 Hi, this is Christen from the University of Tübingen and I'm excited to share with you our recent work, patient quadrature on remaining data manifolds. To give you a short overview of the paper, we begin with the assumption that remaining manifolds are a suitable model when data has inherent nonlinear geometric structure. However, the associated operations are computationally rather expensive. Hence, we advocate the use of probabilistic numerical methods in remaining statistics. These methods take a probabilistic perspective on numerical tasks and thus enable reasoning about informative choices of the algorithm. This makes them suitable in context where the computational budget is limited and computations have to be allocated in a smart manner. In particular, our focus lies on adaptive patient quadrature to efficiently normalize distributions on remaining manifolds. This word is embedded in the broader context of making remaining methods in machine learning more practical to further their adoption. To motivate our approach, here's a dataset from molecular dynamics where protein, adenylate kinase undergoes a large-scale closed-to-open transition. We use the simple PCA to obtain a two-dimensional representation of the data, which makes it more suitable for analysis. However, it is not clear that the resulting coordinates carry any meaning. A simple normal distribution places the mean outside the data support and we also see that the eigendirections are not at all well aligned with the intrinsic trend of the data. This is due to the Mahalanovis distance, which is based on a Euclidean metric. In search for a better model, we replace the Euclidean assumption with a flexible remaining metric that we learn in a data-driven fashion. Such a metric is smoothly changing in a product and specifies how volume is distorted locally. Shortest paths, so-called geodesics, are consequently attracted to follow the data. In the figure, darker color means smaller volume and thus shorter distances. In this framework, a Gaussian on the manifold can be learned from the data. This is called a locally adaptive normal distribution. As you can see in the figure, it provides us with sensible mean and eigendirections. The normalization constant of this distribution is needed to keep the mass near the data support. It is, however, an intractable integral as evaluations require solving a system of ordinary differential equations, called the geodesic equations. Thus we must acquire information carefully, which renders a naive Monte Carlo approach prohibitive. Instead, we employ adaptive Bayesian quadrature to compute the integral. Specifically, we use WASABI, or sequential active Bayesian integration. This method returns a distribution over the integral and comes with an adaptive scheme to select informative nodes, so that we can reduce the number of expensive geodesic computations to a minimum. We customize it by including prime metric knowledge in the Gaussian process and also propose a novel acquisition function, which is tailored to the manifold setting. Our experiments show that this approach enables faster integration than Monte Carlo, without sacrificing on accuracy. In summary, this project is a further case study for the following proposition. Probabilistic numerical methods, when they are tailored to a problem at hand, make more efficient use of our limited computational resources by exploiting prior domain knowledge and combining that with intelligent acquisition strategies. Thanks for listening. If you got curious, please have a look at our paper and feel free to contact us.