Abstract
Group Equivariant Operators (GEOs) are a fundamental tool in the research on neural networks, since they make available a new kind of geometric knowledge engineering for deep learning, which can exploit symmetries in artificial intelligence and reduce the number of parameters required in the learning process. In this paper we introduce a new method to build non-linear GEOs and non-linear Group Equivariant Non-Expansive Operators (GENEOs), based on the concepts of symmetric function and permutant. This method is particularly interesting because of the good theoretical properties of GENEOs and the ease of use of permutants to build equivariant operators, compared to the direct use of the equivariance groups we are interested in. In our paper, we prove that the technique we propose works for any symmetric function, and benefits from the approximability of continuous symmetric functions by symmetric polynomials. A possible use in Topological Data Analysis of the GENEOs obtained by this new method is illustrated.
Highlights
In this paper we introduce a new method to build non-linear Group Equivariant Operators (GEOs) and non-linear Group Equivariant Non-Expansive Operators (GENEOs), based on the concepts of symmetric function and permutant
This paper opens the way to the research about the natural conjecture that each non-linear Gequivariant non-expansive operator can be produced by applying our new technique to suitable symmetric functions and permutants, provided that the group G transitively acts on a finite signal domain
We have introduced a new method to build GENEOs, grounded on the concepts of symmetric function and permutant
Summary
The theory of equivariant operators has become a topic of great interest to the scientific community, since these operators allow to make explicit the use of symmetries in deep learning and artificial intelligence (Mallat, 2012, 2016; Bengio et al, 2013; Zhang et al, 2015; Anselmi et al, 2016, 2019; Cohen and Welling, 2016; Worrall et al, 2017), thereby reducing the number of parameters required in the learning process. This paper opens the way to the research about the natural conjecture that each non-linear Gequivariant non-expansive operator can be produced (or at least well approximated) by applying our new technique to suitable symmetric functions and permutants, provided that the group G transitively acts on a finite signal domain. This probably nontrivial problem will be attacked in following papers, grounding on the results obtained in this article. The other sections present our new results about the construction of non-linear GENEOs via symmetric functions and permutants
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