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},  volume={TR02},  url={https://api.semanticscholar.org/CorpusID:9582328}}
For any c = c(m) &rhoe; 1, a lower bound of &OHgr;(m2 log2c m) is obtained for the size of any arithmetic circuit for the product of two matrices, as long as the circuit doesn't use products with field elements of absolute value larger than c.

143 Citations

Kronecker products, low-depth circuits, and matrix rigidity

A new rigidity upper bound is shown, showing that the following classes of matrices are not rigid enough to prove circuit lower bounds using Valiant’s approach, which generalizes recent results on non-rigidity.

Smaller Low-Depth Circuits for Kronecker Powers

New, smaller constructions of constant-depth linear circuits for computing any matrix which is the Kronecker power of a fixed matrix, and generalize to improving the standard construction for any depth $\leq O(\log N)$.

A super-quadratic lower bound for depth four arithmetic circuits

An Ω(n2.5) lower bound is shown for general depth four arithmetic circuits computing an explicit n-variate degree-Θ(n) multilinear polynomial over any field of characteristic zero, inspired by a well-known greedy approximation algorithm for the weighted set-cover problem.

Towards an Almost Quadratic Lower Bound on the Monotone Circuit Complexity of the Boolean Convolution

We study the monotone circuit complexity of the so called semi-disjoint bilinear forms over the Boolean semi-ring, in particular the n-dimensional Boolean vector convolution. Besides the size of a…

Multiplying matrices using n arithmetic operations

This work presents a novel mathematically inspired algorithm while pointing towards an alternative research path, which opens the possibility of novel algorithms and a taxonomy of matrix multiplications and associated complexities.

Orbits and Arithmetical Circuit Lower Bounds

The orbit of a regular function f over a field F under action by a matrix group G is the collection of functions f(Ex) for E ∈ G. We show that some lower bounds of Bürgisser and Lotz [BL03] and…

Removing the log Factor from (min, +)-Products on Bounded Range Integer Matrices

We revisit the problem of multiplying two square matrices over the (min , +) semi-ring, where all entries are integers from a bounded range [ − M : M] ∪ {∞} . The current state of the art for this…

Elusive functions and lower bounds for arithmetic circuits

It is shown that for many settings of the parameters n,m,s,r, explicit constructions of elusive polynomial-mappings imply strong (up to exponential) lower bounds for general arithmetic circuits.

Lower bound frontiers in arithmetical circuit complexity

The lower bound frontier in arithmetical circuit complexity for several recently considered restricted computational models is expanded and a quantative lower bound on the expected value of the determinant of certain random Vandermonde matrices with nodes on the unit circle in the complex plane is proved.

Optimal Complexity Lower Bound for Polynomial Multiplication

Lower bounds of order n log n are proved for both the problem of multiplying polynomials of degree n, and of dividing polynomers with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers, to establish a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix.
...

19 References

Lower bounds for matrix product

We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two n /spl times/ n matrices over finite fields. In particular we obtain the…

Lower bounds for matrix product, in bounded depth circuits with arbitrary gates

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

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

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

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.

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 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…

Note on a Lower Bound on the Linear Complexity of the Fast Fourier Transform

A lower bound for the number of additions necessary to compute a family of linear functions by a linear algorithm is given when an upper bound <italic>c</italic> can be assigned to the modulus of the…

The Smallest Eigenvalue of a Large Dimensional Wishart Matrix

For each s : 1,2... let n: r?(s) be a posit ive integer such that n/s --+ y > 0 as s --+ cc. Let V" be an n X s matrix whose entries are i.i.d. N(0, 1) random variables and let M": (I/s)V"V:.The…

Related Papers

Showing 1 through 3 of 0 Related Papers