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Expected real and normalized complex Gaussian power-rangefinder error

Statement

Assume Countable Choice for the finite-dimensional Gaussian probability law. Let $A \in F^{m \times n}$ , $F = R$ or $C$ , with the same $k \geq 2$ , $p \geq 2$ , $ℓ = k + p \leq \min (m, n)$ and field-normalized Gaussian draws as in Expected spectral error for real and normalized complex Gaussian range finders. For an integer $q \geq 0$ , let $Q$ be the basis returned by Reorthogonalized randomized power subspace iteration on $A, Ω, q$ . Equivalently, for $B = (A A^{*})^{q} A$ , $Ran (Q) = Ran (B Ω)$ . The corresponding singular-value bound $E ∥ (I - Q Q^{*}) A ∥_{2} \leq {[(1 + \sqrt{\frac{k}{p - 1}}) σ_{k + 1}^{2 q + 1} + \frac{e \sqrt{k + p}}{p} {(\sum_{j > k} σ_{j}^{2 (2 q + 1)})}^{1 / 2}]}^{1 / (2 q + 1)} .$

Facts & Assumptions

Given: Countable Choice for the finite Gaussian law, and the fixed matrix, Gaussian field convention, $k, p, ℓ$ constraints, and an integer $q \geq 0$ .

[F1]

The real-field powered bound is proved under these hypotheses (Expected real Gaussian power-rangefinder error).

[F2]

The shared real and normalized-complex base theorem applies to every fixed matrix in the stated field and parameter range (Expected spectral error for real and normalized complex Gaussian range finders).

[F3]

The QR-reorthogonalized procedure returns a measurable basis whose range is $Ran ((A A^{*})^{q} A Ω)$ (Measurability of the reorthogonalized power range finder, with range equality proved in Power-rangefinder error transfer under finite expected powered error).

[F4]

For a measurable orthogonal projector onto the output range of that power procedure, the expected error for $A$ is at most the $(2 q + 1)$ th root of its finite expected powered error (Power-rangefinder error transfer under finite expected powered error).

[F5]

If $A = U Σ V^{*}$ is an SVD, then $B = (A A^{*})^{q} A = U Σ^{2 q + 1} V^{*}$ ; this includes $q = 0$ and zero singular values (Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition).

[F6]

The field-normalized Gaussian draw has iid standard real entries or iid complex entries $(X + i Y) / \sqrt{2}$ with $E ∣ g ∣^{2} = 1$ (Standard real and normalized complex Gaussian matrix laws).

[A1]

Countable Choice is the axiom used for the finite-dimensional Gaussian laws (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1givenF1

If $F = R$ , [F1] is exactly the displayed bound under the stated hypotheses.

1.2givenF2F3F4F5F6A1algebra

Suppose $F = C$ and put $B = (A A^{*})^{q} A$ . By [F3], the output projector $P = Q Q^{*}$ is measurable and projects onto $Ran (B Ω)$ . By [F5], $B$ has the same dimensions as $A$ and its singular values are $σ_{j} (B) = σ_{j} (A)^{2 q + 1}$ , including zeros. Thus [F2] applied to the fixed matrix $B$ and the same normalized-complex sketch gives $E ∥ (I - P) B ∥_{2} \leq (1 + \sqrt{\frac{k}{p - 1}}) σ_{k + 1}^{2 q + 1} + \frac{e \sqrt{k + p}}{p} {(\sum_{j > k} σ_{j}^{2 (2 q + 1)})}^{1 / 2} .$ This is the powered-error expectation needed by [F4], and it is finite.

2.1step 1.2F2F4F5algebra

Apply the transfer inequality [F4] to the finite bound in step 1.2 and take the nonnegative $1 / (2 q + 1)$ power. This gives exactly the displayed complex-field bound. The case $q = 0$ has exponent $1$ . If $A = 0$ or $rank (A) \leq k$ , [F5] makes the powered matrix tails zero; the right side of the [F2] estimate in step 1.2 is then zero, and [F4] gives zero expected target error.

3.1step 1.1step 2.1∎

The real and complex field branches are steps 1.1 and 2.1, respectively, and together prove the Statement.

Depends on

Used by

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Sources

Halko, Martinsson, and Tropp, Corollary 10.10 (standard reference, not scraped)
Martinsson and Tropp, Corollary 11.9 (standard reference, not scraped)