Abstract
This paper deals with a problem of matrix completion in which each column vector of the matrix belongs to a low-dimensional differentiable manifold (LDDM), with the target matrix being high or full rank. To solve this problem, algorithms based on polynomial mapping and matrix-rank minimization (MRM) have been proposed; such methods assume that each column vector of the target matrix is generated as a vector in a low-dimensional linear subspace (LDLS) and mapped to a pth order polynomial and that the rank of a matrix whose column vectors are dth monomial features of target column vectors is deficient. However, a large number of columns and observed values are needed to strictly solve the MRM problem using this method when p is large; therefore, this paper proposes a new method for obtaining the solution by minimizing the rank of the submatrix without transforming the target matrix, so as to obtain high estimation accuracy even when the number of columns is small. This method is based on the assumption that an LDDM can be approximated locally as an LDLS to achieve high completion accuracy without transforming the target matrix. Numerical examples show that the proposed method has a higher accuracy than other low-rank approaches.
Highlights
This paper deals with the following completion problem for a matrix X ∈ RM×N on a low-dimensional differentiable manifold (LDDM) Mr: FindX =[ x1 x2 · · · xN ]subject to (X)m,n = X(0) m,n for (m, n) ∈ (1)xi ∈ Mr for all i ∈ I, where the (m, n)th element of a matrix is denoted by (·)m,n, I is an index set defined as I = {1, 2, · · ·, N}, and Mr ⊂ RM, and X(0) denote an unknown r-dimensional differential manifold, a given index set, and a given observed matrix, respectively
[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 low-dimensional linear subspace (LDLS) when the column vectors belong to a union of linear subspaces (UoLS)
2 Related works Here, we focus on some matrix completion algorithms based on matrix rank minimization (MRM) on an unknown manifold, Mr First, this paper introduces the algorithms for the case where Mr is an r-dimensional linear subspace in Section 2.1; Section 2.2 shows the algorithms using the polynomial kernel for a UoLS and an LDDM
Summary
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 methods have been proposed to solve this high-rank matrix completion problem [11,12,13,14,15,16], all of which are based on subspace clustering [17]. [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 This approach solves the rank minimization problem about the Gram matrix of the monomial features by relaxing the problem into one of rank minimization of a polynomial kernel matrix. These algorithms recover a matrix only when Mr can be approximately divided into some LDLSs and do not work well otherwise
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
