Splat Regression Models

TL;DR

The paper introduces Splat Regression Models, a versatile class of function approximators that express outputs as mixtures of localized, anisotropic bumps (splats) with learnable positions, scales, and orientations. By viewing splat parameters as a distribution over components (splat measures) and optimizing via Wasserstein-Fisher-Rao gradient flows, the authors provide a principled training framework that unifies forward modeling, inverse problems, and optimization, while recovering Gaussian splatting as a special case. Theoretical results establish regularity and universal approximation for finite splat models, and practical algorithms are derived from the measure-space gradient flow perspective. Empirically, splat models achieve strong performance on multiscale interpolation, 2D regression, and physics-informed PDE fitting, often with far fewer parameters and faster convergence than baselines like KAN and MLP, highlighting their potential for diverse, low-dimensional applications and inverse problems.

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.

Figures (5)

• Figure 1: A representative approximation problem for the function $f^*(x) = \sin(20 \pi x(2-x))$, $d=p=1$. We fit a $k=30$ splat model using least squares with $n=200$ noiseless samples and we compare to the performance of Chebyshev polynomial interpolation and Haar wavelet approximation. By learning an 'adaptive grid' interpolation, the splat regression model drastically outperforms a Haar wavelet approximation, and competes with the gold-standard Chebyshev polynomial interpolation. (Left). Training iterates of a $k=30$ splat model (blue) as it fits $f^*$ (green) by minimizing squared error with respect to $n=200$ uniform samples (orange). (Right). Validation MSE of the splat model over training. We also show the validation MSE of a Chebyshev approximation with $m=30,45$ gridpoints and of a Haar wavelet approximation at scales $2^l$, $l = 1, \ldots, 8$. See Section \ref{['sec:expts-interpolation']} for more details.
• Figure 2: We compare splat regression, Kolmogorov-Arnold Networks liu_kan_2025, and fully connected Multi-layer Perceptron in a noisy regression task. We observe that splat models achieve order of magnitude lower fitting error while using a small fraction of the parameters of MLP and KAN networks. The notation $[m_1, m_2, \ldots, m_L]$ denotes the widths of an $L$-layer network.
• Figure 3: We compare splat regression, Kolmogorov-Arnold Networks liu_kan_2025, and fully connected Multi-layer Perceptron in a physics informed regression task. Models are fit to solve the Allen-Cahn equation on $[0,1]^2$. (Left). True solution under this parameter regime. (Middle). Validation error for each model class as a function of the number of training iterations. (Right). Validation error relative to total number of model parameters. Among the test pool, a $k=50$ splat model outperforms all KAN and MLP architectures by an order of magnitude while using significantly fewer parameters.
• Figure 4: Fitting a sawtooth function in the setting of \ref{['fig:splat-training']}. This is a much harder function to fit with interpolation methods. Perhaps surprisingly, splats outperforms the Haar wavelet decomposition, which can exactly fit vertical discontinuities.
• Figure 5: In the setting of \ref{['fig:splat-training']}, we test the effect of Chebyshev initialization, which is slightly less performant.

Theorems & Definitions (28)

• Proposition 1: Splats are a generalized Bures-Wasserstein manifold
• Definition 1: Splat Measures and Splat Models
• Proposition 2: Sufficient conditions for regularity
• Proposition 3: Corollary of Cybenko_1989, Definition 1 and Theorem 1
• Definition 2: First variation
• Theorem 1: Wasserstein-Fisher-Rao gradient of $\mu \mapsto \mathcal{F}(f_\mu)$
• Example 1: Empirical risk minimization
• Example 2: Inverse problems and physics-informed training
• proof
• Theorem 2: Definition 1 and Theorem 1, Cybenko_1989