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In many applications of shape analysis, lengths between some landmarks are constrained. For instance, biomolecules often have some bond lengths and some bond angles constrained, and variation occurs only along unconstrained bonds and constrained bonds' torsions where the latter are conveniently modelled by dihedral angles. Our work has been motivated by low resolution biomolecular chain RNA where only some prominent atomic bonds can be well identified. Here, we propose a new modelling strategy for such constrained shape analysis starting with a product of polar coordinates (polypolars), where, due to constraints, for example, some radial coordinates should be omitted, leaving products of spheres (polyspheres). We give insight into these coordinates for particular cases such as five landmarks which are motivated by a practical RNA application. We also discuss distributions for polypolar coordinates and give a specific methodology with illustration when the constrained size-and-shape variables are concentrated. There are applications of this in clustering and we give some insight into a modified version of the MINT-AGE algorithm., Comment: 33 pages, 9 figures

Samples of phylogenetic trees arise in a variety of evolutionary and biomedical applications, and the Fr\'echet mean in Billera-Holmes-Vogtmann tree space is a summary tree shown to have advantages over other mean or consensus trees. However, use of the Fr\'echet mean raises computational and statistical issues which we explore in this paper. The Fr\'echet sample mean is known often to contain fewer internal edges than the trees in the sample, and in this circumstance calculating the mean by iterative schemes can be problematic due to slow convergence. We present new methods for identifying edges which must lie in the Fr\'echet sample mean and apply these to a data set of gene trees relating organisms from the apicomplexa which cause a variety of parasitic infections. When a sample of trees contains a significant level of heterogeneity in the branching patterns, or topologies, displayed by the trees then the Fr\'echet mean is often a star tree, lacking any internal edges. Not only in this situation, the population Fr\'echet mean is affected by a non-Euclidean phenomenon called stickness which impacts upon asymptotics, and we examine two data sets for which the mean tree is a star tree. The first consists of trees representing the physical shape of artery structures in a sample of medical images of human brains in which the branching patterns are very diverse. The second consists of gene trees from a population of baboons in which there is evidence of substantial hybridization. We develop hypothesis tests which work in the presence of stickiness. The first is a test for the presence of a given edge in the Fr\'echet population mean; the second is a two-sample test for differences in two distributions which share the same sticky population mean., Comment: 37 pages, 16 figures

Advanced experiments using THz pump and X-ray probe pulses at modern free-electron lasers (FELs) like the European X-ray FEL require a frequency-tunable, high-power, narrow-band THz source maintaining the repetition rate and pulse structure of the X-ray pulses. This paper reports the first results from a THz source, that is based on a single-pass high-gain THz FEL operating with a central wavelength of 100 micrometers. The THz FEL prototype is currently in operation at the Photo Injector Test facility at DESY in Zeuthen (PITZ) and uses the same type of electron source as the European XFEL photo injector. A self-amplified spontaneous emission (SASE) FEL was envisioned as the main mechanism for generating the THz pulses. Although the THz FEL at PITZ is supposed to use the same mechanism as at X-ray facilities, it cannot be considered as a simple scaling of the radiation wavelength because there is a large difference in the number of electrons per radiation wavelength, which is five orders of magnitude higher for the THz case. The bunching factor arising from the electron beam current profile contributes strongly to the initial spontaneous emission starting the FEL process. Proof-of-principle experiments were done at PITZ using an LCLS-I undulator to generate the first high-power, high-repetition-rate single-pass THz FEL radiation. Electron bunches with a beam energy of ~17 MeV and a bunch charge of up to several nC are used to generate THz pulses with a pulse energy of several tens of microjoules. For example, for an electron beam with a charge of ~2.4 nC, more than 100 microjoules were generated at a central wavelength of 100 micrometers. The narrowband spectrum was also demonstrated by spectral measurements. These proof-of-principle experiments pave the way for a tunable, high-repetition-rate THz source providing pulses with energies in the millijoule range., Comment: (unpublished) for submission to PRAB

Fr\'echet means, conceptually appealing, generalize the Euclidean expectation to general metric spaces. We explore how well Fr\'echet means can be estimated from independent and identically distributed samples and uncover a fundamental limitation: In the vicinity of a probability distribution $P$ with nonunique means, independent of sample size, it is not possible to uniformly estimate Fr\'echet means below a precision determined by the diameter of the set of Fr\'echet means of $P$. Implications were previously identified for empirical plug-in estimators as part of the phenomenon \emph{finite sample smeariness}. Our findings thus confirm inevitable statistical challenges in the estimation of Fr\'echet means on metric spaces for which there exist distributions with nonunique means. Illustrating the relevance of our lower bound, examples of extrinsic, intrinsic, Procrustes, diffusion and Wasserstein means showcase either deteriorating constants or slow convergence rates of empirical Fr\'echet means for samples near the regime of nonunique means., Comment: 24 pages, 1 figure

