Fast Fixed Dimension L2-Subspace Embeddings of Arbitrary Accuracy, With Application to L1 and L2 Tasks
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We give a fast oblivious L2-embedding of A∈R^n x d to B∈R^r x d satisfying (1-ε)A x_2^2 <B x_2^2 <= (1+ε) Ax_2^2. Our embedding dimension r equals d, a constant independent of the distortion ε. We use as a black-box any L2-embedding Π^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^1+η d) and distortion O((d d)/ 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.
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