Curvature and Concentration of Hamiltonian Monte Carlo in High Dimensions

In this article, we analyze Hamiltonian Monte Carlo (HMC) by placing it in the setting of Riemannian geometry using the Jacobi metric, so that each step corresponds to a geodesic on a suitable Riemannian manifold. We then combine the notion of curvature of a Markov chain due to Joulin and Ollivier with the classical sectional curvature from Riemannian geometry to derive error bounds for HMC in important cases, where we have positive curvature. These cases include several classical distributions such as multivariate Gaussians, and also distributions arising in the study of Bayesian image registration. The theoretical development suggests the sectional curvature as a new diagnostic tool for convergence for certain Markov chains.Comment: Comments welcom

Conformal Regression in Calorie Prediction for Team Jumbo-Visma

UCI WorldTour races, the premier men's elite road cycling tour, are grueling events that put riders' physical fitness and endurance to the test. The coaches of Team Jumbo-Visma have long been responsible for predicting the energy needs of each rider of the Dutch team for every race on the calendar. Those must be estimated to ensure riders have the energy and resources necessary to maintain a high level of performance throughout a race. This task, however, is both time-consuming and challenging, as it requires precise estimates of race speed and power output. Traditionally, the approach to predicting energy needs has relied on coaches' judgement and experience, but this method has its limitations and often leads to inaccurate predictions. In this paper, we propose a new, more effective approach to predicting energy needs for cycling races. By predicting the speed and power with regression models, we provide the coaches with calorie needs estimate for each individual rider per stage instantly. In addition, we compare methods to quantify uncertainty in estimating the speed and power of Team Jumbo-Visma riders for cycling races. The empirical analysis of the jackknife+, jackknife-minmax, jackknife-minmax-after-bootstrap, CV+, CV-minmax, conformalized quantile regression (CQR) and inductive conformal prediction (ICP) methods in conformal prediction reveals all methods except minmax based methods achieve valid prediction intervals while producing prediction intervals tight enough to be used for decision making. Furthermore, methods computing prediction intervals of fixed size produce significantly tighter intervals for low significance value. Among the methods computing intervals of varying length across the input space, namely the CQR and ICP methods, ICP computes tighter prediction intervals at larger significance level.Comment: 12 pages, 5 figure

Mesh-based vs. Image-based Statistical Appearance Model of the Human Femur: a Preliminary Comparison Study for the Creation of Finite Element Meshes

Statistical models have been recently introduced in computational orthopaedics to investigate the bone mechanical properties across several populations. A fundamental aspect for the construction of statistical models concerns the establishment of accurate anatomical correspondences among the objects of the training dataset. Various methods have been proposed to solve this problem such as mesh morphing or image registration algorithms. The objective of this study is to compare a mesh-based and an image-based statistical appearance model approaches for the creation of nite element(FE) meshes. A computer tomography (CT) dataset of 157 human left femurs was used for the comparison. For each approach, 30 finite element meshes were generated with the models. The quality of the obtained FE meshes was evaluated in terms of volume, size and shape of the elements. Results showed that the quality of the meshes obtained with the image-based approach was higher than the quality of the mesh-based approach. Future studies are required to evaluate the impact of this finding on the final mechanical simulations

Simple Sorting Criteria Help Find the Causal Order in Additive Noise Models

Additive Noise Models (ANM) encode a popular functional assumption that enables learning causal structure from observational data. Due to a lack of real-world data meeting the assumptions, synthetic ANM data are often used to evaluate causal discovery algorithms. Reisach et al. (2021) show that, for common simulation parameters, a variable ordering by increasing variance is closely aligned with a causal order and introduce var-sortability to quantify the alignment. Here, we show that not only variance, but also the fraction of a variable's variance explained by all others, as captured by the coefficient of determination $R^2$, tends to increase along the causal order. Simple baseline algorithms can use $R^2$-sortability to match the performance of established methods. Since $R^2$-sortability is invariant under data rescaling, these algorithms perform equally well on standardized or rescaled data, addressing a key limitation of algorithms exploiting var-sortability. We characterize and empirically assess $R^2$-sortability for different simulation parameters. We show that all simulation parameters can affect $R^2$-sortability and must be chosen deliberately to control the difficulty of the causal discovery task and the real-world plausibility of the simulated data. We provide an implementation of the sortability measures and sortability-based algorithms in our library CausalDisco (https://github.com/CausalDisco/CausalDisco).Comment: See https://github.com/CausalDisco/CausalDisco for implementation

