My inquiry pertains to the exploration and understanding of academic literature, a meta-question by nature. Specifically, I have come across a multitude of machine learning research papers that leverage topological properties and curvature for data analysis, incorporating these into an end-to-end neural network architecture.
To elucidate, in the realm of Topological Data Analysis (TDA), given a set of data points or a point cloud, one can approximate it by a simplex and subsequently compute its persistent homologies. The persistent homologies serve as unique signatures or distinctive features of the given point cloud. These features can then be incorporated into a neural network to enhance the precision of data classification tasks.
Prominent examples of this concept can be found in literature such as "Topological Graph Neural Networks" and "Understanding Oversquashing and Bottleneck on Graphs via Curvature" (Fey, 2020; Bronstein et al., 2021).
Hitherto, we have discussed machine learning methods that leverage the geometric attributes of point clouds. However, let us consider a scenario where we are dealing with a collection of point clouds and a set of graphs, each of genus 'g', with 'v' vertices, and 'n' marked boundaries. For this scenario, we assume that the degree of the unmarked boundaries is fixed by an integer 'd'. We denote this set as $M_{g,n}^{v}(d)$. This set is finite and exhibits a recursion similar to the Tutte recursion.
The central question that arises is: can this information be beneficial within the framework of machine learning?
References:
Fey, M. (2020). Topological Graph Neural Networks. Bronstein, M. et al. (2021). Understanding Oversquashing and Bottleneck on Graphs via Curvature.