Mattia G Bergomi - Academia.edu (original) (raw)

Most Recent Papers by Mattia G Bergomi

Understanding of animal collectives is limited by the ability to track each individual. We descri... more Understanding of animal collectives is limited by the ability to track each individual. We describe an algorithm and software that extract all trajectories from video, with high identification accuracy for collectives of up to 100 individuals. idtracker.ai uses two convolutional networks: one that detects when animals touch or cross and another for animal identification. The tool is trained with a protocol that adapts to video conditions and tracking difficulty.

Can music be represented as a meaningful geometric and topological object? In this paper, we prop... more Can music be represented as a meaningful geometric and topological object? In this paper, we propose a strategy to describe some music features as a polyhedral surface obtained by a simplicial interpretation of the Tonnetz. The Tonnetz is a graph largely used in computational musicology to describe the harmonic relationships of notes in equal tuning. In particular, we use persistent homology in order to describe the persistent properties of music encoded in the aforementioned model. Both the relevance and the characteristics of this approach are discussed by analyzing some paradigmatic compositional styles. Eventually, the task of automatic music style classification is addressed by computing the hierarchical clustering of the topological fingerprints associated with some collections of compositions.

This article proposes some thoughts on formal and computational models in and for popular music b... more This article proposes some thoughts on formal and computational models in and for popular music by focusing on Beatles songs. After a brief presentation of some systematic approaches in the analysis of musical form and of some theoretical tools used in the geometric representation of musical structures and processes (the Tonnetz and other Neo-Riemannian constructions), the authors deal with the questions raised by the analysis of a collection of Beatles songs once they are studied either from a formal or a computational viewpoint. Even though the form and the structure of Beatles songs can be studied without using mathematical tools, the computer-aided modelling of the segmentation process of a musical piece, as well as the techniques belonging to the field of Music Information Retrieval, allow to give a quantitative, computational-oriented interpretation of Pop songs. At the same time, this approach opens the question of the singularity of this repertoire with respect to other popular music pieces.

Volume ! by Mattia G Bergomi

Nearly half a century after Luciano Berio praised the Beatles in his “Commenti al Rock” (1967), t... more Nearly half a century after Luciano Berio praised the Beatles in his “Commenti al Rock” (1967), this special issue of Volume! surveys the research carried out on the band that was, according to John Lennon, “more popular than Jesus”. In light of an impressive bibliography covering the first 50 years of what we now call “Beatles Studies”, one learns, for example, that the British Invasion originated in Paris, that Popular Music Studies began with the musicological study of popular music, that the theory of harmonic vectors can help analyze pop music or that Marshall McLuhan's concepts shed an interesting light on albums such as Abbey Road.

Papers by Mattia G Bergomi

Proceedings of the ... AAAI Conference on Artificial Intelligence, Jun 26, 2023

We provide a unifying framework where artificial neural networks and their architectures can be f... more We provide a unifying framework where artificial neural networks and their architectures can be formally described as particular cases of a general mathematical constructionmachines of finite depth. Unlike neural networks, machines have a precise definition, from which several properties follow naturally. Machines of finite depth are modular (they can be combined), efficiently computable, and differentiable. The backward pass of a machine is again a machine and can be computed without overhead using the same procedure as the forward pass. We prove this statement theoretically and practically, via a unified implementation that generalizes several classical architectures-dense, convolutional, and recurrent neural networks with a rich shortcut structure-and their respective backpropagation rules.

arXiv (Cornell University), Dec 31, 2018

The aim of this paper is to provide a general mathematical framework for group equivariance in th... more The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group equivariant non-expansive operators (GENEOs), which are maps between function spaces associated with groups of transformations. We study the topological and metric properties of the space of GENEOs to evaluate their approximating power and set the basis for general strategies to initialise and compose operators. We begin by defining suitable pseudo-metrics for the function spaces, the equivariance groups, and the set of non-expansive operators. Basing on these pseudo-metrics, we prove that the space of GENEOs is compact and convex, under the assumption that the function spaces are compact and convex. These results provide fundamental guarantees in a machine learning perspective. We show examples on the MNIST and fashion-MNIST datasets. By considering isometry-equivariant non-expansive operators, we describe a simple strategy to select and sample operators, and show how the selected and sampled operators can be used to perform both classical metric learning and an effective initialisation of the kernels of a convolutional neural network.

