用可调方向和尺度的函数块建模，提升低维数据拟合精度与可解释性。
Splat Regression Models
- 以可变方向和尺度的异质函数块（splats）混合表示模型输出
- 通过Wasserstein-Fisher-Rao梯度流优化，实现高精度拟合
- 统一解释高斯点云方法，适合图像重建与逆问题研究者
我们提出一类高度表达的函数逼近器，称为点云回归模型（Splat Regression Models）。模型输出由异质且各向异性的尖峰函数（称为splat）构成的混合体，每个splat由输出向量加权。点云建模的核心优势在于可局部调整每个splat的尺度与方向，兼具高可解释性与准确性。拟合过程转化为对混合测度空间的优化，可通过Wasserstein-Fisher-Rao梯度流实现。作为副产品，我们重新推导出流行的高斯点云方法（Gaussian Splatting），为其提供统一理论框架，清晰区分反问题、模型形式与优化算法。数值实验表明，该模型与算法在涉及低维数据的多种逼近、估计与逆问题中表现出灵活性与潜力。
原文摘要 · Abstract (English)
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.
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