WRF-GS: Weighted Residual Gaussian Splatting

• The paper introduces a novel WRF-GS approach that unifies Gaussian splatting with a weighted-residual least-squares loss optimized via Wasserstein–Fisher–Rao gradient flows.
• It employs a discrete-time gradient descent algorithm to update Gaussian mixture parameters, ensuring computational tractability and convergence.
• The framework enhances interpretability by linking particle descent in a geometric space to localized feature fitting, offering a robust method for inverse problems.

The Weighted Residual Formulation for Gaussian Splatting (WRF-GS) provides a principled, theoretically grounded methodology for function approximation and inverse problems, especially in low-dimensional settings. The WRF-GS unifies the Gaussian Splatting paradigm—where models are constructed as finite mixtures of multivariate Gaussians—with a weighted-residual, least-squares loss that is minimized via Wasserstein–Fisher–Rao (WFR) gradient flows. This synthesis allows the direct interpretation of splat model optimization as a particle descent in a geometric space of mixing measures, yielding a tractable, expressive, and interpretable approach to model fitting and estimation (Daniels et al., 18 Nov 2025).

1. Gaussian Splat Model Definition

The core of the WRF-GS is the Gaussian Splat model, an extension of the classical mixture-of-Gaussians paradigm. A target function $y = f^*(x)$ is approximated by

$$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$

where

• $w_k \in \mathbb{R}$ (or $\mathbb{R}^p$ for vector-valued output) are mixture weights,
• $\mu_k \in \mathbb{R}^d$ are Gaussian centers,
• $\Sigma_k \in \mathrm{PD}(d)$ are symmetric positive-definite covariance matrices.

The model parameters are collectively denoted as $\Theta = \{(w_k, \mu_k, \Sigma_k)\}_{k=1}^K$, and each component $\mathcal{N}(x; \mu, \Sigma)$ is the standard multivariate Gaussian density. Heterogeneous and anisotropic splats are supported, allowing arbitrary scale and orientation for each basis function.

Mixtures of Gaussians constitute a dense subset in $C(\Omega)$ for any compact domain $\Omega \subset \mathbb{R}^d$, thus forms a universal approximator with established rates for the required number of components as a function of approximation error and dimension.

2. Weighted Residual Loss and Empirical Risk

The fitting objective in WRF-GS is a weighted least-squares loss over the data $$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$0:

$$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$1

where the weighting function $$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$2 can encode sampling density, importance, or region-specific emphasis. The residuals at each point are $$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$3.

Alternatively, the loss is interpretable as an $$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$4 risk under a measure $$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$5 with density $$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$6:

$$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$7

In the limit of $$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$8 and uniform sampling, this recovers the continuous risk functional.

3. Wasserstein–Fisher–Rao Gradient Flow Interpretation

Each Gaussian splat can be seen as a "particle" state $$f(x; \Theta) = \sum_{k=1}^K w_k\, \mathcal{N}(x; \mu_k, \Sigma_k)$$9 and collected into an empirical mixing measure:

$w_k \in \mathbb{R}$0

Model evaluation is equivalent to an expectation over the measure $w_k \in \mathbb{R}$1:

$w_k \in \mathbb{R}$2

and the risk $w_k \in \mathbb{R}$3 is interpretable as $w_k \in \mathbb{R}$4 for $w_k \in \mathbb{R}$5.

Minimization of this empirical risk is cast as a gradient flow under the WFR metric using the JKO variational update:

$w_k \in \mathbb{R}$6

The associated Euler–Lagrange equation leads, as $w_k \in \mathbb{R}$7, to a continuity–reaction PDE on the mixing measure:

$w_k \in \mathbb{R}$8

where $w_k \in \mathbb{R}$9 are the WFR gradients of the risk functional.

When pulled back to the finite-dimensional parameter space, this gives explicit ODEs for the model parameters. For scalar outputs, the parameter gradients are:

$\mathbb{R}^p$0

$\mathbb{R}^p$1

$\mathbb{R}^p$2

4. Discrete-Time Optimization Algorithm

Parameter updates in WRF-GS are performed via discrete-time gradient descent, optionally using momentum or Adam. One iteration involves:

• Computing residuals $\mathbb{R}^p$3 at all data locations.
• For each Gaussian component:
  • Compute gradients for weight, mean, and covariance as above.
  • Update parameters with appropriate step sizes ($\mathbb{R}^p$4), optional weight regularization ($\mathbb{R}^p$5), and isotropic covariance regularization ($\mathbb{R}^p$6).

The update step can be summarized as follows:

$\Sigma_k \in \mathrm{PD}(d)$8

This iterative scheme is computationally explicit and leverages the analytic structure of the objective and its gradients.

5. Analytical and Theoretical Properties

The WRF-GS framework admits several rigorous properties:

• Regularity: If the Gaussian kernel is $\mathbb{R}^p$7, then the mixture $\mathbb{R}^p$8 inherits $\mathbb{R}^p$9 bounded derivatives.
• Universal Approximation: On compact domains in $\mu_k \in \mathbb{R}^d$0, Gaussian mixtures are dense in the space of continuous functions; for any $\mu_k \in \mathbb{R}^d$1, a number $\mu_k \in \mathbb{R}^d$2 of Gaussians suffice for sup-norm accuracy.
• Energy Dissipation: Along a WFR gradient flow $\mu_k \in \mathbb{R}^d$3, the objective satisfies a dissipation law:

$\mu_k \in \mathbb{R}^d$4

ensuring non-increasing risk over time.

• Convergence Under Convexity: If $\mu_k \in \mathbb{R}^d$5 is geodesically convex in the WFR metric, the flow converges exponentially to the minimizer.

The translation of functional gradient flows to explicit parametric ODEs establishes both theoretical guarantees and a foundation for practical algorithm design.

6. Illustrative One-Dimensional Example

A concrete 1D example demonstrates the sequence of computations and updates:

• Given data $\mu_k \in \mathbb{R}^d$6, $\mu_k \in \mathbb{R}^d$7, $\mu_k \in \mathbb{R}^d$8.
• Model: $\mu_k \in \mathbb{R}^d$9 components, initialized with $\Sigma_k \in \mathrm{PD}(d)$0, $\Sigma_k \in \mathrm{PD}(d)$1, $\Sigma_k \in \mathrm{PD}(d)$2, $\Sigma_k \in \mathrm{PD}(d)$3.
• Kernel matrix computation, model prediction, and residuals as per the formulas above.
• Gradients computed numerically, e.g., $\Sigma_k \in \mathrm{PD}(d)$4, $\Sigma_k \in \mathrm{PD}(d)$5.
• Parameter update (e.g., with step size 0.1): $\Sigma_k \in \mathrm{PD}(d)$6, $\Sigma_k \in \mathrm{PD}(d)$7, etc.

This iteration illustrates how WRF-GS jointly tunes the Gaussian parameters to fit multi-scale, localized structure in the data using analytically determined gradients.

7. Significance and Interpretation

The WRF-GS scheme enables interpretability via direct correspondence between model components (particles/splats) and localized features in the input space. The identification of the weighted-residual loss with a WFR gradient flow situates Gaussian Splatting within a rigorous geometric framework, clarifying the relationship between the model, loss, and optimization routine. This perspective distinguishes the WRF-GS approach from heuristic kernel density and mixture modeling, endowing it with a unifying theory and robust analytic tools suitable for diverse approximation, estimation, and inverse problems (Daniels et al., 18 Nov 2025).