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DOI: 10.4230/LIPIcs.SoCG.2017.50

A Spectral Gap Precludes Low-Dimensional Embeddings

Authors: Assaf Naor

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract

We prove that if an n-vertex O(1)-expander embeds with average distortion D into a finite dimensional normed space X, then necessarily the dimension of X is at least n^{c/D} for some universal constant c>0. This is sharp up to the value of the constant c, and it improves over the previously best-known estimate dim(X)> c(log n)^2/D^2 of Linial, London and Rabinovich, strengthens a theorem of Matousek, and answers a question of Andoni, Nikolov, Razenshteyn and Waingarten.

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Assaf Naor. A Spectral Gap Precludes Low-Dimensional Embeddings. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{naor:LIPIcs.SoCG.2017.50,
  author =	{Naor, Assaf},
  title =	{{A Spectral Gap Precludes Low-Dimensional Embeddings}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{50:1--50:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.50},
  URN =		{urn:nbn:de:0030-drops-71822},
  doi =		{10.4230/LIPIcs.SoCG.2017.50},
  annote =	{Keywords: Metric embeddings, dimensionality reduction, expander graphs, nonlinear spectral gaps, nearest neighbor search, complex interpolation, Markov type.}
}

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DOI: 10.4230/LIPIcs.ICALP.2016.83

Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost

Authors: Alexandr Andoni, Assaf Naor, and Ofer Neiman

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract

Transportation cost metrics, also known as the Wasserstein distances W_p, are a natural choice for defining distances between two pointsets, or distributions, and have been applied in numerous fields. From the computational perspective, there has been an intensive research effort for understanding the W_p metrics over R^k, with work on the W_1 metric (a.k.a earth mover distance) being most successful in terms of theoretical guarantees. However, the W_2 metric, also known as the root-mean square (RMS) bipartite matching distance, is often a more suitable choice in many application areas, e.g. in graphics. Yet, the geometry of this metric space is currently poorly understood, and efficient algorithms have been elusive. For example, there are no known non-trivial algorithms for nearest-neighbor search or sketching for this metric. In this paper we take the first step towards explaining the lack of efficient algorithms for the W_2 metric, even over the three-dimensional Euclidean space R^3. We prove that there are no meaningful embeddings of W_2 over R^3 into a wide class of normed spaces, as well as that there are no efficient sketching algorithms for W_2 over R^3 achieving constant approximation. For example, our results imply that: 1) any embedding into L1 must incur a distortion of Omega(sqrt(log(n))) for pointsets of size n equipped with the W_2 metric; and 2) any sketching algorithm of size s must incur Omega(sqrt(log(n))/sqrt(s)) approximation. Our results follow from a more general statement, asserting that W_2 over R^3 contains the 1/2-snowflake of all finite metric spaces with a uniformly bounded distortion. These are the first non-embeddability/non-sketchability results for W_2.

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Alexandr Andoni, Assaf Naor, and Ofer Neiman. Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 83:1-83:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{andoni_et_al:LIPIcs.ICALP.2016.83,
  author =	{Andoni, Alexandr and Naor, Assaf and Neiman, Ofer},
  title =	{{Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{83:1--83:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.83},
  URN =		{urn:nbn:de:0030-drops-62028},
  doi =		{10.4230/LIPIcs.ICALP.2016.83},
  annote =	{Keywords: Transportation metric, embedding, snowflake, sketching}
}

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A Spectral Gap Precludes Low-Dimensional Embeddings

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Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost

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