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Orthogonal-projector power error inequality

Statement

Let $A \in F^{m \times n}$ , let $P$ be an orthogonal projector on $F^{m}$ , and let $q \geq 0$ be an integer. Then $∥ (I - P) A ∥_{2} \leq ∥ (I - P) (A A^{*})^{q} A ∥_{2}^{1 / (2 q + 1)} .$

Facts & Assumptions

Given: A finite real or complex matrix $A$ , an orthogonal projector $P$ , and an integer $q \geq 0$ .

[F1]

$A$ has an SVD, and its squared singular values diagonalize $A A^{*}$ (Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition).

[F2]

Positive semidefinite finite-dimensional operators have an orthonormal eigenbasis (A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S).

[F3]

The spectral norm of a positive semidefinite operator is its largest eigenvalue (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||, The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector).

[F4]

Jensen's inequality holds for finite probability distributions (Jensen's inequality for expectation).

Proof

1.1F1F2F3algebra

Choose an SVD $A = U Σ V^{*}$ and put $D = Σ Σ^{*}$ and $R = U^{*} (I - P) U$ . Then $D$ is diagonal positive semidefinite and $R$ is an orthogonal projector. By [F1]–[F3], $∥ (I - P) A ∥_{2}^{2} = ∥ R D R ∥_{2}$ and $∥ (I - P) (A A^{*})^{q} A ∥_{2}^{2} = ∥ R D^{2 q + 1} R ∥_{2}$ .

2.1step 1.1F2F3F4choosealgebra

If $λ = ∥ R D R ∥_{2} = 0$ , [step 1.1] gives $∥ (I - P) A ∥_{2} = 0$ , so the asserted inequality follows. If $λ > 0$ , choose a unit eigenvector $x$ for the top eigenvalue of $R D R$ ; then $R x = x$ and $λ = x^{*} D x = \sum_{i} d_{i} ∣ x_{i} ∣^{2}$ , with $d_{i} \geq 0$ . The weights $∣ x_{i} ∣^{2}$ sum to $1$ , so [F4] for $t \mapsto t^{2 q + 1}$ gives $λ^{2 q + 1} \leq \sum_{i} d_{i}^{2 q + 1} ∣ x_{i} ∣^{2} = x^{*} D^{2 q + 1} x \leq ∥ R D^{2 q + 1} R ∥_{2}$ .

3.1step 1.1step 2.1algebra∎

Substitute $λ = ∥ (I - P) A ∥_{2}^{2}$ from step 1.1 into step 2.1 and take the nonnegative $(2 q + 1)$ st root. This yields the asserted inequality, also for $q = 0$ .

Depends on

Used by

Power-rangefinder error transfer under finite expected powered error Theorem

Dependency tree · two levels

17 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, Proposition 8.6 and Theorem 9.2 (standard reference, not scraped)
Martinsson and Tropp, Randomized Numerical Linear Algebra: Foundations & Algorithms, Lemma 11.8 (standard reference, not scraped)