Science and Engineering（保罗·G·艾伦计算机科学与工程学院）；University of Washington（华盛顿大学）
We study the asymmetric low-rank factorization problem: \[\min_{\mathbf{U} \in \mathbb{R}^{m \times d}, \mathbf{V} \in \mathbb{R}^{n \times d}} \frac{1}{2}\|\mathbf{U}\mathbf{V}^\top -\mathbfΣ\|_F^2\] where $\mathbfΣ$ is a given matrix of size $m \times n$ and rank $d$. This is a canonical problem that admits two difficulties in optimization: 1) non-convexity and 2) non-smoothness (due to unbalancedness of $\mathbf{U}$ and $\mathbf{V}$). This is also a prototype for more complex problems such as asymmetric matrix sensing and matrix completion. Despite being non-convex and non-smooth, it has be…
University College London（伦敦大学学院）；Computational Statistics and Machine Learning - Istituto Italiano di Tecnologia（意大利技术研究院计算统计与机器学习）
Trace norm regularization is a widely used approach for learning low rank matrices. A standard optimization strategy is based on formulating the problem as one of low rank matrix factorization which, however, leads to a non-convex problem. In practice this approach works well, and it is often computationally faster than standard convex solvers such as proximal gradient methods. Nevertheless, it is not guaranteed to converge to a global optimum, and the optimization can be trapped at poor stationary points. In this paper we show that it is possible to characterize all critical points of the non…
In this letter, we propose a new identification criterion that guarantees the recovery of the low-rank latent factors in the nonnegative matrix factorization (NMF) model, under mild conditions. Specifically, using the proposed criterion, it suffices to identify the latent factors if the rows of one factor are \emph{sufficiently scattered} over the nonnegative orthant, while no structural assumption is imposed on the other factor except being full-rank. This is by far the mildest condition under which the latent factors are provably identifiable from the NMF model.
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimiz…
Graphical model for factorization and completion of relatively high rank tensors by sparse sampling
D3 Center, Osaka University（大阪大学D3中心）；Department of Physics, Graduate School of Science, Osaka University（大阪大学理学研究生院物理学科）；Department of Systems Science, Graduate School of Informatics, Kyoto University（京都大学信息学研究生院系统科学学科）；RIKEN center for AIP（RIKEN AIP研究中心）
In this paper we introduce a new column selection strategy, named here ``Deviation Maximization", and apply it to compute rank-revealing QR factorizations as an alternative to the well known block version of the QR factorization with the column pivoting method, called QP3 and currently implemented in LAPACK's xgeqp3 routine. We show that the resulting algorithm, named QRDM, has similar rank-revealing properties of QP3 and better execution times. We present numerical test results on a wide data set of numerically singular matrices, which has become a reference in the recent literature.
We introduce a Generalized Randomized QR-decomposition that may be applied to arbitrary products of matrices and their inverses, without needing to explicitly compute the products or inverses. This factorization is a critical part of a communication-optimal spectral divide-and-conquer algorithm for the nonsymmetric eigenvalue problem. In this paper, we establish that this randomized QR-factorization satisfies the strong rank-revealing properties. We also formally prove its stability, making it suitable in applications. Finally, we present numerical experiments which demonstrate that our theore…
We present Flip-Flop Spectrum-Revealing QR (Flip-Flop SRQR) factorization, a significantly faster and more reliable variant of the QLP factorization of Stewart, for low-rank matrix approximations. Flip-Flop SRQR uses SRQR factorization to initialize a partial column pivoted QR factorization and then compute a partial LQ factorization. As observed by Stewart in his original QLP work, Flip-Flop SRQR tracks the exact singular values with "considerable fidelity". We develop singular value lower bounds and residual error upper bounds for Flip-Flop SRQR factorization. In situations where singular va…
Many applications in scientific computing and data science require the computation of a rank-revealing factorization of a large matrix. In many of these instances the classical algorithms for computing the singular value decomposition are prohibitively computationally expensive. The randomized singular value decomposition can often be helpful, but is not effective unless the numerical rank of the matrix is substantially smaller than the dimensions of the matrix. We introduce a new randomized algorithm for producing rank-revealing factorizations based on existing work by Demmel, Dumitriu and Ho…
The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank decomposition is often seen as an advantageous method to interpret and analyze data. In this work, we examine the factorizations of binary matrices using standard arithmetic (real and nonnegative) and logical operations (Boolean and $\mathbb{Z}_2$). We examine the relationships between the different ranks, and discuss when factorization is unique. In particular, we characterize when a Boolean factorization $X = W \land H$ has a unique $W$, a unique $H$ (for a…
In [2] a new factorization for infinite Hessenberg banded matrices was introduced. In this note we prove that this kind of factorization can also be used for finite matrices. In addition, a new method for solving banded linear systems is provided.
