Splat Regression Models
Authors: Mara Daniels
Philippe Rigollet
Venue: ICLR 2026 Poster (2026)
Keywords: Wasserstein
Fisher-Rao
gradient flow
gaussian splatting
scientific machine learning
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We propose Splat Regression Models, a new trainable function representation that is well-suited for low-dimensional statistical problems. We develop theory and algorithms for gradient-based training and recover Gaussian Splatting as a special case.

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.

Published as a conference paper at ICLR 2026
SPLAT REGRESSION MODELS
Mara Daniels
Philippe Rigollet
Department of Mathematics
Department of Mathematics
Massachusetts Institute of Technology ma radan @mit eddu
Massachusetts Institute of Technology rigollet @math. mit edu
ABSTRACT
We introduce highly expressive class of function approximators called Splat Re- gression 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 imple- mented using Wasserstein- Seann--iihe---eee Rasher-Rao gradient flows. As byproduct, we recover special case, providing uni-
the popular Gaussian Splatting methodology fied theoretical framework for this state-of-the-ar technique that clearly disam- biguates the inverse problem, the model, and the optimization algorithm. Through numerical experiments, we demonstrate that the resulting models and algorithms constitute flexible and promising approach for solving diverse approximation, estimation, and inverse problems involving low-dimensional data.
INTRODUCTION
In the recent history of the deep machine learning discipline, certain problem areas have enjoyed "inflection points" wherein the attainable performance and scale have rapidly improved by many orders of magnitude. These inflection points are often directly coupled to the discovery of the right model architecture for the problem at hand. To name a few examples: deep convolutional networks and ResNets for image classification (Krizhevsky et al. 2012: He et al. 2016); the U- Net architecture for image segmentation and generation (Ronneberger et al. 2015 Song & Ermon, 2019] and the transformer architecture for language modeling Vaswani al. (2017). ). This motivates the search for new parsimonious architectures for different problem domains, and in this work, target low-dimensional modeling problems such as those lying at the intersection of computational science and machine learning.
We introduce new candidate that we call the 'Splat Regression Model.' In its simplest form, the model can be written as,
f(x) Σ A.A))
1a E EP 6, Rd. A E Rddd
where (Zx; (4, 2) is the Gaussian density function. This rather simple architecture can be seen as a two-layer neural network with atypical activation function, alternatively, as generaliza- tion of the classical Nadaraya- Watson estimator for nonparametric regression to use heterogeneous mixture weights. Toward the goal of developing general theory for these heterogeneous mixture models, we abstract them into the form fa(x) == Bm-p, [v). where μt, are conditionals of proba- bility distribution v) ~ μ.. We develop a theoretical framework for optimizing these functions and understanding their expressivity. We further demonstrate that the splat regression model is performant architecture for low dimensional approximation, regression, and physics-infome ting problems, typically outperforming Kolmogorov-Arool Network (KAN) Liu et al. (2025) and Multilayer Perceptron (MLP) models by 10-100x even with far fewer parameters.
fit-
Another major success story of large-scals machine learning is in the computer graphics literature on solving Novel View Synthesis, where the goal is to learn a 3D scene from 2D snapshots annotated by camera position. A major breakthrough came with the introduction of Neural Radiance Fields

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Splat Function Training
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Figure 1: A representative approximation problem for the function f"(x sin(20Tæ(2 - x)), = p = We fit a 30 splat model using least squares with n = 200 noiseless samples and we compare to the performance of Chebyshev polynomial interpolation and Haar wavelet approxima- tion. By learning an "adaptive grid' interpolation, the splat regression model drastically outperforms Haar wavelet approximations, and competes with the gold-standard Chebyshev polynomial inter- polation. (Left). Training iterates of a 30 splat model (blue) as fits f (green) by minimizing squared error with respect to 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 Haar wavelet approximation scales 2 1,...,8.
Mildenhall et al. (2021a). A few years later, the so-called "3D Gaussian Splatting" methodology
took over premier architecture for graphics and reconstruction problems. There parallels between developments in Novel View Synthesis and recent progress in physics-informed learning,
which has arguably not yet enjoyed its inflection point. essential component NeRF modeling is the use of sinusoidal positional encoding, which drastically improves the modeling capabilities across different 'scales' of variation in the target modeling problem, and which is paralleled by the use of positional encoding schemes in physics informed PDE-fitting (Tancik et al. 2020 Zeng al. 2024). to moderate success. In both cases, rendering solutions by pointwise evaluation of an MI.P across the spatial domain can be prohibitively slow. Recognizing that Novel View Synthesis is inherently a spatial inverse problem, a major goal of our work replicate the successes 3D Gaussian Splatting across wide variety of physical modeling and inverse problems. We intend for this work to provide an instruction manual for deploying and training splat regression models in these settings. Our main contributions the following.
1. We introduce the Splat Regression Model. We prove some preliminary structural theorems about this model and we show that is a universal approximator.
2. We develop principled optimization algorithms for gradient-based training, which cru- cial to our empirical success. To do this, we leverage an interpretation of the splat model parameters hierarchical 'distribution over distributions. and we invoke the theory of Wasserstein--iitte Rao gradient flows to compute gradient updates in parameter space.
3. We recover 3D Gaussian Splatting as an instance of splat regression modeling. In Example detail clean formulation of it, where different aspects of the pipeline split into clear, 'modular' parts.
4. We test the performance of splat modeling in few representative modeling problems in- cluding low-dimensional regression (Figure 2 and physics-informed fitting.
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2 RELATED WORK
Novel View Synthesis. The Novel View Synthesis in computer graphics has a long history. Input data is typically sourced using Structure From Motion (SfM) methods (Ullman 1979 Ozyeşil et al., 2017). In the idealized setting, scenes are rendered using the Radiative Transfer Equation (Chan- drasekhar 2013). Much work has gone into developing fast approximations to the RTE, such as the widely used a-blending technique introduced in (Porter & Dufi 1984. Carpenter] 1984). For 3D scene representation, Zwicker et al.] (2002b) introduced the "elliptical weighted average" method, although these ideas were not applied at scale for about two decades (Kerbl et al., 2023a). Major interest in this problem was sparked with the release of Neural Radiance Fields Mildenhall et al. (202T)) an approach based on optimizing MLP parameters by differentiating through (an approxi- mation of) the RTE.
Physics-informed Modeling, Following the introduction of PINNs (Raissi et al. 2019), much work has gone into developing ML methods for solving computational science problems. As documented by KKiihhappiiann et al. (2021). PINN training can have surprising failure modes and requires careful tuning and hand-crafted architectures. More recently, adding positional encoding schemes [Taniik et al. (2020); Huang et al. (202T) through either RBF or sinusoidal encoding was shown to improve performance. Alternatively, the Kolmogorov-Arool Network architecture et al. (2025); Rigas & Papachristou (2025) was introduced
competitor, and some comparisons between KAN and RBF interpolation methods are explored in Li (2024). One way to view splat regression modeling is as learning 'adaptive interpolation grid," and our experimental results suggest that 'smart positional encoding is all you need' for oiw-dimesioonal modeling problems.
