A reduced rank model for spatial categorical data with many classes

2026-03-24

AI Summary

This work addresses the challenges of parameter explosion and difficulty in modeling inter-class joint dependencies in high-dimensional categorical space classification. The authors propose an identifiable reduced-rank spatial multinomial model that captures class-specific spatial effects through shared low-dimensional latent factors, substantially reducing parameter dimensionality while preserving the dependence structure among categories. To overcome the failure of conventional conjugate priors and Pólya-Gamma augmentation under this factorized formulation, they develop a Gibbs sampling algorithm incorporating Metropolis–Hastings updates based on Laplace approximations. Simulations demonstrate the effectiveness of the proposed dimension selection and proposal distributions, and the method enables scalable inference in mapping dominant tree species across the Blue Ridge Mountains, supporting flexible predictions for individual classes, unions of classes, and aggregated regional summaries.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We develop an identifiable reduced-rank spatial multinomial model for categorical data with many classes. The model represents class-specific spatial effects through a low-dimensional set of shared latent factors, substantially reducing parameter dimension while preserving joint dependence across classes. Because standard conjugate and Pólya-Gamma methods fail under this factorization, we propose a Gibbs sampler using Laplace-approximation proposals within Metropolis-Hastings updates. Simulation studies examine dimension selection and the accuracy of the Laplace proposals. An application to dominant tree species mapping in the Blue Ridge Mountains demonstrates scalable inference and flexible joint predictions for individual classes, class unions, and area-level summaries.

Problem

• spatial categorical data
• many classes
• reduced rank
• multinomial model
• dimension reduction

Innovation

• reduced-rank model
• spatial multinomial
• latent factors
• Laplace approximation
• Gibbs sampling