Group-invariant max filtering
This provides a new tool for classification tasks in machine learning, particularly for handling group symmetries, but it is incremental as it builds on concepts like the scattering transform.
The paper introduces max filters, a family of group-invariant functions for real inner product spaces, and shows they separate orbits with bilipschitz properties for finite groups in ℝᵈ and exhibit stability to diffeomorphic distortion in L²(ℝᵈ) similar to the scattering transform.
Given a real inner product space $V$ and a group $G$ of linear isometries, we construct a family of $G$-invariant real-valued functions on $V$ that we call max filters. In the case where $V=\mathbb{R}^d$ and $G$ is finite, a suitable max filter bank separates orbits, and is even bilipschitz in the quotient metric. In the case where $V=L^2(\mathbb{R}^d)$ and $G$ is the group of translation operators, a max filter exhibits stability to diffeomorphic distortion like that of the scattering transform introduced by Mallat. We establish that max filters are well suited for various classification tasks, both in theory and in practice.