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Fast Structured Matrix Computations: Tensor Rank and Cohn–Umans Method

Abstract

We discuss a generalization of the Cohn–Umans method, a potent technique developed for studying the bilinear complexity of matrix multiplication by embedding matrices into an appropriate group algebra. We investigate how the Cohn–Umans method may be used for bilinear operations other than matrix multiplication, with algebras other than group algebras, and we relate it to Strassen’s tensor rank approach, the traditional framework for investigating bilinear complexity. To demonstrate the utility of the generalized method, we apply it to find the fastest algorithms for forming structured matrix–vector product, the basic operation underlying iterative algorithms for structured matrices. The structures we study include Toeplitz, Hankel, circulant, symmetric, skew-symmetric, f-circulant, block Toeplitz–Toeplitz block, triangular Toeplitz matrices, Toeplitz-plus-Hankel, sparse/banded/triangular. Except for the case of skew-symmetric matrices, for which we have only upper bounds, the algorithms derived using the generalized Cohn–Umans method in all other instances are the fastest possible in the sense of having minimum bilinear complexity. We also apply this framework to a few other bilinear operations including matrix–matrix, commutator, simultaneous matrix products, and briefly discuss the relation between tensor nuclear norm and numerical stability.

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Notes

  1. Even if the exponent of matrix multiplication turns out to be 2; note that this is asymptotic.

  2. This is essential; (7) cannot be replaced by \(\bigl (\sum _{i=1}^{r}|\lambda _{i} |^p \bigr )^{1/p}\) for \(p > 1\) or \(\max _{i=1,\ldots ,r} |\lambda _i |\). See [17, Section 3].

  3. Later on in the article we will consider embedding of vector spaces into algebras.

  4. We do not distinguish between an irreducible representation of G and its irreducible \(\mathbb {C}[G]\)-submodule.

  5. The reader is reminded that scalar multiplications by a constant like \(\omega ^i\) are not counted in bilinear complexity.

  6. The result is, however, coordinate independent, i.e., it does not depend on our choice of the bases.

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Acknowledgments

We thank Henry Cohn for very helpful discussions that initiated this work. We are also grateful to Andrew Chien, Nikos Pitsianis, and Xiaobai Sun for answering our questions about energy costs and circuit complexity of various integer and floating point operations; to Mike Stein for suggesting that we examine bttb matrices; and to Chris Umans for prompting Construction 6. We thank the two anonymous referees and the handling editor for their exceptionally helpful comments and constructive suggestions. In particular, we included Sects. 1.2 and 3.2 at the handling editor’s urging, which in retrospect were glaring omissions. LHL and KY are partially supported by AFOSR FA9550-13-1-0133, DARPA D15AP00109, NSF IIS 1546413, DMS 1209136, and DMS 1057064. In addition, KY’s work is also partially supported by NSF CCF 1017760.

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Correspondence to Lek-Heng Lim.

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Communicated by Nicholas Higham.

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Ye, K., Lim, LH. Fast Structured Matrix Computations: Tensor Rank and Cohn–Umans Method. Found Comput Math 18, 45–95 (2018). https://doi.org/10.1007/s10208-016-9332-x

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Keywords

  • Bilinear complexity
  • Tensor rank
  • Tensor nuclear norm
  • Cohn–Umans method
  • Structured matrix–vector product
  • Stability
  • Sparse and structured matrices

Mathematics Subject Classification

  • 15B05
  • 65F50
  • 65Y20
  • 13P25
  • 22D20