rudelson_selection_gram_spectral_bound

Rudelson selection one-sided Gram spectral bound (rank-one form). For a finite family of vectors $y : \iota \to \mathbb{R}^d$ with $\|y_c\|^2 = y_c \cdot y_c \le M$ for all $c \in s$ (and $0 \le M$), the spectral norm of the energy-weighted Gram operator is dominated by $M$ times the spectral norm of the plain Gram operator:

Here $y_c \otimes y_c =$ Matrix.vecMulVec (y c) (y c) is the rank-one outer product and the norm is the $\ell^2$ operator norm ‖Matrix.toEuclideanCLM (𝕜 := ℝ) X‖. This is the spectral consequence of the rank-one squaring identity $(yy^*)^2 = \|y\|^2 (yy^*)$ combined with Loewner monotonicity, exactly the bound the noncommutative-Khintchine (Lust-Picquard) step of Rudelson's selection lemma applies to the one-sided Gram operator $\sum_c X_c^2$ with $X_c = y_c \otimes y_c$ self-adjoint. Proof (reduction): $\sum_c \|y_c\|^2 (y_c\otimes y_c) \preceq M \sum_c (y_c\otimes y_c)$ in the Loewner order (each summand is a nonnegative-scalar multiple of a PSD rank-one), then apply spectral-norm Loewner monotonicity and pull the scalar $M \ge 0$ out of the operator norm.