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 = `1 ⊗ 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 `1 ⊗X into a lower-dimensional normed space ` ′ 1 ⊗ 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 anO(1/ )-distortion embedding from the earth-mover metric EMD∆ on the grid [∆] to ` O( ) 1 ⊗EEMD∆ (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 EMD∆ withO(1/ ) approximation into sketches of size ∆ .