  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.
