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