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Karen Habermann

Publications and source records attributed to Karen Habermann.

17 recordsLinked to original sources
arXiv2026

Learning the Geometry of Data: A Mathematical Review of Shape Space Analysis

A central objective of machine learning is to identify structure and patterns in data. Advances in data acquisition have increasingly produced datasets whose observations possess rich geometric form, giving rise to shape spaces that encode variability in object geometry. Such datasets arise across a wide range of disciplines, including biology, medicine, anthropology, and computer vision, where subtle geometric differences often carry important scientific information. Traditional machine learning methods, however, are frequently ill-equipped to account for the nonlinear geometric structure underlying these data. This survey synthesizes a rapidly growing body of work on shape space analysis, which provides a mathematical and computational framework for the study of geometric data. Drawing on ideas from differential geometry, statistics, and machine learning, we organize the literature around a common analytical pipeline: shape representation and parameterization, the rigorous construction of robust geodesic metrics, statistical analysis on shape spaces, and geometry-aware learning methods. We discuss how these tools enable the characterization of shape variability, the comparison of geometric objects, and the analysis of structural trajectories across populations and time. To illustrate the breadth of the field, we highlight applications spanning multiple scales of biological organization, including studies of subcellular morphology and primate tooth evolution. Across these and many other domains, researchers face common challenges arising from complex, nonlinear, and often unaligned geometric variation. The review concludes by identifying key theoretical and computational challenges, as well as emerging opportunities driven by increasingly large and diverse geometric datasets.

math.ST↗
arXiv2026

Stochastic completeness for landmark space

We study stochastic completeness for landmark spaces equipped with Riemannian metrics induced by right-invariant metrics on subgroups of the diffeomorphism group of the shape domain. We extend a previous stochastic completeness result, which only covers the case of exactly two landmarks, to landmark spaces with any number of landmarks. This succeeds the characterization of geodesic completeness for landmark spaces with arbitrary numbers of landmarks, and thus finishes the completeness characterization for landmark spaces by covering the stochastic case. The proof makes use of Grigor'yan's volume growth criterion for stochastic completeness, which requires a suitable upper bound for the volume of growing geodesic balls. We obtain quantitative controls for geodesic balls in the landmark space by bounding both its Euclidean size and the rate at which pairwise landmark distances can approach zero. We then combine this with a lower bound on the minimal eigenvalue of the landmark cometric in terms of the Fourier transform of the kernel to yield volume growth bounds sufficient to prove stochastic completeness of landmark spaces for wide classes of kernels, including Matérn kernels.

math.PR↗
arXiv2025

Brownian motion on spaces of discrete regular curves

We introduce and study Brownian motion on spaces of discrete regular curves in Euclidean space equipped with discrete Sobolev-type metrics. It has been established that these spaces of discrete regular curves are geodesically complete if and only if the Sobolev-type metric is of order two or higher. By relying on a general result by Grigor'yan and controlling the volume growth of geodesic balls, we show that all spaces of discrete regular curves that are geodesically complete are also stochastically complete, that is, the associated Brownian motion exists for all times. This provides a rigorous footing for performing data statistics, such as data inference and data imputation, on these spaces. Our result is the first stochastic completeness result in shape analysis that applies to the full shape space of interest. For illustrative purposes, we include simulations for sample paths of Brownian motion on spaces of discrete regular curves. For the space of triangles in the plane modulo rotation, translation and scaling, we further provide heuristics which suggest that this space remains stochastically complete even for Sobolev-type metrics of order zero and one.

math.PR↗
arXiv2025

Asymptotic error in the eigenfunction expansion for the Green's function of a Sturm-Liouville problem

We study the asymptotic error arising when approximating the Green's function of a Sturm-Liouville problem through a truncation of its eigenfunction expansion, both for the Green's function of a regular Sturm-Liouville problem and for the Green's function associated with the Hermite polynomials, the associated Laguerre polynomials, and the Jacobi polynomials, respectively. We prove that the asymptotic error obtained on the diagonal can be expressed in terms of the coefficients of the related second-order Sturm-Liouville differential equation, and that the suitable scaling exponent which yields a non-degenerate limit on the diagonal depends on the asymptotic behaviour of the corresponding eigenvalues. We further consider the asymptotic error away from the diagonal and analyse which scaling exponents ensure that it remains at zero. For the Hermite polynomials, the associated Laguerre polynomials, and the Jacobi polynomials, a Christoffel-Darboux type formula, which we establish for all classical orthogonal polynomial systems, allows us to obtain a better control away from the diagonal than what a sole application of known asymptotic formulae gives. As a consequence of our study for regular Sturm-Liouville problems, we identify the fluctuations for the Karhunen-Loève expansion of Brownian motion.

