Injectivity, stability, and positive definiteness of max filtering

TL;DR

This paper investigates the mathematical properties of max filtering in inner product spaces, focusing on conditions for injectivity, stability, and positive definiteness, with implications for invariant representations and kernel methods.

Contribution

It provides nearly sharp conditions for injective embedding of orbit spaces and characterizes when max filtering yields positive definite kernels.

Findings

Identifies conditions for injective embedding of V/G into Euclidean space.

Estimates distortion of the quotient metric when G is finite.

Characterizes when max filtering produces positive definite kernels.

Abstract

Given a real inner product space V and a group G of linear isometries, max filtering offers a rich class of G-invariant maps. In this paper, we identify nearly sharp conditions under which these maps injectively embed the orbit space V/G into Euclidean space, and when G is finite, we estimate the map's distortion of the quotient metric. We also characterize when max filtering is a positive definite kernel.