Oblivious Subspace Embeddings

Years and Authors of Summarized Original Work

• 2006; Sarlós

Problem Definition

This entry surveys some of the applications of “oblivious subspace embeddings,” introduced by Sarlós in [19], to problems in linear algebra.

Definition 1 ([19])

Given 0 < ɛ < 1∕2 and a d-dimensional subspace $E \subseteq \mathbb{R}^{n}$, we say an m × n matrix Π is an ɛ-subspace embedding for E if

$$\displaystyle{\forall x \in E\ (1-\varepsilon )\|x\|_{2}^{2} \leq \|\varPi x\|_{ 2}^{2} \leq (1+\varepsilon )\|x\|_{ 2}^{2}.}$$

The goal is to have m small so that Π provides dimensionality reduction for E.

Given 0 < ɛ, δ < 1∕2, and integers 1 ≤ d ≤ n, an (ɛ,δ,d,n)-oblivious subspace embedding (OSE) is a distribution $\mathcal{D}$ over m × n matrices such that for every d-dimensional linear subspace $E \subseteq \mathbb{R}^{N}$ of dimension d,

$$\displaystyle{\mathop{\mathbb{P}}\limits_{\varPi \sim \mathcal{D}}\!\!\!\left (\varPi \!\text{ is an }\varepsilon \text{-subspace embedding for }\!E\right )\! >\!\!...