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Documents authored by Naor, Assaf

Document
DOI: 10.4230/LIPIcs.SoCG.2026.56
Optimal Randomized Clustering of Matrices

Authors: Mustafa Alper Gunes and Assaf Naor

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
If X = (𝖬_n(ℝ),‖⋅‖_X) is a unitarily invariant normed space, i.e., ‖𝖴𝖠𝖵‖_X = ‖𝖠‖_X for every matrix 𝖠 ∈ 𝖬_n(ℝ) and every two orthogonal matrices 𝖴,𝖵 ∈ 𝖬_n(ℝ), then we evaluate up to universal constant factors the smallest σ > 0 for which there is a probability distribution over partitions of X into clusters of diameter at most 1 yet for every two matrices 𝖠,𝖡 ∈ 𝖬_n(ℝ) the probability that they fall into distinct clusters is at most σ times the X-distance between 𝖠 and 𝖡. Specifically, we prove that this infimal σ, which is called the separation modulus of X and is denoted SEP(X), satisfies: (1) SEP(X) = Θ(√n⋅ ‖𝖨_n‖_X⋅ diam(B_X)), where 𝖨_n is the n-by-n identity matrix and diam(B_X) is the diameter with respect to the standard Euclidean metric on 𝖬_n(ℝ) of the unit ball B_ X of X. Our proof of (1) proceeds through an asymptotic evaluation of the spectral gap of the Laplacian with Dirichlet boundary conditions on B_ X, which we achieve by exact computations for a Jacobi orthogonal random matrix ensemble. Assuming oracle access to norm evaluations in X, by combining (1) with a new deterministic algorithm for a O(1)-approximation of the diameter of convex bodies in ℝⁿ that are given by a weak membership oracle and are symmetric with respect to coordinate permutations and reflections about the standard axes (this task is famously known to be impossible in the absence of such symmetries), we get an oracle polynomial time algorithm whose output is the separation modulus of X up to universal constant factors. Another example of a consequence of (1) is that for each m ∈ {1,…,n} the separation modulus of the m'th Ky Fan norm on 𝖬_n(ℝ) is bounded from above and from below by universal constant multiples of m√n if m ⩾ √n, and of n if m ⩽ √n. We also deduce from (1) an upper bound on the Lipschitz extension modulus of X that improves over the previously best-known bound even in the special case when X is 𝖬_n(ℝ) equipped with the 𝓁₂ⁿ → 𝓁₂ⁿ operator norm.
Cite as

Mustafa Alper Gunes and Assaf Naor. Optimal Randomized Clustering of Matrices. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 56:1-56:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{gunes_et_al:LIPIcs.SoCG.2026.56,
  author =	{Gunes, Mustafa Alper and Naor, Assaf},
  title =	{{Optimal Randomized Clustering of Matrices}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{56:1--56:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.56},
  URN =		{urn:nbn:de:0030-drops-258624},
  doi =		{10.4230/LIPIcs.SoCG.2026.56},
  annote =	{Keywords: Clustering, Unitarily Invariant Matrix Norms, Oracle Polynomial Time Approximation Algorithms for Radii of Convex Bodies, Extension of Lipschitz Functions, Random Matrices, Spectrum of the Laplacian with Dirichlet Boundary Conditions, Reverse Isoperimetry}
}
Document
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.}
}
Document
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.
Cite as

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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