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The aim of this paper is to provide a general mathematical framework forgroup equivariance in the machine learning context. The framework builds on asynergy between persistent homology and the theory of group actions. We definegroup-equivariant non-expansive operators (GENEOs), which are maps betweenfunction spaces associated with groups of transformations. We study thetopological and metric properties of the space of GENEOs to evaluate theirapproximating power and set the basis for general strategies to initialise andcompose operators. We begin by defining suitable pseudo-metrics for thefunction spaces, the equivariance groups, and the set of non-expansiveoperators. Basing on these pseudo-metrics, we prove that the space of GENEOs iscompact and convex, under the assumption that the function spaces are compactand convex. These results provide fundamental guarantees in a machine learningperspective. We show examples on the MNIST and fashion-MNIST datasets. Byconsidering isometry-equivariant non-expansive operators, we describe a simplestrategy to select and sample operators, and show how the selected and sampledoperators can be used to perform both classical metric learning and aneffective initialisation of the kernels of a convolutional neural network.

Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning