2014/11/10 by Andoni, Alexandr, Krauthgamer, Robert, Razenshteyn, Ilya · 1 citation
#Computational Complexity (cs.CC) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1411.2577
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 ℓp for p∈(0,2] also admits an efficient sketching scheme. But is the converse true, i.e., is embedding into ℓp 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 ℓ1-ε with constant distortion. We further prove that for norms that are closed under sum-product, efficient sketching is equivalent to embedding into ℓ1 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 ℓ1 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.