rademacher_matrix_operator_norm_first_moment_log_window_from_2p

This is the formal dimension-window bridge for the Rudelson/Lust-Picquard noncommutative-Khintchine step used in the Candes--Recht tangent-sampling estimate.

Source: Candes--Recht, Exact Matrix Completion via Convex Optimization, PDF p. 19, Section 4.2, Theorem 4.2, equation (4.9), together with Rudelson, Random vectors in the isotropic position, JFA 164 (1999), Theorem 1, proof Steps 1--2, and the Lust-Picquard/Pisier noncommutative Khintchine inequality.

Mathematical statement. Let $\iota$ be a finite index set and let $H_c\in\mathbb R^{d\times d}$ be Hermitian matrices. If

with $\nu\ge0$, $d\le N^2$, and $N\ge2$, then there is a universal constant $C_{\log}>0$ such that

Variables and notation. Here $n = N$ is the ambient maximum-dimension parameter used downstream in Candes--Recht, while this abstract bridge writes the matrix dimension as $d$. In the downstream matrix-completion application, $p = m/(n_1n_2)$ is the Bernoulli sampling rate, $\Omega$ is one fixed Bernoulli sample realization, $H_c$ is the rank-one tensor $\mathbf 1_{c\in\Omega}(y_c\otimes y_c)$, and $\nu$ is later supplied by a separate variance-proxy bridge from $\sum_c H_c^2\preceq R^2G_\Omega$. The probability model in this node is only the auxiliary Rademacher sign average over $\varepsilon_c\in\{-1,1\}$; the Bernoulli probability model appears in the parent Rudelson route.

Formalization note. This is a formal bridge. It does not appear verbatim in Candes--Recht. It bridges source-backed parent/import theorem rademacher_matrix_operator_norm_2p_moment_bound (0cafa5a3) plus the Rademacher power-mean bridge rademacher_expectation_power_mean (9a9634f8) to the source-backed child route for inner_sign_average_khintchine_variance_proxy_bound_of_two_le_max (f4806ebd). The proof should choose an even moment exponent $2q$ with $q\asymp\log N$, use $d\le N^2$ and $N\ge2$ to absorb $d^{1/(2q)}$, and then pass from the $2q$ moment to the first moment.