Convolutional Filtering and Neural Networks with Non Commutative Algebras

Parada-Mayorga, Alejandro; Butler, Landon; Ribeiro, Alejandro

doi:10.48550/arxiv.2108.09923

Cited by 2 publications

(11 citation statements)

References 18 publications

(50 reference statements)

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“…Given that the realized filter is a non commutative ASP model on multigraphs, it follows that the stability results of non commutative algebras hold as well. In particular, by Corollary 1 of [30], the…”

Section: Stability Of the Relaxed Filtermentioning

confidence: 96%

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Algebraic Convolutional Filters on Lie Group Algebras

Kumar¹,

Parada-Mayorga²,

Ribeiro³

2022

Preprint

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Group convolutional neural networks are a useful tool for utilizing symmetries known to be in a signal; however, they require that the signal is defined on the group itself. Existing approaches either work directly with group signals, or they impose a lifting step with heuristics to compute the convolution which can be computationally costly. Taking an algebraic signal processing perspective, we propose a novel convolutional filter from the Lie group algebra directly, thereby removing the need to lift altogether. Furthermore, we establish stability of the filter by drawing connections to multigraph signal processing. The proposed filter is evaluated on a classification problem on two datasets with SO(3) group symmetries.

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Section: Stability Of the Relaxed Filtermentioning

confidence: 96%

“…4: return neighbors n 0,...,k−1 and weights λ λ λ stability (see [30,Definition 10]) takes the from of…”

Section: Algorithm 1 Offline Transformation Operator Generationmentioning

confidence: 99%

Algebraic Convolutional Filters on Lie Group Algebras

Kumar¹,

Parada-Mayorga²,

Ribeiro³

2022

Preprint

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“…As stated in [22] there exists a direct integral of Hilbert spaces A = A(λ)dm(λ) and a unitary operator F : L 2 (G) → A which 1 Notice that this is a standard approach for defining bandlimited signals on graphs [34], graphons [35]- [37] and other domains [21], [25], [38]. The Hilbert space H contains the signals transformed according to the symmetries in G, and its properties do not depend on the properties of…”

Section: A Bandlimited Filtersmentioning

confidence: 99%

“…We additionally consider the notion of filter stability -when the deformation of the operator is proportional to the size of the deformation. Stability has been explored through the lens of graph signal processing [1], [2] and recently generalized to ASMs [25]- [27]. In section V-B.…”

Section: Equivariance and Stabilitymentioning

confidence: 99%

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Algebraic Convolutional Filters on Lie Group Algebras

Kumar

Parada-Mayorga

Ribeiro

2023

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

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In this paper we propose a framework to leverage Lie group symmetries on arbitrary spaces exploiting algebraic signal processing (ASP). We show that traditional group convolutions are one particular instantiation of a more general Lie group algebra homomorphism associated to an algebraic signal model rooted in the Lie group algebra L 1 (G) for given Lie group G. Exploiting this fact, we decouple the discretization of the Lie group convolution elucidating two separate sampling instances: the filter and the signal. To discretize the filters, we exploit the exponential map that links a Lie group with its associated Lie algebra. We show that the discrete Lie group filter learned from the data determines a unique filter in L 1 (G), and we show how this uniqueness of representation is defined by the bandwidth of the filter given a spectral representation. We also derive error bounds for the approximations of the filters in L 1 (G) with respect to its learned discrete representations. The proposed framework allows the processing of signals on spaces of arbitrary dimension and where the actions of some elements of the group are not necessarily well defined. Finally, we show that multigraph convolutional signal models come as the natural discrete realization of Lie group signal processing models, and we use this connection to establish stability results for Lie group algebra filters. To evaluate numerically our results, we build neural networks with these filters and we apply them in multiple datasets, including a knot classification problem.

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Convolutional Filtering and Neural Networks with Non Commutative Algebras

Cited by 2 publications

References 18 publications

Algebraic Convolutional Filters on Lie Group Algebras

Algebraic Convolutional Filters on Lie Group Algebras

Algebraic Convolutional Filters on Lie Group Algebras

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