Subspace segmentation assumes that data comes from the union of different subspaces and the purpose of segmentation is to partition the data into the corresponding subspace. Low-rank representation (LRR) is a classic spectral-type method for solving subspace segmentation problems, that is, one first obtains an affinity matrix by solving a LRR model and then performs spectral clustering for segmentation. This paper proposes a group norm regularized factorization model (GNRFM) inspired by the LRR model for subspace segmentation and then designs an Accelerated Augmented Lagrangian Method (AALM) algorithm to solve this model. Specifically, we adopt group norm regularization to make the columns of the factor matrix sparse, thereby achieving a purpose of low rank, which means no Singular Value Decompositions (SVD) are required and the computational complexity of each step is greatly reduced. We obtain affinity matrices by using different LRR models and then performing cluster testing on different sets of synthetic noisy data and real data, respectively. Compared with traditional models and algorithms, the proposed method is faster and more robust to noise, so the final clustering results are better. Moreover, the numerical results show that our algorithm converges fast and only requires approximately ten iterations.
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 …
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…
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…
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…
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.
Ruder Bošković Institute（鲁杰尔·博什科维奇研究所）；ETH Zürich, Swiss Federal Institute of Technology（苏黎世联邦理工学院）
A recent theoretical analysis shows the equivalence between non-negative matrix factorization (NMF) and spectral clustering based approach to subspace clustering. As NMF and many of its variants are essentially linear, we introduce a nonlinear NMF with explicit orthogonality and derive general kernel-based orthogonal multiplicative update rules to solve the subspace clustering problem. In nonlinear orthogonal NMF framework, we propose two subspace clustering algorithms, named kernel-based non-negative subspace clustering KNSC-Ncut and KNSC-Rcut and establish their connection with spectral norm…
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…
Continuous learning seeks to perform the learning on the data that arrives from time to time. While prior works have demonstrated several possible solutions, these approaches require excessive training time as well as memory usage. This is impractical for applications where time and storage are constrained, such as edge computing. In this work, a novel training algorithm, regularized subspace approximation classifier (RSAC), is proposed to achieve lightweight continuous learning. RSAC contains a feature reduction module and classifier module with regularization. Extensive experiments show that…
Krylov Cubic Regularized Newton: A Subspace Second-Order Method with Dimension-Free Convergence Rate
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Amazon Web Services（亚马逊云科技）；Technion – Israel Institute of Technology（以色列理工学院）；University of Minnesota（明尼苏达大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Low-Rank Representation (LRR) highly suffers from discarding the locality information of data points in subspace clustering, as it may not incorporate the data structure nonlinearity and the non-uniform distribution of observations over the ambient space. Thus, the information of the observational density is lost by the state-of-art LRR models, as they take a constant number of adjacent neighbors into account. This, as a result, degrades the subspace clustering accuracy in such situations. To cope with deficiency, in this paper, we propose to consider a hypergraph model to facilitate having a …
School of Software Engineering, Xi’an Jiaotong University（西安交通大学软件学院）；School of Computer Science and Engineering, Nanjing University of Science and Technology（南京理工大学计算机科学与工程学院）
Multi-view clustering is an important and fundamental problem. Many multi-view subspace clustering methods have been proposed, and most of them assume that all views share a same coefficient matrix. However, the underlying information of multi-view data are not fully exploited under this assumption, since the coefficient matrices of different views should have the same clustering properties rather than be uniform among multiple views. To this end, this paper proposes a novel Constrained Bilinear Factorization Multi-view Subspace Clustering (CBF-MSC) method. Specifically, the bilinear factoriza…
Xi’an Jiaotong University（西安交通大学）；Shanghai University（上海大学）；Nanjing University of Posts and Telecommunications（南京邮电大学）
This study investigates the problem of multi-view clustering, where multiple views contain consistent information and each view also includes complementary information. Exploration of all information is crucial for good multi-view clustering. However, most traditional methods blindly or crudely combine multiple views for clustering and are unable to fully exploit the valuable information. Therefore, we propose a method that involves consistent and complementary graph-regularized multi-view subspace clustering (GRMSC), which simultaneously integrates a consistent graph regularizer with a comple…
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). T…
College of Informatics, Huazhong Agricultural University（华中农业大学信息学院）
