Impossibility of dimension reduction in the nuclear norm

Assaf Naor, Gilles Pisier, Gideon Schechtman

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations

Abstract

Let S1 (the Schatten von Neumann trace class) denote the Banach space of all compact linear operators T : '2 ! '2 whose nuclear norm ||T||S1j=1 σj (T) is finite, where fj(T)g1j =1 are the singular values of T. We prove that for arbitrarily large n 2 N there exists a subset C S1 with jCj = n that cannot be embedded with bi-Lipschitz distortion O(1) into any no(1)-dimensional linear subspace of S1. C is not even a O(1)-Lipschitz quotient of any subset of any no(1)-dimensional linear subspace of S1. Thus, S1 does not admit a dimension reduction result á la Johnson and Lindenstrauss (1984), which complements the work of Harrow Montanaro and Short (2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with S1 replaced by the Banach space '1 of absolutely summable sequences via the work of Brinkman and Charikar (2003). In fact, the above set C can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in S1. The challenge is to demonstrate that C cannot be faithfully realized in an arbitrary low-dimensional subspace of S1, while Brinkman and Charikar obtained such an assertion only for subspaces of S1 that consist of diagonal operators (i.e., subspaces of '1). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace X of S1 is at most a universal constant multiple of p log dim(X).

Original languageEnglish (US)
Title of host publication29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
EditorsArtur Czumaj
PublisherAssociation for Computing Machinery
Pages1345-1352
Number of pages8
ISBN (Electronic)9781611975031
DOIs
StatePublished - Jan 1 2018
Event29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States
Duration: Jan 7 2018Jan 10 2018

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
CountryUnited States
CityNew Orleans
Period1/7/181/10/18

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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    Naor, A., Pisier, G., & Schechtman, G. (2018). Impossibility of dimension reduction in the nuclear norm. In A. Czumaj (Ed.), 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 (pp. 1345-1352). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery. https://doi.org/10.1137/1.9781611975031.88