The Factor Width Rank of a Matrix
Nathaniel Johnston, Shirin Moein, Sarah Plosker
TL;DR
The paper develops a detailed theory of factor width and factor-width-k rank for PSD matrices, showing that factor width bounds relate to bandwidth and chordal-graph structure (factor width ≤ bandwidth, with equality for small k) and providing explicit results for special matrix classes (banded, arrowhead). It further establishes lower and upper bounds via covering designs and k-clique coverings, and analyzes how Hadamard products and powers affect factor width, including both integer and non-integer powers and asymptotic behavior for large powers. The work integrates convex-geometry insights with combinatorial designs to bound fran_k and reveals deep connections between matrix factorization structure and graph-theoretic properties, with implications for optimization and quantum-information-inspired decompositions. Overall, the results yield a framework for estimating and computing factor width ranks, identify regimes where fran_k equals the usual rank, and illuminate how elementwise operations transform factor-width structure.
Abstract
A matrix is said to have factor width at most $k$ if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single $k \times k$ principal submatrix. We explore the ``factor-width-$k$ rank'' of a matrix, which is the minimum number of rank-$1$ matrices that can be used in such a factor-width-at-most-$k$ decomposition. We show that the factor width rank of a banded or arrowhead matrix equals its usual rank, but for other matrices they can differ. We also establish several bounds on the factor width rank of a matrix, including a tight connection between factor-width-$k$ rank and the $k$-clique covering number of a graph, and we discuss how the factor width and factor width rank change when taking Hadamard products and Hadamard powers.