Low-rank matrix approximations over canonical subspaces

Abstract

In this paper we derive closed form expressions for the nearest rank-\(k\) matrix on canonical subspaces. 
 
 We start by studying three kinds of subspaces. Let \(X\) and \(Y\) be a pair of given matrices. The first subspace contains all the \(m\times n\) matrices \(A\) that satisfy \(AX=O\). The second subspace contains all the \(m \times n\) matrices \(A\) that satisfy \(Y^TA = O\), while the matrices in the third subspace satisfy both \(AX =O\) and \(Y^TA = 0\).
 
 The second part of the paper considers a subspace that contains all the symmetric matrices \(S\) that satisfy \(SX =O\). In this case, in addition to the nearest rank-\(k\) matrix we also provide the nearest rank-\(k\) positive approximant on that subspace. 
 
 A further insight is gained by showing that the related cones of positive semidefinite matrices, and negative semidefinite matrices, constitute a polar decomposition of this subspace.
 The paper ends with two examples of applications. The first one regards the problem of computing the nearest rank-\(k\) centered matrix, and adds new insight into the PCA of a matrix.
 The second application comes from the field of Euclidean distance matrices. The new results on low-rank positive approximants are used to derive an explicit expression for the nearest source matrix. This opens a direct way for computing the related positions matrix.