Fast Algorithms for a New Relaxation of Optimal Transport

07/14/2023

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by   Moses Charikar, et al.

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We introduce a new class of objectives for optimal transport computations of datasets in high-dimensional Euclidean spaces. The new objectives are parametrized by ρ≥ 1, and provide a metric space ℛ_ρ(·, ·) for discrete probability distributions in ℝ^d. As ρ approaches 1, the metric approaches the Earth Mover's distance, but for ρ larger than (but close to) 1, admits significantly faster algorithms. Namely, for distributions μ and ν supported on n and m vectors in ℝ^d of norm at most r and any ϵ > 0, we give an algorithm which outputs an additive ϵ r-approximation to ℛ_ρ(μ, ν) in time (n+m) ·poly((nm)^(ρ-1)/ρ· 2^ρ / (ρ-1) / ϵ).