TY - JOUR
AU - Dax, Achiya
PY - 2020/09/08
Y2 - 2024/06/17
TI - Low-rank matrix approximations over canonical subspaces
JF - J. Numer. Anal. Approx. Theory
JA - J. Numer. Anal. Approx. Theory
VL - 49
IS - 1
SE - Articles
DO - 10.33993/jnaat491-1195
UR - https://ictp.acad.ro/jnaat/journal/article/view/1195
SP - 22-44
AB - <div><span style="background-color: #ffffff;">In this paper we derive closed form expressions for the nearest rank-\(k\) matrix on canonical subspaces. </span></div><div> </div><div><span style="background-color: #ffffff;">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 \(Y^TA = 0\).</span></div><div> </div><div><span style="background-color: #ffffff;">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. </span></div><div> </div><div><span style="background-color: #ffffff;">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.</span></div><div><span style="background-color: #ffffff;">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.</span></div><div><span style="background-color: #ffffff;">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.</span></div>
ER -