Tensor Methods for Nonlinear Matrix Completion

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

Sparse Subspace Clustering: Algorithm, Theory, and Applications

In many real-world problems, we are dealing with collections of high-dimensional data, such as images, videos, text and web documents, DNA microarray data, and more. Often, high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories the data belongs to. In this paper, we propose and study an algorithm, called Sparse Subspace Clustering (SSC), to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of data into subspaces. Since solving the sparse optimization program is in general NP-hard, we consider a convex relaxation and show that, under appropriate conditions on the arrangement of subspaces and the distribution of data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm can be solved efficiently and can handle data points near the intersections of subspaces. Another key advantage of the proposed algorithm with respect to the state of the art is that it can deal with data nuisances, such as noise, sparse outlying entries, and missing entries, directly by incorporating the model of the data into the sparse optimization program. We demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real-world problems of motion segmentation and face clustering