Rademacher-sketch: a dimensionality-reducing embedding for sum-product norms, with an application to earth-mover distance

• Authors: Elad Verbin;Qin Zhang
• Venue: ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
• Year: 2012

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/ε)-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/ε) approximation into sketches of size ΔO(ε).