# Group-invariant max filtering

arXiv: https://arxiv.org/abs/2205.14039 · HTML: https://mlsanity.com/paper/2205.14039/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 81 references that are in the map.
- First posted on arXiv 27 May 2022.

## Builds on (most cited first, 16 of 16)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Recent Advances in Convolutional Neural Networks](https://mlsanity.com/paper/1512.07108/) (2015)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [A Survey on The Expressive Power of Graph Neural Networks](https://mlsanity.com/paper/2003.04078/) (2020)
- [A Short Tutorial on The Weisfeiler-Lehman Test And Its Variants](https://mlsanity.com/paper/2201.07083/) (2022)
- [Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems](https://mlsanity.com/paper/1505.05114/) (2015)
- [Geometric Scattering for Graph Data Analysis](https://mlsanity.com/paper/1810.03068/) (2018)
- [Low Dimensional Invariant Embeddings for Universal Geometric Learning](https://mlsanity.com/paper/2205.02956/) (2022)
- [On Lipschitz Bounds of General Convolutional Neural Networks](https://mlsanity.com/paper/1808.01415/) (2018)
- [Permutation Invariant Representations with Applications to Graph Deep Learning](https://mlsanity.com/paper/2203.07546/) (2022)
- [Using Filter Banks in Convolutional Neural Networks for Texture Classification](https://mlsanity.com/paper/1601.02919/) (2016)
- [Estimation under group actions: recovering orbits from invariants](https://mlsanity.com/paper/1712.10163/) (2017)
- [Invertibility and Robustness of Phaseless Reconstruction](https://mlsanity.com/paper/1308.4718/) (2013)
- [Dihedral multi-reference alignment](https://mlsanity.com/paper/2107.05262/) (2021)

## Built on by (most cited first, 18 of 18)

- [Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theorem](https://mlsanity.com/paper/2306.06529/) (2023)
- [Approximately Equivariant Graph Networks](https://mlsanity.com/paper/2308.10436/) (2023)
- [Machine learning and invariant theory](https://mlsanity.com/paper/2209.14991/) (2022)
- [Towards fully covariant machine learning](https://mlsanity.com/paper/2301.13724/) (2023)
- [Fourier Sliced-Wasserstein Embedding for Multisets and Measures](https://mlsanity.com/paper/2504.02544/) (2025)
- [On the H"older Stability of Multiset and Graph Neural Networks](https://mlsanity.com/paper/2406.06984/) (2024)
- [Stability of sorting based embeddings](https://mlsanity.com/paper/2410.05446/) (2024)
- [Quick ViTs: Speeding up Vision Transformers through Equivariance](https://mlsanity.com/paper/2505.15441/) (2025)
- [On the Shift Invariance of Max Pooling Feature Maps in Convolutional Neural Networks](https://mlsanity.com/paper/2209.11740/) (2022)
- [Quantitative Bounds for Sorting-Based Permutation-Invariant Embeddings](https://mlsanity.com/paper/2510.22186/) (2025)
- [On the (Non) Injectivity of Piecewise Linear Janossy Pooling](https://mlsanity.com/paper/2505.20150/) (2025)
- [Toward bilipshiz geometric models](https://mlsanity.com/paper/2511.11735/) (2025)
- [Quantitative Approximation Rates for Group Equivariant Learning](https://mlsanity.com/paper/2602.20370/) (2026)
- [Invariant Shape Analysis of Surfaces with Spherical Topology](https://mlsanity.com/paper/2609.39567/) (2026)
- [Fourier Sliced-Wasserstein Embedding for Multisets and Measures](https://mlsanity.com/paper/2405.16519/) (2024)
- [Classification of Firn Data via Topological Features](https://mlsanity.com/paper/2504.16150/) (2025)
- [If you can distinguish, you can express: Galois theory, Stone--Weierstrass, machine learning, and linguistics](https://mlsanity.com/paper/2510.09902/) (2025)
- [Transversal Pooling Neural Networks](https://mlsanity.com/paper/2609.36237/) (2026)

## Abstract (arXiv)

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=ℝd and G is finite, a suitable max filter bank separates orbits, and is even bilipschitz in the quotient metric. In the case where V=L2(ℝd) 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.