Fast sparse matrix multiplication
ACM Transactions on Algorithms (TALG)
Elusive functions and lower bounds for arithmetic circuits
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Theoretical Computer Science
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Graph expansion and communication costs of fast matrix multiplication: regular submission
Proceedings of the twenty-third annual ACM symposium on Parallelism in algorithms and architectures
Communication-optimal Parallel and Sequential QR and LU Factorizations
SIAM Journal on Scientific Computing
Graph expansion and communication costs of fast matrix multiplication
Journal of the ACM (JACM)
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Our main result is a lower bound of $\Omega(m^2 \log m)$ for the size of any arithmetic circuit for the product of two matrices, over the real or complex numbers, as long as the circuit does not use products with field elements of absolute value larger than 1 (where m × m is the size of each matrix). That is, our lower bound is superlinear in the number of inputs and is applied for circuits that use addition gates, product gates, and products with field elements of absolute value up to 1. We also prove size-depth tradeoffs for such circuits: We show that if a circuit, as above, is of depth d, then its size is $\Omega(m^{2+ 1/O(d)})$.