Finslerian aspects of random geometry

The performance of machine learning algorithms relies on their ability to learn meaningful features. To achieve this, a common hypothesis is to consider that real world data lie on a non-linear manifold embedded in a low dimensional space. Because of this lack of linearity, operations are not trivial, and they have to be defined by taking into account the curvature of the underlying structure. Imagine for example that we want to compute the shortest path between two data points lying on a sphere. We cannot just take the straightest line between those two points since the path would not follow the manifold. Instead, we have to take into account the curvature of the sphere, given here by its radius, in our calculations. For more complex curvy structures, this can be done by borrowing tools developed in Riemannian geometry, and in particular by using a Riemannian metric: a well-defined linear and symmetric product that locally captures the curvature.

The real world is also generally noisy, since data points can be missing or wrongly labelled, and so we need to use stochastic methods to model this noise and quantify data uncertainty. This quantification is actually essential to better apprehend the geometry of a data manifold. However, by studying the manifold from a statistical point of view, some properties don’t longer hold. This is the case for instance when we want to compute the shortest path between two points by minimizing the path energy; we can easily do it in a deterministic setting, but we can’t if we consider average values.

In the previous example, this issue arose because of the non-linearity of the length with respect to the metric. A clever hack would be to dismiss the linearity of the Riemannian metric and design a specific one for the desired task, while keeping the classical properties of Riemannian geometry. It turns out that Finslerian metrics are perfectly fitted for this: they can be neither linear nor symmetric, but they still locally capture the curvature of the manifold, which can help us to perform statistical operations. The aim of the project is to further develop the theory of stochastic geometry through a Finslerian lens.