Mean-field Theory of Two-layer Networks. Wasserstein Missserstiin-Fisher-PP Rao gradient flows can be used to study the optimization of two-layer ReLU neural networks. This approach was pioneered by Chizat & Bach (2018), who prove conditional global convergence result for two-layer network optimization. Further work in this direction includes (Mei et al.] 2019a The main difference between optimizing ReLU networks and Splat models is that our model has fundamentally differ- ent geometry of its parameter space, since first-layer output neurons are represented as elements of a Bures- Wasserstein manifold. This also major difference between our work and Chewi et al. (0255),, which introduces a modified ReLU network architecture whose neurons can be "superpo- sitions' of ReLU functions.
3 SPLAT REGRESSION MODELS
We now state a few essential definitions and basic properties of splat regression models. Readers who are uninterested in the abstract formulation may skip to equation[]] which is the concrete object that appears in algorithms. In Example D we discuss how to fit these models by minimizing loss on a training dataset.
We write L (TRd; to denote the functions f Tidd -> Rp for which f [lf())) || π (dx) < co, where is a measure on Idd, When 2, we endow it with the inner product (f,g) = J g(x)) rdxx.. We write (R(( to denote the real-valued s-times continuously differen- tiable functions on gd with s bounded derivatives. For function & p: Rdd + R. we write Vo(x)) its gradient, v²Φ(z) its Hessian, and AΦ(x) = Ei= 22, o(x) its Laplacian. For th : Rd + RP write Dib(1)) its Jacobian and (when p d) write div (ar) Σd_ Oz (xx) its diver- gence. We denote by P(2)) the space of probability distributions on Ω by P.(2) the distributions with s s> finite moments, and by (!) the distributions with density. For μ E P(2) we write (f
fx) (dxx) and, if µ has a density, we denote it as p(z) Ω + R. Last, we denote by T# the pushforward of p. that is, the distribution of T(X) when the identity map, / = n..
μ. We write id(x)
We begin by defining the family of splat models that we consider in this work. We first define the family of splats and then explain how they are used in splat modeling. Proposition 1 (Splats are generalized Bures- Wasserstein manifold). Let p E P(Rd) be a centered isotropic mother splat. We denote the set of all splats as,
(R.) = =f {A(() + b) # p: : A e dxd bb (R)
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Then BW (RR) is geodesically convex subset of W2 (R and on this space the Wasserstein metric reduces to the Bures- Wasserstein metric (Modin 2016 Bhatia et al. 2019).
W2 (AAb: PR,s) = ||b - s1l2 ||A + || 22 RlI.
A, E Rdxd b, Rd
where || ||| the Frobenius norm and || || is the nuclear norm.
We give a short, self-contained proof this proposition in Appendix B.2
We occasionally write PA.b = (A(.) + # when wish call attention to the parameters (A, b) of the splat. These parameters can be thought of as global coordinatization of BW (R... We discuss further the Wasserstein distance and its associated geometry in Section 3.2
We envision P (I2 an isotropic 'bump' function and call each individual copy splar in homage to the computer graphics literature where an instance this model was first widely popularized. We call p the mother splat in analogy to the mother wavelets used in wavelet theory (Mallai 2008). Splat models are defined as follows.
Definition 1 (Splat Measures and Splat Models). Let p (Raa(R have zero mean and identity covariance, and let PA.6 be the distribution of AX + b when X p. We say that E P(RP x BW, (R*)) a splat measure and the associated splat model fu(()) is,
[x(() == E[upA,a(>)]
U,PA,
If the support of u has elements, we say μ is k-splat measure, Sy. is k-splat model, and we write
fx(z) Σ ut PA.. (z) A ((..PA..).)) supp p.
(1)
Σ U/ p(A b,]) det |
We write Sp.a == P(R x BW, (Rd)) (resp. s(()) to denote the set of splat measures (resp. k-splat measures).
As we later discuss, Sp.d may be endowed with natural metric structure, allowing us to perform gradient-based training of k-splat models.
3.1 REGULARITY AND UNIVERSAL APPROXIMATION
We note the following basic condition for (xx) to be continuously differentiable. Proposition 2 (Sufficient conditions for regularity). Lei μ E Sp.d a and suppose that the mother splat has 2 0 bounded derivatives. Then fz) also has bounded derivatives.
also helpful to understand the expressiveness of the class of splat models, and in particular, to understand what kinds of functions can be approximated by k-splat models for finite k. As we show Appendix [A] the finite splat model equation D already the scope of Cybenko's 's early result on universal approximation of real-valued functions by two-layer neural networks.
Proposition 3 (Corollary of Cybenko (1989), Definition and Theorem 1). Let Q be compact subset of Rdd and suppose is a continuous density with marginal pi(z1) Jax p( Ø2 I.) dx2 .......0 dxg. Then for any f E C. (1;;)) and e > there exists a k-splar measure μ (with mother splat such that
sup - - fx()) <e. zEft
We also prove in Appendix A quantitative bound on the number of splats required to approximate any Lipschitz function on a nice' domain using any 'nice" mother splat.
Theorem (Informal quantitative universal approximation). For any bounded, "nice" Ω C Idd, any bounded, Lipschitz Ω + IR,, and any bounded, Lipschitz mother splat with finite first moment, there exists for each 0 a k-splat measure μ with & S 22(()) and,
sup [] (x: - fx))112 =E
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To complement this, give the following lower bound the error of uniform approximation over the family of L-Lipschitz functions, whose proof is also contained in Appendix
Theorem 4 (Informal universal approximation lower bound). Let p be -p-liscchitz Bp- bounded mother splat density. Fix & 2 1 and h > O, and set
Sn = /
U. (A.()) + b : ET,
F F
[0, 11 R: f(0) 0. 1-Lipschitz)
where I' contains all splat parameters (v, A) E IR Rddd xR" with v, bounded and miin 2 h, where minn (A) is the smallest singular value of A. it holds that,
sup inf || foo Se SEF fESk
(2)
then (up to logarithmic factors) - kdR.
The lower bound rate essentially tight up to logarithmic factors: any I E F admits a piecewise constant approximation on sets of the form 80 (s, + 1)c) where each (s;)i=1...d has 1 < 8, S d and where
- 1/d Such approximation is representable by a k by taking mother splat = --c, c)dt for an appropriate с > Taken together, Theorem and iimply that there exists (non-Lipschitz) mother splat that achieves the minimax optimal rate k-1/d of approximation of Lipschitz functions, and also, that for any 'nice" mother splat, the approximation rate is at worst ~ -=122((+2)) However, much like the corresponding universal approximation rates for multilayer perceptrons, these worst case approximation bounds typically do not describe the number of parameters required to achieve a high quality fit for real data. We discuss further in Section
splat model
Theorem 4
3.2 GEOMETRY OF SPLAT MODELS
Ignoring some technical caveats, the space P(RR)) of prohability distributions be viewed manifold by assigming it a distance metric such as W2(H,,), v/), the 2-Wasserstein distance, or H (p, v), the Hellinger distance. In both cases the distance induces metric in the geometric sense, and tangent space structure, which together can be used to construct gradient-descent-like algorithms over the space of probability distributions. In Wasserstein space, moving between points Po, Hl E Pa (R is accomplished by applying "infinitesimal" transport maps and the geodesics follow the straight line interpolations geometry of the Hellinger metric, known as the Fisher- or Information geometry Amari (1983), moving between points is accomplished by "mass teleportation." " which means directly scaling the density up or down, pointwise over the domain, in mass paassprreerrvig way. Its geodesics are ue [e( (or//H((z) + (x))(()x)) where 04, 3r are spherical linear interpolation coordinates, as can be seen from the _ form of H² 11) [()o((z> )") dx for densities fo, E Pac (RRd). Finally, is also possible to consider the geometry that arises when allowing movements by transport and teleportation at the same time, which is the Wasserstein- Fistnr-iieerree Rao geometry (Chewi et al.) 2025a).
