The Mean Shape under the Relative Curvature Condition

Mohsen Taheri; Stephen M. Pizer; Jörn Schulz

Home/Archive/arXiv

arXiv · 2404.01043

The Mean Shape under the Relative Curvature Condition

Mohsen Taheri·Stephen M. Pizer·Jörn Schulz

View original source ↗Read source document ↗Export metadata ↓

Abstract

Guaranteeing that Fréchet means of object populations do not locally self-intersect or are thereby affected is a serious challenge for object representations because the objects' shape space typically includes elements corresponding to geometrically invalid objects. We show how to produce a shape space guaranteeing no local self-intersections for specific but important cases where objects are represented by swept elliptical disks. This representation can model a variety of anatomic objects, such as the colon and hippocampus. Our approach of computing geodesic paths in this shape space enables detailed comparisons of structural variations between groups, such as patients and controls. The guarantee is met by constraining the shape space using the Relative Curvature Condition (RCC) of swept regions. This study introduces the Elliptical Tube Representation (ETRep) framework to provide a systematic approach to ensure valid mean shapes, effectively addressing the challenges of complex non-convex spaces while adhering to the RCC. The ETRep shape space incorporates an intrinsic distance metric defined based on the skeletal coordinate system of the shape space. The proposed methodology is applied to statistical shape analysis, facilitating the development of both global and partial hypothesis testing methods, which were employed to investigate hippocampal structures in early Parkinson's disease.

Explore related subjects

stat.ME

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Mohsen Taheri, Stephen M. Pizer, Jörn Schulz. 2024-12-30. The Mean Shape under the Relative Curvature Condition. https://arxiv.org/abs/2404.01043

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bias-Correction for Privacy-Protected Spatial Autoregressive Models with Application to Restaurant Network Analysis

Spatial autoregressive (SAR) models and their extensions are important tools for studying network effects. However, with an increasing emphasis on data privacy, data providers often implement protection measures that render standard SAR models inapplicable. In this study, we introduce a privacy-protected SAR model that incorporates noise into both the response and covariates to meet privacy requirements. With noise present in both components, the traditional quasi-maximum likelihood estimator becomes difficult to compute because the likelihood function cannot be directly formulated. To bypass this hurdle, we begin with a pseudo-likelihood approach, initially omitting the noise in the covariates. A Newton-Raphson algorithm is then applied to compute the estimator; however, the estimator is biased. To address this, we propose a bias-corrected Newton-Raphson-type algorithm that simultaneously accounts for noise in both the response and covariates. We further show, under appropriate regularity conditions, that the resulting estimator is consistent and asymptotically normal. To further enhance computational efficiency, we also develop a bias-corrected least squares estimator. Several extensions are discussed, and the finite-sample performance of the proposed methods is evaluated through extensive simulations. We apply the proposed methodology to restaurant transaction data from a third-party payment platform. Our method identifies a statistically significant competitive network effect among restaurants and further reveals meaningful restaurant-customer interaction patterns.

stat.ME↗

arXiv2026

A variational framework for modal estimation

Multivariate mode estimation arises in many statistical problems such as inverse problems, multimodal sampling, and density-based clustering, but becomes challenging in moderate to high dimensions, especially when the underlying density is not directly evaluable. We introduce GERVE (Gibbs-measure Entropy-Regularized Variational Estimation), a sample-based method for estimating multivariate modes by approximating Gibbs distributions directly from samples, without estimating or evaluating the density. GERVE uses Gaussian-mixture variational annealing and natural-gradient optimization, producing a mixture concentrated in high-density regions whose component responsibilities also provide a clustering of the observations. We prove theoretical guarantees in two regimes: as the Gibbs temperature goes to zero, the optimal variational mixture concentrates around the global modes of the population density; at fixed positive temperature, we prove existence, consistency, and asymptotic normality of empirical maximizers and propose a bootstrap procedure for uncertainty quantification. Simulations and a real-data experiment show that GERVE accurately recovers modes and produces meaningful clusters.

stat.ME↗

arXiv2026

Objective Model Prior Probabilities in Variable Selection

For many years it was routine to use equal model prior probabilities in Bayesian model uncertainty analysis. At least twenty years ago it became clear that this was problematic, leading to support of much too large models in the increasingly huge model spaces being considered in genomics and other fields. A popular replacement was to adopt a suggestion of Harold Jeffreys for the variable selection problem in which a total of $k$ possible variables are being considered for inclusion in the model: give the collection of all models containing $d$ variables ($d = 0, . . . , k$) prior probability $1/(k + 1)$ and then divide this prior probability equally among the models in the collection. Many other choices of model prior probabilities that impose severe parsimony have also been introduced. We begin by reviewing the problems with using equal model prior probabilities and then discuss some serious problems with the Jeffreys choice. Finally, we introduce and study a number of objective alternative choices of model prior probabilities, from both numerical and theoretical perspectives.

stat.ME↗