VC dimensions of group convolutional neural networks

Area of Science:

• Machine Learning
• Artificial Intelligence
• Deep Learning Theory

Background:

• Group convolutional neural networks (GCNNs) are designed for data with symmetries.
• Understanding the generalization capacity of GCNNs is crucial for their effective application.

Purpose of the Study:

• To precisely estimate the VC dimensions of simple sets of GCNNs.
• To analyze the generalization capacity of GCNNs, particularly in the context of infinite groups.

Main Methods:

• Theoretical analysis of VC dimensions for GCNNs.
• Identification of precise estimates for specific GCNN architectures.

Main Results:

• Precise VC dimension estimates were identified for simple sets of GCNNs.
• Two-parameter families of GCNNs with infinite groups and specific kernels exhibit infinite VC dimension.
• This holds true despite the networks being invariant to the group's action.

Conclusions:

• The generalization capacity of GCNNs can be surprisingly high, even for simple models.
• Invariance properties do not necessarily limit the VC dimension of GCNNs.
• These findings have implications for the design and understanding of symmetric deep learning models.