optimal transport maps, μr: = (1 (TTTgg=> + (1 1) id) y poo Under the
The Wasserstein and Fisher-Rao geometric perspectives are widely used in the design and analy- sis of probabilistic algorithms, such as for dynamical sampling (Chewi al. 2025a), variational inference (Lambert al. 2022), barycentric interpolation (Gouic al. 2019 Chewi et al.) 2020). and lineage tracing (Schiebinger et al.. 2019). We apply both to design a principled gradient-base training method for splat regression. As byproduct, we recover the heuristic methods used to op- timize Gaussian Splat models, which has a clean interpretation as regularized risk minimization via Wasserstein -iin--Fih-aaaa gradient descent. We first explain the Wasserstein structure of the space of splats, then the Wasserstein- Feehin--Fihee--ee Rao structure of the space of splat measures.
Relative to partiele methods, lifting Sp.d allows one to implement 'smoothed particles' on P(R" R thus providing computationally tractable way flow measures that everywhere positive and absolutely continuous. We comment in Appendix [ on the relationship between gradient flows in Sp.d and in P(RP Rd). This generalizes the concept of 'Gaussian particles' introduced in
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because particles represented by any affine pushforward of an arbitrary density p. Such a particle representation was also hinted at in the computer graphics literature on "volume splatting' (Zwicker et al. 2002a), and has also appeared in early work on hydrodynamics simulations (Gingold & Monaghan, 1977..
Lambert al. 2022).
We endow Sp.d with the Wasserstein and/or Fisher-Rao metric, similar the 'Wasserstein Wasserstein' approach taken in (Lambert et al. 2022 Bonet et al.) (025). After stating our results,
give Examples 1 that illustrate these objects concretely in two practical settings.
Definition 2 (First variation), Let (R R2) R be a functional. Its first variation (when it exists) is the function (F) which satisfies for every f. E (RR,
0,F(1 (=0 l=
(((x), dr.
is convenient to express the Wasserstein gradient W (.F(fu)) : RP x (W, + I2, (R)) in the global coordinate system (v, b) = (u. P..b). This coordinate system is convenient for so that the particle dynamics
writing the gradient W Wr, ((f)) (W.. W (fF))
W (.F((()),, b) A = AF A,
b WWFW A,b).
implement a Wasserstein gradient flow in Sp.d- Our main theorem is the following.
Theorem (Informal - Wasserstein-Fistee Rao gradient of p F(fu)). Let E Spd P(R" x BW and let F (R ( - R be a functional. Assume that has a sub-exponential density. Then the Fisher-Rao gradient is given by,
F(n)(r, A,b) xx=xx.116F(XX
(Bx.A,a ((-
and the Wasserstein gradient given by.
W (.F((a)U,,A.)) EX~pA.L60FF
W WAF(fA(V, A, -ExmPA (8F((X), v) [a V VologgAb(XX] b)²) A-7] WIF(FW(O.A..) -EXxpAA ((FF(X), v) V log (A)
where the argument oF -F([(() was suppressed.
The condition p is required apply integration by parts without a boundary term, but when this does not hold (such as when p(x) = |[||=]|2 < 1}) one can derive the appropriate corrections on case-by-case basis. We conclude with a few concrete examples of F which are relevant to applications.
Example (Empirical risk minimization). Suppose #1, #1,02.- $2. In ~ 2(22) are i.i.d. samples, where f [0, 1]dd for simplicity, and y1 = (x;) with
F()) = ((f(aa).))
for loss function L(ū. y). To caleulate the variation
1 I (E(x,), (oL(f(()) I
F E() = d==
u(f(z)) c((x)),
and so (F,)( Σ"
(x.))))),)))) xx), which otherwise undefined for i E Rd outside the sample. We can importance sampling to estimate the gradients,
1=1 VgL(S(x,),
Ex-PA (an (xF Σ (aso(=))F1((=)))
This estimator is unbiased because ( 2( 1)9). For samples drawn from unknown distribution, one could instead estimate {F) by interpolating the available data, then approxi- mating the average over X PA. using either Monte-Carlo quadrature rule.

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Example 2 (Inverse problems and physics-informed training). We can also take
F(f) -(][((())))))))))))))))))))))))) - (()))
(3)
where is known integro-differentaa operator, g a known forcing function, and we again take ? = 00, 1J for simplicity. To calculate the variation,
De= + + a.=0 { | + +z(()) - -
= - (Als)())) ( (1 de
= = (0)))) (DDA))
dx.
where A is the linearization of A at f and (D5A)" is its adjoint. So dFFIf = AD) A"[gl [g] (x). Two illustrative instances of this setup are,
1. A "hyysies-inormed' splat solver for the Poisson equation. Take = 1, A[fal = Afp- The Fisher-Rao gradient is,
V # F(FA(v,A.b)
100---e(ffatt o(Af(a) g(x) (pA,b(z) dx.
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the same way, the coordinates of the Wasserstein gradient can be expressed as integrals with respect to Ap(x) and (x)) For simple p. these functions can be precomputed to accelerate gradient computations, and the integral approximated by Monte-Carlo.
2. Splat regression modeling for Novel View Synthesis (NVS). NVS can be viewed as an inverse problem whose forward operator A is the Radiative Transfer Equation (RTE) (Chandrasekhar, 2013), (Zwicker et al. 2002a Equation (7)), whose unknown parameters are: the "emission function," R X S R, and the "extinction function' R + R. The emission s(x, is the amount of light radiating from æ E TRd outwards direction v, and the extinction (xx) represents the degree which point 'occludes' points behind it. Following (Kerbl et al. 2023a), we parametrize splat models (xx) = (x) and we parameterize s(x, v) as,
s(x, ) Σ 1 (x)):(v))
where {oi are spherical harmonic basis functions and f. R23 - R2 is another splat model (p = 20). Given these two fields, rendering the scene involves evaluating the Radiative Transfer Equation.
Als, (I = f. s(xr + tv, v)o(( (v) exp (-f, o(x sv) ss) dt.
