Tangent Bundle Convolutional Learning: Manifolds to Cellular Sheaves

핵심 개념

Introducing a novel convolution operation over tangent bundles of Riemann manifolds, leading to Tangent Bundle Neural Networks (TNNs) operating on vector fields over manifolds.

초록

The article introduces a new convolution operation over the tangent bundle of Riemann manifolds using the Connection Laplacian operator. It defines tangent bundle filters and TNNs based on this operation, providing a spectral representation that generalizes existing filters. The discretization procedure makes TNNs implementable, converging to the continuous architecture. The effectiveness of the proposed architecture is numerically evaluated on various learning tasks, showing superiority against benchmark architectures.

통계

"We define tangent bundle filters and tangent bundle neural networks (TNNs) based on this convolution operation."
"Finally, we numerically evaluate the effectiveness of the proposed architecture on various learning tasks."

인용구

"We formally prove that this discretized architecture converges to the underlying continuous TNN."
"The proposed procedures implementable, we formally describe and leverage the link between tangent bundles and orthogonal cellular sheaves."

핵심 통찰 요약

Tangent Bundle Convolutional Learning

by Claudio Batt... 게시일 arxiv.org 03-19-2024

https://arxiv.org/pdf/2303.11323.pdf

더 깊은 질문

How can lowpass filter constraints be practically enforced during training

To enforce lowpass filter constraints during training, one approach is to incorporate regularization techniques that penalize high-frequency components in the learned filters. This can be achieved by adding a penalty term to the loss function that encourages the filter coefficients to have a smoother frequency response. For example, L2 regularization on the filter weights or constraining the spectral norm of the filters can help promote lowpass characteristics. Additionally, using specific activation functions with inherent smoothing properties, such as sigmoid or tanh functions, can also aid in enforcing lowpass behavior during training.

What are potential implications if filters do not satisfy lowpass conditions in practice

If filters do not satisfy lowpass conditions in practice, it could lead to several implications. Firstly, without proper filtering constraints, there may be an increased risk of overfitting to high-frequency noise present in the data. This could result in suboptimal generalization performance and reduced model robustness. Moreover, non-lowpass filters might introduce unwanted artifacts or distortions into the processed signals due to amplification of high frequencies. In applications where preserving signal integrity and minimizing noise are crucial (such as image processing or audio analysis), failure to enforce lowpass conditions could significantly impact the accuracy and reliability of the results.

How does the convergence of discrete architectures to continuous ones impact real-world applications

The convergence of discrete architectures to continuous ones has significant implications for real-world applications. One key advantage is that it provides a theoretical foundation for justifying the use of discretized models in practical settings where continuous operations are not feasible computationally. By demonstrating convergence guarantees between discrete and continuous architectures, practitioners can have confidence that their implemented algorithms will approximate ideal continuous solutions as sample sizes increase or time steps decrease.
This convergence also enables seamless integration between theoretical frameworks based on continuous mathematics and practical implementations based on discrete computations. It ensures consistency across different levels of abstraction and allows for more accurate modeling and simulation of complex systems governed by differential equations or manifold structures.
In real-world applications such as signal processing tasks involving geometric data analysis or neural network operations on irregular domains like graphs or manifolds, this convergence assures researchers and engineers that their algorithmic designs will uphold fundamental principles even when discretized for computational efficiency purposes.