Fast Fixed Dimension L2-Subspace Embeddings of Arbitrary Accuracy, With Application to L1 and L2 Tasks

We give a fast oblivious L2-embedding of $A\in \mathbb{R}^{n x d}$ to $B\in \mathbb{R}^{r x d}$ satisfying $(1-\varepsilon)\|A x\|_2^2 \le \|B x\|_2^2 <= (1+\varepsilon) \|Ax\|_2^2.$ Our embedding dimension $r$ equals $d$, a constant independent of the distortion $\varepsilon$. We use as a black-box any L2-embedding $\Pi^T A$ and inherit its runtime and accuracy, effectively decoupling the dimension $r$ from runtime and accuracy, allowing downstream machine learning applications to benefit from both a low dimension and high accuracy (in prior embeddings higher accuracy means higher dimension). We give applications of our L2-embedding to regression, PCA and statistical leverage scores. We also give applications to L1: 1.) An oblivious L1-embedding with dimension $d+O(d\ln^{1+\eta} d)$ and distortion $O((d\ln d)/\ln\ln d)$, with application to constructing well-conditioned bases; 2.) Fast approximation of L1-Lewis weights using our L2 embedding to quickly approximate L2-leverage scores.