# Lie Group Algebra Convolutional Filters

arXiv: https://arxiv.org/abs/2305.04431 · HTML: https://mlsanity.com/paper/2305.04431/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2023.

## Built on by (most cited first, 7 of 7)

- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [LieRE: Lie Rotational Positional Encodings](https://mlsanity.com/paper/2406.10322/) (2024)
- [Convolutional Filtering with RKHS Algebras](https://mlsanity.com/paper/2411.01341/) (2024)
- [Geometric sparsification in recurrent neural networks](https://mlsanity.com/paper/2406.06290/) (2024)
- [RKHS Representation of Algebraic Convolutional Filters with Integral Operators](https://mlsanity.com/paper/2602.19094/) (2026)
- [Non Commutative Convolutional Signal Models in Neural Networks: Stability to Small Deformations](https://mlsanity.com/paper/2310.03879/) (2023)
- [Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensions](https://mlsanity.com/paper/2609.36106/) (2026)

## Abstract (arXiv)

In this paper we propose a framework to leverage Lie group symmetries on arbitrary spaces exploiting textitalgebraic signal processing (ASP). We show that traditional group convolutions are one particular instantiation of a more general Lie group algebra homomorphism associated to an algebraic signal model rooted in the Lie group algebra L1(G) for given Lie group G. Exploiting this fact, we decouple the discretization of the Lie group convolution elucidating two separate sampling instances: the filter and the signal. To discretize the filters, we exploit the exponential map that links a Lie group with its associated Lie algebra. We show that the discrete Lie group filter learned from the data determines a unique filter in L1(G), and we show how this uniqueness of representation is defined by the bandwidth of the filter given a spectral representation. We also derive error bounds for the approximations of the filters in L1(G) with respect to its learned discrete representations. The proposed framework allows the processing of signals on spaces of arbitrary dimension and where the actions of some elements of the group are not necessarily well defined. Finally, we show that multigraph convolutional signal models come as the natural discrete realization of Lie group signal processing models, and we use this connection to establish stability results for Lie group algebra filters. To evaluate numerically our results, we build neural networks with these filters and we apply them in multiple datasets, including a knot classification problem.