On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operators
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866910283812831232 |
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| author | Cascarano, Pasquale Frosini, Patrizio Quercioli, Nicola Saki, Amir |
| author_facet | Cascarano, Pasquale Frosini, Patrizio Quercioli, Nicola Saki, Amir |
| contents | Group equivariant non-expansive operators have been recently proposed as basic components in topological data analysis and deep learning. In this paper we study some geometric properties of the spaces of group equivariant operators and show how a space $\mathcal{F}$ of group equivariant non-expansive operators can be endowed with the structure of a Riemannian manifold, so making available the use of gradient descent methods for the minimization of cost functions on $\mathcal{F}$. As an application of this approach, we also describe a procedure to select a finite set of representative group equivariant non-expansive operators in the considered manifold. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2103_02543 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operators Cascarano, Pasquale Frosini, Patrizio Quercioli, Nicola Saki, Amir Differential Geometry Machine Learning Primary: 55N31, Secondary: 58D20, 58D30, 62R40, 65D18, 68T09 Group equivariant non-expansive operators have been recently proposed as basic components in topological data analysis and deep learning. In this paper we study some geometric properties of the spaces of group equivariant operators and show how a space $\mathcal{F}$ of group equivariant non-expansive operators can be endowed with the structure of a Riemannian manifold, so making available the use of gradient descent methods for the minimization of cost functions on $\mathcal{F}$. As an application of this approach, we also describe a procedure to select a finite set of representative group equivariant non-expansive operators in the considered manifold. |
| title | On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operators |
| topic | Differential Geometry Machine Learning Primary: 55N31, Secondary: 58D20, 58D30, 62R40, 65D18, 68T09 |
| url | https://arxiv.org/abs/2103.02543 |