Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back

• Published in: IEEE transactions on signal processing Vol. 72; pp. 1 - 17
• Main Authors: Battiloro, Claudio, Wang, Zhiyang, Riess, Hans, Lorenzo, Paolo Di, Ribeiro, Alejandro
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Cellular Sheaves; Cognitive tasks; Computer architecture; Convolution; Discretization; Fields (mathematics); Graph Signal Processing; Laplace equations; Learning; Manifolds; Maximum likelihood detection; Neural networks; Operators (mathematics); Riemann manifold; Sheaf Neural Networks; Sheaves; Tangent Bundle Neural Networks; Tangent Bundle Signal Processing; Vectors
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2024.3379862

In this work we introduce a convolution operation over the tangent bundle of Riemann manifolds in terms of exponentials of the Connection Laplacian operator. We define tangent bundle filters and tangent bundle neural networks (TNNs) based on this convolution operation, which are novel continuous architectures operating on tangent bundle signals, i.e. vector fields over the manifolds. Tangent bundle filters admit a spectral representation that generalizes the ones of scalar manifold filters, graph filters and standard convolutional filters in continuous time. We then introduce a discretization procedure, both in the space and time domains, to make TNNs implementable, showing that their discrete counterpart is a novel principled variant of the very recently introduced sheaf neural networks. We formally prove that this discretized architecture converges to the underlying continuous TNN. Finally, we numerically evaluate the effectiveness of the proposed architecture on various learning tasks, both on synthetic and real data, comparing it against other state-of-the-art and benchmark architectures.