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Bounded-Christoffel sampling stabilizes the empirical Gram matrix

Statement

Let $Λ$ , $N = ∣ Λ ∣ \geq 1$ , and the independent Christoffel samples $X_{1}, \dots, X_{M}$ be as in Christoffel-weighted least-squares polynomial collocation. Let $G$ be its weighted empirical Gram matrix. For $0 < δ < 1$ , if $M \geq 12 N \log (2 N / δ),$ then $P {spec (G) \subseteq [1 / 2, 3 / 2]} \geq 1 - δ .$

Facts & Assumptions

Given: A finite real orthonormal basis of size $N \geq 1$ , independent Christoffel samples, and $0 < δ < 1$ .

[F1]

Christoffel sampling is normalized and its weighted Gram has mean identity gives rank-one matrices $A_{i} ⪰ 0$ with $E A_{i} = I_{N}$ and $∥ A_{i} ∥_{o p} \leq N$ .

[F2]

Christoffel-weighted least-squares polynomial collocation specifies independent samples, hence the matrices formed from them are independent.

[F3]

Bounded independent matrix Bernstein bound from the trace exponential gives, for centered independent Hermitian summands bounded by $R$ with variance bound $v$ , the two-sided tail $P {∥ \sum_{i} B_{i} ∥_{o p} \geq t} \leq 2 N \exp (- t^{2} / (2 (M v + R t / 3)))$ .

[F4]

Christoffel-weighted least-squares polynomial collocation defines $G = M^{- 1} \sum_{i} A_{i}$ .

Proof

1.1F1F2algebra

Put $B_{i} = A_{i} - I_{N}$ . Then $E B_{i} = 0$ . Since $0 ⪯ A_{i} ⪯ N I_{N}$ , spectral calculus gives $A_{i}^{2} ⪯ N A_{i}$ , and hence $0 ⪯ E B_{i}^{2} = E A_{i}^{2} - I_{N} ⪯ (N - 1) I_{N} ⪯ N I_{N}$ . Also every eigenvalue of $A_{i}$ lies in $[0, N]$ , so $∥ B_{i} ∥_{o p} \leq N$ . These are the hypotheses of [F2] with $R = v = N$ .

2.1F2F3F4step 1.1algebra

Apply [F3] to $Y = \sum_{i = 1}^{M} B_{i}$ with $t = M / 2$ . Direct substitution gives $P {∥ G - I_{N} ∥_{o p} > 1 / 2} \leq 2 N \exp (- 3 M / (28 N))$ , because $G - I_{N} = Y / M$ and $(M / 2)^{2} / [2 (M N + N (M / 2) / 3)] = 3 M / (28 N)$ .

3.1givenstep 2.1algebra∎

The sample-size condition makes the last bound at most $2 N \exp [- (9 / 7) \log (2 N / δ)] = δ (δ / (2 N))^{2 / 7} \leq δ$ , since $0 < δ / (2 N) < 1$ . On the complementary event, for every unit vector $x$ , $∣ x^{T} (G - I_{N}) x ∣ \leq ∥ G - I_{N} ∥_{o p} \leq 1 / 2$ ; the Rayleigh quotient characterization therefore puts every eigenvalue of $G$ in $[1 / 2, 3 / 2]$ .

Depends on

Used by

Weighted least squares is stable on the good Gram event Theorem

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Sources

Albert Cohen and Giovanni Migliorati, Optimal Weighted Least-Squares Methods, §§2–3 (standard reference, not scraped)
Joel A. Tropp, An Introduction to Matrix Concentration Inequalities, §§3.2–3.3 and §6.6 (standard reference, not scraped)