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Expected real Gaussian power-rangefinder error

Statement

Under the real-field hypotheses of Expected spectral error of the real Gaussian range finder, let $q \geq 0$ be an integer, draw the same Gaussian sketch $Ω$ , and let $Q$ be the basis returned by Reorthogonalized randomized power subspace iteration on $A, Ω, q$ . Equivalently, if $B = (A A^{*})^{q} A$ , then $Ran (Q) = Ran (B Ω)$ . Then $E ∥ (I - Q Q^{*}) A ∥_{2} \leq {[(1 + \sqrt{\frac{k}{p - 1}}) σ_{k + 1}^{2 q + 1} + \frac{e \sqrt{k + p}}{p} {(\sum_{j > k} σ_{j}^{2 (2 q + 1)})}^{1 / 2}]}^{1 / (2 q + 1)} .$

Facts & Assumptions

Given: The inherited Countable Choice assumption for the Gaussian law, and a fixed real matrix, the Gaussian range-finder hypotheses, and an integer power count $q \geq 0$ .

[F1]

The base real Gaussian range finder satisfies the expected spectral-error bound under $k \geq 2$ , $p \geq 2$ , $k + p \leq \min (m, n)$ (Expected spectral error of the real Gaussian range finder).

[F2]

The QR-reorthogonalized procedure spans $Ran ((A A^{*})^{q} A Ω)$ and its original-matrix error transfers from the powered error (Power-rangefinder error transfer under finite expected powered error).

[F3]

If $A = U Σ V^{*}$ , then $B = (A A^{*})^{q} A = U Σ^{2 q + 1} V^{*}$ (Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition).

[F4]

The power procedure returns a Borel orthogonal projector, including rank-deficient samples (Measurability of the reorthogonalized power range finder).

Proof

1.1givenF1F3

Apply [F1] to the fixed matrix $B = (A A^{*})^{q} A$ , which has the same dimensions as $A$ and uses the same Gaussian sketch dimensions. By [F3], its singular values are $σ_{j} (B) = σ_{j} (A)^{2 q + 1}$ , including zero singular values.

2.1step 1.1F2F4algebra∎

By [F4], the output projector is measurable, so the transfer hypothesis is satisfied. By [F2], its range is the sample range for $B$ . Substitute the resulting bound for $E ∥ (I - Q Q^{*}) B ∥_{2}$ into the transfer inequality [F2]. Taking the nonnegative $1 / (2 q + 1)$ power gives exactly the displayed expression.

Depends on

Used by

Expected real and normalized complex Gaussian power-rangefinder error Theorem

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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