arXiv subjects

Dominic Schuhmacher

Publications and source records attributed to Dominic Schuhmacher.

At least 19 recordsLinked to original sources

arXiv2025

Sharp Convergence Rates of Empirical Unbalanced Optimal Transport for Spatio-Temporal Point Processes

We statistically analyze empirical plug-in estimators for unbalanced optimal transport (UOT) formalisms, focusing on the Kantorovich-Rubinstein distance, between general intensity measures based on observations from spatio-temporal point processes. Specifically, we model the observations by two weakly time-stationary point processes with spatial intensity measures $μ$ and $ν$ over the expanding window $(0,t]$ as $t$ increases to infinity, and establish sharp convergence rates of the empirical UOT in terms of the intrinsic dimensions of the measures. We assume a sub-quadratic temporal growth condition of the variance of the process, which allows for a wide range of temporal dependencies. As the growth approaches quadratic, the convergence rate becomes slower. This variance assumption is related to the time-reduced factorial covariance measure, and we exemplify its validity for various point processes, including the Poisson cluster, Hawkes, Neyman-Scott, and log-Gaussian Cox processes. Complementary to our upper bounds, we also derive matching lower bounds for various spatio-temporal point processes of interest and establish near minimax rate optimality of the empirical Kantorovich-Rubinstein distance.

math.ST↗

arXiv2025

Block Graph Neural Networks for tumor heterogeneity prediction

Accurate tumor classification is essential for selecting effective treatments, but current methods have limitations. Standard tumor grading, which categorizes tumors based on cell differentiation, is not recommended as a stand-alone procedure, as some well-differentiated tumors can be malignant. Tumor heterogeneity assessment via single-cell sequencing offers profound insights but can be costly and may still require significant manual intervention. Many existing statistical machine learning methods for tumor data still require complex pre-processing of MRI and histopathological data. In this paper, we propose to build on a mathematical model that simulates tumor evolution (Ożański (2017)) and generate artificial datasets for tumor classification. Tumor heterogeneity is estimated using normalized entropy, with a threshold to classify tumors as having high or low heterogeneity. Our contributions are threefold: (1) the cut and graph generation processes from the artificial data, (2) the design of tumor features, and (3) the construction of Block Graph Neural Networks (BGNN), a Graph Neural Network-based approach to predict tumor heterogeneity. The experimental results reveal that the combination of the proposed features and models yields excellent results on artificially generated data ($89.67\%$ accuracy on the test data). In particular, in alignment with the emerging trends in AI-assisted grading and spatial transcriptomics, our results suggest that enriching traditional grading methods with birth (e.g., Ki-67 proliferation index) and death markers can improve heterogeneity prediction and enhance tumor classification.

cs.CV↗

arXiv2024

Stein's Method for Spatial Random Graphs

In this article, we derive Stein's method for approximating a spatial random graph by a generalised random geometric graph, which has vertices given by a finite Gibbs point process and edges based on a general connection function. Our main theorems provide explicit upper bounds for integral probability metrics and, at improved rates, a recently introduced Wasserstein metric for random graph distributions. The bounds are in terms of a vertex error term based on the Papangelou kernels of the vertex point processes and two edge error terms based on conditional edge probabilities. In addition to providing new tools for spatial random graphs along the way, such as a graph-based Georgii--Nguyen--Zessin formula, we also give applications of our bounds to the percolation graph of large balls in a Boolean model and to discretising a generalised random geometric graph.

math.PR↗

arXiv2023

Assignment Based Metrics for Attributed Graphs

We introduce the Graph TT (GTT) and Graph OSPA (GOSPA) metrics based on optimal assignment, which allow us to compare not only the edge structures but also general vertex and edge attributes of graphs of possibly different sizes. We argue that this provides an intuitive and universal way to measure the distance between finite simple attributed graphs. Our paper discusses useful equivalences and inequalities as well as the relation of the new metrics to various existing quantifications of distance between graphs. By deriving a representation of a graph as a pair of point processes, we are able to formulate and study a new type of (finite) random graph convergence and demonstrate its applicability using general point processes of vertices with independent random edges. Computational aspects of the new metrics are studied in the form of an exact and two heuristic algorithms that are derived from previous algorithms for similar tasks. As an application, we perform a statistical test based on the GOSPA metric for functional differences in olfactory neurons of Drosophila flies.

