Learning Boolean functions with concentrated spectra

TL;DR

This paper introduces a new approach for learning Boolean functions concentrated in the Fourier domain, estimating their VC dimension, and applying an efficient empirical risk minimization method to classify handwritten digits effectively.

Contribution

It provides the first VC dimension estimate for this class of functions and proposes a computationally efficient learning method for practical classification tasks.

Findings

Effective classification of MNIST digits using the proposed method

Small sample complexity due to VC dimension bounds

Open problems for further research in Fourier-concentrated Boolean functions

Abstract

This paper discusses the theory and application of learning Boolean functions that are concentrated in the Fourier domain. We first estimate the VC dimension of this function class in order to establish a small sample complexity of learning in this case. Next, we propose a computationally efficient method of empirical risk minimization, and we apply this method to the MNIST database of handwritten digits. These results demonstrate the effectiveness of our model for modern classification tasks. We conclude with a list of open problems for future investigation.