Kronecker products, low-depth circuits, and matrix rigidity

@article{Alman2021KroneckerPL,
  title={Kronecker products, low-depth circuits, and matrix rigidity},
  author={Josh Alman},
  journal={Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing},
  year={2021}
}
  • Josh Alman
  • Published 24 February 2021
  • Mathematics
  • Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
For a matrix M and a positive integer r, the rank r rigidity of M is the smallest number of entries of M which one must change to make its rank at most r. There are many known applications of rigidity lower bounds to a variety of areas in complexity theory, but fewer known applications of rigidity upper bounds. In this paper, we use rigidity upper bounds to prove new upper bounds in a few different models of computation. Our results include: - For any d>1, and over any field F, the N × N Walsh… 
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References

SHOWING 1-10 OF 38 REFERENCES
Complexity of Linear Boolean Operators
TLDR
A thorough survey of the research in this direction is provided, and some new results are proved to fill out the picture.
Communication in bounded depth circuits
TLDR
It is shown that rigidity of matrices can be used to prove lower bounds on depth 2 circuits and communication graphs and a general nonlinear bound on a certain type of circuits is proved.
On the complexity of matrix product
  • R. Raz
  • Computer Science, Mathematics
    STOC '02
  • 2002
TLDR
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.
Probabilistic rank and matrix rigidity
TLDR
A notion of Probabilistic rank and probabilistic sign-rank of a matrix, which measure the extent to which a matrix can be probabilistically represented by low-rank matrices, is considered, which shows that for every function f which is randomly self-reducible in a natural way, Bounding the communication complexity of f is equivalent to bounding the rigidity of the matrix of f, via an equivalence with probabilism rank.
RAPID MULTIPLICATION OF RECTANGULAR MATRICES *
The number of essential multiplications required to multiply matrices of size N N and N N is bounded by CN log N. Key words, matrix multiplication, tensor rank, algebraic complexity Introduction. Let
More Applications of the Polynomial Method to Algorithm Design
TLDR
This paper extends the polynomial method to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods.
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
Faster all-pairs shortest paths via circuit complexity
TLDR
A new randomized method for computing the min-plus product of two n × n matrices is presented, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense n-node directed graphs with arbitrary edge weights.
A Refined Laser Method and Faster Matrix Multiplication
TLDR
This paper is a refinement of the laser method that improves the resulting value bound for most sufficiently large tensors, and obtains the best bound on $\omega$ to date.
Equivalence of Systematic Linear Data Structures and Matrix Rigidity
TLDR
This work derives an equivalence between rigidity and the systematic linear model of data structures, and proves that lower bounds on the query time imply rigidity lower bounds for the query set itself.
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