Recent years have witnessed a growing academic interest in multi-view subspace clustering. In this paper, we propose a novel Double Graphs Regularized Multi-view Subspace Clustering (DGRMSC) method, which aims to harness both global and local structural information of multi-view data in a unified framework. Specifically, DGRMSC firstly learns a latent representation to exploit the global complementary information of multiple views. Based on the learned latent representation, we learn a self-representation to explore its global cluster structure. Further, Double Graphs Regularization (DGR) is p…
the Hong Kong Polytechnic University（香港理工大学）；Harbin Institute of Technology（哈尔滨工业大学）
Multi-view subspace learning (MSL) aims to find a low-dimensional subspace of the data obtained from multiple views. Different from single view case, MSL should take both common and specific knowledge among different views into consideration. To enhance the robustness of model, the complexity, non-consistency and similarity of noise in multi-view data should be fully taken into consideration. Most current MSL methods only assume a simple Gaussian or Laplacian distribution for the noise while neglect the complex noise configurations in each view and noise correlations among different views of p…
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…
In this paper, we extend the recently proposed multivariate rank energy distance, based on the theory of optimal transport, for statistical testing of distributional similarity, to soft rank energy distance. Being differentiable, this in turn allows us to extend the rank energy to a subspace robust rank energy distance, dubbed Projected soft-Rank Energy distance, which can be computed via optimization over the Stiefel manifold. We show via experiments that using projected soft rank energy one can trade-off the detection power vs the false alarm via projections onto an appropriately selected lo…
We study a natural extension of classical empirical risk minimization, where the hypothesis space is a random subspace of a given space. In particular, we consider possibly data dependent subspaces spanned by a random subset of the data, recovering as a special case Nystrom approaches for kernel methods. Considering random subspaces naturally leads to computational savings, but the question is whether the corresponding learning accuracy is degraded. These statistical-computational tradeoffs have been recently explored for the least squares loss and self-concordant loss functions, such as the l…
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…
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）
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…
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）
In many multilingual text classification problems, the documents in different languages often share the same set of categories. To reduce the labeling cost of training a classification model for each individual language, it is important to transfer the label knowledge gained from one language to another language by conducting cross language classification. In this paper we develop a novel subspace co-regularized multi-view learning method for cross language text classification. This method is built on parallel corpora produced by machine translation. It jointly minimizes the training error of …
The inverse-free extreme learning machine (ELM) algorithm proposed in [4] was based on an inverse-free algorithm to compute the regularized pseudo-inverse, which was deduced from an inverse-free recursive algorithm to update the inverse of a Hermitian matrix. Before that recursive algorithm was applied in [4], its improved version had been utilized in previous literatures [9], [10]. Accordingly from the improved recursive algorithm [9], [10], we deduce a more efficient inverse-free algorithm to update the regularized pseudo-inverse, from which we develop the proposed inverse-free ELM algorithm…
The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）
Subspace clustering (SC) aims to cluster data lying in a union of low-dimensional subspaces. Usually, SC learns an affinity matrix and then performs spectral clustering. Both steps suffer from high time and space complexity, which leads to difficulty in clustering large datasets. This paper presents a method called k-Factorization Subspace Clustering (k-FSC) for large-scale subspace clustering. K-FSC directly factorizes the data into k groups via pursuing structured sparsity in the matrix factorization model. Thus, k-FSC avoids learning affinity matrix and performing eigenvalue decomposition, …
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…
Guangdong University of Technology（广东工业大学）；RIKEN（理化学研究所）
For the high dimensional data representation, nonnegative tensor ring (NTR) decomposition equipped with manifold learning has become a promising model to exploit the multi-dimensional structure and extract the feature from tensor data. However, the existing methods such as graph regularized tensor ring decomposition (GNTR) only models the pair-wise similarities of objects. For tensor data with complex manifold structure, the graph can not exactly construct similarity relationships. In this paper, in order to effectively utilize the higher-dimensional and complicated similarities among objects,…
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…
In this paper, we revisit implicit regularization from the ground up using notions from dynamical systems and invariant subspaces of Morse functions. The key contributions are a new criterion for implicit regularization---a leading contender to explain the generalization power of deep models such as neural networks---and a general blueprint to study it. We apply these techniques to settle a conjecture on implicit regularization in matrix factorization.
