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.

翻译：我们引入了一类高度表达性的函数逼近器，称为Splat回归模型。模型输出是异质且各向异性的凸起函数（称为splat）的混合，每个splat由一个输出向量加权。Splat建模的强大之处在于其能够局部调整每个splat的尺度和方向，从而实现高可解释性和高精度。拟合splat模型可简化为在混合测度空间上的优化问题，可通过Wasserstein-Fisher-Rao梯度流实现。作为副产品，我们将流行的Gaussian Splatting方法恢复为特例，为这一先进技术提供了统一的理论框架，清晰地区分了逆问题、模型和优化算法。通过数值实验，我们证明所得模型和算法构成了一种灵活且有前景的方法，可用于解决涉及低维数据的各种逼近、估计和逆问题。