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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…

Geometric and Topological Deep Learning are rapidly growing research areas that enhance machine learning through the use of geometric and topological structures. Within this framework, Group Equivariant Non-Expansive Operators (GENEOs) have…

In this paper we establish a bridge between Topological Data Analysis and Geometric Deep Learning, adapting the topological theory of group equivariant non-expansive operators (GENEOs) to act on the space of all graphs weighted on vertices…

Group equivariant operators are playing a more and more relevant role in machine learning and topological data analysis. In this paper we present some new results concerning the construction of $G$-equivariant non-expansive operators…

The theory of Group Equivariant Non-Expansive Operators (GENEOs) was initially developed in Topological Data Analysis for the geometric approximation of data observers, including their invariances and symmetries. This paper departs from…

This paper introduces a rigorous mathematical framework for neural network explainability, and more broadly for the explainability of equivariant operators called Group Equivariant Operators (GEOs) based on Group Equivariant Non-Expansive…

Recent work in equivariant deep learning bears strong similarities to physics. Fields over a base space are fundamental entities in both subjects, as are equivariant maps between these fields. In deep learning, however, these maps are…

Recent research has shown that incorporating equivariance into neural network architectures is very helpful, and there have been some works investigating the equivariance of networks under group actions. However, as digital images and…

Several recent papers have proposed increasing the expressive power of graph neural networks by exploiting subgraphs or other topological structures. In parallel, researchers have investigated higher order permutation equivariant networks.…

Reliable training of generative adversarial networks (GANs) typically require massive datasets in order to model complicated distributions. However, in several applications, training samples obey invariances that are \textit{a priori}…

Learning solution operators of partial differential equations (PDEs) from data has emerged as a promising route to fast surrogate models in multi-query scientific workflows. However, for geometric PDEs whose inputs and outputs transform…

We perform topological data analysis on the internal states of convolutional deep neural networks to develop an understanding of the computations that they perform. We apply this understanding to modify the computations so as to (a) speed…

We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable…

In this work we investigate how to achieve equivariance to input transformations in deep networks, purely from data, without being given a model of those transformations. Convolutional Neural Networks (CNNs), for example, are equivariant to…

In this work we seek to bridge the concepts of topographic organization and equivariance in neural networks. To accomplish this, we introduce the Topographic VAE: a novel method for efficiently training deep generative models with…

The translation equivariance of convolutions can make convolutional neural networks translation equivariant or invariant. Equivariance to other transformations (e.g. rotations, affine transformations, scalings) may also be desirable as soon…

Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and…

This paper introduces a meta-learning approach for parameterized pseudo-differential operators with deep neural networks. With the help of the nonstandard wavelet form, the pseudo-differential operators can be approximated in a compressed…

This paper introduces a new Convolutional Neural Network (ConvNet) architecture inspired by a class of partial differential equations (PDEs) called quasi-linear hyperbolic systems. With comparable performance on the image classification…

Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and…