Convolutional Neural Networks on Manifolds: From Graphs and Back

Wang, Zhiyang; Ruiz, Luana; Ribeiro, Alejandro

doi:10.1109/ieeeconf56349.2022.10051964

Cited by 9 publications

(15 citation statements)

References 22 publications

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“…The Laplacian Eigenmap framework is based on the approximation of manifolds by weighted undirected graphs constructed with k-nearest neighbors or proximity radius heuristics, with the key assumption being that a set of sampled points of the manifold is available [21]- [23]. Formal connections between GNNs and Manifold Neural Networks (MNNs) are established in [24], [25]. Most of the previous works focused on scalar signals, e.g.…”

Section: A Related Workmentioning

confidence: 99%

“…Our definition is derived from the vector diffusion equation over manifolds; this choice is crucial to make the convolution operation consistent. Convolution on the tangent bundle reduces to manifold convolution [24] in the scalar bundle case (R-valued signals), and standard convolution if the manifold is the real line. Leveraging this operation, we introduce Tangent Bundle Convolutional Filters to process tangent bundle signals (vector fields).…”

Section: B Contributionsmentioning

confidence: 99%

“…1: An example of tangent vector via a recurrent version of our architecture. We empirically demonstrate the advantage of incorporating the tangent bundle structure into our model by comparing TNNs against Manifold Neural Networks from [24], architectures taking into account the manifold structure, but not the tangent spaces.…”

Section: B Contributionsmentioning

confidence: 99%

“…Linear filtering operations are historically synonymous (under appropriate assumptions) with convolution. Time signals are filtered by computing the continuous time convolution of the input signal and the filter impulse response [17]; images are filtered by computing multidimensional convolutions [34]; graph signals are filtered by computing graph convolutions [5]; scalar manifold signals are filtered by computing manifold convolutions [24]. In this paper, we define a tangent bundle filter as the convolution of the filter impulse response h and the tangent bundle signal F. To do so, we exploit the Connection Laplacian Operator.…”

Section: Tangent Bundle Convolutional Filtersmentioning

confidence: 99%

“…Remark 1 . The exponential term e t∆ can be seen as a diffusion or shift operator similiar to a time delay in a linear time-invariant (LTI) filter [36], or to a graph shift operator in a linear shift-invariant (LSI) graph filter [37], or to a manifold shift operator based on the Laplace-Beltrami operator [24]. The reason for this resemblance is that tangent bundle filters are linear combinations of the elements of the tangent bundle diffusion sequence, such as graph filters are linear combinations of the elements of the graph diffusion sequence and manifold filters are linear combinations of the elements of the manifold diffusion sequence.…”

Section: B Tangent Bundle Filtersmentioning

confidence: 99%

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

Battiloro¹,

Wang²,

Riess³

et al. 2023

Preprint

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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.

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Section: A Related Workmentioning

confidence: 99%

Section: B Contributionsmentioning

confidence: 99%

Section: B Contributionsmentioning

confidence: 99%

Section: Tangent Bundle Convolutional Filtersmentioning

confidence: 99%

Section: B Tangent Bundle Filtersmentioning

confidence: 99%

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

Battiloro¹,

Wang²,

Riess³

et al. 2023

Preprint

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Tangent Bundle Filters and Neural Networks: From Manifolds to Cellular Sheaves and Back

Battiloro

Wang

Riess

et al. 2023

ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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In this work we introduce a convolution operation over the tangent bundle of Riemannian manifolds exploiting the Connection Laplacian operator. We use this convolution operation to define tangent bundle filters and tangent bundle neural networks (TNNs), novel continuous architectures operating on tangent bundle signals, i.e. vector fields over manifolds. We discretize TNNs both in space and time domains, showing that their discrete counterpart is a principled variant of the recently introduced Sheaf Neural Networks. We formally prove that this discrete architecture converges to the underlying continuous TNN. We numerically evaluate the effectiveness of the proposed architecture on a denoising task of a tangent vector field over the unit 2-sphere.

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