(4)
The RTE is typically evaluated using discrete approximation called a-blending Zwicker et al. 2002b Kerbl al. 2023a) and splat parameters are trained via SGD with particle birth-death dynamics to minimize residuals (equation 3
Finally, remark that the performance of Gaussian splatting in large-scale Novel View Synthe- sis depends heavily on the use of many training heuristics, such as initialization of splat locations Kerbl et al. (2023b), selective noising of splat locations during training [Keradmand et al.) (2044), and on 'pruning strategies' to move spurious splats either by deleting them or teleporting them else- where in space (Kerbl al. 2023b Hanson al. 2025). By providing principled optimization perspective on fitting splat models, our results pave the way to interpret these heuristics through the lens of regularized risk minimization. For instance, adjusted Langevin Algorithm can be interpreted as Wasserstein gradient descent of the functional F(u) DKI(μ ( [hunge)) divergence, which enjoys strong convexity whenever Marget strongly log concave. We anticipate that the selective noising heuristic introduced by KKherdmand et al.) (2024) can be interpreted as adding a convex entropic regularizer to the objective (3). improving convergence. Similarly, particle birth-death dynamics are a well-known discretization algorithm for Fisher-Rao gradient flows Lu et al. (202)). With an appropriate discretization, Theorem [ thus prescribes pruning criterion that is guaranteed (in the continuous-time limit) to decrease the loss. While it is outside the scope of the present manuscript, principled regularization methods for splats very promising direction for future work.
sampling context, the stochastic Un-
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4 EXPERIMENTS
In this section, demonstrate the power of the splat regression modeling by directly comparing it on approximation, regression, and physics-informed informed modeling problems.
4.1 MULTISCALE APPROXIMATION
First, we show in Figure 1 representative 1D example of training splat regression model, which we compare to two standard function approximation algorithms: chebyshev polynomial interpola- tion and wavelet basis projection.
In the left subpanel, we show snapshots of splat training arranged from right to left, top to bottom. We train a
30 splat model initialized with v; 0, b, (i 1)/k, and A; 1/2k for i = 1... k. Training is performed by least squares fitting via Wasserstein gradient descent with learning rate 10-1 and no momentum. The loss function is ((),) = E== (f(x)) ) y1)² for = 200 noiseless samples of the splat model. Aneedotally, the exponentially fast convergence which appears in Figure is robust to different initializations and functions. We show in Appendix C Figure 5 version of this experiment where the {b: are initialized as a k-point Chebyshev grid as well as a version where data is sampled from discontinuous sawtooth function.
((0, 1J). In the right subpanel, we show the validation log-MSE
We select this example as simple one dimensional picture of how splat models are well suited to fit 'multi-scale' features of the observed data. For comparison, horizontal lines indicate the validation MSE the Chebyshev and wavelet approximations. The splat model with A = 30 splats signifi- cantly outperforms the Chebyshev interpolation with m 30 interpolation nodes. Setting in order to control for the total number of model parameters ((3 x = 90) for splat vs. (2 x m = 90) for Chebyshev), we see that the splat model achieves slightly worse approximation error. Relative to wavelet approximation, the splat model outperforms Haar wavelet projection with m 255 parameters (corresponding to level encouraging, particularly given that very simple optimization and initialization schemes are enough to successfully train the splat model.
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of the basis hierarchy). We find these results extremely
Va kodien VSL dalnng To FiNe
Froi elatin ArdMocure Sir
6.8
5.00
12
(arambrr)
Figure 2: We compare splat regression, Kolmogorov-Amold Networks Liu et al. (025), 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 small fraction of the parameters of MLP and KAN networks.
4.2 REGRESSION WITH KAN, MLP, AND SRM
We compare the performance of splat regression modeling to Kolmogorov-Amold networks Liu al. (2025) and fully connected MI. architectures on a regression task. We sample x; 2( ( L..dd and set 14 f(x) + 6 where & ~ (00 0.01) and,
f(x, 3) sin (z) cos (3Ty).
Each model is trained using 3000 iterations using Adam with learning rate 10-4
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We attribute the improved performance of splat models to their spatially localized nature, which be viewed as learned positional encoding scheme. However, due to their expressivity, splat models may be more susceptible overfitting, requiring regularization to achieve good fits.
4.3 PHYSICS--NFORMED MODELING
131
023
12
64 10
20.0 C000 1222
Figure 3: We compare splat regression, Kolmogorov-Amolt Networks Liu et al. (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 function of the number of training iterations. (Right). Validation error relative to total number of model parameters. Among the test pool, splat model outperforms all KAN and MLP architectures by order of magnitude while using significantly fewer parameters.
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We further test the splat regression model in a two dimensional physics-informed learning problem where the goal is estimate the solution of the Allen Cahn equation,
2 (u))) + u(x) u (x)) =
x E [0, 1]
This equation is well known example of PDE whose solutions can have 'boundary interfaces' that converge to discontinuities as e > 0. We take e = 0.1 and train each model to minimize a weighted sum of the loss.
C(ue)
(uu((z;) f(x:))1 f(x))) +
Σ
(1o)) u (22)
Piint
Tindy 1=1
where H2 ~ 2[0, 1]d) and where 2j ~ u((00, 1]d) for == 1,... 10 and = 1,..., 5 10a both iid. We observe that the splat equation converges very quickly and highly accurate solution, even when limited to use orders of magnitude fewer parameters the KAN and MLP models.
5 CONCLUSION
We have shown that splat models are highly effective in solving a variety of representative low dimensional modeling problems. We believe that these results extremely promising from prac- tical perspective. Furthermore, we have drawn novel connection between 3D Gaussian Splatting and Busserstein--iisher between the optimal transport, computer graphics, and statistics communities.
gradient flows, which we hope will lead to many symbiotic interactions
There is much future work be done. First, as we discuss Section 3.2 splat models are highly expressive and therefore susceptible to overfitting, warranting a larger scale computational study beyond the scope of our work. The splatting community has developed many heuristics for regular- izing and pruning splat models, which we expect will be instrumental to large-scale splat regression modeling. As a second direction, we remark that the proposed model can be interpreted as a new neural network layer, opening the door to compositions of splat models ('deep splat networks') and/or integrations into existing deep architectures. We look forward to investigating these direc- tions in followup work.

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ACKNOWLEDGMENTS
The authors wish to thank Ziang Chen, Tudor Manole, Youssef Marzouk, and Max Simchowitz for many helpful and friendly discussions. This material based upon work supported by the U.S. Department of Energy, Office of Science, Office Advanced Scientific Computing Research, Department of Energy Computational Science Graduate Fellowship under Award Number(s) DE- SC0023112.
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A WELL-POSEDNESS, REGULARITY, AND UNIVERSAL APPROXIMATION
Here we consolidate the proofs that are related to understanding structural properties of splat mea- sures μ E P(R BW p (TR)) and splat functions. This includes 2 on the pose-poss and differentiability of splat functions and also the proofs of two universal approximation theorems, Proposition 3 and Theorem
3
We restate these claims for reading convenience.
Proposition 2 (Sufficient conditions for regularity). Let μ. E Sp.d and suppose that the mother splat p has 8 bounded derivatives. Then (x) also has bounded derivatives.
Proof. Since p E (RR) and for any A E Rddd E R x + Ax + b is C (RR), the function (v,x) v ((A.)) + (x)) has bounded and continuous derivatives in æ and dominated by the v-integrable function v + C, supresup(v)) |l where C₂ is a universal constant. Derivatives of (-) are therefore given by differentiation under the integral.
Next, we show that under a wide range of splat models satisfy the conditions of Cybenko's s theorem universal approximation by two-layer neural networks. We restate the universal approx- imation theorem and check that its conditions hold.
