A Flexible Optimization Framework for Regularized Matrix-Tensor Factorizations with Linear Couplings
Simula Metropolitan Center for Digital Engineering（Simula城市数字工程中心）；Oslo Metropolitan University（奥斯陆城市大学）；CNRS（法国国家科学研究中心）；Université de Rennes（雷恩大学）；Inria（法国国家信息与自动化研究所）；IRISA（IRISA研究所）
Coupled matrix and tensor factorizations (CMTF) are frequently used to jointly analyze data from multiple sources, also called data fusion. However, different characteristics of datasets stemming from multiple sources pose many challenges in data fusion and require to employ various regularizations, constraints, loss functions and different types of coupling structures between datasets. In this paper, we propose a flexible algorithmic framework for coupled matrix and tensor factorizations which utilizes Alternating Optimization (AO) and the Alternating Direction Method of Multipliers (ADMM). The framework facilitates the use of a variety of constraints, loss functions and couplings with linear transformations in a seamless way. Numerical experiments on simulated and real datasets demonstrate that the proposed approach is accurate, and computationally efficient with comparable or better performance than available CMTF methods for Frobenius norm loss, while being more flexible. Using Kullback-Leibler divergence on count data, we demonstrate that the algorithm yields accurate results also for other loss functions.
We present a general-purpose data compression algorithm, Regularized L21 Semi-NonNegative Matrix Factorization (L21 SNF). L21 SNF provides robust, parts-based compression applicable to mixed-sign data for which high fidelity, individualdata point reconstruction is paramount. We derive a rigorous proof of convergenceof our algorithm. Through experiments, we show the use-case advantages presentedby L21 SNF, including application to the compression of highly overdeterminedsystems encountered broadly across many general machine learning processes.
Manifold regularization methods for matrix factorization rely on the cluster assumption, whereby the neighborhood structure of data in the input space is preserved in the factorization space. We argue that using the k-neighborhoods of all data points as regularization constraints can negatively affect the quality of the factorization, and propose an unsupervised and selective regularized matrix factorization algorithm to tackle this problem. Our approach jointly learns a sparse set of representatives and their neighbor affinities, and the data factorization. We further propose a fast approxima…
University of Michigan（密歇根大学）；University of Florida（佛罗里达大学）；University of Florida Informatics Institute（佛罗里达大学信息学研究所）
We introduce a general tensor model suitable for data analytic tasks for {\em heterogeneous} datasets, wherein there are joint low-rank structures within groups of observations, but also discriminative structures across different groups. To capture such complex structures, a double core tensor (DCOT) factorization model is introduced together with a family of smoothing loss functions. By leveraging the proposed smoothing function, the model accurately estimates the model factors, even in the presence of missing entries. A linearized ADMM method is employed to solve regularized versions of DCOT…
We present a converged algorithm for Tikhonov regularized nonnegative matrix factorization (NMF). We specially choose this regularization because it is known that Tikhonov regularized least square (LS) is the more preferable form in solving linear inverse problems than the conventional LS. Because an NMF problem can be decomposed into LS subproblems, it can be expected that Tikhonov regularized NMF will be the more appropriate approach in solving NMF problems. The algorithm is derived using additive update rules which have been shown to have convergence guarantee. We equip the algorithm with a…
A Complete Loss Landscape Analysis of Regularized Deep Matrix Factorization
School of Data Science, Fudan University, Shanghai, China（复旦大学数据科学学院，上海，中国）；Department of Computer and Information Science, University of Macau, Macau SAR, China（澳门大学计算机与信息科学系，澳门特别行政区，中国）
Efficient algorithms for regularized Poisson Non-negative Matrix Factorization
Swiss Data Science Center（瑞士数据科学中心）；EPFL（洛桑联邦理工学院）；ETH Zürich（苏黎世联邦理工学院）；Unité Matériaux et Transformations（材料与转化单元）；Université de Lille（里尔大学）；CNRS（法国国家科学研究中心）；Institute of Physics（物理研究所）；Electron Spectrometry and Microscopy Laboratory（电子光谱与显微镜实验室）
Inria Saclay（法国国家信息与自动化研究所萨克莱分所）；Univ. Paris Saclay（巴黎萨克雷大学）；IIIT - Delhi（德里英迪拉·甘地信息技术学院）
Co-administration of two or more drugs simultaneously can result in adverse drug reactions. Identifying drug-drug interactions (DDIs) is necessary, especially for drug development and for repurposing old drugs. DDI prediction can be viewed as a matrix completion task, for which matrix factorization (MF) appears as a suitable solution. This paper presents a novel Graph Regularized Probabilistic Matrix Factorization (GRPMF) method, which incorporates expert knowledge through a novel graph-based regularization strategy within an MF framework. An efficient and sounded optimization algorithm is pro…
