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Gaussian sketches embed a fixed subspace at the additive dimension rate

Statement

Assume Countable Choice for the finite-dimensional Gaussian probability law. Let $1 \leq d \leq n$ , $K \in {R, C}$ , and let $U \subseteq K^{n}$ be a fixed $d$ -dimensional subspace. Let $G \in K^{m \times n}$ have iid standard real Gaussian entries when $K = R$ , or iid normalized complex Gaussian entries with $E ∣ g ∣^{2} = 1$ when $K = C$ . Put $S = G / \sqrt{m}$ . For $0 < ε, δ < 1$ , if

m \geq 192 ε^{- 2} (d + \log (1 / δ)),

then with probability at least $1 - δ$ , $S$ is an $ε$ -subspace embedding for $U$ . Thus this Gaussian distribution is an $(ε, δ)$ oblivious $ℓ_{2}$ subspace embedding for dimension $d$ .

Facts & Assumptions

Given: Countable Choice for the finite Gaussian law, and a fixed subspace $U$ of dimension $d$ , the stated Gaussian law, $m \geq 1$ , $0 < ε, δ < 1$ , and the displayed sample-size condition.

[F1]

An $ε$ -subspace embedding preserves every squared norm on the fixed subspace within factors $1 \pm ε$ ; an $(ε, δ)$ oblivious embedding has this probability for each fixed subspace (Oblivious squared-norm subspace embeddings).

[F2]

The real standard and normalized complex Gaussian entries have the densities and independent product law stated in Standard real and normalized complex Gaussian matrix laws. Their unit scalar masses, normalized second moments and all finite-product normalizations and independence are proved under Countable Choice in Normalization and second moments of the Gaussian matrix laws.

[F3]

Right orthogonal or unitary multiplication preserves the iid Gaussian matrix law, including independence of the resulting entries (Orthogonal and unitary invariance of Gaussian matrix blocks).

[F4]

A finite-dimensional real or complex inner-product space has an orthonormal basis (Every finite-dimensional real or complex inner product space has an orthonormal basis).

[F5]

For independent standard real or normalized complex Gaussian scalars, the squared-norm deviation probability is at most $2 e^{- m t^{2} / 8}$ for $0 < t < 1$ (Squared-norm concentration for real and normalized complex Gaussians).

[F6]

A complex unit sphere is a real sphere of twice the dimension, and its $1 / 4$ -net can be chosen with at most $9^{2 d}$ points; the Hermitian quadratic net controls the operator norm by factor two (A sphere net controls a real or complex Hermitian operator norm).

[F7]

The operator norm of $H$ is $m a x_{∥ y ∥_{2} = 1} ∥ H y ∥_{2}$ (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||).

[F8]

The inner product obeys Cauchy--Schwarz over $R$ and $C$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

[F9]

The probability of a finite union of measurable events is at most the sum of their probabilities (The finite union bound).

[A1]

Countable Choice supplies the finite-dimensional Lebesgue probability laws used in [F2] (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

Given: The hypotheses in the Statement.

1.1F2F3F4F6A1choose

Choose an orthonormal basis matrix $Q \in K^{n \times d}$ for $U$ , and let $N$ be a $1 / 4$ -net of its unit sphere with $∣ N ∣ \leq 9^{2 d}$ , which covers the real case too. For each fixed $v \in N$ , $Q v$ is a unit vector. Extend $Q v$ to an orthonormal basis of $K^{n}$ by applying [F4] to its orthogonal complement; [F3] then shows that $G Q v$ has $m$ independent standard Gaussian coordinates.

2.1F5F6F9step 1.1givenalgebra

For each fixed $v \in N$ , apply [F5] with $t = ε / 2$ to obtain $\Pr {∣ ∥ S Q v ∥_{2}^{2} - 1 ∣ \geq ε / 2} \leq 2 e^{- m ε^{2} / 32}$ . The finite union bound [F9] for the fixed finite net gives $\Pr {\exists v \in N : ∣ ∥ S Q v ∥_{2}^{2} - 1 ∣ \geq ε / 2} \leq 2 9^{2 d} e^{- m ε^{2} / 32} \leq 2 e^{2 d \log 9 - 6 d - 6 \log (1 / δ)} \leq δ$ . Here the sample-size hypothesis gives $m ε^{2} / 32 \geq 6 (d + \log (1 / δ))$ , and $\log 2 + 2 d \log 9 < 6 d$ for $d \geq 1$ .

3.1F1F6F7F8step 2.1algebra∎

On the complementary event, put $H = Q^{*} S^{*} S Q - I_{d}$ . This is Hermitian and $∣ v^{*} H v ∣ = ∣ ∥ S Q v ∥_{2}^{2} - 1 ∣ < ε / 2$ for every $v \in N$ . By [F6], $∥ H ∥_{2} \leq ε$ . For nonzero $y$ , [F7] and [F8] give $∣ y^{*} H y ∣ \leq ∥ y ∥_{2} ∥ H y ∥_{2} \leq ∥ H ∥_{2} ∥ y ∥_{2}^{2}$ ; the zero vector satisfies the same inequality. Therefore $(1 - ε) ∥ y ∥_{2}^{2} \leq ∥ S Q y ∥_{2}^{2} \leq (1 + ε) ∥ y ∥_{2}^{2}$ . Since $Q$ is an isometry onto $U$ , [F1] identifies this as the required embedding inequality for every $x = Q y \in U$ .

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Sources

David P. Woodruff, Sketching as a Tool for Numerical Linear Algebra, §2.1, Theorem 6 (standard reference, not scraped)
Michael W. Mahoney, Randomized Algorithms for Matrices and Data, §3.1 (standard reference, not scraped)