Theorem (Definition and Theorem Cybenko (1989).. We say that a is discriminatory if for any measure E J == 10,
In ooy x + 0) du(r) =
for all y E R" and 0 E R implies that μ 0. is any continuous discriminatory function, then the finite sums of the form
G(x) Σ a +0,)
are dense in (Ia)
Corollary 1. Let Q C Rdd compact and suppose p continuous density with marginal pi (21) fga-1 p( .d) dxz... .dd. Then for any f E CS; and >> 0, there exists k-splat measure p (with mother splat p) such that
sup [f(x) - fu(t)] zElt
Proof. By scaling we may assume Ω C [0, 1]d We claim that p is discriminatory function and
that the approximator G(x) with a = P) and with parameters {(aj: Mjs 0,) j -1...), k.. which guaranteed by Theorem 2 for some h: > 1, can be realized as splat model. It is easy to see the latter: to realize G(x) as splat, take Au, = Euv5. b., = El 0,. and eg = aj.
Let be nonzero measure on [0, 1]a and let E supp(u). Assume for contradiction that for all y E Rd. 0 e
0 Jann pily" 0) p(dx)= 2 t tP (8) (8" x x) t).
Thus pl (y (1) for all sets (l, = (a [j p (y + 0) > 1} and all > 0. By continuity, the set PT - oo)) is open, so that by affine rescaling we can construct from the basis (t), : E Rd. t,0 0 R} the entire Borel o-algebra on [0, 1]d, which would imply fr == 0.
Finally, we give a more constructive proof universal approximation theorem on sets C Rd whose uniform measure U(?) has finite Poincaré constant. This constant is the smallest Cj for which any f E H (2.) satisfies the inequality
(f(X) V
X-U(()
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This is satisfied by most 'nice' sets, namely whenever f2 is open connected set with Lipschitz boundary (Evans) 2010, $5.8 Theorem 1).
Theorem 3. Let Ω C Ie compact and suppose I (22; R.). Suppose further that the measure (S2; (L) has finite Poincaré constant Cn oc. For e > O there exists a k-splatt measure p. at most & S Co 22(422 (Ly + ByLp)² splats and for which
sup ll.((z) fu(a))l (1 + M,). sef
Where the constants are:
Bp ||
By ||
(p(x) p(u)]
[l - f(y)l22
Lp sup
Ly sup
VVRR
|| yll2
z
|| y/12
Mo = Exmpll [1|2||
Proof. Set fx(x) Ef + cz)], then
llf()) - (x))l2 - / [11(=)) + Ez)1l2 pldz) S L,
We now show that, for k >>>>>> 1 also to be determined, there exist k-splat measure μ so that [lf(()) - (1))) Sc
sup
=e0
Take random $1.... Is ~ Sok and take v = u(dv, x)dx = v(dv)k(u; 2). Then we have
1 & Et 8pen k(v,z) = <(x/cc). and
P Ø1 I1. + sup [z) - fx(x)l > e) x
Var (01 (01... 0k + sup [lfu(z) (x))
2²
sen
B[Var,(01 .x sup [15x()) (x()))l2)
sen
B[Var (21 ... sup (r, fz(2) - f (x)) zei
2
rES"
where Var, (f xx)) is the variance of f(-) holding I ... Ty fixed and resampling xę ~ w independently. In advance of applying Poincaré's inequality, invoke the envelope theorem to see,
k
= = (r, - (wm)) (())
-1((==x:)) Mwi)e ((x))-"p((ee) - xz)/e)
F1
Th H sup E
K
rES
ked (r* DT( xi)/e)
kedtI (r' br ((=)) (())=)))
1
where x", rr attain the supremum. Its norm is bounded by
.... S kedtI B, + B,Lp)
where |I == supren [lf(x) ||2, and C, = sup supz,yERd (((x)~p(y)/// / - - so that by Poincaré
inequality,
Cp
P( + sup 1up [x) > +) 2dd+21k (T. B,e + ByT,,)
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Consequently, for k (over 21,...;; Tg ~ w that seP
2011" c 2)
LJB,c+Bylyy it occurs with probability at least one half 2 c. By the triangle inequality
sup (1f(()) f(x)]l2 I sup - (z))la + (1 #eQ
(())l) & (1 + M..
zE@
Theorem (Uniform approximation lower bound). Let p be an piplisschit Bp Bp-bounded mother splat density, Fix k 2 1 and h and set
ET, Sp = { f Σ (A.()) + +b) (0,, A.,
h
F= {s [0, 11 + R: f(0) 0, f 1-Lipschitz)
where I' contains all splat parameters (v, A.b) E x Rdxd x R satisfying the bounds lu| S B.. mmin (A) 2 h, ||b||2 S 1, and where mmn (A) the smallest singular value of A. If holds that,
sup inf - Se fEF FESa
(5)
then (up to logarithmic factors) c- S kd? when h, c are sufficiently small.
Proof. We compare the metric entropies of F and Sn, Let P(e, (e, Sp) be the packing and covering numbers of J. S, respectively, in the supremum norm on [0, []. It is well known that
log (ee, F) 2 e-d (Wainwright) 2019 Chapter 5). By assumption of equation 2 we also have
log (c, S log. (c)22
For any μ E & the associated fu Sh is given by
fa(x)= = Σ v (A(-) + p(xx)
Σ o)de((A-)))p(A-1)) b)).
(o.AMM)H
(v,A.bjEи
can be rewritten in the form fa(x) Σ u, p(W;(z b,)) for some parameters f(u,, W,, 6,)) with ||u/|22 S B, h-d l|Wallop S h and | 1. Call i R Rddd Rd the set of all (u, W, b) satisfying these bounds and call Sn the functions,
S = 90) -== u. pw, b:)) (u,, W., b.) ET, i= 1,... k
Thus S₁ S S, and it is enough to bound log (e, As before, we denote by Ip the element of S, with parameters j E pok For v, v' E pok v = [(u,, Wr, b v/ = {(ul,, Wt. b))) we have
11g. 1lo < sup zERd i=1
u pl b)) v' (x -'
< sup Rd
[(u ()
p(W b)))
&
< sup
B,lu; - uil + Bob 1
2E3d 1=1
For any x E 1jd
[ (y-xx00 - b> 8')l2 - II(W +)) -')ll2
< 32VdIW-W'll
- W'llop +h || - b1112-
Thus
llg. loc 5 5
Bp luu uiil + 2h 1 Vdd IW - Will + 1- (d+1)) B l|b: 61112-
1=1
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By tensorization, can crudely bound
(e, s ke log.V
kB BBBBoh- 1- [0.1]
+ log.V
2k || 'lop, Bop (1, Baddd)
(BoLLh-(4t1) 11-112, [0. 111))
+ N
S k V log
(c kBB.BBb Bo + d² Volog (VV
+ Vol Va hh
S kd lol
(E kvhh (d+1) B.B. BoBoLo)
Here, (1 Rdxd) is the operator norm ball radius in Rdxd Putting these parts together, we conclude that the minimax rate only hold when
ed S kdf VV log Wk - B. (,
B GRADIENT FLOWS ON MEASURE SPACES
We provide in this section some of the mathematical background that is required to state our main results. We then give the proofs of Theorem Proposition [ and some helpful calculation rules for computing gradients with respect splat measures.
Santersttin-Fihher--r Rao calculus
B. REVIEW OF WASSERSTEIN AND FISHER-RAN RA CALCULUS
First, define the Hellinger and the 'static' Wasserstein distances.