When recommending personalized top-$k$ items to users, how can we recommend the items diversely to them while satisfying their needs? Aggregately diversified recommender systems aim to recommend a variety of items across whole users without sacrificing the recommendation accuracy. They increase the exposure opportunities of various items, which in turn increase potential revenue of sellers as well as user satisfaction. However, it is challenging to tackle aggregate-level diversity with a matrix factorization (MF), one of the most common recommendation model, since skewed real world data lead t…
Southern University of Science and Technology（南方科技大学）；Sun Yat-sen University（中山大学）
Tensor decomposition is an effective tool for learning multi-way structures and heterogeneous features from high-dimensional data, such as the multi-view images and multichannel electroencephalography (EEG) signals, are often represented by tensors. However, most of tensor decomposition methods are the linear feature extraction techniques, which are unable to reveal the nonlinear structure within high-dimensional data. To address such problem, a lot of algorithms have been proposed for simultaneously performs linear and non-linear feature extraction. A representative algorithm is the Graph Reg…
Community describes the functional mechanism of a network, making community detection serve as a fundamental graph tool for various real applications like discovery of social circle. To date, a Symmetric and Non-negative Matrix Factorization (SNMF) model has been frequently adopted to address this issue owing to its high interpretability and scalability. However, most existing SNMF-based community detection methods neglect the high-order connection patterns in a network. Motivated by this discovery, in this paper, we propose a High-Order Proximity (HOP)-incorporated, Symmetry and Graph-regular…
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 eva…
Urban areas provide us with a treasure trove of available data capturing almost every aspect of a population's life. This work focuses on mobility data and how it will help improve our understanding of urban mobility patterns. Readily available and sizable farecard data captures trips in a public transportation network. However, such data typically lacks temporal modalities and as such the task of inferring trip semantic, station function, and user profile is quite challenging. As existing approaches either focus on station-level or user-level signals, they are prone to overfitting and generat…
In this paper, we introduce a probabilistic model for learning nonnegative matrix factorization (NMF) that is commonly used for predicting missing values and finding hidden patterns in the data, in which the matrix factors are latent variables associated with each data dimension. The nonnegativity constraint for the latent factors is handled by choosing priors with support on the nonnegative subspace. Bayesian inference procedure based on Gibbs sampling is employed. We evaluate the model on several real-world datasets including MovieLens 100K and MovieLens 1M with different sizes and dimension…
This work explores non-negative low-rank matrix factorization based on regularized Poisson models (PF or "Poisson factorization" for short) for recommender systems with implicit-feedback data. The properties of Poisson likelihood allow a shortcut for very fast computations over zero-valued inputs, and oftentimes results in very sparse factors for both users and items. Compared to HPF (a popular Bayesian formulation of the problem with hierarchical priors), the frequentist optimization-based approach presented here tends to produce better top-N recommendations with significantly shorter fitting…
IIIT - Delhi（德里印度理工学院）；Inria Saclay（法国国家信息与自动化研究所萨克雷分部）；Univ. Paris Saclay（巴黎萨克雷大学）
This work formulates antiviral repositioning as a matrix completion problem where the antiviral drugs are along the rows and the viruses along the columns. The input matrix is partially filled, with ones in positions where the antiviral has been known to be effective against a virus. The curated metadata for antivirals (chemical structure and pathways) and viruses (genomic structure and symptoms) is encoded into our matrix completion framework as graph Laplacian regularization. We then frame the resulting multiple graph regularized matrix completion problem as deep matrix factorization. This i…
Soochow University（苏州大学）；Hefei University of Technology（合肥工业大学）；Adobe Research（奥多比研究院）；Nanjing University of Information Science and Technology（南京信息工程大学）；National University of Singapore（新加坡国立大学）
We investigate the high-dimensional data clustering problem by proposing a novel and unsupervised representation learning model called Robust Flexible Auto-weighted Local-coordinate Concept Factorization (RFA-LCF). RFA-LCF integrates the robust flexible CF, robust sparse local-coordinate coding and the adaptive reconstruction weighting learning into a unified model. The adaptive weighting is driven by including the joint manifold preserving constraints on the recovered clean data, basis concepts and new representation. Specifically, our RFA-LCF uses a L2,1-norm based flexible residue to encode…
Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳））；The University of Queensland（昆士兰大学）
