The Representation Theory of Neural Networks

Armenta, Marco; Jodoin, Pierre-Marc

doi:10.3390/math9243216

Cited by 12 publications

(13 citation statements)

References 42 publications

(103 reference statements)

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“…Echoing what was said in the Introduction, the graph network (GN) development of §3 leading to defining the category

\mathbf {CCCD}

, affords further applications to ANNs and VAEs (cf. [ 104 ] in terms of quiver representations). Such GNs are embraced by the general theory of cell complexes [ 105 ] to which these apply (see e.g.…”

Section: Conclusion and Outlooksmentioning

confidence: 99%

Sequential Measurements, Topological Quantum Field Theories, and Topological Quantum Neural Networks

Fields

Glazebrook

Marcianò

2022

Fortschritte der Physik

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We introduce novel methods for implementing generic quantum information within a scale-free architecture. For a given observable system, we show how observational outcomes are taken to be finite bit strings induced by measurement operators derived from a holographic screen bounding the system. In this framework, measurements of identified systems with respect to defined reference frames are represented by semantically-regulated information flows through distributed systems of finite sets of binary-valued Barwise-Seligman classifiers. Specifically, we construct a functor from the category of cone-cocone diagrams (CCCDs) over finite sets of classifiers, to the category of finite cobordisms of Hilbert spaces. We show that finite CCCDs provide a generic representation of finite quantum reference frames (QRFs). Hence the constructed functor shows how sequential finite measurements can induce TQFTs. The only requirement is that each measurement in a sequence, by itself, satisfies Bayesian coherence, hence that the probabilities it assigns satisfy the Kolmogorov axioms. We extend the analysis too develop topological quantum neural networks (TQNNs), which enable machine learning with functorial evolution of quantum neural 2-complexes (TQN2Cs) governed by TQFTs amplitudes, and resort to the Atiyah-Singer theorems in order to classify topological data processed by TQN2Cs. We then comment about the quiver representation of CCCDs and generalized spin-networks, a basis of the Hilbert spaces of both TQNNs and TQFTs. We finally review potential implementations of this framework in solid state physics and suggest applications to quantum simulation and biological information processing.

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“…Echoing what was said in the Introduction, the graph network (GN) development of §3 leading to defining the category

\mathbf {CCCD}

Section: Conclusion and Outlooksmentioning

confidence: 99%

Sequential Measurements, Topological Quantum Field Theories, and Topological Quantum Neural Networks

Fields

Glazebrook

Marcianò

2022

Fortschritte der Physik

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“…8. Quiver representations appear in the context of neural network architectures in [2,30], though these are not usual quiver representations due to the presence of nonlinear activation functions.…”

Section: By a Corresponding Zigzag Of The Formmentioning

confidence: 99%

Principal Components Along Quiver Representations

2022

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Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections and are eigenvectors of an associated matrix pencil.

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“…Echoing what was said in the Introduction, the graph network (GN) development of §3 leading to defining the category CCCD, affords further applications to ANNs and VAEs (cf. [104] in terms of quiver representations). Such GNs are embraced by the general theory of cell complexes [105] to which these apply (see e.g.…”

Section: Conclusion and Outlooksmentioning

confidence: 99%

Sequential measurements, TQFTs, and TQNNs

Fields¹,

Glazebrook²,

Marcianò³

2022

Preprint

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We introduce novel methods for implementing generic quantum information within a scalefree architecture. For a given observable system, we show how observational outcomes are taken to be finite bit strings induced by measurement operators derived from a holographic screen bounding the system. In this framework, measurements of identified systems with respect to defined reference frames are represented by semantically-regulated information flows through distributed systems of finite sets of binary-valued Barwise-Seligman classifiers. Specifically, we construct a functor from the category of cone-cocone diagrams (CCCDs) over finite sets of classifiers, to the category of finite cobordisms of Hilbert spaces. We show that finite CCCDs provide a generic representation of finite quantum reference frames (QRFs). Hence the constructed functor shows how sequential finite measurements can induce TQFTs. The only requirement is that each measurement in a sequence, by itself, satisfies Bayesian coherence, hence that the probabilities it assigns satisfy the Kolmogorov axioms. We extend the analysis so develop topological quantum neural networks (TQNNs), which enable machine learning with functorial evolution of quantum neural 2-complexes (TQN2Cs) governed by TQFTs amplitudes, and resort to the Atiyah-Singer theorems in order to classify topological data processed by TQN2Cs. We then comment about the quiver representation of CCCDs and generalized spin-networks, a basis of the Hilbert spaces of both TQNNs and TQFTs. We finally review potential implementations of this framework in solid state physics and suggest applications to quantum simulation and biological information processing.7.2.4 Quiver representations, gauge-networks and noncommutative geometry 39

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The Representation Theory of Neural Networks

Cited by 12 publications

References 42 publications

Sequential Measurements, Topological Quantum Field Theories, and Topological Quantum Neural Networks

Sequential Measurements, Topological Quantum Field Theories, and Topological Quantum Neural Networks

Principal Components Along Quiver Representations

Sequential measurements, TQFTs, and TQNNs

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