rademacher_matrix_operator_norm_2p_moment_bound
Provedcandes-rechtmatrix-completionmatrix-concentrationreferencerudelson
B1 — the matrix-Khintchine operator-norm moment bound (trace-moment to operator-norm bridge). For a Hermitian family () with variance proxy bounded by (every eigenvalue of is ), the Rademacher-average -th operator-norm moment satisfies, for ,
where is the uniform average over the sign patterns (written as ). This is the standard Tropp/van Handel matrix-Khintchine operator-norm moment, obtained from the trace-moment engine general_rademacher_matrix_2p_trace_moment plus the crux , the constant bound , Markov/monotonicity of , and . Needs and .
Preamble
import Definitions.Def_matrix_completion_tangent import Mathlib.Analysis.SpecialFunctions.Pow.Real open Matrix MatrixCompletion open scoped BigOperators
Formal statement
theorem rademacher_matrix_operator_norm_2p_moment_bound
{ι : Type*} [Fintype ι] [DecidableEq ι] {d : ℕ} (hd : 0 < d)
(H : ι → Matrix (Fin d) (Fin d) ℝ)
(hHerm : ∀ c, (H c).IsHermitian)
(normV : ℝ) (hnormVnn : 0 ≤ normV)
(hVHerm : (∑ c : ι, H c * H c).IsHermitian)
(hnormV : ∀ i, hVHerm.eigenvalues i ≤ normV)
(p : ℕ) (hp : 1 ≤ p) :
(∑ eps : Finset ι, ((1 : ℝ) / 2) ^ (Fintype.card ι)
* spectralNorm (∑ c : ι, (if c ∈ eps then (1 : ℝ) else -1) • H c) ^ (2 * p))
^ ((1 : ℝ) / (2 * p))
≤ Real.sqrt (2 * p) * Real.sqrt normV * (d : ℝ) ^ ((1 : ℝ) / (2 * p)) := by sorrySource
Tropp 2015 (An Introduction to Matrix Concentration Inequalities) Thm 4.1; van Handel arXiv:1610.05200 §3; CR2009 arXiv:0805.4471 §4.2.