On the complexity of matrix product

@article{Raz2002OnTC,
  title={On the complexity of matrix product},
  author={Ran Raz},
  journal={Electron. Colloquium Comput. Complex.},
  year={2002}
}
  • R. Raz
  • Published 19 May 2002
  • Computer Science, Mathematics
  • Electron. Colloquium Comput. Complex.
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… 
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References

SHOWING 1-10 OF 33 REFERENCES
Lower Bounds for Matrix Product in Bounded Depth Circuits with Arbitrary Gates
TLDR
The lower bound for the number of edges in any constant depth arithmetic circuit for matrix product (over any field) is superlinear in m2 (where m × m is the size of each matrix) and the lower bound is super linear in theNumber of input variables.
Lower Bounds for Matrix Product
  • Amir Shpilka
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 2003
TLDR
The number of product gates in any bilinear (or quadratic) circuit that computes the product of two n × n matrices over ${\rm GF}(2)$ is at least 3n2 - o(n2).
Lower bounds for matrix product, in bounded depth circuits with arbitrary gates
TLDR
Lower bounds for the number of edges in any constant depth arithmetic circuit for matrix product (over any field is super-linear in m^2), where m is the size of each matrix, are proved.
Spectral methods for matrix rigidity with applications to size-depth tradeoffs and communication complexity
  • Satyanarayana V. Lokam
  • Computer Science, Mathematics
    Proceedings of IEEE 36th Annual Foundations of Computer Science
  • 1995
TLDR
It is shown that the lower bound on a variant of rigidity implies lower bounds on size-depth tradeoffs for arithmetic circuits with bounded coefficients computing linear transformations, and this results complement and strengthen a result of Razborov.
Lower bounds on the bounded coefficient complexity of bilinear maps
  • Peter Bürgisser, Martin Lotz
  • Mathematics, Computer Science
    The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
  • 2002
TLDR
Lower bounds of order n log n are proved for both the problem to multiply polynomials of degree n, and to divide polynmials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers, which establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix.
A Spectral Approach to Lower Bounds with Applications to Geometric Searching
  • B. Chazelle
  • Mathematics, Computer Science
    SIAM J. Comput.
  • 1998
TLDR
It is shown that summing up the weights of n (weighted) points within n halfplanes requires $\Omega(n\log n)$ additions and subtractions, the first nontrivial lower bound for range searching over a group.
Algebraic complexity theory
TLDR
Algebraic complexity theory investigates the computational cost of solving problems with an algebraic flavor, and takes its questions from computer science, mainly numerical and symbolic computation.
Gaussian elimination is not optimal
t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4 . 7 n l°g7 arithmetical
A note on the use of determinant for proving lower bounds on the size of linear circuits
  • P. Pudlák
  • Mathematics, Computer Science
    Inf. Process. Lett.
  • 1998
A 5/2 n2-Lower Bound for the Rank of n×n Matrix Multiplication over Arbitrary Fields
We prove a lower bound of 5/2n/sup 2/-3n for the rank of n/spl times/n-matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division
...
1
2
3
4
...