# Hybrid Lie semi-group and cascade structures for the generalized Gaussian derivative model for visual receptive fields

arXiv: https://arxiv.org/abs/2509.15748 · HTML: https://mlsanity.com/paper/2509.15748/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 101 references that are in the map.
- First posted on arXiv 19 Sep 2025.

## Builds on (most cited first, 29 of 29)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [DISCO: accurate Discrete Scale Convolutions](https://mlsanity.com/paper/2106.02733/) (2021)
- [Scale-covariant and scale-invariant Gaussian derivative networks](https://mlsanity.com/paper/2011.14759/) (2020)
- [Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scales](https://mlsanity.com/paper/2106.06418/) (2021)
- [Normative theory of visual receptive fields](https://mlsanity.com/paper/1701.06333/) (2017)
- [Scale Equivariant U-Net](https://mlsanity.com/paper/2210.04508/) (2022)
- [Scale-Equivariant UNet for Histopathology Image Segmentation](https://mlsanity.com/paper/2304.04595/) (2023)
- [Exploring the ability of CNNs to generalise to previously unseen scales over wide scale ranges](https://mlsanity.com/paper/2004.01536/) (2020)
- [Fully trainable Gaussian derivative convolutional layer](https://mlsanity.com/paper/2207.08424/) (2022)
- [Provably scale-covariant continuous hierarchical networks based on scale-normalized differential expressions coupled in cascade](https://mlsanity.com/paper/1905.13555/) (2019)
- [A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time](https://mlsanity.com/paper/2202.09209/) (2022)
- [Covariance properties under natural image transformations for the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2303.09803/) (2023)
- [Riesz networks: scale invariant neural networks in a single forward pass](https://mlsanity.com/paper/2305.04665/) (2023)
- [Time-causal and time-recursive spatio-temporal receptive fields](https://mlsanity.com/paper/1504.02648/) (2015)
- [Discrete approximations of Gaussian smoothing and Gaussian derivatives](https://mlsanity.com/paper/2311.11317/) (2023)
- [Scale-Equivariant Deep Learning for 3D Data](https://mlsanity.com/paper/2304.05864/) (2023)
- [Unified theory for joint covariance properties under geometric image transformations for spatio-temporal receptive fields according to the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2311.10543/) (2023)
- [A geometric model of multi-scale orientation preference maps via Gabor functions](https://mlsanity.com/paper/1712.04329/) (2017)
- [Riesz feature representation: scale equivariant scattering network for classification tasks](https://mlsanity.com/paper/2307.08467/) (2023)
- [A cortical-inspired geometry for contour perception and motion integration](https://mlsanity.com/paper/1301.3433/) (2013)
- [Scale generalisation properties of extended scale-covariant and scale-invariant Gaussian derivative networks on image datasets with spatial scaling variations](https://mlsanity.com/paper/2409.11140/) (2024)
- [A sub-Riemannian model of the visual cortex with frequency and phase](https://mlsanity.com/paper/1910.04992/) (2019)
- [Orientation selectivity properties for the affine Gaussian derivative and the affine Gabor models for visual receptive fields](https://mlsanity.com/paper/2304.11920/) (2023)
- [Relationships between the degrees of freedom in the affine Gaussian derivative model for visual receptive fields and 2-D affine image transformations, with application to covariance properties of simple cells in the primary visual cortex](https://mlsanity.com/paper/2411.05673/) (2024)
- [On sources to variabilities of simple cells in the primary visual cortex: A principled theory for the interaction between geometric image transformations and receptive field responses](https://mlsanity.com/paper/2509.02139/) (2025)

## Abstract (arXiv)

Because of the variabilities of real-world image structures under the natural image transformations that arise when observing similar objects or spatio-temporal events under different viewing conditions, the receptive field responses computed in the earliest layers of the visual hierarchy may be strongly influenced by such geometric image transformations. One way of handling this variability is by basing the vision system on covariant receptive field families, which expand the receptive field shapes over the degrees of freedom in the image transformations. This paper addresses the problem of deriving relationships between spatial and spatio-temporal receptive field responses obtained for different values of the shape parameters in the resulting multi-parameter families of receptive fields. For this purpose, we derive both (i) infinitesimal relationships, roughly corresponding to a combination of notions from semi-groups and Lie groups, as well as (ii) macroscopic cascade smoothing properties, which describe how receptive field responses at coarser spatial and temporal scales can be computed by applying smaller support incremental filters to the output from corresponding receptive fields at finer spatial and temporal scales, structurally related to the notion of Lie algebras, although with directional preferences. The presented results provide (i) a deeper understanding of the relationships between spatial and spatio-temporal receptive field responses for different values of the filter parameters, which can be used for both (ii) designing more efficient schemes for computing receptive field responses over populations of multi-parameter families of receptive fields, as well as (iii)~formulating idealized theoretical models of the computations of simple cells in biological vision.