Clustering with incomplete views is a challenge in multi-view clustering. In this paper, we provide a novel and simple method to address this issue. Specifically, the proposed method simultaneously exploits the local information of each view and the complementary information among views to learn the common latent representation for all samples, which can greatly improve the compactness and discriminability of the obtained representation. Compared with the conventional graph embedding methods, the proposed method does not introduce any extra regularization term and corresponding penalty paramet…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Technicolor Research（特艺集团研究院）
High-dimensional time series prediction is needed in applications as diverse as demand forecasting and climatology. Often, such applications require methods that are both highly scalable, and deal with noisy data in terms of corruptions or missing values. Classical time series methods usually fall short of handling both these issues. In this paper, we propose to adapt matrix matrix completion approaches that have previously been successfully applied to large scale noisy data, but which fail to adequately model high-dimensional time series due to temporal dependencies. We present a novel tempor…
Beihang University（北京航空航天大学）；Microsoft Research Asia（微软亚洲研究院）
Recent years have witnessed the world-wide emergence of mega-metropolises with incredibly huge populations. Understanding residents mobility patterns, or urban dynamics, thus becomes crucial for building modern smart cities. In this paper, we propose a Neighbor-Regularized and context-aware Non-negative Tensor Factorization model (NR-cNTF) to discover interpretable urban dynamics from urban heterogeneous data. Different from many existing studies concerned with prediction tasks via tensor completion, NR-cNTF focuses on gaining urban managerial insights from spatial, temporal, and spatio-tempor…
Deep matrix factorizations (deep MFs) are recent unsupervised data mining techniques inspired by constrained low-rank approximations. They aim to extract complex hierarchies of features within high-dimensional datasets. Most of the loss functions proposed in the literature to evaluate the quality of deep MF models and the underlying optimization frameworks are not consistent because different losses are used at different layers. In this paper, we introduce two meaningful loss functions for deep MF and present a generic framework to solve the corresponding optimization problems. We illustrate t…
Purdue University（普渡大学）；Lawrence Berkeley Lab（劳伦斯伯克利实验室）；Yahoo! Research（雅虎研究院）；University of Electronic Science and Technology of China（电子科技大学）；University of Cambridge（剑桥大学）
Tensor factorization is a powerful tool to analyse multi-way data. Compared with traditional multi-linear methods, nonlinear tensor factorization models are capable of capturing more complex relationships in the data. However, they are computationally expensive and may suffer severe learning bias in case of extreme data sparsity. To overcome these limitations, in this paper we propose a distributed, flexible nonlinear tensor factorization model. Our model can effectively avoid the expensive computations and structural restrictions of the Kronecker-product in existing TGP formulations, allowing…
Matrix factorization is a popular framework for modeling low-rank data matrices. Motivated by manifold learning problems, this paper proposes a quadratic matrix factorization (QMF) framework to learn the curved manifold on which the dataset lies. Unlike local linear methods such as the local principal component analysis, QMF can better exploit the curved structure of the underlying manifold. Algorithmically, we propose an alternating minimization algorithm to optimize QMF and establish its theoretical convergence properties. Moreover, to avoid possible over-fitting, we then propose a regulariz…
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）
Inthischapterwediscusshowtolearnanoptimalmanifoldpresentationto regularize nonegative matrix factorization (NMF) for data representation problems. NMF,whichtriestorepresentanonnegativedatamatrixasaproductoftwolowrank nonnegative matrices, has been a popular method for data representation due to its ability to explore the latent part-based structure of data. Recent study shows that lots of data distributions have manifold structures, and we should respect the manifold structure when the data are represented. Recently, manifold regularized NMF used a nearest neighbor graph to regulate the learni…
United States Census Bureau（美国人口普查局）；Colorado State University（科罗拉多州立大学）
The nonnegative matrix factorization is a widely used, flexible matrix decomposition, finding applications in biology, image and signal processing and information retrieval, among other areas. Here we present a related matrix factorization. A multi-objective optimization problem finds conical combinations of templates that approximate a given data matrix. The templates are chosen so that as far as possible only the initial data set can be represented this way. However, the templates are not required to be nonnegative nor convex combinations of the original data.