The Fr\'echet mean, a generalization to a metric space of the expectation of a random variable in a vector space, can exhibit unexpected behavior for a wide class of random variables. For instance, it can stick to a point (more generally to a closed set) under resampling: sample stickiness. It can stick to a point for topologically nearby distributions: topological stickiness, such as total variation or Wasserstein stickiness. It can stick to a point for slight but arbitrary perturbations: perturbation stickiness. Here, we explore these and various other flavors of stickiness and their relationship in varying scenarios, for instance on CAT($\kappa$) spaces, $\kappa\in \mathbb{R}$. Interestingly, modulation stickiness (faster asymptotic rate than $\sqrt{n}$) and directional stickiness (a generalization of moment stickiness from the literature) allow for the development of new statistical methods building on an asymptotic fluctuation, where, due to stickiness, the mean itself features no asymptotic fluctuation. Also, we rule out sticky flavors on manifolds in scenarios with curvature bounds.

It is well known, that Fr\'echet means on non-Euclidean spaces may exhibit nonstandard asymptotic rates depending on curvature. Even for distributions featuring standard asymptotic rates, there are non-Euclidean effects, altering finite sampling rates up to considerable sample sizes. These effects can be measured by the variance modulation function proposed by Pennec (2019). Among others, in view of statistical inference, it is important to bound this function on intervals of sampling sizes. In a first step into this direction, for the special case of a K-spider we give such an interval, based only on folded moments and total probabilities of spider legs and illustrate the method by simulations., Comment: 9 pages, 3 figures

It has been observed that the sample mean of certain probability distributions in Billera-Holmes-Vogtmann (BHV) phylogenetic spaces is confined to a lower-dimensional subspace for large enough sample size. This non-standard behavior has been called stickiness and poses difficulties in statistical applications when comparing samples of sticky distributions. We extend previous results on stickiness to show the equivalence of this sampling behavior to topological conditions in the special case of BHV spaces. Furthermore, we propose to alleviate statistical comparision of sticky distributions by including the directional derivatives of the Fr\'echet function: the degree of stickiness., Comment: 8 Pages, 1 Figure, conference submission to GSI 2023

Evolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order to properly quantify this uncertainty during the statistical analysis of collections of possible evolutionary trees inferred from biological data. Recently, the wald space has been introduced: a length space for trees which is a certain subset of the manifold of symmetric positive definite matrices. In this work, the wald space is introduced formally and its topology and structure is studied in detail. In particular, we show that wald space has the topology of a disjoint union of open cubes, it is contractible, and by careful characterization of cube boundaries, we demonstrate that wald space is a Whitney stratified space of type (A). Imposing the metric induced by the affine invariant metric on symmetric positive definite matrices, we prove that wald space is a geodesic Riemann stratified space. A new numerical method is proposed and investigated for construction of geodesics, computation of Fr\'echet means and calculation of curvature in wald space. This work is intended to serve as a mathematical foundation for further geometric and statistical research on this space., Comment: 42 pages, 15 figures

A three-cell 1.3 GHz, Normal Conducting (NC) buncher is designed for a possible future upgrade of the European XFEL to operate in a continuous-wave (CW) / long pulse (LP) mode. The RF geometry of the buncher is optimized for high shunt impedance, large mode separation as well as multipacting free in the operation range. The bunchersupport cavity voltage of 400 kV with an RF power dissipation of 14 kW. A tapered waveguide-based RF power coupler is designed to feed the RF power to the buncher. The RF power coupler port is optimized for field asymmetry compensation. The thermal load due to RF power dissipation is analyzed using Multiphysics simulations in CST and a simplified cooling scheme is designed.

A promising approach to accurate positioning of robots is ground texture based localization. It is based on the observation that visual features of ground images enable fingerprint-like place recognition. We tackle the issue of efficient parametrization of such methods, deriving a prediction model for localization performance, which requires only a small collection of sample images of an application area. In a first step, we examine whether the model can predict the effects of changing one of the most important parameters of feature-based localization methods: the number of extracted features. We examine two localization methods, and in both cases our evaluation shows that the predictions are sufficiently accurate. Since this model can be used to find suitable values for any parameter, we then present a holistic parameter optimization framework, which finds suitable texture-specific parameter configurations, using only the model to evaluate the considered parameter configurations.

