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Documents authored by Bartal, Yair

Document
DOI: 10.4230/LIPIcs.SoCG.2022.13
Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures

Authors: Yair Bartal, Ora Nova Fandina, and Kasper Green Larsen

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in [Yair Bartal et al., 2019] (NeurIPS'19). They provided upper bounds on its quality for a wide range of practical measures and showed that indeed these are best possible in many cases. Yet, some of the most important cases, including the fundamental case of average distortion were left open. In particular, they show that the JL transform has 1+ε average distortion for embedding into k-dimensional Euclidean space, where k = O(1/ε²), and for more general q-norms of distortion, k = O(max{1/ε²,q/ε}), whereas tight lower bounds were established only for large values of q via reduction to the worst case. In this paper we prove that these bounds are best possible for any dimensionality reduction method, for any 1 ≤ q ≤ O((log (2ε² n))/ε) and ε ≥ 1/(√n), where n is the size of the subset of Euclidean space. Our results also imply that the JL method is optimal for various distortion measures commonly used in practice, such as stress, energy and relative error. We prove that if any of these measures is bounded by ε then k = Ω(1/ε²), for any ε ≥ 1/(√n), matching the upper bounds of [Yair Bartal et al., 2019] and extending their tightness results for the full range moment analysis. Our results may indicate that the JL dimensionality reduction method should be considered more often in practical applications, and the bounds we provide for its quality should be served as a measure for comparison when evaluating the performance of other methods and heuristics.
Cite as

Yair Bartal, Ora Nova Fandina, and Kasper Green Larsen. Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bartal_et_al:LIPIcs.SoCG.2022.13,
  author =	{Bartal, Yair and Fandina, Ora Nova and Larsen, Kasper Green},
  title =	{{Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.13},
  URN =		{urn:nbn:de:0030-drops-160219},
  doi =		{10.4230/LIPIcs.SoCG.2022.13},
  annote =	{Keywords: average distortion, practical dimensionality reduction, JL transform}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.20
Covering Metric Spaces by Few Trees

Authors: Yair Bartal, Nova Fandina, and Ofer Neiman

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
A tree cover of a metric space (X,d) is a collection of trees, so that every pair x,y in X has a low distortion path in one of the trees. If it has the stronger property that every point x in X has a single tree with low distortion paths to all other points, we call this a Ramsey tree cover. Tree covers and Ramsey tree covers have been studied by [Yair Bartal et al., 2005; Anupam Gupta et al., 2004; T-H. Hubert Chan et al., 2005; Gupta et al., 2006; Mendel and Naor, 2007], and have found several important algorithmic applications, e.g. routing and distance oracles. The union of trees in a tree cover also serves as a special type of spanner, that can be decomposed into a few trees with low distortion paths contained in a single tree; Such spanners for Euclidean pointsets were presented by [S. Arya et al., 1995]. In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.
Cite as

Yair Bartal, Nova Fandina, and Ofer Neiman. Covering Metric Spaces by Few Trees. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bartal_et_al:LIPIcs.ICALP.2019.20,
  author =	{Bartal, Yair and Fandina, Nova and Neiman, Ofer},
  title =	{{Covering Metric Spaces by Few Trees}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{20:1--20:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.20},
  URN =		{urn:nbn:de:0030-drops-105967},
  doi =		{10.4230/LIPIcs.ICALP.2019.20},
  annote =	{Keywords: tree cover, Ramsey tree cover, probabilistic hierarchical family}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.16
Dimension Reduction Techniques for l_p (1<p<2), with Applications

Authors: Yair Bartal and Lee-Ad Gottlieb

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
For Euclidean space (l_2), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss [Conf. in modern analysis and probability, AMS 1984], with a host of known applications. Here, we consider the problem of dimension reduction for all l_p spaces 1<p<2. Although strong lower bounds are known for dimension reduction in l_1, Ostrovsky and Rabani [JACM 2002] successfully circumvented these by presenting an l_1 embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to l_1 and do not naturally extend to other norms. In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of 1<p<2, thereby demonstrating that the approach initiated by Ostrovsky and Rabani for l_1 can be extended to a much more general framework. We also obtain improved bounds in terms of the intrinsic dimensionality. As a result we achieve improved bounds for proximity problems including snowflake embeddings and clustering.
Cite as

Yair Bartal and Lee-Ad Gottlieb. Dimension Reduction Techniques for l_p (1<p<2), with Applications. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bartal_et_al:LIPIcs.SoCG.2016.16,
  author =	{Bartal, Yair and Gottlieb, Lee-Ad},
  title =	{{Dimension Reduction Techniques for l\underlinep (1\langlep\langle2), with Applications}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{16:1--16:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.16},
  URN =		{urn:nbn:de:0030-drops-59081},
  doi =		{10.4230/LIPIcs.SoCG.2016.16},
  annote =	{Keywords: Dimension reduction, embeddings}
}
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