spatial data analysts

Abstract: Gaussian processes (GPs) are widely used across machine learning, spatial statistics, time-series analysis, optimization, Bayesian statistics, and scientific applications. A central component of a GP model is its kernel, which is typically specified through a parametric family. Among the most widely used choices is the radial basis function (RBF), also known as the squared exponential or Gaussian kernel, owing to its simple form, smoothness, and flexibility. In practice, the kernel parameters are routinely estimated by the maximum likelihood estimators (MLEs), as implemented by standard GP software. Despite this widespread use, the asymptotic behavior of the MLEs remains poorly understood under fixed-domain asymptotics, even for the RBF kernel. The main difficulty arises from the increasingly strong dependence among densely sampled observations and the nonlinear dependence of the covariance matrix on the kernel parameters. In this paper, we address this gap by providing, to the best of our knowledge, the first complete asymptotic characterization of the joint MLE of the spatial variance, lengthscale, and nugget variance under fixed-domain asymptotics. We establish consistency, derive convergence rates for all three parameters, prove joint asymptotic normality, and show that these rates are minimax optimal.

Abstract: In this paper, we revisit the concept of entropy in the spatial context with the aim of deriving computable and interpretable metrics for point pattern analysis in domains such as histopathology. We discuss stochastic point process assumptions like Poisson homogeneity and Simple Sequential Inhibition (SSI) and review established results for Voronoi and Delaunay tilings of point sets for their potential to provide null distributions for rigorous empirical hypothesis testing. We present (1) a novel "dense basin entropy" defined in terms of the Ord distribution supported on Voronoi basins, which is shown to be sensitive to clustering, and (2) a related Markov chain designed as a simplification or approximation of Brownian motion in the underlying planar domain. We investigate bounds on the dense basin entropy in the SSI regime and in empirical data, and show that the entropy rate and spectral gap of the Markov chain lead to sharp discrimination metrics. Finally, we demonstrate a generalization to multiple-set analysis via aggregation of one set over the Markov invariant measure for another. Simulations and experiments in histopathology show that our proposed metrics offer insights complementary to other instantiations of entropy in spatial analysis. The code can be found on our SMProfiler GitHub: https://github.com/nadeemlab/SMProfiler.

Abstract: Packing and covering problems for geometric regions have been studied under many notions of complexity, including VC-dimension, union complexity, shallow-cell complexity, and fatness. Although these restrictions often yield constant-factor approximation algorithms, they do not by themselves generally lead to PTASs. A recurring feature of known hardness constructions is that one region may be \emph{pierced} by many others: a region $B$ pierces $A$ when $A\setminus B$ is disconnected. We study geometric instances through the \emph{piercing degree}. Since piercing is symmetric for Jordan regions, this is the maximum degree of the corresponding piercing graph. Our main result is that, for every fixed piercing degree, the standard local-search algorithms give PTASs for the unweighted \emph{Discrete Independent Set} and \emph{Set Cover} problems. The proof constructs a sublinear balanced separator for an appropriate locality graph and applies it adaptively throughout the recursive local-search analysis. This guarantee depends only on the piercing degree; in particular, it places no bound on the number of components created by an individual piercing pair. We also prove a polynomial shallow-trace bound depending only on the piercing degree. As consequences, for every fixed piercing degree, weighted Set Cover admits a deterministic $C_r$-approximation and weighted Discrete Independent Set admits a deterministic $O(r+1)$-approximation. These results extend the known guarantees for non-piercing families and apply, for example, to axis-parallel rectangles when every rectangle is pierced by only a bounded number of other rectangles.