Lectures on the complexity of bilinear problems
Lectures on the complexity of bilinear problems
A lower bound for matrix multiplication
SIAM Journal on Computing
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Lower bounds for the multiplicative complexity of matrix multiplication
Computational Complexity
Lower bounds for the bilinear complexity of associative algebras
Computational Complexity
On associative algebras of minimal rank
AAECC-2 Proceedings of the 2nd International Conference on Applied Algebra, Algorithms and Error-Correcting Codes
The complexity of bivariate power series arithmetic
Theoretical Computer Science - Mathematical foundations of computer science
Nearly tight bounds on the learnability of evolution
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Lower Bounds for Matrix Product
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Optimization techniques for small matrix multiplication
Theoretical Computer Science
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We prove a lower bound of 5/2n2 - 3n for the multiplicative complexity of n × n-matrix multiplication over arbitrary fields. More general, we show that for any finite dimensional semisimple algebra A with unity, the multiplicative complexity C (A) of the multiplication in A is bounded from below by 5/2 dim A - 3(n1 +...+ nt) if the decomposition of A ≃ A1 × ... × At into simple algebras Aτ ≃ Dτnτ×nτ contains only noncommutative factors, that is, the division algebra Dτ is noncommutative or nτ≥2.We also deal with the complexity of multiplication in algebras with nonzero radical. We present an example that shows that our methods in the semisimple case cannot be applied directly to this problem. We exhibit lower bound techniques for C(A) that yield bounds still significantly above the Alder-Strassen bound. The main application is the lower bound C (Tn(k)) ≥ (21/8-o(1)) dim Tn(k) for the multiplicative complexity of multiplication in the algebra Tn(k) of upper triangular n × n-matrices.