2024/07/23 by Frits Verhagen, Marco Tomamichel, Verhagen, Frits +3 · 2 citations
Computer Science · #Face and Expression Recognition
paper · pdf · doi:10.1109/tit.2025.3585062
We say that a matrix <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> with non-negative entries majorizes another such matrix <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> if there is a stochastic matrix <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> such that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> = <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">TP</i>. We study matrix majorization in large samples and in the catalytic regime in the case where the columns of the matrices need not have equal support, as has been assumed in earlier works. We focus on two cases: either there are no support restrictions (except for requiring a non-empty intersection for the supports) or the final column dominates the others. Using real-algebraic methods, we identify sufficient and almost necessary conditions for majorization in large samples or when using catalytic states under these support conditions. These conditions are given in terms of multivariate divergences that generalize the Rényi divergences. We notice that varying support conditions dramatically affect the relevant set of divergences. Our results find an application in the theory of catalytic state transformation in quantum thermodynamics.