Geometric deep learning and equivariant neural networks

Gerken, Jan E.; Aronsson, Jimmy; Carlsson, Oscar; Linander, Hampus; Ohlsson, Fredrik; Petersson, Christoffer; Persson, Daniel

doi:10.1007/s10462-023-10502-7

Cited by 18 publications

(8 citation statements)

References 70 publications

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“…Our results indicate that meta-representations of neural networks capture the modality specific characteristics in the weight matrix learned by the first-order networks. One possible hypothesis is that weight matrices may have learned to gain the invariances/equivariances specific to each modality (Bronstein et al, 2017;Gerken et al, 2023). It is known that vision and audition exhibit different types of invariances.…”

Section: Why Can We Predict the Modality Neural Network Were Trained On?mentioning

confidence: 99%

Meta-Representations as Representations of Processes

Kanai,

Takatsuki,

Fujisawa

2024

Preprint

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In this study, we explore how the notion of meta-representations in Higher-Order Theories (HOT) of consciousness can be implemented in computational models. HOT suggests that consciousness emerges from meta-representations, which are representations of first-order sensory representations. However, translating this abstract concept into a concrete computational model, such as those used in artificial intelligence, presents a theoretical challenge. For example, a simplistic interpretation of meta-representation as a representation of representation makes the notion rather trivial and ubiquitous. Here, we propose a refined interpretation of meta-representations. Contrary to the simplistic view of meta-representations as mere transformations of the first-order representational states or confidence estimates, we argue that meta-representations are representations of the processes that generate first-order representations. This presents a process-oriented view whereby meta-representations capture the qualitative aspect of how sensory information is transformed into first-order representations. To concretely illustrate and operationalize thus formulated notion of meta-representation, we constructed "meta-networks" designed to explicitly model meta-representations within deep learning architectures. Specifically, we constructed meta-networks by implementing autoencoders of first-order neural networks. In this architecture, the latent spaces embedding those first-order networks correspond to the meta-representations of first-order networks. By applying meta-networks to embed neural networks trained to encode visual and auditory datasets, we show that the meta-representations of first-order networks successfully capture the qualitative aspects of those networks by separating the visual and auditory networks in the meta-representation space. We argue that such meta-representations would be useful for quantitatively compare and contrast the qualitative differences of computational processes. While whether such meta-representational systems exist in the human brain remains an open question, this formulation of meta-representation offers a new empirically testable hypothesis that there are brain regions that represent the processes of transforming a representation in one brain region to a representation in another brain region. Furthermore, this form of meta-representations might underlie our ability to describe the qualitative aspect of sensory experience or qualia.

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Section: Why Can We Predict the Modality Neural Network Were Trained On?mentioning

confidence: 99%

Meta-Representations as Representations of Processes

Kanai,

Takatsuki,

Fujisawa

2024

Preprint

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“…Lemma 2.4 to follow is a special instance of geometric Frobenius reciprocity. It can be found in the excellent but challenging book by Cap and Slovák, [29, Theorem 1.4.4], 3 where they use it to understand G-invariant geometric structures on G/H .…”

Section: Lemma 23 Let V Be a Representation Of G And W A Representati...mentioning

confidence: 99%

Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras

Knibbeler

2024

Adv Comput Math

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We develop an elementary method to compute spaces of equivariant maps from a homogeneous space G/H of a Lie group G to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras.

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“…The importance of equivariance with respect to a group is becoming clear and widespread in many machine learning applications used for drug design, traffic forecasting, object recognition, and detection (see, e.g., Bronstein et al, 2021;Gerken et al, 2023). In some situations, however, requiring equivariance with respect to a whole group could even become an obstacle in the correct learning process of an equivariant neural network.…”

Section: P-geneos In Applicationsmentioning

confidence: 99%

A topological model for partial equivariance in deep learning and data analysis

Ferrari,

Frosini,

Quercioli

et al. 2023

Front. Artif. Intell.

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In this article, we propose a topological model to encode partial equivariance in neural networks. To this end, we introduce a class of operators, called P-GENEOs, that change data expressed by measurements, respecting the action of certain sets of transformations, in a non-expansive way. If the set of transformations acting is a group, we obtain the so-called GENEOs. We then study the spaces of measurements, whose domains are subjected to the action of certain self-maps and the space of P-GENEOs between these spaces. We define pseudo-metrics on them and show some properties of the resulting spaces. In particular, we show how such spaces have convenient approximation and convexity properties.

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Geometric deep learning and equivariant neural networks

Cited by 18 publications

References 70 publications

Meta-Representations as Representations of Processes

Meta-Representations as Representations of Processes

Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras

A topological model for partial equivariance in deep learning and data analysis

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