Machine learning and knowledge extraction, Mar 24, 2023

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Communications in Applied and Industrial Mathematics, 2020

Graphs are a basic tool for the representation of modern data. The richness of the topological in... more Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do that by extending previous work of other researchers in homological persistence, by proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.

2015 iv I read in a book that the objectivity of human thought can be expressed by using the verb... more 2015 iv I read in a book that the objectivity of human thought can be expressed by using the verb to think in impersonal form. Could we ever say "it plays" as we say "it rains", or "today it is windy"? [...] And may we also say "it listens" as we say "it rains"?-Freely translated and adapted by the author from Se una notte d'inverno un viaggiatore, Italo Calvino.

arXiv (Cornell University), Aug 1, 2022

Properties such as composability and automatic differentiation made artificial neural networks a ... more Properties such as composability and automatic differentiation made artificial neural networks a pervasive tool in applications. Tackling more challenging problems caused neural networks to progressively become more complex and thus difficult to define from a mathematical perspective. We present a general definition of linear layer arising from a categorical framework based on the notions of integration theory and parametric spans. is definition generalizes and encompasses classical layers (e.g., dense, convolutional), while guaranteeing existence and computability of the layer's derivatives for backpropagation.

Journal of Mathematics and Music, May 3, 2020

Meaningful low-dimensional representations of dynamical processes are essential to better underst... more Meaningful low-dimensional representations of dynamical processes are essential to better understand the mechanisms underlying complex systems, from music composition to learning in both biological and artificial intelligence. We suggest to describe time-varying systems by considering the evolution of their geometrical and topological properties in time, by using a method based on persistent homology. In the static case, persistent homology allows one to provide a representation of a manifold paired with a continuous function as a collection of multisets of points and lines called persistence diagrams. The idea is to fingerprint the change of a variable-geometry space as a time series of persistence diagrams, and afterwards compare such time series by using dynamic time warping. As an application, we express some music features and their time dependency by updating the values of a function defined on a polyhedral surface, called the Tonnetz. Thereafter, we use this time-based representation to automatically classify three collections of compositions according to their style, and discuss the optimal time-granularity for the analysis of different musical genres.

arXiv (Cornell University), Jul 6, 2020

Using tools from topology and functional analysis, we provide a framework where arti cial neural ... more Using tools from topology and functional analysis, we provide a framework where arti cial neural networks, and their architectures, can be formally described. We de ne the notion of machine in a general topological context and show how simple machines can be combined into more complex ones. We explore nite-and in nite-depth machines, which generalize neural networks and neural ordinary di erential equations. Borrowing ideas from functional analysis and kernel methods, we build complete, normed, in nite-dimensional spaces of machines, and we discuss how to nd optimal architectures and parameters-within those spacesto solve a given computational problem. In our numerical experiments, these kernel-inspired networks can outperform classical neural networks when the training dataset is small.

HAL (Le Centre pour la Communication Scientifique Directe), Sep 12, 2016

Journal of applied and computational topology, Sep 23, 2022

Topological data analysis can provide insight on the structure of weighted graphs and digraphs. H... more Topological data analysis can provide insight on the structure of weighted graphs and digraphs. However, some properties underlying a given (di)graph are hardly mappable to simplicial complexes. We introduce steady and ranging sets: two standardized ways of producing persistence diagrams directly from graph-theoretical features. The two constructions are framed in the context of indexing-aware persistence functions. Furthermore, we introduce a sufficient condition for stability. Finally, we apply the steady-and ranging-based persistence constructions to toy examples and real-world applications.