math.CA↗
arXiv2025

Characterization of geodesic completeness for landmark space

We provide a full characterization of geodesic completeness for spaces of configurations of landmarks with smooth Riemannian metrics that satisfy a rotational and translation invariance and which are induced from metrics on subgroups of the diffeomorphism group for the shape domain. These spaces are widely used for applications in shape analysis, for example, for measuring shape changes in medical imaging and morphometrics in biology. For statistics of such data to be well-defined, it is imperative to know if geodesics exist for all times. We extend previously known sufficient conditions for geodesic completeness based on the regularity of the metric to give a full characterization for smooth Riemannian metrics with a rotational and translation invariance by means of an integrability criterion that involves only the behavior of the cometric kernel as landmarks approach collision. We further use the integrability criterion for geodesic completeness and previous work on stochastic completeness to construct a family of Riemannian landmark manifolds that are geodesically complete but stochastically incomplete.

math.DG↗
arXiv2024

Jump stochastic differential equations for the characterisation of the Bragg peak in proton beam radiotherapy

Proton beam radiotherapy stands at the forefront of precision cancer treatment, leveraging the unique physical interactions of proton beams with human tissue to deliver minimal dose upon entry and deposit the therapeutic dose precisely at the so-called Bragg peak, with no residual dose beyond this point. The Bragg peak is the characteristic maximum that occurs when plotting the curve describing the rate of energy deposition along the length of the proton beam. Moreover, as a natural phenomenon, it is caused by an increase in the rate of nuclear interactions of protons as their energy decreases. From an analytical perspective, Bortfeld proposed a parametric family of curves that can be accurately calibrated to data replicating the Bragg peak in one dimension. We build, from first principles, the very first mathematical model describing the energy deposition of protons. Our approach uses stochastic differential equations and affords us the luxury of defining the natural analogue of the Bragg curve in two or three dimensions. This work is purely theoretical and provides a new mathematical framework which is capable of encompassing models built using Geant4 Monte Carlo, at one extreme, to pencil beam calculations with Bortfeld curves at the other.

physics.med-ph↗
arXiv2024

Score matching for sub-Riemannian bridge sampling

Simulation of conditioned diffusion processes is an essential tool in inference for stochastic processes, data imputation, generative modelling, and geometric statistics. Whilst simulating diffusion bridge processes is already difficult on Euclidean spaces, when considering diffusion processes on Riemannian manifolds the geometry brings in further complications. In even higher generality, advancing from Riemannian to sub-Riemannian geometries introduces hypoellipticity, and the possibility of finding appropriate explicit approximations for the score of the diffusion process is removed. We handle these challenges and construct a method for bridge simulation on sub-Riemannian manifolds by demonstrating how recent progress in machine learning can be modified to allow for training of score approximators on sub-Riemannian manifolds. Since gradients dependent on the horizontal distribution, we generalise the usual notion of denoising loss to work with non-holonomic frames using a stochastic Taylor expansion, and we demonstrate the resulting scheme both explicitly on the Heisenberg group and more generally using adapted coordinates. We perform numerical experiments exemplifying samples from the bridge process on the Heisenberg group and the concentration of this process for small time.

math.PR↗
arXiv2023

Long-time existence of Brownian motion on configurations of two landmarks

We study Brownian motion on the space of distinct landmarks in $\mathbb{R}^d$, considered as a homogeneous space with a Riemannian metric inherited from a right-invariant metric on the diffeomorphism group. As of yet, there is no proof of long-time existence of this process, despite its fundamental importance in statistical shape analysis, where it is used to model stochastic shape evolutions. We make some first progress in this direction by providing a full classification of long-time existence for configurations of exactly two landmarks, governed by a radial kernel. For low-order Sobolev kernels, we show that the landmarks collide with positive probability in finite time, whilst for higher-order Sobolev and Gaussian kernels, the landmark Brownian motion exists for all times. We illustrate our theoretical results by numerical simulations.

math.PR↗
arXiv2022

Brownian bridge expansions for Lévy area approximations and particular values of the Riemann zeta function

We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden-Platen-Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen-Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

math.PR↗
arXiv2022

Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold

We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace-Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points.

math.DG↗
arXiv2020

Cartan connections for stochastic developments on sub-Riemannian manifolds

Analogous to the characterisation of Brownian motion on a Riemannian manifold as the development of Brownian motion on a Euclidean space, we construct sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by developing a canonical stochastic process arising as the lift of Brownian motion to an associated model space. The notion of stochastic development we introduce for equinilpotentisable sub-Riemannian manifolds uses Cartan connections, which take the place of the Levi-Civita connection in Riemannian geometry. We first derive a general expression for the generator of the stochastic process which is the stochastic development with respect to a Cartan connection of the lift of Brownian motion to the model space. We further provide a necessary and sufficient condition for the existence of a Cartan connection which develops the canonical stochastic process to the sub-Riemannian diffusion associated with the sub-Laplacian defined with respect to the Popp volume. We illustrate the construction of a suitable Cartan connection for free sub-Riemannian structures with two generators and we discuss an example where the condition is not satisfied.

