Squared Wasserstein distance is a frequently used tool to measure discrepancy between probability distributions. This distance is typically computed between empirical measures of size $n$ from two underlying random samples. Unfortunately, even in lower dimensional Euclidean space problems $\left( d \in \{2,3\} \right)$, algorithms for Wasserstein distance computation with approximate or exact precision guarantees scale poorly in the runtime as a function of $n$ and the desired precision. In response, we consider the computational-statistical runtime, where the goal is to estimate from samples the Wasserstein distance between potentially smooth measures up to $ε$-additive error in expectation with respect to the sampling; we allow $O(1)$ computational cost for collecting a sample. Towards this, we develop a Sample-Sketch-Solve paradigm where we introduce a regular cartesian grid sketch of the samples. We show that (especially under $α$-Hölder smooth distributions) this can compress the data without increasing asymptotic error, and also regularizes the structure which enables faster exact algorithms. Ultimately, we approximate $W_2^2(P,Q)$ within $ε$ error in $ε^{-\max(2,\frac{d+1+o(1)}{1+α})}$ time for $0 < α< 1$ Hölder smooth distributions $P,Q$ on $(0,1)^{d}$; an optimal $Θ(ε^{-2})$ for $α> 1/2$ when $d=2$ and nearly optimal as $α\to 1$ when $d = 3$.

We revisit extending the Kolmogorov-Smirnov distance between probability distributions to the multidimensional setting and make new arguments about the proper way to approach this generalization. Our proposed formulation maximizes the difference over orthogonal dominating rectangular ranges (d-sided rectangles in R^d), and is an integral probability metric. We also prove that the distance between a distribution and a sample from the distribution converges to 0 as the sample size grows, and bound this rate. Moreover, we show that one can, up to this same approximation error, compute the distance efficiently in 4 or fewer dimensions; specifically the runtime is near-linear in the size of the sample needed for that error. With this, we derive a delta-precision two-sample hypothesis test using this distance. Finally, we show these metric and approximation properties do not hold for other popular variants.

We study Bayesian histograms for distribution estimation on $[0,1]^d$ under the Wasserstein $W_v, 1 \leq v < \infty$ distance in the i.i.d sampling regime. We newly show that when $d < 2v$, histograms possess a special \textit{memory efficiency} property, whereby in reference to the sample size $n$, order $n^{d/2v}$ bins are needed to obtain minimax rate optimality. This result holds for the posterior mean histogram and with respect to posterior contraction: under the class of Borel probability measures and some classes of smooth densities. The attained memory footprint overcomes existing minimax optimal procedures by a polynomial factor in $n$; for example an $n^{1 - d/2v}$ factor reduction in the footprint when compared to the empirical measure, a minimax estimator in the Borel probability measure class. Additionally constructing both the posterior mean histogram and the posterior itself can be done super--linearly in $n$. Due to the popularity of the $W_1,W_2$ metrics and the coverage provided by the $d < 2v$ case, our results are of most practical interest in the $(d=1,v =1,2), (d=2,v=2), (d=3,v=2)$ settings and we provide simulations demonstrating the theory in several of these instances.