Sketching and embedding are equivalent for norms

An outstanding open question posed by Guha and Indyk in 2006 asks us to characterize metric spaces in which distances can be estimated using e cient sketches. Specifically, we say that a sketching algorithm is e cient if it achieves constant approximation using constant sketch size. A well-known result of Indyk [J. ACM, 53 (2006), pp. 307–323] implies that a metric that admits a constant-distortion embedding into p for p ∈ (0, 2] also admits an e cient sketching scheme. But is the converse true, i.e., is embedding into p the only way to achieve e cient 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 e cient sketches i it embeds (linearly) into ₁−_ε with constant distortion. We further prove that for norms that are closed under sum-product, e cient sketching is equivalent to embedding into ₁ with constant distortion. Examples of such norms include the earth mover’s distance (specifically its norm variant, called the Kantorovich–Rubinstein norm), and the trace norm (a.k.a. the Schatten 1-norm or the nuclear norm). Using known nonembeddability theorems for these norms by Naor and Schechtman [SIAM J. Comput., 37 (2007), pp. 804–826] and by Pisier [Compos. Math., 37 (1978), pp. 3–19], we then conclude that these spaces do not admit e cient sketches either, making progress toward answering another open question posed by Indyk in 2006. Finally, we observe that resolving whether “sketching is equivalent to embedding into 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.