We introduce a location statistic for distributions on non-linear geometric spaces, the diffusion mean, serving as an extension and an alternative to the Fr\'echet mean. The diffusion mean arises as the generalization of Gaussian maximum likelihood analysis to non-linear spaces by maximizing the likelihood of a Brownian motion. The diffusion mean depends on a time parameter $t$, which admits the interpretation of the allowed variance of the diffusion. The diffusion $t$-mean of a distribution $X$ is the most likely origin of a Brownian motion at time $t$, given the end-point distribution $X$. We give a detailed description of the asymptotic behavior of the diffusion estimator and provide sufficient conditions for the diffusion estimator to be strongly consistent. Particularly, we present a smeary central limit theorem for diffusion means and we show that joint estimation of the mean and diffusion variance rules out smeariness in all directions simultaneously in general situations. Furthermore, we investigate properties of the diffusion mean for distributions on the sphere $\mathbb S^n$. Experimentally, we consider simulated data and data from magnetic pole reversals, all indicating similar or improved convergence rate compared to the Fr\'echet mean. Here, we additionally estimate $t$ and consider its effects on smeariness and uniqueness of the diffusion mean for distributions on the sphere.

Molecular structures of RNA molecules reconstructed from X-ray crystallography frequently contain errors. Motivated by this problem we examine clustering on a torus since RNA shapes can be described by dihedral angles. A previously developed clustering method for torus data involves two tuning parameters and we assess clustering results for different parameter values in relation to the problem of so-called RNA clashes. This clustering problem is part of the dynamically evolving field of statistics on manifolds. Statistical problems on the torus highlight general challenges for statistics on manifolds. Therefore, the torus PCA and clustering methods we propose make an important contribution to directional statistics and statistics on manifolds in general., Comment: 8 pages, 4 figures, conference submission to GSI 2021

Finite Sample Smeariness (FSS) has been recently discovered. It means that the distribution of sample Fr\'echet means of underlying rather unsuspicious random variables can behave as if it were smeary for quite large regimes of finite sample sizes. In effect classical quantile-based statistical testing procedures do not preserve nominal size, they reject too often under the null hypothesis. Suitably designed bootstrap tests, however, amend for FSS. On the circle it has been known that arbitrarily sized FSS is possible, and that all distributions with a nonvanishing density feature FSS. These results are extended to spheres of arbitrary dimension. In particular all rotationally symmetric distributions, not necessarily supported on the entire sphere feature FSS of Type I. While on the circle there is also FSS of Type II it is conjectured that this is not possible on higher-dimensional spheres., Comment: 8 pages, 4 figures, conference paper, GSI 2021

In image processing, classical methods minimize a suitable functional that balances between computational feasibility (convexity of the functional is ideal) and suitable penalties reflecting the desired image decomposition. The fact that algorithms derived from such minimization problems can be used to construct (deep) learning architectures has spurred the development of algorithms that can be trained for a specifically desired image decomposition, e.g. into cartoon and texture. While many such methods are very successful, theoretical guarantees are only scarcely available. To this end, in this contribution, we formalize a general class of intersection point problems encompassing a wide range of (learned) image decomposition models, and we give an existence result for a large subclass of such problems, i.e. giving the existence of a fixpoint of the corresponding algorithm. This class generalizes classical model-based variational problems, such as the TV-l2 -model or the more general TV-Hilbert model. To illustrate the potential for learned algorithms, novel (non learned) choices within our class show comparable results in denoising and texture removal., Comment: 30 pages, 4 figures

Fr\'echet means on non-Euclidean spaces may exhibit nonstandard asymptotic rates rendering quantile-based asymptotic inference inapplicable. We show here that this affects, among others, all circular distributions whose support exceeds a half circle. We exhaustively describe this phenomenon and introduce a new concept which we call finite samples smeariness (FSS). In the presence of FSS, it turns out that quantile-based tests for equality of Fr\'echet means systematically feature effective levels higher than their nominal level which perseveres asymptotically in case of Type I FSS. In contrast, suitable bootstrap-based tests correct for FSS and asymptotically attain the correct level. For illustration of the relevance of FSS in real data, we apply our method to directional wind data from two European cities. It turns out that quantile based tests, not correcting for FSS, find a multitude of significant wind changes. This multitude condenses to a few years featuring significant wind changes, when our bootstrap tests are applied, correcting for FSS., Comment: 38 pages, 9 figures