Mathematics, Nov 29, 2021

Persistent homology enables fast and computable comparison of topological objects. We give some i... more Persistent homology enables fast and computable comparison of topological objects. We give some instances of a recent extension of the theory of persistence, guaranteeing robustness and computability for relevant data types, like simple graphs and digraphs. We focus on categorical persistence functions that allow us to study in full generality strong kinds of connectedness-clique communities, k-vertex, and k-edge connectedness-directly on simple graphs and strong connectedness in digraphs.

Springer eBooks, 2014

ABSTRACT The reliability of commercial non-invasive BCI (Brain Computer Interface) devices and th... more ABSTRACT The reliability of commercial non-invasive BCI (Brain Computer Interface) devices and the lower cost of these EEG-based systems, determined the increasing interest in their application in different research fields, also thanks to the portability of the equipment. The latter feature makes BCI devices particularly suited for entertainment applications especially due to the possibility to detect the mental state of the users. The relationship between emotions and entertainment is obvious, as the influence of music in human emotional states. While BCI devices represent a challenge in gaming motion control, they have been successfully applied in music production [5] and composition [7]. In our previous work [6] we focused on conscious production of music notes with the aim of developing a prototype for applications in entertainment. In this work we trace the state-of-the art of our research and present our opinion on possible applications of the preliminary obtained results.

arXiv (Cornell University), Jul 30, 2017

Graphs are a basic tool for the representation of modern data. The richness of the topological in... more Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do that by extending previous work of other researchers in homological persistence, by proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.

arXiv (Cornell University), May 22, 2019

Persistence has proved to be a valuable tool to analyze real world data robustly. Several approac... more Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been a empted over time, some topological in avor, based on the vector spacevalued homology functor, other combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category so that functors to that category induce persistence functions. We port the interleaving and bo leneck distances to this novel framework and generalize classical equalities and inequalities. Unlike sets and vector spaces, in many categories the rank of an object does not identify it up to isomorphism: to preserve information about the structure of persistence modules, we de ne colorable ranks, persistence diagrams and prove the equality between multicolored bo leneck distance and interleaving distance in semisimple Abelian categories. To illustrate our framework in practice, we give examples of multicolored persistent homology on ltered topological spaces with a group action and labeled point cloud data.

Lecture Notes in Computer Science, 2016

Can music be represented as a meaningful geometric and topological object? In this paper, we prop... more Can music be represented as a meaningful geometric and topological object? In this paper, we propose a strategy to describe some music features as a polyhedral surface obtained by a simplicial interpretation of the Tonnetz. The Tonnetz is a graph largely used in computational musicology to describe the harmonic relationships of notes in equal tuning. In particular, we use persistent homology in order to describe the persistent properties of music encoded in the aforementioned model. Both the relevance and the characteristics of this approach are discussed by analyzing some paradigmatic compositional styles. Eventually, the task of automatic music style classification is addressed by computing the hierarchical clustering of the topological fingerprints associated with some collections of compositions.

2022 Conference on Cognitive Computational Neuroscience

Understanding of animal collectives is limited by the ability to track each individual. We descri... more Understanding of animal collectives is limited by the ability to track each individual. We describe an algorithm and software that extract all trajectories from video, with high identification accuracy for collectives of up to 100 individuals. idtracker.ai uses two convolutional networks: one that detects when animals touch or cross and another for animal identification. The tool is trained with a protocol that adapts to video conditions and tracking difficulty.

Can music be represented as a meaningful geometric and topological object? In this paper, we prop... more Can music be represented as a meaningful geometric and topological object? In this paper, we propose a strategy to describe some music features as a polyhedral surface obtained by a simplicial interpretation of the Tonnetz. The Tonnetz is a graph largely used in computational musicology to describe the harmonic relationships of notes in equal tuning. In particular, we use persistent homology in order to describe the persistent properties of music encoded in the aforementioned model. Both the relevance and the characteristics of this approach are discussed by analyzing some paradigmatic compositional styles. Eventually, the task of automatic music style classification is addressed by computing the hierarchical clustering of the topological fingerprints associated with some collections of compositions.

