Entropy numbers of embeddings of Schatten classes

Abstract

Let 0 < p, q ≤ ∞ and denote by S N p and S N q the corresponding finite-dimensional Schatten classes. We prove optimal bounds, up to constants only depending on p and q, for the entropy numbers of natural embeddings between S N p and S N q . This complements the known results in the classical setting of natural embeddings between finite-dimensional ℓ p spaces due to Schütt, Edmunds-Triebel, Triebel and Guédon-Litvak/Kühn. We present a rather short proof that uses all the known techniques as well as a constructive proof of the upper bound in the range N ≤ n ≤ N 2 that allows deeper structural insight and is therefore interesting in its own right. Our main result can also be used to provide an alternative proof of recent lower bounds in the area of low-rank matrix recovery.

Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings

“…For instance, Carl and Defant [7] proved a Garnaev-Gluskin result for Gelfand numbers of Schatten class embeddings S N p → S N 2 for 1 ≤ p ≤ 2, König, Meyer, and Pajor [30] obtained that the isotropic constants of S N p unit balls are bounded above by absolute constants for all 1 ≤ p ≤ ∞, and Guédon and Paouris studied their concentration of mass properties in [19]. Even more recently, Radke and Vritsiou succeeded in confirming the thin-shell conjecture for S N ∞ [35], Hinrichs, Prochno, and Vy-bíral computed the entropy numbers for identity mappings S N p → S N q for all 0 < p, q ≤ ∞ [23] and recently obtained asymptotically sharp estimates for Gelfand numbers in almost all regimes [24], Vritsiou confirmed the variance conjecture for S N ∞ [44], and, in a series of papers, Kabluchko, Prochno, and Thäle computed the exact asymptotic volume and volume ratio of S N p unit balls for 0 < p ≤ ∞ [26], studied the threshold behavior of the volume of intersections of unit balls [27], and obtained large deviation principles for the empirical spectral measures of random matrices in Schatten unit balls [28].…”

“…We refer the reader to [10, Theorem 5.5] for details and to [23] for an alternative proof of the lower bound via entropy numbers and Carl's inequality, which was obtained there in the extended regime 0 < p ≤ q ≤ ∞. For general background on compressed sensing, we refer the reader to [9,11,15].…”

A New Framework for Matrix Discrepancy: Partial Coloring Bounds via Mirror Descent

“…We note that[HPV17] claims the bound only up to a constant depending on p and q, but their argument readily gives a universal constant in the regime of p, q ≥ 1.…”

“…We use an operator norm net for the Schatten-1 ball from [HPV17] to construct a relative entropy net with error O(log(m 2 /n)) for the spectraplex S m (Lemma 3.8). When restricted to block diagonal matrices with block size h × h, we use a hybrid of this argument and the earlier approximate Caratheodory argument to find a refined relative entropy net with error O(log(hm/n)) (Theorem 3.9).…”

Revisiting Clustering as Matrix Factorisation on the Stiefel Manifold

“…The covering number of the Stiefel manifold in the operator norm was computed in Hinrichs et al (2017) and is given by…”