On the complexity of bilinear forms: dedicated to the memory of Jacques Morgenstern
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Matrix computations (3rd ed.)
Introduction to matrix analysis (2nd ed.)
Introduction to matrix analysis (2nd ed.)
A Spectral Approach to Lower Bounds with Applications to Geometric Searching
SIAM Journal on Computing
Note on a Lower Bound on the Linear Complexity of the Fast Fourier Transform
Journal of the ACM (JACM)
The Linear Complexity of Computation
Journal of the ACM (JACM)
On the complexity of matrix product
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Theoretical Computer Science
Complexity Lower Bounds using Linear Algebra
Foundations and Trends® in Theoretical Computer Science
Faster polynomial multiplication via discrete fourier transforms
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
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We prove lower bounds of order n log n for both the problem of multiplying polynomials of degree n, and of dividing 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 of multiplying 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.