This article proposes some thoughts on formal and computational models in and for popular music b... more This article proposes some thoughts on formal and computational models in and for popular music by focusing on Beatles songs. After a brief presentation of some systematic approaches in the analysis of musical form and of some theoretical tools used in the geometric representation of musical structures and processes (the Tonnetz and other Neo-Riemannian constructions), the authors deal with the questions raised by the analysis of a collection of Beatles songs once they are studied either from a formal or a computational viewpoint. Even though the form and the structure of Beatles songs can be studied without using mathematical tools, the computer-aided modelling of the segmentation process of a musical piece, as well as the techniques belonging to the field of Music Information Retrieval, allow to give a quantitative, computational-oriented interpretation of Pop songs. At the same time, this approach opens the question of the singularity of this repertoire with respect to other popular music pieces.

Nearly half a century after Luciano Berio praised the Beatles in his “Commenti al Rock” (1967), t... more Nearly half a century after Luciano Berio praised the Beatles in his “Commenti al Rock” (1967), this special issue of Volume! surveys the research carried out on the band that was, according to John Lennon, “more popular than Jesus”. In light of an impressive bibliography covering the first 50 years of what we now call “Beatles Studies”, one learns, for example, that the British Invasion originated in Paris, that Popular Music Studies began with the musicological study of popular music, that the theory of harmonic vectors can help analyze pop music or that Marshall McLuhan's concepts shed an interesting light on albums such as Abbey Road.

Proceedings of the ... AAAI Conference on Artificial Intelligence, Jun 26, 2023

We provide a unifying framework where artificial neural networks and their architectures can be f... more We provide a unifying framework where artificial neural networks and their architectures can be formally described as particular cases of a general mathematical constructionmachines of finite depth. Unlike neural networks, machines have a precise definition, from which several properties follow naturally. Machines of finite depth are modular (they can be combined), efficiently computable, and differentiable. The backward pass of a machine is again a machine and can be computed without overhead using the same procedure as the forward pass. We prove this statement theoretically and practically, via a unified implementation that generalizes several classical architectures-dense, convolutional, and recurrent neural networks with a rich shortcut structure-and their respective backpropagation rules.

arXiv (Cornell University), Dec 31, 2018

The aim of this paper is to provide a general mathematical framework for group equivariance in th... more The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group equivariant non-expansive operators (GENEOs), which are maps between function spaces associated with groups of transformations. We study the topological and metric properties of the space of GENEOs to evaluate their approximating power and set the basis for general strategies to initialise and compose operators. We begin by defining suitable pseudo-metrics for the function spaces, the equivariance groups, and the set of non-expansive operators. Basing on these pseudo-metrics, we prove that the space of GENEOs is compact and convex, under the assumption that the function spaces are compact and convex. These results provide fundamental guarantees in a machine learning perspective. We show examples on the MNIST and fashion-MNIST datasets. By considering isometry-equivariant non-expansive operators, we describe a simple strategy to select and sample operators, and show how the selected and sampled operators can be used to perform both classical metric learning and an effective initialisation of the kernels of a convolutional neural network.

Machine learning and knowledge extraction, Mar 24, 2023

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Communications in Applied and Industrial Mathematics, 2020

Graphs are a basic tool for the representation of modern data. The richness of the topological in... more Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do that by extending previous work of other researchers in homological persistence, by proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.

