Sketching and Embedding are Equivalent for Norms

TL;DR

This paper characterizes when finite-dimensional normed spaces admit efficient sketching algorithms, showing they are equivalent to embedding into certain ℓ_p spaces, and connects this to longstanding open problems in functional analysis.

Contribution

It provides an almost complete characterization of efficient sketching for normed spaces in terms of embeddings into ℓ_{1-ε} and ℓ_1, linking sketching complexity to geometric properties.

Findings

Efficient sketches exist iff the space embeds into ℓ_{1-ε} with constant distortion.

Norms closed under sum-product admit efficient sketching iff they embed into ℓ_1.

Certain norms like Earth Mover's Distance and trace norm do not admit efficient sketches.

Abstract

An outstanding open question posed by Guha and Indyk in 2006 asks to characterize metric spaces in which distances can be estimated using efficient sketches. Specifically, we say that a sketching algorithm is efficient if it achieves constant approximation using constant sketch size. A well-known result of Indyk (J. ACM, 2006) implies that a metric that admits a constant-distortion embedding into $\ell_p$ for $p\in(0,2]$ also admits an efficient sketching scheme. But is the converse true, i.e., is embedding into $\ell_p$ the only way to achieve efficient sketching? We address these questions for the important special case of normed spaces, by providing an almost complete characterization of sketching in terms of embeddings. In particular, we prove that a finite-dimensional normed space allows efficient sketches if and only if it embeds (linearly) into $\ell_{1-\varepsilon}$ with constant distortion. We further prove that for norms that are closed under sum-product, efficient sketching is equivalent to embedding into $\ell_1$ with constant distortion. Examples of such norms include the Earth Mover's Distance (specifically its norm variant, called Kantorovich-Rubinstein norm), and the trace norm (a.k.a. Schatten $1$-norm or the nuclear norm). Using known non-embeddability theorems for these norms by Naor and Schechtman (SICOMP, 2007) and by Pisier (Compositio. Math., 1978), we then conclude that these spaces do not admit efficient sketches either, making progress towards answering another open question posed by Indyk in 2006. Finally, we observe that resolving whether "sketching is equivalent to embedding into $\ell_1$ for general norms" (i.e., without the above restriction) is equivalent to resolving a well-known open problem in Functional Analysis posed by Kwapien in 1969.