# 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} }

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…

## 5 Citations

Arithmetic Circuits, Structured Matrices and (not so) Deep Learning

- Computer Science
- 2022

In this survey, a recent work that combines arithmetic circuit complexity, structured matrices and deep learning essentially answers the research question of replacing unstructured weight matrices in neural networks by structured ones.

Fast, algebraic multivariate multipoint evaluation in small characteristic and applications

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2021

It is shown that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size and Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant’s program.

Matrix rigidity depends on the target field

- MathematicsComputational Complexity Conference
- 2021

The lower-bound method combines elementary arguments from algebraic geometry with "untouched minors" arguments and establishes a gap of a factor of 3/2 − o(1) between strict and absolute rigidities.

Improved upper bounds for the rigidity of Kronecker products

- MathematicsMFCS
- 2021

The class of Hadamard matrices that are known not to be Valiant-rigid are significantly expanded; these now include the Kronecker products of Paley-Hadamards matrices and Hadamards of bounded size.

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