Exponentially Improved Dimensionality Reduction for ℓ_1: Subspace Embeddings and Independence Testing
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Despite many applications, dimensionality reduction in the ℓ_1-norm is much less understood than in the Euclidean norm. We give two new oblivious dimensionality reduction techniques for the ℓ_1-norm which improve exponentially over prior ones: 1. We design a distribution over random matrices S ∈ℝ^r × n, where r = 2^poly(d/(εδ)), such that given any matrix A ∈ℝ^n × d, with probability at least 1-δ, simultaneously for all x, SAx_1 = (1 ±ε)Ax_1. Note that S is linear, does not depend on A, and maps ℓ_1 into ℓ_1. Our distribution provides an exponential improvement on the previous best known map of Wang and Woodruff (SODA, 2019), which required r = 2^2^Ω(d), even for constant ε and δ. Our bound is optimal, up to a polynomial factor in the exponent, given a known 2^√(d) lower bound for constant ε and δ. 2. We design a distribution over matrices S ∈ℝ^k × n, where k = 2^O(q^2)(ε^-1 q log d)^O(q), such that given any q-mode tensor A ∈ (ℝ^d)^⊗ q, one can estimate the entrywise ℓ_1-norm A_1 from S(A). Moreover, S = S^1 ⊗ S^2 ⊗⋯⊗ S^q and so given vectors u_1, …, u_q ∈ℝ^d, one can compute S(u_1 ⊗ u_2 ⊗⋯⊗ u_q) in time 2^O(q^2)(ε^-1 q log d)^O(q), which is much faster than the d^q time required to form u_1 ⊗ u_2 ⊗⋯⊗ u_q. Our linear map gives a streaming algorithm for independence testing using space 2^O(q^2)(ε^-1 q log d)^O(q), improving the previous doubly exponential (ε^-1log d)^q^O(q) space bound of Braverman and Ostrovsky (STOC, 2010).
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