Splat Regression Models
This work provides a unified theoretical framework for Gaussian Splatting, addressing function approximation problems in machine learning and related fields, though it appears incremental as it builds upon existing methodology.
The authors tackled the problem of function approximation by introducing Splat Regression Models, which use mixtures of anisotropic bump functions to achieve both high interpretability and accuracy, and demonstrated through numerical experiments that this approach is flexible for solving diverse approximation and inverse problems with low-dimensional data.
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.