3 Search Results for "Eskenazis, Alexandros"

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.13

Learning Low-Degree Quantum Objects

Authors: Srinivasan Arunachalam, Arkopal Dutt, Francisco Escudero Gutiérrez, and Carlos Palazuelos

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

We consider the problem of learning low-degree quantum objects up to ε-error in 𝓁₂-distance. We show the following results: (i) unknown n-qubit degree-d (in the Pauli basis) quantum channels and unitaries can be learned using O(1/ε^d) queries (which is independent of n), (ii) polynomials p:{-1,1}ⁿ → [-1,1] arising from d-query quantum algorithms can be learned from O((1/ε)^d ⋅ log n) many random examples (x,p(x)) (which implies learnability even for d = O(log n)), and (iii) degree-d polynomials p:{-1,1}ⁿ → [-1,1] can be learned through O(1/ε^d) queries to a quantum unitary U_p that block-encodes p. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded polynomials.

Cite as

Srinivasan Arunachalam, Arkopal Dutt, Francisco Escudero Gutiérrez, and Carlos Palazuelos. Learning Low-Degree Quantum Objects. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arunachalam_et_al:LIPIcs.ICALP.2024.13,
  author =	{Arunachalam, Srinivasan and Dutt, Arkopal and Escudero Guti\'{e}rrez, Francisco and Palazuelos, Carlos},
  title =	{{Learning Low-Degree Quantum Objects}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{13:1--13:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.13},
  URN =		{urn:nbn:de:0030-drops-201563},
  doi =		{10.4230/LIPIcs.ICALP.2024.13},
  annote =	{Keywords: Tomography}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2024.55

Dimensionality of Hamming Metrics and Rademacher Type

Authors: Alexandros Eskenazis

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract

Let X be a finite-dimensional normed space. We prove that if the Hamming cube {-1,1}ⁿ embeds into X with bi-Lipschitz distortion at most D ≥ 1, then dim(X) ≳ sup_{p ∈ [1,2]} n^p/(D^p 𝖳_p(X)^p), where 𝖳_p(X) is the Rademacher type p constant of X. This estimate yields a mutual refinement of distortion lower bounds which follow from works of Oleszkiewicz (1996) and Ivanisvili, van Handel and Volberg (2020). The proof relies on a combination of semigroup techniques on the biased hypercube with the Borsuk-Ulam theorem from algebraic topology.

Cite as

Alexandros Eskenazis. Dimensionality of Hamming Metrics and Rademacher Type. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{eskenazis:LIPIcs.SoCG.2024.55,
  author =	{Eskenazis, Alexandros},
  title =	{{Dimensionality of Hamming Metrics and Rademacher Type}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{55:1--55:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.55},
  URN =		{urn:nbn:de:0030-drops-200004},
  doi =		{10.4230/LIPIcs.SoCG.2024.55},
  annote =	{Keywords: Hamming cube, Rademacher type, metric embeddings, Borsuk-Ulam theorem}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2022.40

ε-Isometric Dimension Reduction for Incompressible Subsets of 𝓁_p

Authors: Alexandros Eskenazis

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

Abstract

Fix p ∈ [1,∞), K ∈ (0,∞) and a probability measure μ. We prove that for every n ∈ ℕ, ε ∈ (0,1) and x₁,…,x_n ∈ L_p(μ) with ‖max_{i ∈ {1,…,n}}|x_i|‖_{L_p(μ)} ≤ K, there exists d ≤ (32e² (2K)^{2p}log n)/ε² and vectors y₁,…, y_n ∈ 𝓁_p^d such that ∀i,j∈{1,…,n}, ‖x_i-x_j‖^p_{L_p(μ)}-ε ≤ ‖y_i-y_j‖_{𝓁_p^d}^p ≤ ‖x_i-x_j‖^p_{L_p(μ)}+ε. Moreover, the argument implies the existence of a greedy algorithm which outputs {y_i}_{i = 1}ⁿ after receiving {x_i}_{i = 1}ⁿ as input. The proof relies on a derandomized version of Maurey’s empirical method (1981) combined with a combinatorial idea of Ball (1990) and a suitable change of measure. Motivated by the above embedding, we introduce the notion of ε-isometric dimension reduction of the unit ball B_E of a normed space (E,‖⋅‖_E) and we prove that B_{𝓁_p} does not admit ε-isometric dimension reduction by linear operators for any value of p≠2.

Cite as

Alexandros Eskenazis. ε-Isometric Dimension Reduction for Incompressible Subsets of 𝓁_p. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{eskenazis:LIPIcs.SoCG.2022.40,
  author =	{Eskenazis, Alexandros},
  title =	{{\epsilon-Isometric Dimension Reduction for Incompressible Subsets of 𝓁\underlinep}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{40:1--40:14},
  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.2022.40},
  URN =		{urn:nbn:de:0030-drops-160486},
  doi =		{10.4230/LIPIcs.SoCG.2022.40},
  annote =	{Keywords: Dimension reduction, \epsilon-isometric embedding, Maurey’s empirical method, change of measure}
}

Refine by Author
2 Eskenazis, Alexandros
1 Arunachalam, Srinivasan
1 Dutt, Arkopal
1 Escudero Gutiérrez, Francisco
1 Palazuelos, Carlos

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2 Theory of computation → Random projections and metric embeddings
1 Mathematics of computing → Approximation
1 Mathematics of computing → Functional analysis
1 Mathematics of computing → Probabilistic algorithms
1 Theory of computation → Quantum complexity theory

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1 Borsuk-Ulam theorem
1 Dimension reduction
1 Hamming cube
1 Maurey’s empirical method
1 Rademacher type
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