Hadamard Powers and Kernel Perceptrons

TL;DR

This paper explores the mathematical relationship between Hadamard powers and polynomial kernel perceptrons, providing explicit rank calculations for Boolean and real matrices and linking these to perceptron classification capacities.

Contribution

It introduces explicit formulas for the rank of Hadamard powers in specific cases and interprets these results in the context of perceptron classification capabilities.

Findings

Explicit rank formulas for Hadamard powers of Boolean matrices

Explicit rank formulas for Hadamard powers of real matrices

Insights into perceptron classification capacities

Abstract

We study a relation between Hadamard powers and polynomial kernel perceptrons. The rank of Hadamard powers for the special case of a Boolean matrix and for the generic case of a real matrix is computed explicitly. These results are interpreted in terms of the classification capacities of perceptrons.