The Andoni–Krauthgamer–Razenshteyn characterization of sketchable norms fails for sketchable metrics

Subhash Khot, Assaf Naor

Research output: Contribution to conferencePaperpeer-review

1 Scopus citations

Abstract

Andoni, Krauthgamer and Razenshteyn (AKR) proved (STOC’15) that a finite-dimensional normed space (X, k·kX) admits a O(1) sketching algorithm (namely, with O(1) sketch size and O(1) approximation) if and only if for every ε ∈ (0, 1) there exist α > 1 and an embedding f : X → `1−ε such that kx − ykX 6 kf(x) − f(y)k1ε 6 αkx − ykX for all x, y ∈ X. The”if part” of this theorem follows from a sketching algorithm of Indyk (FOCS 2000). The contribution of AKR is therefore to demonstrate that the mere availability of a sketching algorithm implies the existence of the aforementioned geometric realization. Indyk’s algorithm shows that the”if part” of the AKR characterization holds true for any metric space whatsoever, i.e., the existence of an embedding as above implies sketchability even when X is not a normed space. Due to this, a natural question that AKR posed was whether the assumption that the underlying space is a normed space is needed for their characterization of sketchability. We resolve this question by proving that for arbitrarily large n ∈ N there is an n-point metric space (M(n), dM(n)) which is O(1)-sketchable yet for every ε ∈ (0, 12 ), if α(n) > 1 and fn : M(n) → `1ε are such that dM(n)(x, y) 6 kfn(x) − fn(y)k1ε 6 α(n)dM(n)(x, y) for all x, y ∈ M(n), then necessarily limn→∞ α(n) = ∞.

Original languageEnglish (US)
Pages1814-1824
Number of pages11
DOIs
StatePublished - 2019
Event30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States
Duration: Jan 6 2019Jan 9 2019

Conference

Conference30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019
CountryUnited States
CitySan Diego
Period1/6/191/9/19

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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