Traditional NMF-based signal decomposition relies on the factorization of spectral data, which is typically computed by means of short-time frequency transform. In this paper we propose to relax the choice of a pre-fixed transform and learn a short-time orthogonal transform together with the factorization. To this end, we formulate a regularized optimization problem reminiscent of conventional NMF, yet with the transform as additional unknown parameters, and design a novel block-descent algorithm enabling to find stationary points of this objective function. The proposed joint transform learni…
This paper is concerned with the factorization form of the rank regularized loss minimization problem. To cater for the scenario in which only a coarse estimation is available for the rank of the true matrix, an $\ell_{2,0}$-norm regularized term is added to the factored loss function to reduce the rank adaptively; and account for the ambiguities in the factorization, a balanced term is then introduced. For the least squares loss, under a restricted condition number assumption on the sampling operator, we establish the KL property of exponent $1/2$ of the nonsmooth factored composite function …
This paper presents a sequential randomized lowrank matrix factorization approach for incrementally predicting values of an unknown function at test points using the Gaussian Processes framework. It is well-known that in the Gaussian processes framework, the computational bottlenecks are the inversion of the (regularized) kernel matrix and the computation of the hyper-parameters defining the kernel. The main contributions of this paper are two-fold. First, we formalize an approach to compute the inverse of the kernel matrix using randomized matrix factorization algorithms in a streaming scenar…
This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian for the loss function, we derive an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, under the noisy and full sample setting we establish its KL property of exponent $1/2$ on its global minimizer set, and under the noisy and partial sample s…
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…
Istituto Italiano di Tecnologia（意大利技术研究院）；Johns Hopkins University（约翰斯·霍普金斯大学）
Dropout is a simple yet effective algorithm for regularizing neural networks by randomly dropping out units through Bernoulli multiplicative noise, and for some restricted problem classes, such as linear or logistic regression, several theoretical studies have demonstrated the equivalence between dropout and a fully deterministic optimization problem with data-dependent Tikhonov regularization. This work presents a theoretical analysis of dropout for matrix factorization, where Bernoulli random variables are used to drop a factor, thereby attempting to control the size of the factorization. Wh…
Motivated by the needs of online large-scale recommender systems, we specialize the decoupled extended Kalman filter (DEKF) to factorization models, including factorization machines, matrix and tensor factorization, and illustrate the effectiveness of the approach through numerical experiments on synthetic and on real-world data. Online learning of model parameters through the DEKF makes factorization models more broadly useful by (i) allowing for more flexible observations through the entire exponential family, (ii) modeling parameter drift, and (iii) producing parameter uncertainty estimates…
University of Florida（佛罗里达大学）；Johns Hopkins University（约翰霍普金斯大学）
With the recent success of representation learning methods, which includes deep learning as a special case, there has been considerable interest in developing representation learning techniques that can incorporate known physical constraints into the learned representation. As one example, in many applications that involve a signal propagating through physical media (e.g., optics, acoustics, fluid dynamics, etc), it is known that the dynamics of the signal must satisfy constraints imposed by the wave equation. Here we propose a matrix factorization technique that decomposes such signals into a…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所 HIIT）；Aalto University（阿尔托大学）；Tampere University（坦佩雷大学）；Kyoto University（京都大学）
In matrix factorization, available graph side-information may not be well suited for the matrix completion problem, having edges that disagree with the latent-feature relations learnt from the incomplete data matrix. We show that removing these $\textit{contested}$ edges improves prediction accuracy and scalability. We identify the contested edges through a highly-efficient graphical lasso approximation. The identification and removal of contested edges adds no computational complexity to state-of-the-art graph-regularized matrix factorization, remaining linear with respect to the number of no…
Brain networks in fMRI are typically identified using spatial independent component analysis (ICA), yet mathematical constraints such as sparse coding and positivity both provide alternate biologically-plausible frameworks for generating brain networks. Non-negative Matrix Factorization (NMF) would suppress negative BOLD signal by enforcing positivity. Spatial sparse coding algorithms ($L1$ Regularized Learning and K-SVD) would impose local specialization and a discouragement of multitasking, where the total observed activity in a single voxel originates from a restricted number of possible br…
Nonnegative Matrix Factorization (NMF) has been continuously evolving in several areas like pattern recognition and information retrieval methods. It factorizes a matrix into a product of 2 low-rank non-negative matrices that will define parts-based, and linear representation of nonnegative data. Recently, Graph regularized NMF (GrNMF) is proposed to find a compact representation,which uncovers the hidden semantics and simultaneously respects the intrinsic geometric structure. In GNMF, an affinity graph is constructed from the original data space to encode the geometrical information. In this …
