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Impossibility of Dimension Reduction in the Nuclear Norm

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Abstract

Let \({\mathsf {S}}_1\) (the Schatten–von Neumann trace class) denote the Banach space of all compact linear operators \(T:\ell _2\rightarrow \ell _2\) whose nuclear norm \(\Vert T\Vert _{{\mathsf {S}}_1}=\sum _{j=1}^\infty \upsigma _j(T)\) is finite, where \(\{\upsigma _j(T)\}_{j=1}^\infty \) are the singular values of T. We prove that for arbitrarily large \(n\in {\mathbb {N}}\) there exists a subset \({\mathscr {C}}\subseteq {\mathsf {S}}_1\) with \(|{\mathscr {C}}|=n\) that cannot be embedded with bi-Lipschitz distortion O(1) into any \(n^{o(1)}\)-dimensional linear subspace of \({\mathsf {S}}_1\). \({\mathscr {C}}\) is not even a O(1)-Lipschitz quotient of any subset of any \(n^{o(1)}\)-dimensional linear subspace of \({\mathsf {S}}_1\). Thus, \({\mathsf {S}}_1\) 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 \({\mathsf {S}}_1\) replaced by the Banach space \(\ell _1\) of absolutely summable sequences via the work of Brinkman and Charikar (J ACM 52(5):766–788, 2005). In fact, the above set \({\mathscr {C}}\) can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in \({\mathsf {S}}_1\). The challenge is to demonstrate that \({\mathscr {C}}\) cannot be faithfully realized in an arbitrary low-dimensional subspace of \({\mathsf {S}}_1\), while Brinkman and Charikar obtained such an assertion only for subspaces of \({\mathsf {S}}_1\) that consist of diagonal operators (i.e., subspaces of \(\ell _1\)). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace X of \({\mathsf {S}}_1\) is at most a universal constant multiple of \(\sqrt{\log \dim (X)}\).

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Notes

  1. Formally, for the purpose of efficient approximate nearest neighbor search one cannot use a dimension reduction statement like (2) as a “black box” without additional information about the low-dimensional embedding itself rather than its mere existence. One would want the embedding to be fast to compute and “data oblivious,” as in the classical Johnson–Lindenstrauss lemma [45]. There is no need to give a precise formulation here because the present article is devoted to ruling out any low-dimensional low-distortion embedding whatsoever.

  2. We shall use throughout this article the following (standard) asymptotic notation. Given two quantities \(Q,Q'>0\), the notations \(Q\lesssim Q'\) and \(Q'\gtrsim Q\) mean that \(Q\leqslant {\mathsf {K}}Q'\) for some universal constant \({\mathsf {K}}>0\). The notation \(Q\asymp Q'\) stands for \((Q\lesssim Q') \wedge (Q'\lesssim Q)\). If we need to allow for dependence on certain parameters, we indicate this by subscripts. For example, in the presence of an auxiliary parameter \(\uppsi \), the notation \(Q\lesssim _\uppsi Q'\) means that \(Q\leqslant c(\uppsi )Q' \), where \(c(\uppsi ) >0\) is allowed to depend only on \(\uppsi \), and similarly for the notations \(Q\gtrsim _\uppsi Q'\) and \(Q\asymp _\uppsi Q'\).

  3. Those who prefer to consider the nuclear norm on \(m\times m\) matrices can do so throughout, since all of our results are equivalent to their matricial counterparts; see Lemma 2.1 below for a formulation of this (straightforward) statement.

  4. One should note here that \({\mathsf {S}}_1\) does not admit a bi-Lipschitz embedding into any \(L_1(\upmu )\) space, as follows by combining the corresponding linear result of [66, 85] with a classical differentiation argument [12], or directly by using a bi-Lipschitz invariant that is introduced in the forthcoming work [79].

  5. For many purposes it is important to work with complex scalars, and correspondingly complex matrices. However, for the purpose of the ensuing metric results, statements over \({\mathbb {R}}\) are equivalent to their complex counterparts.

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Acknowledgements

We thank A. Andoni, R. Krauthgamer and M. Mendel for helpful input.

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  1. Mathematics Department, Princeton University, Princeton, NJ, 08544-1000, USA

    Assaf Naor

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Gilles Pisier

  3. Department of Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel

    Gideon Schechtman

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A. Naor was supported by the BSF, the NSF, the Packard Foundation and the Simons Foundation. G. Schechtman was supported by the ISF. The research that is presented here was conducted under the auspices of the Simons Algorithms and Geometry (A&G) Think Tank. An extended abstract that announces the results that are obtained here appeared in the proceedings of the 29th annual ACM–SIAM Symposium on Discrete Algorithms (SODA 2018).

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Naor, A., Pisier, G. & Schechtman, G. Impossibility of Dimension Reduction in the Nuclear Norm. Discrete Comput Geom 63, 319–345 (2020). https://doi.org/10.1007/s00454-019-00162-2

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