Entropy Estimates on Tensor Products of Banach Spaces and Applications to Low-Rank Recovery

Kiryung Lee, Rakshith Sharma Srinivasa, Marius Junge, Justin Romberg

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Low-rank matrix models have been universally useful for numerous applications starting from classical system identification to more modern matrix completion in signal processing and statistics. The Schatten-1 norm, also known as the nuclear norm, has been used as a convex surrogate of the low-rankness since it induces a low-rank solution to inverse problems. While the Schatten-1 norm for low-rankness has a nice analogy with the ℓ1 norm for sparsity through the singular value decomposition, other matrix norms also induce low-rankness. Particularly as one interprets a matrix as a linear operator between Banach spaces, various tensor product norms generalize the role of the Schatten-1 norm. Inspired by a recent work on the max-norm-based matrix completion, we provide a unified view on a class of tensor product norms and their interlacing relations on low-rank operators. Furthermore we derive entropy estimates between the injective and projective tensor products of a family of Banach space pairs and demonstrate their applications to matrix completion and decentralized subspace sketching.

Original languageEnglish (US)
Title of host publication2019 13th International Conference on Sampling Theory and Applications, SampTA 2019
PublisherInstitute of Electrical and Electronics Engineers Inc.
ISBN (Electronic)9781728137414
DOIs
StatePublished - Jul 2019
Event13th International Conference on Sampling Theory and Applications, SampTA 2019 - Bordeaux, France
Duration: Jul 8 2019Jul 12 2019

Publication series

Name2019 13th International Conference on Sampling Theory and Applications, SampTA 2019

Conference

Conference13th International Conference on Sampling Theory and Applications, SampTA 2019
CountryFrance
CityBordeaux
Period7/8/197/12/19

Keywords

  • Banach spaces
  • entropy number
  • Low-rank matrix
  • matrix completion
  • sketching
  • tensor product

ASJC Scopus subject areas

  • Statistics and Probability
  • Signal Processing
  • Analysis
  • Statistics, Probability and Uncertainty
  • Applied Mathematics

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  • Cite this

    Lee, K., Srinivasa, R. S., Junge, M., & Romberg, J. (2019). Entropy Estimates on Tensor Products of Banach Spaces and Applications to Low-Rank Recovery. In 2019 13th International Conference on Sampling Theory and Applications, SampTA 2019 [9030989] (2019 13th International Conference on Sampling Theory and Applications, SampTA 2019). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/SampTA45681.2019.9030989