Describing the shape and chemistry: persistent homology As an alternative to the aforementioned pore geometry descriptors, the use of topological data analysis has been proposed to capture the shape of materials. Topological data analysis can be used to obtain features that are invariant to a continuous transformation of the material structure. Persistent homology, a branch of topological data analysis, captures all the topological information underlying a given point cloud, such as the geometric coordinates of a material. Given a set of points (i.e., atom positions), we can obtain a so-called filtration (e.g., Vietoris-Rips) by continuously increasing the radius of these points to get a family of nested unions of spheres. Persistent homology then tracks the appearance and disappearance of topological features (such as channels and voids) in this filtration. The radius at which a feature appears is the "birth," while the radius at which a feature disappears is the "death." The persistence of a topological feature is the difference between birth and death, which acts as a measure of how prominent a given topological feature is. The set of all birth-death points is called a persistence diagram. As reticular materials have many channels and voids, persistent homology provides a holistic approach to capturing these topological features. Persistence diagrams are a multiset of birth-death points in the extended plane, and each material can have a different number of points in its persistence diagram. Since most machine learning models operate on points in a fixed-dimensional Euclidean space, one needs to vectorize these diagrams into fixed-size arrays. One can accomplish this, for instance, by computing persistence images, which can be thought of as smoothed versions of persistence diagrams and have been used before in materials science. A challenge with this representation, however, is that it is often very high-dimensional. This can, due to the curse of dimensionality, lead to learning problems (in particular in low-data regimes). As an alternative, we implemented a vectorization method that approximates persistence diagrams using Gaussian mixture models. This allows for low-dimensional representations that can still provide approximate Wasserstein distances (that are conventionally used to measure distances, i.e., a proxy for the difference between persistence diagrams). Additionally, we also implement a simple vectorization as a 2D histogram.