# Modelling and analysis of the 8 filters from the "master key filters hypothesis" for depthwise-separable deep networks in relation to idealized receptive fields based on scale-space theory

arXiv: https://arxiv.org/abs/2509.12746 · HTML: https://mlsanity.com/paper/2509.12746/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 38 references that are in the map.
- First posted on arXiv 16 Sep 2025.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications](https://mlsanity.com/paper/1704.04861/) (2017)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [ConvNeXt V2: Co-designing and Scaling ConvNets with Masked Autoencoders](https://mlsanity.com/paper/2301.00808/) (2023)
- [Structured Receptive Fields in CNNs](https://mlsanity.com/paper/1605.02971/) (2016)
- [PDE-based Group Equivariant Convolutional Neural Networks](https://mlsanity.com/paper/2001.09046/) (2020)
- [Resolution learning in deep convolutional networks using scale-space theory](https://mlsanity.com/paper/2106.03412/) (2021)
- [Scale-covariant and scale-invariant Gaussian derivative networks](https://mlsanity.com/paper/2011.14759/) (2020)
- [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)
- [Fully trainable Gaussian derivative convolutional layer](https://mlsanity.com/paper/2207.08424/) (2022)
- [A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time](https://mlsanity.com/paper/2202.09209/) (2022)
- [Discrete approximations of Gaussian smoothing and Gaussian derivatives](https://mlsanity.com/paper/2311.11317/) (2023)
- [Frequency learning for structured CNN filters with Gaussian fractional derivatives](https://mlsanity.com/paper/2111.06660/) (2021)
- [Unveiling the Unseen: Identifiable Clusters in Trained Depthwise Convolutional Kernels](https://mlsanity.com/paper/2401.14469/) (2024)
- [Approximation properties relative to continuous scale space for hybrid discretizations of Gaussian derivative operators](https://mlsanity.com/paper/2405.05095/) (2024)
- [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)
- [The Master Key Filters Hypothesis: Deep Filters Are General](https://mlsanity.com/paper/2412.16751/) (2024)
- [The Quest for Universal Master Key Filters in DS-CNNs](https://mlsanity.com/paper/2509.11711/) (2025)

## Abstract (arXiv)

This paper presents the results of analysing and modelling a set of 8 ``master key filters'', which have been extracted by applying a clustering approach to the receptive fields learned in depthwise-separable deep networks based on the ConvNeXt architecture. For this purpose, we first compute spatial spread measures in terms of weighted mean values and weighted variances of the absolute values of the learned filters, which support the working hypotheses that: (i) the learned filters can be modelled by separable filtering operations over the spatial domain, and that (ii) the spatial offsets of the those learned filters that are non-centered are rather close to half a grid unit. Then, we model the clustered ``master key filters'' in terms of difference operators applied to a spatial smoothing operation in terms of the discrete analogue of the Gaussian kernel, and demonstrate that the resulting idealized models of the receptive fields show good qualitative similarity to the learned filters. This modelling is performed in two different ways: (i) using possibly different values of the scale parameters in the coordinate directions for each filter, and (ii) using the same value of the scale parameter in both coordinate directions. Then, we perform the actual model fitting by either (i) requiring spatial spread measures in terms of spatial variances of the absolute values of the receptive fields to be equal, or (ii) minimizing the discrete l1- or l2-norms between the idealized receptive field models and the learned filters. Complementary experimental results then demonstrate the idealized models of receptive fields have good predictive properties for replacing the learned filters by idealized filters in depthwise-separable deep networks, thus showing that the learned filters in depthwise-separable deep networks can be well approximated by discrete scale-space filters.