Tensor Methods for Nonlinear Matrix Completion

Abstract

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 are an important special case. We propose an LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. For example, a second-order tensorized representation is formed by taking the Kronecker 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 are not in the original representation. We also provide a formal mathematical justification for the success of our method. In particular, we give bounds on the rank of these data in the tensorized representation, and we prove sampling requirements to guarantee uniqueness of the solution. We also provide experimental results showing that the new approach outperforms existing state-of-the-art methods for matrix completion under a union-of-subspaces model.

Polynomial Matrix Completion for Missing Data Imputation and Transductive Learning

“…We compare the proposed PMC methods with LRMC method (nulcear norm minimization), SRMC (Fan and Chow 2017), LADMC (Ongie et al 2018) (the iterative algorithm), VMC-2 (2-order polynomial kernel), VMC-3 (3order polynomial kernel) (Ongie et al 2017), and NLMC (Fan and Chow 2018) (RBF kernel) in matrix completion on synthetic data, subspace clustering on incomplete data, motion capture data recovery, and classification on incomplete data. Note that PMC with R 1 is equivalent to the NLMC method of .…”

“…The models have been proposed for subspace clustering (Elhamifar and Vidal 2013), manifold learning (Roweis and Saul 2000;Van Der Maaten, Postma, and Van den Herik 2009), and deep learning (Hinton and Salakhutdinov 2006). Related work Recently, matrix completion on multiplesubspace data and nonlinear data has drawn many researchers' attention (Eriksson, Balzano, and Nowak 2011;Yang, Robinson, and Vidal 2015;Li and Vidal 2016;Fan and Cheng 2018;Alameda-Pineda et al 2016;Elhamifar 2016;Fan, Zhao, and Chow 2018;Fan and Chow 2017;Ongie et al 2017;Ongie et al 2018). For example, (Elhamifar 2016) proposed a group-sparse optimization with rank-one constraints to complete and cluster multiple-subspace data.…”

Local low-rank approach to nonlinear-matrix completion

“…Note that the truncated nuclear norm with r = 0 is equal to the nuclear norm. In this case, the problem ( 11) is same as the problem (16). When the variables X and D i are constant, the optimal solution for each Z i is obtained by…”

“…In the same way, to solve the problem (11), this paper describes Algorithm 1 using iterative partial matrix shrinkage (IPMS) [9] for the problem (16), which contains the algorithm for (11). Here, 0 M,N ∈ R M ×N denotes a zero matrix, η 1 , η 2 denote lower limits for the termination conditions…”

“…As an example, a matrix is of high rank when its column vectors lie on a union of linear subspaces (UoLS), which the column space of the matrix is high dimension even when the dimension of the linear subspace is low. In this case, several method have been proposed to solve this high-rank matrixcompletion problem [11,12,13,14,15,16], all of which are based on subspace clustering [17]. In particular, [15] proposed an algebraic-variety approach known as variety-based matrix completion (VMC), which is based on the fact that the monomial features of each column vector belong to an LDLS when the column vectors belong to a UoLS.…”