Matrix compression along isogenic blocks

Abstract

A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation that Hadamard entrywise functional calculus preserves isogenic blocks has already proved to be of paramount importance for thresholding large correlation matrices. The proposed isogenic stratification of the set of complex matrices bears similarities to the Schubert cell stratification of a homogeneous algebraic manifold. An array of potential applications to current investigations in computational matrix analysis is briefly mentioned, touching concepts such as symmetric statistical models, hierarchical matrices and coherent matrix organization induced by partition trees.

Totally positive kernels, Pólya frequency functions, and their transforms

“…Define the weight matrix W π ∈ R N ×m to have (i, j) entry 1 if i ∈ I j and 0 otherwise. As verified in [3], we have that…”

“…However, Proposition 4.5 (1) implies that f is continuous on [0, ρ) m , so this independence extends to the whole domain of f and therefore (2) implies (3). Finally, if (3) Finally, that (2) implies (1) in Corollary 5.4 is immediate. For the converse, note again that the conclusion holds for the restriction of f to (0, ρ) and the continuity of f on I = [0, ρ) follows from Proposition 4.5 (1).…”

“…The key idea in the proof is to employ a construction introduced and studied in detail in our recent work [3], which we now recall and study further. π : R m×m → R N ×N as the linear map such that…”