TY - JOUR

T1 - Sketching and embedding are equivalent for norms

AU - Andoni, Alexandr

AU - Krauthgamer, Robert

AU - Razenshteyn, Ilya

N1 - An extended abstract appeared in Proceedings of the 47th ACM Symposium on the Theory of Computing, 2015. This work was done in part while all authors were at Microsoft Research Silicon Valley, as well as when the first author was at the Simons Institute for the Theory of Computing at UC Berkeley. Funding: The second author was supported in part by US–Israel BSF grant 2010418 and Israel Science Foundation grant 897/13. We are grateful to Assaf Naor for pointing us to [1, 42], as well as for numerous very enlightening discussions throughout this project. We also thank Gideon Schechtman for useful discussions and explaining some of the literature. We thank Piotr Indyk for fruitful discussions and for encouraging us to work on this project.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85049421172&partnerID=8YFLogxK

U2 - https://doi.org/10.1137/15M1017958

DO - https://doi.org/10.1137/15M1017958

M3 - مقالة

SN - 0097-5397

VL - 47

SP - 890

EP - 916

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 3

ER -