Impossibility of sketching of the 3D transportation metric with quadratic cost

Transportation cost metrics, also known as the Wasserstein distances W_p, are a natural choice for defining distances between two pointsets, or distributions, and have been applied in numerous fields. From the computational perspective, there has been an intensive research effort for understanding the W_p metrics over R^k, with work on the W₁ metric (a.k.a earth mover distance) being most successful in terms of theoretical guarantees. However, the W₂ metric, also known as the root-mean square (RMS) bipartite matching distance, is often a more suitable choice in many application areas, e.g. in graphics. Yet, the geometry of this metric space is currently poorly understood, and efficient algorithms have been elusive. For example, there are no known non-trivial algorithms for nearest-neighbor search or sketching for this metric. In this paper we take the first step towards explaining the lack of efficient algorithms for the W₂ metric, even over the three-dimensional Euclidean space R³. We prove that there are no meaningful embeddings of W₂ over R³ into a wide class of normed spaces, as well as that there are no efficient sketching algorithms for W₂ over R³ achieving constant approximation. For example, our results imply that: 1) any embedding into L₁ must incur a distortion of Ω(√log n) for pointsets of size n equipped with the W₂ metric; and 2) any sketching algorithm of size s must incur Ω(√log n/√s) approximation. Our results follow from a more general statement, asserting that W₂ over R³ contains the 1/2-snowflake of all finite metric spaces with a uniformly bounded distortion. These are the first non-embeddability/non-sketchability results for W₂.