Definition 3 (Wasserstein Distance (Villani, 2003)). Let M be Riemannian manifold (endowed with the canonical volume measure) and let μ., v (M(M) be measures with finite p-th moments. The p- Wasserstein distance is.
{ dd (dx,ddY)
WP( = inf rET(u.)
where (.-) is the geodesic distance on M and I(u, v) is the set of all joint couplings whose marginals are πx = μ. my v. We denote by (s(M) the space Pp(M) endowed with the W,, metric.
Definition 4 (Hellinger Distance (Amari 1983) ) Let M be as in 3 and let μ, E P(M) be mea- sures with finite second moments. Setting ( + v), the Hellinger distance
2
du (x)
du dA
(x)
X(dx)
where 4x (x)) is the Radon-Nikodym derivative. Here, À can be replaced by any measure X 14, V. We denote by H(M) the space P(M) endowed with the H metric.
For our purposes we only need to consider rather two rather simple manifolds: the Euclidean space (with the canonical || 2| metric) and the Bures- Wasserstein space BW, We postpone the proof of Proposition [ where we formally define the Bures- Wasserstein manifold, and proceed with the review. For further reference and for proofs of the background theorems stated here, wonderful
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reference (Santambrogio 2015) as well the more contemporary (Chewi et al.
2025a). which
emphasizes a statistical perspective.
In their seminal 1998 paper. Jordan, Kinderlehrer, and Otto (Jordan al. 1998 introduced itera- tive 'minimizing movements' scheme as a variational approach to solving the Fokker- Plank equation in fluid dynamics, leading them (largely by accident) to discover the geometric perspective the so-called 'static' Wasserstein distance. Shortly afterwards, Benamou & Brenier (2000) developed the following elegant formulation.
Theorem 5 Bannamou-Brenisee Benamou & Brenier (2000). Given two probability measures Po: 11 (2(M), it holds that
W (10. P1)
(6)
=inf {To Ex-u, (X)I de dep + div (140a) = 0, P4== pa: pt=1 = =11 Re
where [1,((())11 (bx(x), (1(x)) is the langent space norm I E M. Moreover, the optimizer (ve) induces curve (n4} with the following properties.
1. At time O, vo(x)) T Tuo- where -po- (2) is the optimal transportation map from pio to p1-
2. (V1) has zero acceleration, in the sense that v. ex((((z) Tyo (x) is constant in time.
3. For times & > 0, = Law [X + Ho (X) -X) with X ~ po, and the joint law of (X,X + t(T po (X) X)) the optimal coupling of (Jo: pt).
The integrand of equation 6 is called the 'kinetic energy' of the ensemble μr and the minimizing curves (He))ejo.1] are the geodesics of (2MM). One can show by analyzing the Euler-Lagrange equations of equation that the tangent vectors (ve) have the form or (x) = (.b( Con- versely, for any curve of measures (Ve)xEjo.1) that is absolutely continuous in the following sense, there exists de whose gradient weakly solves the transport equation 0,4; div y (v Vo.). Ev- idently Φi itself is the solution of Poisson equation with positive semidefinite coefficents, which already enjoys strong existence, uniqueness, and regularity theorems Evans (2010).
Definition 5 (Absolute continuity). Let (He)sE(o.1) E W2(M) be a curve of measures. The metric derivative (t) is defined,
W2 Ph) (t) lim h-0
h
and μr is absolutely continuous (in W2 metric) at t [o, if [t)(t)
Theorem 6. If (Ho)xe(o.1) is absolutely continuous, then there exists function di : M + R which weakly solves
0ep + div div Vox) =0
and moreover for which Vo E (2:) for every E [0, 1].
The tangent spaces of T, (MM) are defined in the usual way as the linear span of the tangent vector of any absolutely continuous curve passing through μ4 customary to identify tangent vector 0,14 div (M, Vol.) with its drift function Vol. since the metric (x has simple expression in terms of the drifts.
Theorem 7. The space (2(M) is a Riemannian manifold with tangent space structure
T,W((M) [u (²) :6: C2) Ro -0= مام
with inner product (u, v), Ex-n (u(X) v(X)) where (:, )x is the metric on M.
Finally, the Wasserstein gradient WF))) just the representer of & + F + eE) in Tr Wa(M).
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Definition 6 (Wasserstein Gradient). Let F M + R, the Wasserstein gradient is (when it exists) the function WF(()) Tu W2(M) such that for every & E (M).
O (6F, (0F(()) O v(dX).
where (1+))E(-ō55) is any curve with tangent vector & at μ.
We turn our attention the Fisher-Rao geometry, which is more straightforward to explain. A standard reference for this material is the book |Amari (2016). The "admissible" tangent vectors in this geometry are those belonging to the absolutely continuous curves,
Definition 7 (Fisher-Rao absolute continuity). curve (14))E(0.11 is absolutely continuous [0, if its Hellinger metric derivative is finite:
(il(t) = lim ((Uete:He) < <0. c-0
Theorem 8. If (Pe)Ee(0.)) E H(W) is absolutely continuous at [0, then there exists de M + I such that Oeper= = aspee and where 5xxmulaa(Xe =.
The Fisher-Rao geometry essentially based on the fact that for any probability density p(x) M 7 R. the square root density /μ(zz) is an element of the unit sphere S(C) == {or E (M) ||a|| 1}, since it has (Vū(z))) 2 dx = trivially. We endow it with the pullback metric on P(M) induced by the map p vi E (MC). What that means concretely is:
The geodesic connecting Po, A1 is the spherical linear interpolant connecting v/u, /pn: (sin(11 -1)0)
(sin(10) siin())
/4
140 +
M1,
cos (0) = (VHo. v
sin
Each tangent space Tμ H(M) is by definition isomorphic to T / H(M). The isomorphism map is the pushforward or 'differential" di, which is given by
di(E) 0g cl + +EE)
5 2
c=0
and so the inner product on (M)
1 Eeb
(S. xx (di(E), d(())) ya 1 (x)
P
By absolute continuity may write & ap, v By for densities a, M + R, and the Fisher-Roe Rao metric can be rewritten as
f (l)))(=))(ddz))
(ap, Bp)
It is customary to identify the tangent vectors with their densities and to write
TyH(MM = la E (M) E. = 0) c²(w)
with metric 1(.
The Fisher-Rao gradient of functional F is (when it exists) the function FR T, H(M) that satisfies, for E TyH(MM,
O.F(u + cap))
= (VFR)(()))))
==0
Finally, the Wasserstein-Fistt Fishinn-Fiihe-eeRe geometry the one whose tangent spaces are direct sums of the Wasserstein and the Fisher-Rat tangent spaces. In other words, Eisserstein-Fisheer Rao tangent vectors have the form
de div Vo.)
a4/4
Vor T,W(M), as T.H(MM..
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and as usual we identify each tangent vector with its coefficients (Vor, 01). The Sinteerrtein-Fisheer
Rao metric
(el((t) ((Vor. ai), (Ver, 3:)) = $ { (c(=))(()) /FFO u(dz),
and its tangent spaces
(MM) @ (.H(M) [(a, :3, (M) - -, v VΦ) Volco c.)
and the Praserstein--isherrr gradient of J (WFRF((H) (WF(p), ((R)(
B.2 PROOFS OF PROPOSITION AND THEOREM
The proof of Proposition |
follows exactly the same steps the case p NO La).
