Impossibility of Dimension Reduction in the Nuclear Norm

Assaf Naor, Gilles Pisier, Gideon Schechtman

Research output: Contribution to journalArticlepeer-review

Abstract

Let S1 (the Schatten–von Neumann trace class) denote the Banach space of all compact linear operators T: ℓ2→ ℓ2 whose nuclear norm ‖T‖S1=∑j=1∞σj(T) is finite, where {σj(T)}j=1∞ are the singular values of T. We prove that for arbitrarily large n∈ N there exists a subset C⊆ S1 with | C| = 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 (Conference in modern analysis and probability, American Mathematical Society, Providence, 1984), which complements the work of Harrow et al. (Automata, languages and programming (ICALP 2011), Springer, Heidelberg, 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 ℓ1 of absolutely summable sequences via the work of Brinkman and Charikar (J ACM 52(5):766–788, 2005). 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 ℓ1). 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 logdim(X).

Original languageEnglish (US)
Pages (from-to)319-345
Number of pages27
JournalDiscrete and Computational Geometry
Volume63
Issue number2
DOIs
StatePublished - Mar 1 2020

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • Dimension reduction
  • Lipschitz quotients
  • Markov convexity
  • Metric embeddings
  • Nuclear norm
  • Schatten–von Neumann classes

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