math.PR↗

arXiv2023

Distance maps between Japanese kanji characters based on hierarchical optimal transport

We introduce a general framework for assigning distances between kanji based on their dissimilarity. What we mean by this term may depend on the concrete application. The only assumption we make is that the dissimilarity between two kanji is adequately expressed as a weighted mean of penalties obtained from matching nested structures of components in an optimal way. For the cost of matching, we suggest a number of modules that can be freely combined or replaced with other modules, including the relative unbalanced ink transport between registered components, the distance between the transformations required for registration, and the difference in prespecified labels. We give a concrete example of a kanji distance function obtained in this way as a proof of concept. Based on this function, we produce 2D kanji maps by multidimensional scaling and a table of 100 randomly selected Jōjō kanji with their 16 nearest neighbors. Our kanji distance functions can be used to help Japanese learners from non-CJK backgrounds acquire kanji literacy. In addition, they may assist editors of kanji dictionaries in presenting their materials and may serve in text processing and optical character recognition systems for assessing the likelihood of errors.

stat.ME↗

arXiv2022

Location problems with cutoff

In this paper we study a generalized version of the Weber problem of finding a point that minimizes the sum of its distances to a finite number of given points. In our setting these distances may be $cut$ $off$ at a given value $C > 0$, and we allow for the option of an $empty$ solution at a fixed cost $C'$. We analyze under which circumstances these problems can be reduced to the simpler Weber problem, and also when we definitely have to solve the more complex problem with cutoff. We furthermore present adaptions of the algorithm of [Drezner et al., 1991, $Transportation$ $Science$ 25(3), 183--187] to our setting, which in certain situations are able to substantially reduce computation times as demonstrated in a simulation study. The sensitivity with respect to the cutoff value is also studied, which allows us to provide an algorithm that efficiently solves the problem simultaneously for all $C>0$.

math.OC↗

arXiv2022

ANOVA for Data in Metric Spaces, with Applications to Spatial Point Patterns

We give a review of recent ANOVA-like procedures for testing group differences based on data in a metric space and present a new such procedure. Our statistic is based on the classic Levene's test for detecting differences in dispersion. It uses only pairwise distances of data points and and can be computed quickly and precisely in situations where the computation of barycenters ("generalized means") in the data space is slow, only by approximation or even infeasible. We show the asymptotic normality of our test statistic and present simulation studies for spatial point pattern data, in which we compare the various procedures in a 1-way ANOVA setting. As an application, we perform a 2-way ANOVA on a data set of bubbles in a mineral flotation process.

stat.ME↗

arXiv2021

Characteristic and Necessary Minutiae in Fingerprints

Fingerprints feature a ridge pattern with moderately varying ridge frequency (RF), following an orientation field (OF), which usually features some singularities. Additionally at some points, called minutiae, ridge lines end or fork and this point pattern is usually used for fingerprint identification and authentication. Whenever the OF features divergent ridge lines (e.g. near singularities), a nearly constant RF necessitates the generation of more ridge lines, originating at minutiae. We call these the necessary minutiae. It turns out that fingerprints feature additional minutiae which occur at rather arbitrary locations. We call these the random minutiae or, since they may convey fingerprint individuality beyond the OF, the characteristic minutiae. In consequence, the minutiae point pattern is assumed to be a realization of the superposition of two stochastic point processes: a Strauss point process (whose activity function is given by the divergence field) with an additional hard core, and a homogeneous Poisson point process, modelling the necessary and the characteristic minutiae, respectively. We perform Bayesian inference using an MCMC-based minutiae separating algorithm (MiSeal). In simulations, it provides good mixing and good estimation of underlying parameters. In application to fingerprints, we can separate the two minutiae patterns and verify by example of two different prints with similar OF that characteristic minutiae convey fingerprint individuality.

