PDE-CNNs: Axiomatic Derivations and Applications
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arXiv
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| Format: | Preprint |
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2024
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| author | Bellaard, Gijs Sakata, Sei Smets, Bart M. N. Duits, Remco |
| author_facet | Bellaard, Gijs Sakata, Sei Smets, Bart M. N. Duits, Remco |
| contents | PDE-based Group Convolutional Neural Networks (PDE-G-CNNs) use solvers of evolution PDEs as substitutes for the conventional components in G-CNNs. PDE-G-CNNs can offer several benefits simultaneously: fewer parameters, inherent equivariance, better accuracy, and data efficiency.
In this article we focus on Euclidean equivariant PDE-G-CNNs where the feature maps are two-dimensional throughout. We call this variant of the framework a PDE-CNN.
From a machine learning perspective, we list several practically desirable axioms and derive from these which PDEs should be used in a PDE-CNN, this being our main contribution. Our approach to geometric learning via PDEs is inspired by the axioms of scale-space theory, which we generalize by introducing semifield-valued signals.
Our theory reveals new PDEs that can be used in PDE-CNNs and we experimentally examine what impact these have on the accuracy of PDE-CNNs. We also confirm for small networks that PDE-CNNs offer fewer parameters, increased accuracy, and better data efficiency when compared to CNNs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15182 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | PDE-CNNs: Axiomatic Derivations and Applications Bellaard, Gijs Sakata, Sei Smets, Bart M. N. Duits, Remco Machine Learning Computer Vision and Pattern Recognition PDE-based Group Convolutional Neural Networks (PDE-G-CNNs) use solvers of evolution PDEs as substitutes for the conventional components in G-CNNs. PDE-G-CNNs can offer several benefits simultaneously: fewer parameters, inherent equivariance, better accuracy, and data efficiency. In this article we focus on Euclidean equivariant PDE-G-CNNs where the feature maps are two-dimensional throughout. We call this variant of the framework a PDE-CNN. From a machine learning perspective, we list several practically desirable axioms and derive from these which PDEs should be used in a PDE-CNN, this being our main contribution. Our approach to geometric learning via PDEs is inspired by the axioms of scale-space theory, which we generalize by introducing semifield-valued signals. Our theory reveals new PDEs that can be used in PDE-CNNs and we experimentally examine what impact these have on the accuracy of PDE-CNNs. We also confirm for small networks that PDE-CNNs offer fewer parameters, increased accuracy, and better data efficiency when compared to CNNs. |
| title | PDE-CNNs: Axiomatic Derivations and Applications |
| topic | Machine Learning Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2403.15182 |