Fast and efficient dimensionality reduction using Structurally Random Matrices

  • Authors:

  • Thong T. Do;Lu Gan; Yi Chen;Nam Nguyen;Trac D. Tran

  • Affiliations:

  • Department of Electrical and Computer Engineering, The Johns Hopkins University, USA;School of Engineering and Design, Brunel University, UK;Department of Electrical and Computer Engineering, The Johns Hopkins University, USA;Department of Electrical and Computer Engineering, The Johns Hopkins University, USA;Department of Electrical and Computer Engineering, The Johns Hopkins University, USA

  • Venue:
  • ICASSP '09 Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing
  • Year:
  • 2009

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Abstract

Structurally Random Matrices (SRM) are first proposed in [1] as fast and highly efficient measurement operators for large scale compressed sensing applications. Motivated by the bridge between compressed sensing and the Johnson-Lindenstrauss lemma [2] , this paper introduces a related application of SRMs regarding to realizing a fast and highly efficient embedding. In particular, it shows that a SRM is also a promising dimensionality reduction transform that preserves all pairwise distances of high dimensional vectors within an arbitrarily small factor ∈, provided that the projection dimension is on the order of O(∈−2 log3 N), where N denotes the number of d-dimensional vectors. In other words, SRM can be viewed as the sub-optimal Johnson-Lindenstrauss embedding that, however, owns very low computational complexity O(d log d) and highly efficient implementation that uses only O(d) random bits, making it a promising candidate for practical, large scale applications where efficiency and speed of computation are highly critical.