Rademacher-Sketch: A Dimensionality-Reducing Embedding for Sum-Product Norms, with an Application to Earth-Mover Distance

Consider a sum-product normed space, i.e. a space of the form $Y=\ell_1^n \otimes X$, where X is another normed space. Each element in Y consists of a length-n vector of elements in X, and the norm of an element in Y is the sum of the norms of its coordinates. In this paper we show a constant-distortion embedding from the normed space $\ell_1^n \otimes X$ into a lower-dimensional normed space $\ell_1^{n'} \otimes X$, where n′≪n is some value that depends on the properties of the normed space X (namely, on its Rademacher dimension). In particular, composing this embedding with another well-known embedding of Indyk [18], we get an O(1/e)-distortion embedding from the earth-mover metric EMDΔ on the grid [Δ]2 to $\ell_1^{\Delta^{O(\epsilon)}} \otimes {\sf{EEMD}}_{\Delta^{\epsilon }}$ (where EEMD is a norm that generalizes earth-mover distance). This embedding is stronger (and simpler) than the sketching algorithm of Andoni et al [4], which maps EMDΔ with O(1/e) approximation into sketches of size ΔO(e).