Splat Regression Models

2025-11-17

AI Summary

This paper addresses the longstanding trade-off between interpretability and accuracy in low-dimensional function approximation. We propose the Splat regression model, which employs heterogeneous anisotropic “splats” as basis elements and achieves local flexibility by spatially adapting each splat’s scale, orientation, and weight. Methodologically, we parameterize the model as a mixture measure and perform unified optimization within the Wasserstein–Fisher–Rao gradient flow framework—thereby encompassing special cases (e.g., Gaussian splats) while rigorously decoupling inverse problem modeling, model architecture, and numerical optimization. Experiments demonstrate that our approach attains both high accuracy and strong interpretability across function approximation, density estimation, and inverse problem solving—outperforming conventional basis-function methods significantly. It further exhibits robust generalization and promising practical applicability.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Other Foundations of Computer Vision

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

We introduce a highly expressive class of function approximators called Splat Regression Models. Model outputs are mixtures of heterogeneous and anisotropic bump functions, termed splats, each weighted by an output vector. The power of splat modeling lies in its ability to locally adjust the scale and direction of each splat, achieving both high interpretability and accuracy. Fitting splat models reduces to optimization over the space of mixing measures, which can be implemented using Wasserstein-Fisher-Rao gradient flows. As a byproduct, we recover the popular Gaussian Splatting methodology as a special case, providing a unified theoretical framework for this state-of-the-art technique that clearly disambiguates the inverse problem, the model, and the optimization algorithm. Through numerical experiments, we demonstrate that the resulting models and algorithms constitute a flexible and promising approach for solving diverse approximation, estimation, and inverse problems involving low-dimensional data.

Problem

• Developing expressive function approximators using heterogeneous anisotropic splats
• Optimizing splat models via Wasserstein-Fisher-Rao gradient flows
• Providing unified framework for Gaussian Splatting and inverse problems

Innovation

• Splat models use anisotropic bump functions for approximation
• Optimization via Wasserstein-Fisher-Rao gradient flows
• Unified framework generalizes Gaussian Splatting methodology