arXiv - CS - Neural and Evolutionary Computing Pub Date : 2020-04-06 , DOI: arxiv-2004.02881
Luciano Melodia; Richard Lenz

Neural nets have been used in an elusive number of scientific disciplines. Nevertheless, their parameterization is largely unexplored. Dense nets are the coordinate transformations of a manifold from which the data is sampled. After processing through a layer, the representation of the original manifold may change. This is crucial for the preservation of its topological structure and should therefore be parameterized correctly. We discuss a method to determine the smallest topology preserving layer considering the data domain as abelian connected Lie group and observe that it is decomposable into $\mathbb{R}^p \times \mathbb {T}^q$. Persistent homology allows us to count its $k$-th homology groups. Using K\"unneth's theorem, we count the $k$-th Betti numbers. Since we know the embedding dimension of $\mathbb{R}^p$ and $\mathcal{S}^1$, we parameterize the bottleneck layer with the smallest possible matrix group, which can represent a manifold with those homology groups. Resnets guarantee smaller embeddings due to the dimension of their state space representation.

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