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

Approximability of pq Matrix Norms: Generalized Krivine Rounding and Hypercontractive Hardness

Abstract

We study the problem of computing the p → q operator norm of a matrix A in ℝm×n, defined as ‖Apq : = supx∊ℝn\{0}Axq/‖xp. This problem generalizes the spectral norm of a matrix (p = q = 2) and the Grothendieck problem (p = ∞, q = 1), and has been widely studied in various regimes.
When pq, the problem exhibits a dichotomy: constant factor approximation algorithms are known if 2 is in [q, p], and the problem is hard to approximate within almost polynomial factors when 2 is not in [q,p]. For the case when 2 is in [q, p] we prove almost matching approximation and NP-hardness results.
The regime when p < q, known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with p = 2 and q > 2 was studied by [Barak et. al., STOC’12] who gave sub-exponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the Exponential Time Hypothesis. However, no NP-hardness of approximation is known for these problems for any p < q.
We prove the first NP-hardness result for approximating hypercontractive norms. We show that for any 1 < p < q < ∞ with 2 not in [p, q], ‖Apq is hard to approximate within 2O(log1−∊ n) assuming NP is not contained in BPTIME(2logO(1) n)).

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cover image Proceedings
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 1358 - 1368
Editor: Timothy M. Chan, University of Illinois at Urbana-Champaign, USA
ISBN (Online): 978-1-61197-548-2

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Published online: 2 January 2019

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