stat.ME↗

arXiv2019

Discrete versus continuous domain models for disease mapping

The main goal of disease mapping is to estimate disease risk and identify high-risk areas. Such analyses are hampered by the limited geographical resolution of the available data. Typically the available data are counts per spatial unit and the common approach is the Besag--York--Molli{é} (BYM) model. When precise geocodes are available, it is more natural to use Log-Gaussian Cox processes (LGCPs). In a simulation study mimicking childhood leukaemia incidence using actual residential locations of all children in the canton of Zürich, Switzerland, we compare the ability of these models to recover risk surfaces and identify high-risk areas. We then apply both approaches to actual data on childhood leukaemia incidence in the canton of Zürich during 1985-2015. We found that LGCPs outperform BYM models in almost all scenarios considered. Our findings suggest that there are important gains to be made from the use of LGCPs in spatial epidemiology.

stat.AP↗

arXiv2019

Metrics and barycenters for point pattern data

We introduce the transport-transform (TT) and the relative transport-transform (RTT) metrics between finite point patterns on a general space, which provide a unified framework for earlier point pattern metrics, in particular the generalized spike time and the normalized and unnormalized OSPA metrics. Our main focus is on barycenters, i.e. minimizers of a $q$-th order Fréchet functional with respect to these metrics. We present a heuristic algorithm that terminates in a local minimum and is shown to be fast and reliable in a simulation study. The algorithm serves as an umbrella method that can be applied on any state space where an appropriate algorithm for solving the location problem for individual points is available. We present applications to geocoded data of crimes in Euclidean space and on a street network, illustrating that barycenters serve as informative summary statistics. Our work is a first step towards statistical inference in covariate-based models of repeated point pattern observations.

stat.CO↗

arXiv2018

Semi-discrete optimal transport - the case p=1

We consider the problem of finding an optimal transport plan between an absolutely continuous measure $μ$ on $\mathcal{X} \subset \mathbb{R}^d$ and a finitely supported measure $ν$ on $\mathbb{R}^d$ when the transport cost is the Euclidean distance. We may think of this problem as closest distance allocation of some ressource continuously distributed over space to a finite number of processing sites with capacity constraints. This article gives a detailed discussion of the problem, including a comparison with the much better studied case of squared Euclidean cost ("the case $p=2$"). We present an algorithm for computing the optimal transport plan, which is similar to the approach for $p=2$ by Aurenhammer, Hoffmann and Aronov [Algorithmica 20, 61-76, 1998] and Mérigot [Computer Graphics Forum 30, 1583--1592, 2011]. We show the necessary results to make the approach work for the Euclidean cost, evaluate its performance on a set of test cases, and give a number of applications. The later include goodness-of-fit partitions, a novel visual tool for assessing whether a finite sample is consistent with a posited probability density.

math.NA↗

arXiv2018

Convergence Rates for the Degree Distribution in a Dynamic Network Model

In the stochastic network model of Britton and Lindholm [Dynamic random networks in dynamic populations. Journal of Statistical Physics, 2010], the number of individuals evolves according to a supercritical linear birth and death process, and a random social index is assigned to each individual at birth, which controls the rate at which connections to other individuals are created. We derive a rate for the convergence of the degree distribution in this model towards the mixed Poisson distribution determined by Britton and Lindholm based on heuristic arguments. In order to do so, we deduce the degree distribution at finite time and derive an approximation result for mixed Poisson distributions to compute an upper bound for the total variation distance to the asymptotic degree distribution.

