An eigenspace approach to isotypic projections for data on binary trees

TL;DR

This paper introduces an eigenspace method for isotypic projections tailored to data on binary trees, extending Fourier analysis concepts to tree automorphisms with potential computational benefits.

Contribution

It develops a linear algebra-based approach for isotypic projections invariant under binary tree automorphisms, generalizing Fourier transform techniques.

Findings

Method simplifies computation of isotypic projections

Potential for increased computational efficiency

Applicable to data structured as binary trees

Abstract

The classical Fourier transform is, in essence, a way to take data and extract components (in the form of complex exponentials) which are invariant under cyclic shifts. We consider a case in which the components must instead be invariant under automorphisms of a binary tree. We present a technique by which a slightly relaxed form of the generalized Fourier transform in this case can eventually be computed using only simple tools from linear algebra, which has possible advantages in computational efficiency.