Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

The Andoni--Krauthgamer--Razenshteyn characterization of sketchable norms fails for sketchable metrics (1810.04321v1)

Published 10 Oct 2018 in cs.DS, cs.CG, math.FA, and math.MG

Abstract: Andoni, Krauthgamer and Razenshteyn (AKR) proved (STOC 2015) that a finite-dimensional normed space $(X,|\cdot|X)$ admits a $O(1)$ sketching algorithm (namely, with $O(1)$ sketch size and $O(1)$ approximation) if and only if for every $\varepsilon\in (0,1)$ there exist $\alpha\geqslant 1$ and an embedding $f:X\to \ell{1-\varepsilon}$ such that $|x-y|X\leqslant |f(x)-f(y)|{1-\varepsilon}\leqslant \alpha |x-y|X$ for all $x,y\in X$. 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 $n\in \mathbb{N}$ there is an $n$-point metric space $(M(n),d{M(n)})$ which is $O(1)$-sketchable yet for every $\varepsilon\in (0,\frac12)$, if $\alpha(n)\geqslant 1$ and $f_n:M(n)\to \ell_{1-\varepsilon}$ are such that $d_{M(n)}(x,y)\leqslant |f_n(x)-f_n(y)|{1-\varepsilon}\leqslant \alpha(n) d{M(n)}(x,y)$ for all $x,y\in M(n)$, then necessarily $\lim_{n\to \infty} \alpha(n)= \infty$.

Citations (2)

Summary

We haven't generated a summary for this paper yet.