Proposition (Splats are generalized Bures-Wasseree Wasserstein manifold). Let E (Ro( be centered isotropic mother splat. We denote the set of all splats as,
BW, { (A() + b) P A Raxd b Rd
Then BW (RR") a geodesically convex subset of W2 (RR.), and on this space the Wasserstein metric
reduces to the Bures- Wasserstein metric (Modin 2016 Bhatia et al.) 2019).
= ||b 105-sj1 a||2 + || 12 + || 3 - 2 RlI.
A, R E Rad b
where || || is the Frobenius norm and | ||+ is the nuclear norm.
Proof. Fix & (Ao(.) + 2 and PA1. (14+))++))4))) + (Chei et al. 2025a Theorem 1.16), the map
By Brenier's uniqueness theorem
bb) T(x) b1 45 (ApA Ao) 1/2 An' 12 bb)
is the optimal transport map bo = PA,.b; as it is the gradient of a convex function. It follows
that the geodesic connecting Pa to P1 is P: = ((1 t)id =T) E BW, As is centered and isotropic, the distance (2 P1)
W2 P1) = Ex (A + 6o) - T (Aoo X + 60]1121
= Rp 11:0 - b + (Ao - t Ao) 1 x1|
-10bp
bn 12 + |1
AP An) 1/255///11
where Σx = Cov, (X) = la- Thus W2 P1) = W2(N( Ao ), , Noh,, A A1A))) is the Bures-
Wasserstein metric Bhatia et al. (2019).
We now proceed to the proof of Theorem We develop these calculations at formal level the reader who wishes to understand an entirely rigorous proof may compare to Ambrosio et al. (2008 Chapter 11).
Theorem (Wasserstein--isheer Rao gradient of μ + F(f)). Let μ E Sp,a = P(R" x BW, (R)))
and let (R RP) - IR a functional. Assume that p has a sub-exponential density. Then the Fisher-Raa Rao gradient is given (if it exists) by the formula,
Ff, A, b) Ex (OF((), v)] [x w)).
and the Wasserstein gradient is given (if it exists) by the formula,
W. (.F(ff((, A,b) = ExxpA. (8F(X))
b)7) A-) W (V -ExxpA ((F((), v)) (Ia V. (x) - W W.F(FWDu.AA,) -BExppa ((FF()), v) V log (X))
where the argument of 3F()==5F1) was suppressed.
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Proof. We begin by calculating the Fisher-Rat gradient. Along the way, we develop 'chain rule,' which may aid in future calculations. Specifically we view μ fu as a mapping between manifolds, fir H(Sp.a) + (22))))) and we calculate its differential dfp. TyH(Spaa) + (3(R) For a E T.
{f d. Ic=0
Jsp.d va(v, A, A)bA,,) p(dv, dA, db).
Invoking the definition of the first variation, our chain rule takes the form FeFtta) = (fll)))))lllll))3(44e
= // (8 (8FI(ul()))) a(v, A, pA,((*) da p(de, dA, db)
/ alv, a(v,A, A, [Ex-AAIKFFIUI10 o)]) u(du, dA, db).
We identify the Fisher-Rao gradient by matching the integrand to the definition.
Computing the Wasserstein gradient follows largely the same steps. One checks the following aux- iliary calculations.
b)) A to (A-((2) (b)))) det A| -| A - A - A-
p (A b)) det A
-PA,6(>)) V. Vz log P. b)²) A-T
Vo (o -2) det 1| -(Vp)(A- b)) det Al
v. log.log,.. pA,b(2) (A)(())-
Now fix E T, (SSppa) We calculate the differential d fp [u] as,
d [p(u[(r)= = S up).6(x) divs, (d)) dA, db)
spa
/ (Vo,AD (opa)(=)) u(v, A, b)} p(dv, dAA db)
S uuo(,, A, b)),,((())) pldu, db,
A.b); u(de, dA, db)
v
(Vo (pAs) (z). u. (v,AA))) u(dv, dA, db)
where u (ups UA, ub) are the coordinate function representations u. Thus, setting 5
- divss.
O.F F.F(fpte)) = (8F1ffl (f (2
1/ (AFUAI(+)) (., ())))AA()))) da p(du, dA, db)
// (OFILN(O)))) (VA (PAN) (PAM))) (x)) u,,, A,b)) de u(dv, dA, db)
+ // (SFT)))))))) (V. us), A,b)) do u(du, dA, db).
From this we see,
W. W.FUUNe.AAb) / (AM())) di
WAF(FN(U, A,b) = (-1)==)(077)) ((FFLIIC)) of (o "A,((x) da
(WFUWNC,AAD = (10D://.A)--(/0011--- (OFFL v/ log (A.())))))))))) dr.
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There is an interesting relationship between the geometry of Sp.d and that of the space P(R" R.). Intuitively, splats in BW,(R") (IR are like 'smoothed particles,' this relationship bears out concretely when calculating the Wasserstein and Fisher-Rao gradients in P(RP x Rd). Theorem 9 (Wasserstein-Fisher
gradient of 1t F(Sx) for μ P(RP R2)). Let is P(RP x R and let F : R²) - R be functional. Assume that p has sub-exponential density. Then the Fisher-Rao gradient is given (if it exists) by the formula,
μ F(F)()) x) (ōF(x)), u) [xxv=p((F((X) VY).
and the Wasserstein gradient is given (if it exists) by the formula,
W (fw))( = =F(())
W ((,)(e, - DzF(2))
where the argument of 6F() 6F(/(() was suppressed.
Comparing to the Cessserstinn-isherr. Rao gradients in Theorem I we see that the gradients have the same form as Theorem | the same relationship as can be seen by integrating by parts,
but they are averaged with respect to the splat density. The W W has
W (v) (, b) -Exxaa,, (F,,,,)),,FF())) log (A,((>)] = (xxppall")) (v"",,F(D)))
These formulas follow from the following expressions for dw Su, d s which can be derived by the same arguments as for Theorem
(5Flff), d fu("))
/ (8F(fa) @ @ D'DDF((f u(u, u(de, dx)
df (8FFFll dx).
C EXPERIMENTAL DETAILS AND ADDITIONAL EXPERIMENTS
C.I WALL CLOCK RUNTIMES
We report the average wall clock runtime to train each of the models in Figures run on a 2020 Macbook Pro M1. We also include each model's parameter count and final MSE for comparison.
Model Type
Parameter count
Wall Time (s)
Final MSE
KAN
521
24.2853
0.2129
KAN
5201
121.9557
0.1404
KAN
7861
930. 1017
0.0045
KAN
15601
358.9623
0.0728
KAN
20801
529.0655
0.1781
MLP
801
7,6694
0.2266
MLP
2001
8.7870
0.2165
MLP
4001
10.5255
0.2052
MLP
41001
12.1382
0.0390
MLP
252501
22,5081
0.0087
SPLAT
60
51.9213
0.0575
SPLAT
300
104.6052
0.0016
SPLAT
600
149.7740
0.0005
SPLAT
1200
243.9303
0.0006
SPLAT
1800
295.8554
0.0005
SPLAT
2400
378.7603
0.0008
Although they have significantly fewer parameters, training splat models has more expensive wall clock time. We use autodifferentiatioon to compute gradients with respect to the splat model pa- rameters, and the appearance of A1 and det A-1 in equation equation are particularly in our non-optimized implementation. We expect that a faster implementation with hardcoded gradient formulas will be significantly faster. Further, as be seen in Figures 2 and 3] the splat models converge in far fewer iterations than KAN and MLP, so the reported run times include the time required to take many unnecessary training steps for the splat models.
