On the complexity of matrix product

Authors:
Ran Raz
Affiliations:
Weizmann Institute, Rehovot, Israel
Venue:
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Year:
2002

Citing 10
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Abstract

We prove a lower bound of &OHgr;(m2 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 doesn't use products with field elements of absolute value larger than 1 (where mxm is the size of each matrix). That is, our lower bound is super-linear 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.More generally, for any c = c(m) &rhoe; 1, we obtain a lower bound of &OHgr;(m2 log2c 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 doesn't use products with field elements of absolute value larger than c.We also prove size-depth tradeoffs for such circuits.