Splat Regression Models
Mara Daniels, Philippe Rigollet
TL;DR
Splat Regression Models are a new class of highly flexible function approximators that use mixtures of anisotropic bump functions, offering interpretability and accuracy, with a unified framework encompassing Gaussian Splatting.
Contribution
We introduce Splat Regression Models, a novel approach that unifies and generalizes Gaussian Splatting within a theoretical framework for flexible function approximation.
Findings
Achieve high interpretability and accuracy in function approximation.
Unified framework for splat models and Gaussian Splatting.
Effective in diverse approximation 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…
Peer Reviews
Decision·ICLR 2026 Poster
1. Paper provides clean and important "neural network component/layer" that is built upon Gaussian splats. Idea is well principled under measure-theoretic formulation and WFR gradient flow framework. 2. Theoretical results on regularity and WFR gradient derivations are solid and plausible. Geometric perspective is also well-crafted. 3. Paper is well-written as it narrates the topic in a very intuitive way and also provides important theoretical results in a rigorous way. 4. Experiments on low-d
1. The bound on $k$ being $\epsilon^{-2(d+2)}$ is likely loose. Authors also acknowledge that they use comparably smaller $k$'s in practice, which suggests that better theory could be developed to improve this bound. 2. For low-dimensional tasks, more ablations could be better. Given that model has this overfitting effect, it might worth to discuss also add some experiments with regularization tricks as discussed to be future work in the paper. This has been acknowledged but there is not any pri
1. The presented framework is a principled generalization of Gaussian Splatting style approaches allowing for clear interpretable function families as approximators. 2. For physics-based problems and similar low-dimensional spaces, the inductive bias represented by `splats' is a natural match. \ 3. The approach seems to be more parameter efficient than MLPs and KANs with smoother and faster convergence characteristics, at least for the simple, representative problems presented here. 4. The W
1. The current empirical experiments are quite limited with simple settings and baselines. I suggest adding a few more baselines like Random-fourier-features + MLP, and more complex settings (perhaps 3D regression, or higher order PDEs). 2. While Gaussian splatting (NVS) is mentioned theoretically, there are no empirical results validating this. Specifically, even a simple experiment validating theorem 1 under the novel view synthesis setting. 3. I believe the inner matrix transformation map
* This work provides a unified theoretical framework that recovers 3D Gaussian Splatting as a special case of a broader class of function approximators * It introduces a rigorous optimization scheme based on Wasserstein–Fisher–Rao (WFR) gradient flows over the space of mixing measures. This gives a principled justification for heuristic practices in Gaussian Splatting * Each “splat” acts as a learnable, anisotropic, localized basis function—effectively learning an adaptive interpolation grid. T
* All experiments are in 1D or 2D. The theoretical approximation bound scales as $k \le \varepsilon^{-2(d+2)}$, indicating a severe curse of dimensionality. This limits applicability to problems beyond very low-dimensional scientific modeling. Nevertheless, for low dimensional modeling problems It will be helpful to provide comparisons with the same baseline (MLP, KAN) in terms of memory and running time for both training and inference. It is also important the authors to clarify what is their
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Taxonomy
TopicsGenerative Adversarial Networks and Image Synthesis · Markov Chains and Monte Carlo Methods · Gaussian Processes and Bayesian Inference