3D Gaussian Splatting (3DGS) provides a new framework for novel view synthesis and has inspired substantial subsequent research. In this paper, the authors improve its fundamental paradigm and formulation, arguing that as an unnormalized mixture model, it needs to be neither Gaussians nor splatting. They propose Student Splatting and Scooping (SSS), a new non-monotonic mixture model consisting of positive and negative Student's t distributions with learnable tail-fatness parameters, learned by a principled SGHMC sampling approach. Experiments demonstrate that SSS achieves matching or better quality with similar numbers of components, and obtains comparable results while reducing the component number by as much as 82%.

3D Gaussian Splatting (3DGS) provides a new framework for novel view synthesis and has sparked significant research in neural rendering. Since 3DGS is becoming foundational to many models, improvements to it offer substantial benefits. The success of 3DGS lies in its dual strengths: "Gaussians can approximate an arbitrary density function hence good expressivity" and "splatting provides a flexible way of identifying only the relevant Gaussians" for efficient rendering. However, the framework still suffers from insufficient expressivity and low parameter efficiency, i.e. needing a large number of components.

Since 3DGS fundamentally fits "a 3D mixture model to a radiance field," the authors contend it should not be restricted to Gaussians or splatting. They propose Student Splatting and Scooping (SSS), which replaces Gaussians with flexible Student's t distributions with one additional degree of freedom, extends splatting to include both positive and negative densities, and introduces a principled sampling approach based on Stochastic Gradient Hamiltonian Monte Carlo (SGHMC). SSS contains a simple yet strong and non-trivial generalization of 3DGS and its variants, achieving state-of-the-art rendering quality while demonstrating high parameter efficiency.

The key contributions of this work are threefold. First, a new mixture model using flexible Student's t distributions that can adaptively range from Cauchy (fat-tailed) to Gaussian distributions through a learnable degree-of-freedom parameter, providing greater expressivity than fixed Gaussian components. Second, the introduction of negative density components ("scooping") that subtract color densities from the mixture, enabling superior parameter efficiency particularly for complex topologies. Third, a principled SGHMC-based sampling scheme that addresses the tight parameter coupling among the degree of freedom, mean, and covariance.

3D reconstruction and novel view synthesis are long-standing research topics in computer vision, with traditional methods including Multi-View Stereo (MVS) and Structure from Motion (SfM). Deep learning has brought important changes to the field. Neural Radiance Field (NeRF) proposes to implicitly encode the radiance field of a 3D object or scene into a neural network. Many extensions exist: Mip-NeRF for quality improvement, Plenoxels and Instant-NGP for acceleration, D-NeRF for dynamic scenes, and DreamFusion for generation tasks. However, the biggest drawback of NeRF is that the ray casting process for rendering is time consuming.

3DGS solves the above rendering speed problem by replacing volume rendering with a differentiable rasterization method, using 3D Gaussians as the primitive for the splatting method. Subsequent work improves various aspects: quality (GS++, Mip-Splatting), dynamic scenes (4D Gaussian Splatting, Deformable 3D Gaussians), and generation (DreamGaussian). Some research improves the fundamental paradigm, including FreGS, 3DGS-MCMC, and work focusing on optimization and adaptive density control. Recent efforts explore different primitives other than 3D Gaussians, such as 2DGS for surface reconstruction, GES using a generalized exponential kernel, and 3DHGS decomposing one Gaussian into two half-Gaussians.

The present work positions itself "among the few recent efforts in improving the fundamental formulation of 3DGS," but differs from existing approaches in three key ways: it uses more expressive and flexible distributions, the 3D Student's t distribution; it employs both positive and negative densities rather than positive-only monotonic mixtures; and it proposes a principled sampling approach rather than relying on standard gradient-based optimization with heuristic density control strategies.

3DGS essentially fits an unnormalized 3D Gaussian mixture model to a radiance field. The formulation requires that all weights satisfy w_i > 0, with each Gaussian component associated with an opacity o ∈ [0, 1] and color c represented by spherical harmonics. The rendering equation reveals that the projected C(u) can be seen as a 2D Gaussian mixture, except that a component's weight is also a function of other components, introducing additional cross-component interactions. Gaussians are chosen because they are closed under affine transformation and marginalization of variables, enabling fast computation via projection. Critically, 3DGS is a monotonic mixture as it is additive, i.e. w_i > 0, and all existing variants follow this paradigm.

The authors propose an unnormalized t-distribution mixture model, where a t-distribution is defined by a mean (location) μ ∈ R³, a covariance matrix Σ (shape), and a degree of freedom (tail-fatness) ν ∈ [1, +∞), along with opacity and color. The scalar normalization constant can be dropped to facilitate learning. The choice of t-distribution is driven by two factors. First, "t-distribution is a strong generalization of Gaussians": when ν → 1, the t-distribution approaches Cauchy; when ν → ∞, it approaches Gaussian. Since "Cauchy is fat-tailed, it can cover larger areas with higher densities than Gaussians therefore potentially reducing the number of components," and since μ, Σ, and ν are all learnable, SSS becomes a mixture of components learned from an infinite number of distribution families.

