Tight Bounds for ℓ_p Oblivious Subspace Embeddings
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An ℓ_p oblivious subspace embedding is a distribution over r × n matrices Π such that for any fixed n × d matrix A, _Π[for all x, Ax_p ≤Π Ax_p ≤κAx_p] ≥ 9/10, where r is the dimension of the embedding, κ is the distortion of the embedding, and for an n-dimensional vector y, y_p is the ℓ_p-norm. Another important property is the sparsity of Π, that is, the maximum number of non-zero entries per column, as this determines the running time of computing Π· A. While for p = 2 there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparisty, for the important case of 1 ≤ p < 2, much less was known. In this paper we obtain nearly optimal tradeoffs for ℓ_p oblivious subspace embeddings for every 1 ≤ p < 2. We show for every 1 ≤ p < 2, any oblivious subspace embedding with dimension r has distortion κ = Ω(1/(1/d)^1 / p·^2 / pr + (r/n)^1 / p - 1 / 2). When r = poly(d) in applications, this gives a κ = Ω(d^1/p^-2/p d) lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for p = 1 and the oblivious subspace embedding of Meng and Mahoney (STOC, 2013) for 1 < p < 2 are optimal up to poly((d)) factors. We also give sparse oblivious subspace embeddings for every 1 ≤ p < 2 which are optimal in dimension and distortion, up to poly( d) factors. Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise ℓ_p low rank approximation. Our results give improved algorithms for these applications.
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