math.DG↗
arXiv2020

Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds

We are concerned with stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. Employing the Riemannian approximations to the sub-Riemannian manifold which make use of the Reeb vector field, we obtain a second order partial differential operator on the surface arising as the limit of Laplace-Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. We illustrate the results with examples and we identify canonical surfaces in the Heisenberg group, and in ${\rm SU}(2)$ and ${\rm SL}(2,\mathbb{R})$ equipped with the standard sub-Riemannian contact structures as model cases for this setting. Our techniques further allow us to derive an expression for an intrinsic Gaussian curvature of a surface in a general three-dimensional contact sub-Riemannian manifold.

math.PR↗
arXiv2019

Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds

Given a metric measure space $(X,\mathsf{d},\mathfrak{m})$ and a lower semicontinuous, lower bounded function $k\colon X\to\mathbb{R}$, we prove the equivalence of the synthetic approaches to Ricci curvature at $x\in X$ being bounded from below by $k(x)$ in terms of $\bullet$ the Bakry-Émery estimate $ΔΓ(f)/2 - Γ(f,Δf) \geq k\,Γ(f)$ in an appropriate weak formulation, and $\bullet$ the curvature-dimension condition $\mathrm{CD}(k,\infty)$ in the sense Lott-Sturm-Villani with variable $k$. Moreover, for all $p\in(1,\infty)$, these properties hold if and only if the perturbed $p$-transport cost \begin{equation*} W_p^{\underline{k}}(μ_1,μ_2,t):=\inf_{(\mathsf{b}^1,\mathsf{b}^2)} \mathbb{E}\Big[\mathrm{e}^{\int_0^{2t} p \underline{k}\left(\mathsf{b}^1_{r}, \mathsf{b}^2_{r}\right)/2\,\mathrm{d} r} \mathsf{d}^p\!\left(\mathsf{b}^1_{2t},\mathsf{b}^2_{2t} \right)\!\Big]^{1/p} \end{equation*} is nonincreasing in $t$. The infimum here is taken over pairs of coupled Brownian motions $\mathsf{b}^1$ and $\mathsf{b}^2$ on $X$ with given initial distributions $μ_1$ and $μ_2$, respectively, and $\underline{k}(x,y) := \inf_γ\int_0^1 k(γ_s)\,\mathrm{d} s$ denotes the "average" of $k$ along geodesics $γ$ connecting $x$ and $y$. Furthermore, for any pair of initial distributions $μ_1$ and $μ_2$ on $X$, we prove the existence of a pair of coupled Brownian motions $\mathsf{b}^1$ and $\mathsf{b}^2$ such that a.s. for every $s,t\in[0,\infty)$ with $s\leq t$, we have \begin{equation*} \mathsf{d}\!\left(\mathsf{b}_t^1,\mathsf{b}_t^2\right)\leq \mathrm{e}^{-\int_s^t \underline{k}\left(\mathsf{b}_r^1,\mathsf{b}_r^2\right)/2\,\mathrm{d} r} \mathsf{d}\!\left(\mathsf{b}_s^1,\mathsf{b}_s^2\right)\!. \end{equation*}

math.FA↗
arXiv2019

A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

We consider a standard one-dimensional Brownian motion on the time interval $[0,1]$ conditioned to have vanishing iterated time integrals up to order $N$. We show that the resulting processes can be expressed explicitly in terms of shifted Legendre polynomials and the original Brownian motion, and we use these representations to prove that the processes converge weakly as $N\to\infty$ to the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by $\sqrt{N}$ and show that they converge in finite dimensional distributions as $N\to\infty$ to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. The fluctuation result is a consequence of a limit theorem for Legendre polynomials which quantifies their completeness and orthogonality property. In the proof of the latter, we encounter a Catalan triangle.

math.PR↗
arXiv2018

Small-time fluctuations for sub-Riemannian diffusion loops

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and the generator of the diffusion process is non-elliptic at that point, the deterministic Malliavin covariance matrix is always degenerate. We identify, after a suitable rescaling, another limiting Malliavin covariance matrix which is non-degenerate, and we show that, with the same scaling, the diffusion Malliavin covariance matrices are uniformly non-degenerate. We further show that the suitably rescaled fluctuations of the diffusion loop converge to a limiting diffusion loop, which is equal in law to the loop we obtain by taking the limiting process of the unconditioned rescaled diffusion processes and condition it to return to its starting point. The generator of the unconditioned limiting rescaled diffusion process can be described in terms of the original generator.

math.PR↗
arXiv2018

Small-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak Hörmander type

We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions, in a model class of diffusions satisfying a weak Hörmander condition where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly. We explicitly describe the limit fluctuation process in terms of quantities associated to the unconditioned diffusion. In the discussion of examples, we also find an expression for the bridge from 0 to 0 in time 1 of an iterated Kolmogorov diffusion.

math.PR↗