Northeastern University（东北大学）；Stevens Institute of Technology（史蒂文斯理工学院）
Nonnegative matrix factorization (NMF) has been widely used to dimensionality reduction in machine learning. However, the traditional NMF does not properly handle outliers, so that it is sensitive to noise. In order to improve the robustness of NMF, this paper proposes an adaptive weighted NMF, which introduces weights to emphasize the different importance of each data point, thus the algorithmic sensitivity to noisy data is decreased. It is very different from the existing robust NMFs that use a slow growth similarity measure. Specifically, two strategies are proposed to achieve this: fuzzier…
Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization can be solved very efficiently by standard algorithms in practice. However, due to the non-convexity caused by the factorization model, there is a limited theoretical understanding of this formulation. In this paper, we establish a theoretical guarantee for the factorization formulation to correctly recover the underlying low-rank matrix. In particular, we show that under similar conditions to those in previous works, many standard optimization algorithms con…
Valid causal inference in observational studies often requires controlling for confounders. However, in practice measurements of confounders may be noisy, and can lead to biased estimates of causal effects. We show that we can reduce the bias caused by measurement noise using a large number of noisy measurements of the underlying confounders. We propose the use of matrix factorization to infer the confounders from noisy covariates, a flexible and principled framework that adapts to missing values, accommodates a wide variety of data types, and can augment many causal inference methods. We boun…
The Ohio State University（俄亥俄州立大学）；The University of Hong Kong（香港大学）；Carnegie Mellon University（卡内基梅隆大学）
Low-rank matrix completion has achieved great success in many real-world data applications. A matrix factorization model that learns latent features is usually employed and, to improve prediction performance, the similarities between latent variables can be exploited by pairwise learning using the graph regularized matrix factorization (GRMF) method. However, existing GRMF approaches often use the squared loss to measure the pairwise differences, which may be overly influenced by dissimilar pairs and lead to inferior prediction. To fully empower pairwise learning for matrix completion, we prop…
College of Arts and Sciences, National University of Defense Technology（国防科技大学文理学院）
It has been an important approach of using matrix completion to perform image restoration. Most previous works on matrix completion focus on the low-rank property by imposing explicit constraints on the recovered matrix, such as the constraint of the nuclear norm or limiting the dimension of the matrix factorization component. Recently, theoretical works suggest that deep linear neural network has an implicit bias towards low rank on matrix completion. However, low rank is not adequate to reflect the intrinsic characteristics of a natural image. Thus, algorithms with only the constraint of low…
Qingdao University（青岛大学）；Harbin Institute of Technology（哈尔滨工业大学）；University of Electronic Science and Technology of China（电子科技大学）；University of Kentucky（肯塔基大学）
Nonnegative matrix factorization (NMF) has been widely studied in recent years due to its effectiveness in representing nonnegative data with parts-based representations. For NMF, a sparser solution implies better parts-based representation.However, current NMF methods do not always generate sparse solutions.In this paper, we propose a new NMF method with log-norm imposed on the factor matrices to enhance the sparseness.Moreover, we propose a novel column-wisely sparse norm, named $\ell_{2,\log}$-(pseudo) norm to enhance the robustness of the proposed method.The $\ell_{2,\log}$-(pseudo) norm i…
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…
Northwestern Polytechnical University（西北工业大学）；Center for Optical Imagery Analysis and Learning (OPTIMAL)（光学影像分析与学习中心）
Nonnegative Matrix Factorization (NMF) is a widely-used data analysis technique, and has yielded impressive results in many real-world tasks. Generally, existing NMF methods represent each sample with several centroids, and find the optimal centroids by minimizing the sum of the approximation errors. However, the outliers deviating from the normal data distribution may have large residues, and then dominate the objective value seriously. In this study, an Entropy Minimizing Matrix Factorization framework (EMMF) is developed to tackle the above problem. Considering that the outliers are usually…
Predicting the novel effects of drugs based on information about approved drugs can be regarded as a recommendation system. Matrix factorization is one of the most used recommendation systems and various algorithms have been devised for it. A literature survey and summary of existing algorithms for predicting drug effects demonstrated that most such methods, including neighborhood regularized logistic matrix factorization, which was the best performer in benchmark tests, used a binary matrix that considers only the presence or absence of interactions. However, drug effects are known to have tw…
Northeastern University（东北大学）；Stevens Institute of Technology（史蒂文斯理工学院）
Traditional nonnegative matrix factorization (NMF) learns a new feature representation on the whole data space, which means treating all features equally. However, a subspace is often sufficient for accurate representation in practical applications, and redundant features can be invalid or even harmful. For example, if a camera has some sensors destroyed, then the corresponding pixels in the photos from this camera are not helpful to identify the content, which means only the subspace consisting of remaining pixels is worthy of attention. This paper proposes a new NMF method by introducing ada…
