U.S. Food and Drug Administration (FDA) approved Type-1 Diabetes Mellitus Simulator (T1DMS)

This work proposes a geometric insight into equivariant message passing on Riemannian manifolds. As previously proposed, numerical features on Riemannian manifolds are represented as coordinate-independent feature fields on the manifold. To any coordinate-independent feature field on a manifold comes attached an equivariant embedding of the principal bundle to the space of numerical features. We argue that the metric this embedding induces on the numerical feature space should optimally preserve the principal bundle's original metric. This optimality criterion leads to the minimization of a twisted form of the Polyakov action with respect to the graph of this embedding, yielding an equivariant diffusion process on the associated vector bundle. We obtain a message passing scheme on the manifold by discretizing the diffusion equation flow for a fixed time step. We propose a higher-order equivariant diffusion process equivalent to diffusion on the cartesian product of the base manifold. The discretization of the higher-order diffusion process on a graph yields a new general class of equivariant GNN, generalizing the ACE and MACE formalism to data on Riemannian manifolds.

The Support Vector Machine (SVM) is a powerful and widely used classification algorithm. Its performance is well known to be impacted by a tuning parameter which is frequently selected by cross-validation. This paper uses the Karush-Kuhn-Tucker conditions to provide rigorous mathematical proof for new insights into the behavior of SVM in the large and small tuning parameter regimes. These insights provide perhaps unexpected relationships between SVM and naive Bayes and maximal data piling directions. We explore how characteristics of the training data affect the behavior of SVM in many cases including: balanced vs. unbalanced classes, low vs. high dimension, separable vs. non-separable data. These results present a simple explanation of SVM's behavior as a function of the tuning parameter. We also elaborate on the geometry of complete data piling directions in high dimensional space. The results proved in this paper suggest important implications for tuning SVM with cross-validation.