2015 iv I read in a book that the objectivity of human thought can be expressed by using the verb... more 2015 iv I read in a book that the objectivity of human thought can be expressed by using the verb to think in impersonal form. Could we ever say "it plays" as we say "it rains", or "today it is windy"? [...] And may we also say "it listens" as we say "it rains"?-Freely translated and adapted by the author from Se una notte d'inverno un viaggiatore, Italo Calvino.

arXiv (Cornell University), Aug 1, 2022

Properties such as composability and automatic differentiation made artificial neural networks a ... more Properties such as composability and automatic differentiation made artificial neural networks a pervasive tool in applications. Tackling more challenging problems caused neural networks to progressively become more complex and thus difficult to define from a mathematical perspective. We present a general definition of linear layer arising from a categorical framework based on the notions of integration theory and parametric spans. is definition generalizes and encompasses classical layers (e.g., dense, convolutional), while guaranteeing existence and computability of the layer's derivatives for backpropagation.

Journal of Mathematics and Music, May 3, 2020

Meaningful low-dimensional representations of dynamical processes are essential to better underst... more Meaningful low-dimensional representations of dynamical processes are essential to better understand the mechanisms underlying complex systems, from music composition to learning in both biological and artificial intelligence. We suggest to describe time-varying systems by considering the evolution of their geometrical and topological properties in time, by using a method based on persistent homology. In the static case, persistent homology allows one to provide a representation of a manifold paired with a continuous function as a collection of multisets of points and lines called persistence diagrams. The idea is to fingerprint the change of a variable-geometry space as a time series of persistence diagrams, and afterwards compare such time series by using dynamic time warping. As an application, we express some music features and their time dependency by updating the values of a function defined on a polyhedral surface, called the Tonnetz. Thereafter, we use this time-based representation to automatically classify three collections of compositions according to their style, and discuss the optimal time-granularity for the analysis of different musical genres.

arXiv (Cornell University), Jul 6, 2020

Using tools from topology and functional analysis, we provide a framework where arti cial neural ... more Using tools from topology and functional analysis, we provide a framework where arti cial neural networks, and their architectures, can be formally described. We de ne the notion of machine in a general topological context and show how simple machines can be combined into more complex ones. We explore nite-and in nite-depth machines, which generalize neural networks and neural ordinary di erential equations. Borrowing ideas from functional analysis and kernel methods, we build complete, normed, in nite-dimensional spaces of machines, and we discuss how to nd optimal architectures and parameters-within those spacesto solve a given computational problem. In our numerical experiments, these kernel-inspired networks can outperform classical neural networks when the training dataset is small.

HAL (Le Centre pour la Communication Scientifique Directe), Sep 12, 2016

Journal of applied and computational topology, Sep 23, 2022

Topological data analysis can provide insight on the structure of weighted graphs and digraphs. H... more Topological data analysis can provide insight on the structure of weighted graphs and digraphs. However, some properties underlying a given (di)graph are hardly mappable to simplicial complexes. We introduce steady and ranging sets: two standardized ways of producing persistence diagrams directly from graph-theoretical features. The two constructions are framed in the context of indexing-aware persistence functions. Furthermore, we introduce a sufficient condition for stability. Finally, we apply the steady-and ranging-based persistence constructions to toy examples and real-world applications.

Mathematics, Nov 29, 2021

Persistent homology enables fast and computable comparison of topological objects. We give some i... more Persistent homology enables fast and computable comparison of topological objects. We give some instances of a recent extension of the theory of persistence, guaranteeing robustness and computability for relevant data types, like simple graphs and digraphs. We focus on categorical persistence functions that allow us to study in full generality strong kinds of connectedness-clique communities, k-vertex, and k-edge connectedness-directly on simple graphs and strong connectedness in digraphs.

Springer eBooks, 2014

ABSTRACT The reliability of commercial non-invasive BCI (Brain Computer Interface) devices and th... more ABSTRACT The reliability of commercial non-invasive BCI (Brain Computer Interface) devices and the lower cost of these EEG-based systems, determined the increasing interest in their application in different research fields, also thanks to the portability of the equipment. The latter feature makes BCI devices particularly suited for entertainment applications especially due to the possibility to detect the mental state of the users. The relationship between emotions and entertainment is obvious, as the influence of music in human emotional states. While BCI devices represent a challenge in gaming motion control, they have been successfully applied in music production [5] and composition [7]. In our previous work [6] we focused on conscious production of music notes with the aim of developing a prototype for applications in entertainment. In this work we trace the state-of-the art of our research and present our opinion on possible applications of the preliminary obtained results.

