In the low rank matrix completion (LRMC) problem, the low rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces (UoS) are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. For example, a second-order tensorization representation is formed by taking the outer product of each column with itself, and we consider higher order tensorizations as well. This approach will succeed in many cases where traditional LRMC is guaranteed to fail because the data are low-rank in the tensorized representation but not in the original representation. We also provide a formal mathematical justification for the success of our method. In particular, we show bounds of the rank of these data in the tensorized representation, and we prove sampling requirements to guarantee uniqueness of the solution. Interestingly, the sampling requirements of our LADMC algorithm nearly match the information theoretic lower bounds for matrix completion under a UoS model. We also provide experimental results showing that the new approach significantly outperforms existing state-of-the-art methods for matrix completion in many situations.
The present work is closely related to the algebraic subspace clustering (ASC) framework, also known as generalized principal component analysis (GPCA) @cite_41 @cite_35 @cite_5 @cite_6 . Similar to our approach, the ASC framework models unions of subspaces as an algebraic variety, and makes use of tensorizations ( Veronese embeddings) of the data to identify the subspaces. Characterizations of the rank of tensorized data when the original (untensorized) data belongs to a union of subspaces are fundamental to the ASC framework and are based on results in @cite_30 . However, the results in @cite_30 only apply for higher-order tensorizations of the data, and we prove in this work that similar characterizations of the rank hold for lower-order tensorizations. Finally, the ASC framework has not been extended to the missing data matrix completion setting, which is the main focus of this work.
@cite_30 @cite_35 @cite_41 @cite_6 @cite_5
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Recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high-dimensional data that have very complicated structures. From both geometric and statistical points of view, such unsorted data are considered mixed as different parts of the data have significantly different structures which cannot be described by a single model. In this paper we propose to use subspace arrangements—a union of multiple subspaces—for modeling mixed data: each subspace in the arrangement is used to model just a homogeneous subset of the data. Thus, multiple subspaces together can capture the heterogeneous structures within the data set. In this paper, we give a comprehensive introduction to a new approach for the estimation of subspace arrangements. This is known as generalized principal component analysis (GPCA). In particular, we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust, even when the given data are corrupted by noise or contaminated with outliers. This new method in many ways improves and generalizes extant methods for modeling or clustering mixed data. There have been successful applications of this new method to many real-world problems in computer vision, image processing, and system identification. In this paper, we will examine several of those representative applications. This paper is intended to be expository in nature. However, in order that this may serve as a more complete reference for both theoreticians and practitioners, we take the liberty of filling in several gaps between the theory and the practice in the existing literature. In this paper, we address the multi-view subspace clustering problem. Our method utilizes the circulant algebra for tensor, which is constructed by stacking the subspace representation matrices of different views and then rotating, to capture the low rank tensor subspace so that the refinement of the view-specific subspaces can be achieved, as well as the high order correlations underlying multi-view data can be explored. By introducing a recently proposed tensor factorization, namely tensor-Singular Value Decomposition (t-SVD) kilmer13 , we can impose a new type of low-rank tensor constraint on the rotated tensor to capture the complementary information from multiple views. Different from traditional unfolding based tensor norm, this low-rank tensor constraint has optimality properties similar to that of matrix rank derived from SVD, so the complementary information among views can be explored more efficiently and thoroughly. The established model, called t-SVD based Multi-view Subspace Clustering (t-SVD-MSC), falls into the applicable scope of augmented Lagrangian method, and its minimization problem can be efficiently solved with theoretical convergence guarantee and relatively low computational complexity. Extensive experimental testing on eight challenging image dataset shows that the proposed method has achieved highly competent objective performance compared to several state-of-the-art multi-view clustering methods. Subspace learning techniques have been extensively used for dimensionality reduction (DR) in many pattern classification problem domains. Recently, methods like Subclass Discriminant Analysis (SDA) and Clustering-based Discriminant Analysis (CDA), which use subclass information for the discrimination between the data classes, have attracted much attention. In parallel, important work has been accomplished on Graph Embedding (GE), which is a general framework unifying several subspace learning techniques. In this paper, GE has been extended in order to integrate subclass discriminant information resulting to the novel Subclass Graph Embedding (SGE) framework, which is the main contribution of our work. It is proven that SGE encapsulates a diversity of both supervised and unsupervised unimodal methods like Locality Preserving Projections (LPP), Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA). The theoretical link of SDA and CDA methods with SGE is also established. Along these lines, it is shown that SGE comprises a generalization of the typical GE framework including subclass DR methods. Moreover, it allows for an easy utilization of kernels for confronting non-linearly separable data. Employing SGE, in this paper a novel DR algorithm, which uses subclass discriminant information, called Subclass Marginal Fisher Analysis (SMFA) has been proposed. Through a series of experiments on various real-world datasets, it is shown that SMFA outperforms in most of the cases the state-of-the-art demonstrating the efficacy and power of SGE as a platform to develop new methods. HighlightsGraph Embedding is extended in order to integrate subclass informationThe novel Subclass Graph Embedding framework is proposed.The kernelized version of the new framework is presentedSubclass Graph Embedding encapsulates various subspace learning methods.A novel Subclass Marginal Fisher Analysis method is proposed. In this paper, we explore the problem of multiview subspace clustering. We introduce a low-rank tensor constraint to explore the complementary information from multiple views and, accordingly, establish a novel method called Low-rank Tensor constrained Multiview Subspace Clustering (LT-MSC). Our method regards the subspace representation matrices of different views as a tensor, which captures dexterously the high order correlations underlying multiview data. Then the tensor is equipped with a low-rank constraint, which models elegantly the cross information among different views, reduces effectually the redundancy of the learned subspace representations, and improves the accuracy of clustering as well. The inference process of the affinity matrix for clustering is formulated as a tensor nuclear norm minimization problem, constrained with an additional L2,1-norm regularizer and some linear equalities. The minimization problem is convex and thus can be solved efficiently by an Augmented Lagrangian Alternating Direction Minimization (AL-ADM) method. Extensive experimental results on four benchmark datasets show the effectiveness of our proposed LT-MSC method.