Hi, this is Chrisson from the University of Tübingen and I'm excited to share with you our recent work, patient quadrature, and remain in 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 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 firt is embedded in the broader context of making remaining methods and 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 close to open transition. We use the simple PCA to obtain the 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 Mahala novice distance, which is based on a Euclidean metric. In search for a better model, you replace the Euclidean assumption of 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 a sense of a mean and eigendirections. The normalization constant of this distribution is needed to keep the mass near the data support. It is, however, an interactable integral, as if evaluations require solving a system of ordinary differential equations, called the due-design equations. Thus, we must acquire information carefully, which renders an naive Monte Carlo approach prohibitive. Instead, we employ adaptive fashion quadrature to compute the integral, specifically, views Vasabi, what sequential active fashion integration. This method returns the 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 primary 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 further case study for the following proposition. Probabilistic numerical methods, when they are tailored to a problem maintained, 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.