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On Sketching the q to p Norms

Authors Aditya Krishnan, Sidhanth Mohanty, David P. Woodruff



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Aditya Krishnan
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Sidhanth Mohanty
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
David P. Woodruff
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

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Aditya Krishnan, Sidhanth Mohanty, and David P. Woodruff. On Sketching the q to p Norms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 15:1-15:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.15

Abstract

We initiate the study of data dimensionality reduction, or sketching, for the q -> p norms. Given an n x d matrix A, the q -> p norm, denoted |A |_{q -> p} = sup_{x in R^d \ 0} |Ax |_p / |x |_q, is a natural generalization of several matrix and vector norms studied in the data stream and sketching models, with applications to datamining, hardness of approximation, and oblivious routing. We say a distribution S on random matrices L in R^{nd} - > R^k is a (k,alpha)-sketching family if from L(A), one can approximate |A |_{q -> p} up to a factor alpha with constant probability. We provide upper and lower bounds on the sketching dimension k for every p, q in [1, infty], and in a number of cases our bounds are tight. While we mostly focus on constant alpha, we also consider large approximation factors alpha, as well as other variants of the problem such as when A has low rank.

Subject Classification

ACM Subject Classification
  • Theory of computation → Numeric approximation algorithms
Keywords
  • Dimensionality Reduction
  • Norms
  • Sketching
  • Streaming

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