Entropy numbers of embeddings of Schatten classes

1 Introduction and Main Results

• Schatten classes are non-commutative counterparts of ℓp spaces, and their structure is relevant to low-rank matrix recovery.Entropy numbers measure the extent to which an operator is compact; an operator is compact exactly when its entropy numbers converge to zero.Source: §1 · pp. 1–2
• Theorem 1.1 gives entropy-number estimates for natural identities between finite-dimensional Schatten classes across 0<p,q≤∞, with cases depending on n and N.The estimates are up to constants depending only on p and q.Source: §1 · p. 2
• The main result provides an alternative proof of lower bounds for low-rank matrix recovery, including the regime 0<p≤1 and p<q≤2.The paper relates the entropy estimates to Gelfand numbers and Carl’s inequality to obtain the recovery bound.Source: §1 · p. 3
• The proof combines Carl’s inequality, Gelfand-number estimates, Grassmannian volume estimates, and interpolation; a separate construction gives explicit covers achieving optimal upper bounds.The explicit-cover construction is presented for cases where such covers are more instructive than the short proof.Source: §1 · p. 3

2.1 General notation

• The general notation fixes Lebesgue volume, Euclidean structure, bounded operators, and comparison up to constants depending on parameters.The notation a(n) ≍_p b(n) means both bounds hold with constants depending on p.Source: §2.1 · p. 4

2.2 Quasi-Banach spaces

• Quasi-norms relax the triangle inequality, and the Aoki-Rolewicz theorem permits equivalent p-norms for estimates up to multiplicative constants.A complete space with the metric induced by a p-norm is called a p-Banach space.Source: §2.2 · p. 4

2.3 Singular values and Schatten p-classes

• Schatten classes equip N × N matrices with norms or quasi-norms based on singular values, including nuclear, Hilbert-Schmidt, and operator norms.For p < 1, the Schatten p-quasi-norm satisfies the p-triangle inequality.Source: §2.3 · pp. 4–5

2.4 Unitarily invariant and symmetric norms

• Unitarily invariant norms correspond to 1-symmetric norms on singular-value vectors, paralleling the relation between Schatten p-norms and ℓp-norms.For diagonal matrices, the corresponding norm is the norm of the diagonal vector.Source: §2.4 · p. 5

2.5 Geometry of Grassmannians

• The Grassmannian is metrized through orthogonal projections, and its metric-ball volume estimates support packing arguments.For Schatten q metrics, the cited diameter bounds scale with k^(1/q).Source: §2.5 · p. 6

2.6 Entropy numbers and other s-numbers

• Entropy numbers measure coverings of operator images, and their composition and perturbation properties are used alongside Carl's inequality.The preliminaries also introduce Gelfand and Kolmogorov numbers as other s-number tools.Source: §2.6 · pp. 6–7

2.7 Interpolation properties of entropy numbers

• Interpolation properties allow entropy estimates to pass between endpoint spaces when interpolation is in the domain or target, but not simultaneously in both.The stated limitation on simultaneous interpolation is attributed to Edmunds and Netrusov.Source: §2.7 · p. 8

3.1 Upper bound in the case 0 < q ≤p ≤∞

• The factorization argument establishes the optimal upper bound throughout the case 0 < q ≤ p ≤ ∞.The proof uses factorization and a finite-dimensional entropy estimate.Source: §3.1 · p. 9

3.2 Lower bound in the cases 0 < q ≤p ≤∞and 0 < p ≤q ≤∞, N2 ≤n

• Volume comparison gives the lower bound for 0 < q ≤ p throughout n, and the correct order for 0 < p ≤ q when n ≥ N^2.The proof states the lower bound for q ≤ p over the full range of n and N.Source: §3.2 · p. 9

3.3 Upper bound in the case 0 < p ≤q ≤∞, N2 ≤n

• For 0 < p ≤ q and n ≥ N^2, volume arguments establish the upper bound, with monotonicity filling the interval near N^2.The proof uses the estimate at N^2 to close the range N^2 ≤ n ≤ λ_p,q N^2.Source: §3.3 · pp. 9–10

3.4 Upper bound in the case 0 < p ≤q ≤∞, 1 ≤n ≤N2

• For 2N ≤ n ≤ N², the proof uses the Gelfand-number estimate in Lemma 2.6, Carl’s inequality in Lemma 2.5 to obtain an entropy estimate, and interpolation for remaining p and q cases.Source: §3.4 · p. 11

3.5 Lower bound in the case 0 < p ≤q ≤∞, 1 ≤n ≤N2

• For 0 < p ≤ q and n ≤ N^2, Grassmannian packing yields lower bounds at n = kN for q ≥ 1, with monotonicity and prior bounds extending coverage.The separated projections have cardinality 2^(kN); the proof then applies monotonicity and previously established estimates.Source: §3.5 · pp. 12–13

• An operator-norm net construction yields a 3ε-net of the Stiefel manifold V_N^K.The proof combines a net for subspaces with nets for orthogonal matrices.Source: §4 · pp. 13–14
• Lemma 4.2 provides covering-number estimates for rank-constrained matrices in the Schatten q-class, supplying a net ingredient for the construction.Source: §4 · p. 14
• For 0<p≤q≤∞ and N≤n≤N², Theorem 4.3 gives an entropy-number upper bound with a constant depending only on p and q.Source: §4 · p. 15
• The constructive proof decomposes a Schatten-unit-ball matrix by singular value decomposition into low-rank pieces and a residual for the net construction.The pieces have rank at most 2^(j−1).Source: §4 · pp. 15–16