TY - GEN

T1 - Impossibility of dimension reduction in the nuclear norm

AU - Naor, Assaf

AU - Pisier, Gilles

AU - Schechtman, Gideon

N1 - Publisher Copyright:
© Copyright 2018 by SIAM.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2018

Y1 - 2018

N2 - Let S1 (the Schatten von Neumann trace class) denote the Banach space of all compact linear operators T : '2 ! '2 whose nuclear norm ||T||S1 =Σj∞=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).

AB - Let S1 (the Schatten von Neumann trace class) denote the Banach space of all compact linear operators T : '2 ! '2 whose nuclear norm ||T||S1 =Σj∞=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).

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U2 - 10.1137/1.9781611975031.88

DO - 10.1137/1.9781611975031.88

M3 - Conference contribution

AN - SCOPUS:85045557831

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1345

EP - 1352

BT - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

A2 - Czumaj, Artur

PB - Association for Computing Machinery

T2 - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

Y2 - 7 January 2018 through 10 January 2018

ER -