arXiv (Cornell University), Jul 30, 2017

Graphs are a basic tool for the representation of modern data. The richness of the topological in... more Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do that by extending previous work of other researchers in homological persistence, by proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.

arXiv (Cornell University), May 22, 2019

Persistence has proved to be a valuable tool to analyze real world data robustly. Several approac... more Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been a empted over time, some topological in avor, based on the vector spacevalued homology functor, other combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category so that functors to that category induce persistence functions. We port the interleaving and bo leneck distances to this novel framework and generalize classical equalities and inequalities. Unlike sets and vector spaces, in many categories the rank of an object does not identify it up to isomorphism: to preserve information about the structure of persistence modules, we de ne colorable ranks, persistence diagrams and prove the equality between multicolored bo leneck distance and interleaving distance in semisimple Abelian categories. To illustrate our framework in practice, we give examples of multicolored persistent homology on ltered topological spaces with a group action and labeled point cloud data.

Lecture Notes in Computer Science, 2016

Can music be represented as a meaningful geometric and topological object? In this paper, we prop... more Can music be represented as a meaningful geometric and topological object? In this paper, we propose a strategy to describe some music features as a polyhedral surface obtained by a simplicial interpretation of the Tonnetz. The Tonnetz is a graph largely used in computational musicology to describe the harmonic relationships of notes in equal tuning. In particular, we use persistent homology in order to describe the persistent properties of music encoded in the aforementioned model. Both the relevance and the characteristics of this approach are discussed by analyzing some paradigmatic compositional styles. Eventually, the task of automatic music style classification is addressed by computing the hierarchical clustering of the topological fingerprints associated with some collections of compositions.

2022 Conference on Cognitive Computational Neuroscience

Among the various generalizations of persistent topology, the one based on rank functions and lea... more Among the various generalizations of persistent topology, the one based on rank functions and leading to indexing-aware functions appears to be particularly suited to catch graph-theoretical properties without the need for a simplicial construction and a homology computation. This paper defines and studies "simple" and "single-vertex" features in directed and undirected graphs, by which several indexing-aware persistence functions are produced, within the scheme of steady and ranging sets. The implementation of the "sink" feature and its application to trust networks provide an example of the ease of use and meaningfulness of the method.

arXiv (Cornell University), Dec 28, 2022

Artificial neural networks can learn complex, salient data features to achieve a given task. On t... more Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines fully aware of data constraints and symmetries. We introduce a class of persistence-based neural network layers. Persistence-based layers allow the users to easily inject knowledge about symmetries (equivariance) respected by the data, are equipped with learnable weights, and can be composed with state-of-the-art neural architectures.

Machine Learning and Knowledge Extraction

Artificial neural networks can learn complex, salient data features to achieve a given task. On t... more Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines fully aware of data constraints and symmetries. We introduce an original class of neural network layers based on a generalization of topological persistence. The proposed persistence-based layers allow the users to encode specific data properties (e.g., equivariance) easily. Additionally, these layers can be trained through standard optimization procedures (backpropagation) and composed with classical layers. We test the performance of generalized persistence-based layers as pooling operators in convolutional neural networks for image classification on the MNIST, Fashion-MNIST and CIFAR-10 datasets.

Cornell University - arXiv, Aug 1, 2022

Properties such as composability and automatic differentiation made artificial neural networks a ... more Properties such as composability and automatic differentiation made artificial neural networks a pervasive tool in applications. Tackling more challenging problems caused neural networks to progressively become more complex and thus difficult to define from a mathematical perspective. We present a general definition of linear layer arising from a categorical framework based on the notions of integration theory and parametric spans. is definition generalizes and encompasses classical layers (e.g., dense, convolutional), while guaranteeing existence and computability of the layer's derivatives for backpropagation.