Simplicial Complex Networks

Mohammad Firouzi, Sadra Boreiri, Hamed Firouzi

Keywords: topological data analysis, supervised learning, simplicial approximation

TL;DR: A novel method for supervised learning through subdivisioning the input space along with function approximation.

Abstract: Universal approximation property of neural networks is one of the motivations to use these models in various real-world problems. However, this property is not the only characteristic that makes neural networks unique as there is a wide range of other approaches with similar property. Another characteristic which makes these models interesting is that they can be trained with the backpropagation algorithm which allows an efficient gradient computation and gives these universal approximators the ability to efficiently learn complex manifolds from a large amount of data in different domains. Despite their abundant use in practice, neural networks are still not well understood and a broad range of ongoing research is to study the interpretability of neural networks. On the other hand, topological data analysis (TDA) relies on strong theoretical framework of (algebraic) topology along with other mathematical tools for analyzing possibly complex datasets. In this work, we leverage a universal approximation theorem originating from algebraic topology to build a connection between TDA and common neural network training framework. We introduce the notion of automatic subdivisioning and devise a particular type of neural networks for regression tasks: Simplicial Complex Networks (SCNs). SCN's architecture is defined with a set of bias functions along with a particular policy during the forward pass which alternates the common architecture search framework in neural networks. We believe the view of SCNs can be used as a step towards building interpretable deep learning models. Finally, we verify its performance on a set of regression problems.