Capturing the Multiscale Anatomical Shape Variability with Polyaffine Transformation Trees

International audienceMandible fractures are classified depending on their location. In clinical practice, locations are grouped into regions at different scales according to anatomical, functional and esthetic considerations. Implant design aims at defining the optimal implant for each patient. Emerging population-based techniques analyze the anatomical variability across a population and perform statistical analysis to identify an optimal set of implants. Current efforts are focused on finding clusters of patients with similar characteristics and designing one implant for each cluster. Ideally, the description of anatomical variability is directly connected to the clinical regions. This connection is what we present here, by introducing a new registration method that builds upon a tree of locally affine transformations that describes variability at different scales. We assess the accuracy of our method on 146 CT images of femurs. Two medical experts provide the ground truth by manually measuring six landmarks. We illustrate the clinical importance of our method by clustering 43 CT images of mandibles for implant design. The presented method does not require any application-specific input, which makes it attractive for the analysis of other multiscale anatomical structures. At the core of our new method lays the introduction of a new basis for stationary velocity fields. This basis has very close links to anatomical substructures. In the future, this method has the potential to discover the hidden and possibly sparse structure of the anatomy

Discussion of "Geodesic Monte Carlo on Embedded Manifolds"

Contributed discussion and rejoinder to "Geodesic Monte Carlo on Embedded Manifolds" (arXiv:1301.6064)Comment: Discussion of arXiv:1301.6064. To appear in the Scandinavian Journal of Statistics. 18 page

Lattice Regularization of the Chiral Schwinger Model

We analyze the chiral Schwinger model on an infinite lattice using the continuum definition of the fermion determinant and a linear interpolation of the lattice gauge fields. For non-compact and Wilson formulation of the gauge field action it is proven that the effective lattice model is Osterwalder-Schrader positive, which is a sufficient condition for the reconstruction of a physical Hilbert space from the model defined on a Euclidean lattice. For the non-compact model we furthermore establish the existence of critical points where the corresponding continuum theory can be reconstructed. We show that the continuum limit for the two-point functions of field strength and chiral densities can be controlled analytically. The article ends with some remarks on fermionic observables.Comment: 18 page

Wound healing in rabbit corneas after flapless refractive lenticule extraction with a 345 nm ultraviolet femtosecond laser

Purpose To characterize corneal wound healing in a rabbit model after flapless refractive lenticule extraction with a 345 nm ultraviolet femtosecond laser. Setting Departments of Ophthalmology and Anatomy II, University of Erlangen-Nürnberg and Wavelight GmbH, Erlangen, Germany. Design Methods Flapless refractive lenticule extraction was performed in 1 eye each of 20 New Zealand white rabbits (−5.0 diopters). Groups of 4 animals were euthanized after 48 hours, 1 week, 2 weeks, 4 weeks, and 3 months, respectively. Corneal samples were prepared for histology and fluorescence microscopy. To assess corneal cell death, proliferation, and myofibroblastic transdifferentiation, terminal uridine deoxynucleotidyl nick end-labeling (TUNEL) assay as well as immunostaining for Ki67 and α-smooth muscle actin (αSMA) were performed on sagittal cryosections. Results Histology revealed a zone of keratocyte depletion with a thickness of approximately 50 μm around the extraction site. At 48 hours, pronounced TUNEL staining of keratocytes was detected around the interface (159.9 cells/mm ± 18.4 [SD]), which steadily decreased to 74.9 ± 19.8 cells/mm at 1 week and 5.7 ± 4.8 cells/mm at 2 weeks. Ki67 staining of keratocytes was evident at 48 hours (10.0 ± 3.8 cells/mm), which then decreased at 1 week (5.2 ± 1.7 cells/mm) and 2 weeks (0.4 ± 0.5 cells/mm). From 4 weeks onward, no TUNEL or Ki67 staining was detected. The corneal stroma was αSMA-negative at all timepoints. Conclusion Application of the 345 nm laser showed no signs of problematic repair processes in the cornea, which supports the initiation of the clinical phase