Theoretical analysis reveals exact asymptotic volumes and large deviations principles in finite-dimensional Schatten classes, highlighting geometric properties of non-square matrix balls.
The study of Schatten classes has a long tradition in geometric functional analysis and related fields. In this paper we study a variety of geometric and probabilistic aspects of finite-dimensional Schatten classes of not necessarily square matrices. Among the main results are the exact and asymptotic volume of the Schatten-∞ unit ball, the boundedness of its isotropy constant, a Poincar\'e-Maxwell-Borel lemma for the uniform distribution on the Schatten-∞ ball, and Sanov-type large deviations principles for the singular values of matrices sampled uniformly from the Schatten-p unit ball for any 0 < p ≤ ∞.
No takes yet. Share an insight, caveat, or question.
Juhos et al. (2024) studied this question.
Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context: