Fast Algorithms for a New Relaxation of Optimal Transport

We introduce a new class of objectives for optimal transport computations of datasets in high-dimensional Euclidean spaces. The new objectives are parametrized by $ρ\geq 1$, and provide a metric space $\mathcal{R}_ρ(\cdot, \cdot)$ for discrete probability distributions in $\mathbb{R}^d$. 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 $\mathbb{R}^d$ of norm at most $r$ and any $ε> 0$, we give an algorithm which outputs an additive $εr$-approximation to $\mathcal{R}_ρ(μ, ν)$ in time $(n+m) \cdot \mathrm{poly}((nm)^{(ρ-1)/ρ} \cdot 2^{ρ/ (ρ-1)} / ε)$.