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29 March 2024
 
  » arxiv » cs.CC/0301016

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Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps
Peter Buergisser ; Martin Lotz ;
Date 16 Dec 2002
Subject Computational Complexity ACM-class: F1.1; F2.1; I.1.2 | cs.CC
AbstractWe prove lower bounds of order $nlog n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the linear problem to multiply a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern’s bound [J. ACM 20, pp. 305-306, 1973] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix. In addition, we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications.
Source arXiv, cs.CC/0301016
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