math.PR↗

arXiv2018

Rate-Distortion Theory of Finite Point Processes

We study the compression of data in the case where the useful information is contained in a set rather than a vector, i.e., the ordering of the data points is irrelevant and the number of data points is unknown. Our analysis is based on rate-distortion theory and the theory of finite point processes. We introduce fundamental information-theoretic concepts and quantities for point processes and present general lower and upper bounds on the rate-distortion function. To enable a comparison with the vector setting, we concretize our bounds for point processes of fixed cardinality. In particular, we analyze a fixed number of unordered Gaussian data points and show that we can significantly reduce the required rates compared to the best possible compression strategy for Gaussian vectors. As an example of point processes with variable cardinality, we study the best possible compression of Poisson point processes. For the specific case of a Poisson point process with uniform intensity on the unit square, our lower and upper bounds are separated by only a small gap and thus provide a good characterization of the rate-distortion function.

cs.IT↗

arXiv2016

DOTmark - A Benchmark for Discrete Optimal Transport

The Wasserstein metric or earth mover's distance (EMD) is a useful tool in statistics, machine learning and computer science with many applications to biological or medical imaging, among others. Especially in the light of increasingly complex data, the computation of these distances via optimal transport is often the limiting factor. Inspired by this challenge, a variety of new approaches to optimal transport has been proposed in recent years and along with these new methods comes the need for a meaningful comparison. In this paper, we introduce a benchmark for discrete optimal transport, called DOTmark, which is designed to serve as a neutral collection of problems, where discrete optimal transport methods can be tested, compared to one another, and brought to their limits on large-scale instances. It consists of a variety of grayscale images, in various resolutions and classes, such as several types of randomly generated images, classical test images and real data from microscopy. Along with the DOTmark we present a survey and a performance test for a cross section of established methods ranging from more traditional algorithms, such as the transportation simplex, to recently developed approaches, such as the shielding neighborhood method, and including also a comparison with commercial solvers.

math.OC↗

arXiv2016

Wireless network signals with moderately correlated shadowing still appear Poisson

We consider the point process of signal strengths emitted from transmitters in a wireless network and observed at a fixed position. In our model, transmitters are placed deterministically or randomly according to a hard core or Poisson point process and signals are subjected to power law path loss and random propagation effects that may be correlated between transmitters. We provide bounds on the distance between the point process of signal strengths and a Poisson process with the same mean measure, assuming correlated log-normal shadowing. For "strong shadowing" and moderate correlations, we find that the signal strengths are close to a Poisson process, generalizing a recently shown analogous result for independent shadowing.

cs.NI↗

arXiv2015

On qualitative robustness of the Lotka--Nagaev estimator for the offspring mean of a supercritical Galton--Watson process

We characterize the sets of offspring laws on which the Lotka--Nagaev estimator for the mean of a supercritical Galton--Watson process is qualitatively robust. These are exactly the locally uniformly integrating sets of offspring laws, which may be quite large. If the corresponding global property is assumed instead, we obtain uniform robustness as well. We illustrate both results with a number of concrete examples. As a by-product of the proof we obtain that the Lotka--Nagaev estimator is [locally] uniformly weakly consistent on the respective sets of offspring laws, conditionally on non-extinction.

math.ST↗

arXiv2014

On ANOVA decompositions of kernels and Gaussian random field paths

The FANOVA (or "Sobol'-Hoeffding") decomposition of multivariate functions has been used for high-dimensional model representation and global sensitivity analysis. When the objective function f has no simple analytic form and is costly to evaluate, a practical limitation is that computing FANOVA terms may be unaffordable due to numerical integration costs. Several approximate approaches relying on random field models have been proposed to alleviate these costs, where f is substituted by a (kriging) predictor or by conditional simulations. In the present work, we focus on FANOVA decompositions of Gaussian random field sample paths, and we notably introduce an associated kernel decomposition (into 2^{2d} terms) called KANOVA. An interpretation in terms of tensor product projections is obtained, and it is shown that projected kernels control both the sparsity of Gaussian random field sample paths and the dependence structure between FANOVA effects. Applications on simulated data show the relevance of the approach for designing new classes of covariance kernels dedicated to high-dimensional kriging.

math.PR↗

arXiv2014

Gibbs point process approximation: Total variation bounds using Stein's method

We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factors, and employ the Georgii-Nguyen-Zessin equation for the total bound.

math.PR↗