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…
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 (…
Nonnegative matrix factorization (NMF) with group sparsity constraints is formulated as a probabilistic graphical model and, assuming some observed data have been generated by the model, a feasible variational Bayesian algorithm is derived for learning model parameters. When used in a supervised learning scenario, NMF is most often utilized as an unsupervised feature extractor followed by classification in the obtained feature subspace. Having mapped the class labels to a more general concept of groups which underlie sparsity of the coefficients, what the proposed group sparse NMF model allows…
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…
In this paper, we propose a novel approach in order to recover a quantized matrix with missing information. We propose a regularized convex cost function composed of a log-likelihood term and a Trace norm term. The Bi-factorization approach and the Augmented Lagrangian Method (ALM) are applied to find the global minimizer of the cost function in order to recover the genuine data. We provide mathematical convergence analysis for our proposed algorithm. In the Numerical Experiments Section, we show the superiority of our method in accuracy and also its robustness in computational complexity comp…
Regularized M-estimators are used in diverse areas of science and engineering to fit high-dimensional models with some low-dimensional structure. Usually the low-dimensional structure is encoded by the presence of the (unknown) parameters in some low-dimensional model subspace. In such settings, it is desirable for estimates of the model parameters to be \emph{model selection consistent}: the estimates also fall in the model subspace. We develop a general framework for establishing consistency and model selection consistency of regularized M-estimators and show how it applies to some special c…
Southwest University（西南大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；George Mason University（乔治梅森大学）
Multi-view clustering aims at integrating complementary information from multiple heterogeneous views to improve clustering results. Existing multi-view clustering solutions can only output a single clustering of the data. Due to their multiplicity, multi-view data, can have different groupings that are reasonable and interesting from different perspectives. However, how to find multiple, meaningful, and diverse clustering results from multi-view data is still a rarely studied and challenging topic in multi-view clustering and multiple clusterings. In this paper, we introduce a deep matrix fac…
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…
School of Mathematics and Information Sciences, Yantai University（烟台大学数学与信息科学学院）；Department of Mathematics, Shandong Normal University（山东师范大学数学学院）
The implementation of conventional sparse principal component analysis (SPCA) on high-dimensional data sets has become a time consuming work. In this paper, a series of subspace projections are constructed efficiently by using Household QR factorization. With the aid of these subspace projections, a fast deflation method, called SPCA-SP, is developed for SPCA. This method keeps a good tradeoff between various criteria, including sparsity, orthogonality, explained variance, balance of sparsity, and computational cost. Comparative experiments on the benchmark data sets confirm the effectiveness …
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…
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…
iALS is a popular algorithm for learning matrix factorization models from implicit feedback with alternating least squares. This algorithm was invented over a decade ago but still shows competitive quality compared to recent approaches like VAE, EASE, SLIM, or NCF. Due to a computational trick that avoids negative sampling, iALS is very efficient especially for large item catalogues. However, iALS does not scale well with large embedding dimensions, d, due to its cubic runtime dependency on d. Coordinate descent variations, iCD, have been proposed to lower the complexity to quadratic in d. In …
National University of Defense Technology（国防科技大学）；University of Sydney（悉尼大学）；Google（谷歌）
Non-negative matrix factorization (NMF) minimizes the Euclidean distance between the data matrix and its low rank approximation, and it fails when applied to corrupted data because the loss function is sensitive to outliers. In this paper, we propose a Truncated CauchyNMF loss that handle outliers by truncating large errors, and develop a Truncated CauchyNMF to robustly learn the subspace on noisy datasets contaminated by outliers. We theoretically analyze the robustness of Truncated CauchyNMF comparing with the competing models and theoretically prove that Truncated CauchyNMF has a generaliza…
In many applications, high-dimensional data points can be well represented by low-dimensional subspaces. To identify the subspaces, it is important to capture a global and local structure of the data which is achieved by imposing low-rank and sparseness constraints on the data representation matrix. In low-rank sparse subspace clustering (LRSSC), nuclear and $\ell_1$ norms are used to measure rank and sparsity. However, the use of nuclear and $\ell_1$ norms leads to an overpenalized problem and only approximates the original problem. In this paper, we propose two $\ell_0$ quasi-norm based regu…
Sparse subspace clustering methods, such as Sparse Subspace Clustering (SSC) \cite{ElhamifarV13} and $\ell^{1}$-graph \cite{YanW09,ChengYYFH10}, are effective in partitioning the data that lie in a union of subspaces. Most of those methods use $\ell^{1}$-norm or $\ell^{2}$-norm with thresholding to impose the sparsity of the constructed sparse similarity graph, and certain assumptions, e.g. independence or disjointness, on the subspaces are required to obtain the subspace-sparse representation, which is the key to their success. Such assumptions are not guaranteed to hold in practice and they …
The Pennsylvania State University（宾夕法尼亚州立大学）；Google LLC（谷歌有限责任公司）
Real-world social networks and digital platforms are comprised of individuals (nodes) that are linked to other individuals or entities through multiple types of relationships (links). Sub-networks of such a network based on each type of link correspond to distinct views of the underlying network. In real-world applications, each node is typically linked to only a small subset of other nodes. Hence, practical approaches to problems such as node labeling have to cope with the resulting sparse networks. While low-dimensional network embeddings offer a promising approach to this problem, most of t…