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C.2 ADDITIONAL APPROXIMATION EXPERIMENTS
The following plots show versions of our function approximation experiment in two different set- tings. The first shows the same experiment, but for fitiing sawtooth target function. The second shows the same experiment, but with {6e J initialized as k-point chebyshev grid.
So
Progress
1800
Figure 4: Fitting sawtooth function in the setting I This is much harder function to fit with interpolation methods. Perhaps surprisingly, splats outperforms the Haar wavelet decomposition, which can exactly fit vertical discontinuities.
Propress
Mal 2t
16
3.
nãe sãe
Eariation
Figure 5: In the setting of] I test the effect of Chebyshev initialization, which is slightly less performant.
24


Reviews:
[{"review_id": "XakO0DcbHp", "reviewer": "Reviewer_5ihC", "summary": "Paper proposes new function approximators, named as Splat Regression Models (SRMs). It is built as a mixture of finite number of localized splats (e.g. anisotropic Gaussians), where they learn positions, shapes, and weights during the training. SRMs resemble 2 layer neural network architecture with different activation function. It is framed in a measure theoretic way, and splat parameter updates are analyzed/derived carefully with Wasserstein-Fisher-Rao (WFR) gradients. Theory also provides regularity inheritance of mother splat and universal approximation bounds for this proposed \"new layer\". SRMs achieve lower error rates with fewer parameters than MLPs/KANs on low-dimensional regressions, inverse problems, and PDE tasks. Paper further recovers common 3D Gaussian splatting framework as a special case of SRM.", "questions": "1. Can you provide wall-clock time, FLOPs, and/or GPU memory comparisons? SRMs has less number of parameters, but it is interesting to see if optimization process or gradient calculation part has some overhead compared to baselines.\n2. Can you add some initial experiments (i) with regularization techniques; (ii) against better tuned kernels, RBF networks; (iii) on high-dimensions or deep splat idea?\n3. I wonder about the analysis on sensitivity of $k$ and initialization strategies of $(A, b, v)$. Can you provide study on that showing different combinations to clarify why the current choices are best?\n4. For PDE regression tasks, why there are not many baselines as there are plenty of them there with physics-informed ideas and neural operators?\n5. Have authors conducted any experiments on 3D gaussian splats? It is proven to be special case theoretically, but is SRM pipeline scalable to this with WFR gradients?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "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.\n2. Theoretical results on regularity and WFR gradient derivations are solid and plausible. Geometric perspective is also well-crafted.\n3. Paper is well-written as it narrates the topic in a very intuitive way and also provides important theoretical results in a rigorous way. \n4. Experiments on low-dimensions/PDEs outperform main baselines with smaller number of parameters, which shows the expressivity of SRMs.\n5. It also frames 3D gaussian splatting as a special case in a principled way.", "weaknesses": "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.\n2. 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 principled regularizer provided. Moreover, the baselines for these tasks could be extended to tuned classical kernels or more PDE regimes. \n3. There are not experiments on high-dimensional tasks. Given that the bound is loose, the higher dimension scaling of method is very intriguing. Authors propose deep splat networks by stacking proposed layers, but it would be beneficial to have some initial set of experiments on this side.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ruAQgY7JEf", "reviewer": "Reviewer_ojFa", "summary": "The paper presents a new class of function approximators parameterized by a mixture of scaled and shifted bump functions that they call splats. The authors introduce these as generalizations of the popular Gaussian Splatting approach. They show that such functions act as universal approximators and present an optimization approach for fitting such models on data through Wasserstein-Fisher-Rao gradient flows. The paper presents examples of applications including physics informed training and empirical results for multiscale training and a physics solver.", "questions": "See weaknesses above.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "1. The presented framework is a principled generalization of Gaussian Splatting style approaches allowing for clear interpretable function families as approximators.\n\n2. For physics-based problems and similar low-dimensional spaces, the inductive bias represented by `splats' is a natural match. \\\n\n3. 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. \n\n4. The WFR lens for splatting reveals interesting future directions into principled regularization approaches compared to the heuristics used currently.", "weaknesses": "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).\n\n2. 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. \n\n3. I believe the inner matrix transformation map $T(.)$ should be $A_0 A_1 A^T_1 A_0^T$ for it to be PD in order for results from Bhatia, 2019 to hold. Please feel free to correct me if my understanding is wrong.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3BQKYTLB4Q", "reviewer": "Reviewer_D7Ct", "summary": "This paper introduces Splat Regression Models (SRMs), a new class of function approximators based on mixtures of anisotropic, heterogeneous bump functions (“splats”). The authors frame SRMs as a generalization of classical nonparametric regression and show that 3D Gaussian Splatting emerges as a special case. They develop a theoretical foundation using Wasserstein–Fisher–Rao (WFR) gradient flows for optimization over the space of mixing measures and demonstrate strong empirical performance on low-dimensional regression and physics-informed tasks, outperforming MLPs and Kolmogorov–Arnold Networks (KANs).", "questions": "The practical scope of this work is currently narrow, and the broader impact hinges on addressing scalability, regularization, and real-world validation.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* This work provides a unified theoretical framework that recovers 3D Gaussian Splatting as a special case of a broader class of function approximators\n* 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 \n* Each “splat” acts as a learnable, anisotropic, localized basis function—effectively learning an adaptive interpolation grid. This offers better interpretability than black-box MLPs and more flexibility than fixed bases\n*  SRMs are shown to be universal approximators, with a quantitative bound on the number of splats needed to approximate Lipschitz functions.", "weaknesses": "* 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 definition of low dimensionality .\n* Experiments use synthetic or toy PDEs (e.g., Allen–Cahn on a unit square). There is no validation on real-world data from physics, biology, or engineering, which significantly weakens any claims about practical relevance.\n* SRMs are closely related to adaptive RBF networks, mixture density networks, and mesh-free methods, but the authors do not adequately contrast their approach with these methods. \n* The authors acknowledge that SRMs are highly expressive and prone to overfitting but they don't provide concrete regularization strategies in the main experiments, which would be critical for deployment in real-world problems.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zk7HtH9Umh", "reviewer": "Reviewer_MULa", "summary": "The paper presents the idea of a splat model which from my understanding is an extension of RBF's of Kernel neural networks. The paper focus on theory and shows that the method has a potential to do well for certain class of problems.", "questions": "1. None of the experiments is done on \"standard\" data sets. \nThe method seems to be doing better than other methods on simple problems but can you show a single problem that is not a toy problem. I am not looking for SOTA, just to see that you are in the same ball park.\n\n2. The paper is dense with theory. What are the major new results here. Some look rather trivial", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is strong theoretically although I **did not verify the theoretical results**.", "weaknesses": "The paper presents very little numerical evidence. This is not necessarily bad and one has to weigh the novelty in the theory if the paper is to be accepted.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)