Second, the t-distribution also provides good properties similar to Gaussians, specifically being closed under affine transformation and marginalization. The authors derive a closed-form 2D t-distribution for projection, which "enables us to easily derive the key gradients for learning" — "unlike existing research also using alternative mixture components but requiring approximation." This mathematical tractability is critical for maintaining the computational advantages of the splatting framework while switching to a more expressive distribution family.

While monotonic mixture models are powerful, a non-monotonic mixture model has been proposed by introducing negative components, allowing the model to operate with both positive and negative densities. In existing approaches, a squared mixture formulation "increases the model evaluation complexity to O(n²)," making it significantly slower. The authors instead "still use the positive/negative formulation but with w ∈ R instead of w > 0," allowing negative components directly. Though "this might cause issues as the formulation is then not well defined with negative components," the density can be parameterized in an energy-based form which is well defined. During learning, the opacity is parameterized by a tanh function, ensuring positive and negative components can dynamically change signs.

Introducing negative t-distributions can enhance the representation power of the mixture. A negative component is "equivalent to removing its color from the mixture" and is "particularly useful in subtracting colors." This is especially beneficial for complex topologies — for instance, a torus shape that would require many positive-only components to approximate can be represented with "merely two components (one positive and one negative)" using the scooping approach. The negative component effectively "scoops out" a region of density from a positive component, enabling fewer components to fit complex shape topologies, which is the fundamental source of SSS's parameter efficiency.

Training SSS involves learning more tightly coupled parameters compared with 3DGS, namely among ν, μ, and Σ. The authors speculate this occurs "because changing ν in learning is changing the family of distributions within which we optimize μ and Σ." To address this, they propose a sampling scheme based on Stochastic Gradient Hamiltonian Monte Carlo (SGHMC), parameterizing the posterior distribution as an energy-based model where the loss function serves as the energy. Since the positive/negative formulation is not a well-defined distribution, "using an energy function circumvents this issue and prescribes the high density regions of good parameters." The momentum term in SGHMC creates "frictions for each dimension," enabling adaptive learning for each parameter.

The practical implementation uses SGHMC on the mean μ and Adam on the other parameters. The update equation is modified to include adaptive friction F and noise N for μ, with a sigmoid function that switches on/off the friction and noise, activating for components with opacity lower than 0.005. Learning begins by initializing with a sparse set of SfM points, running the update "without friction for burn-in for exploration," then "running the full sampling for exploitation." During burn-in, noise is multiplied by the covariance Σ of the component; after burn-in, anisotropy is maintained by friction due to momentum. Components with near-zero opacity are recycled by relocating them rather than simply removed, with the relocation designed to minimize pixel-wise color changes over the whole domain.

Following existing research, experiments employ 11 scenes from 3 datasets: 7 public scenes from Mip-NeRF 360, 2 outdoor scenes from Tanks & Temples, and 2 indoor scenes from Deep Blending. Evaluation uses three standard metrics: Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Metric (SSIM), and Learned Perceptual Image Patch Similarity (LPIPS). Comparison is conducted against 3DGS, GES, 3DHGS, Scaffold-GS, Fre-GS, 3DGS-MCMC, and Mip-NeRF. SSS achieves overall the best results on 6 of the 9 metrics, and the second best on 2 metrics.

The parameter efficiency of SSS is a key advantage. Across scenes, SSS uses significantly fewer components than baseline methods: for example, 468k vs 1.1M, 364k vs 2.6M, 208k vs 3.4M, and 96k vs 2.5M components compared to original 3DGS. Most notably, "SSS achieves 23.6 PSNR with merely 180k components, already surpassing Mip-NeRF, 3DGS and GES" which use millions of components or expensive ray marching. The component reduction reaches up to 82% while maintaining comparable or superior rendering quality, demonstrating that the combination of Student's t distributions and negative components fundamentally improves the expressivity per component.

On specific benchmarks, SSS demonstrates consistent improvements. For the Train scene from Tanks & Temples, SSS achieves 23.23/0.844/0.170 in PSNR/SSIM/LPIPS, where the second-best method 3DHGS achieves 22.95/0.827/0.197 — representing improvements of 1.22%, 2.05%, and 13.7% respectively. The perceptual quality improvement (LPIPS) is particularly notable, suggesting that the heavier tails of the Student's t distribution better capture fine details and complex textures that are perceptually important. These improvements are consistent across both indoor and outdoor scenes, indicating the robustness of the approach.

The ablation study systematically evaluates each component through three configurations: "SGD + positive t-distribution" (Student's t with standard gradient descent), "SGHMC + positive t-distribution" (Student's t with proposed sampling, no negative components), and the full SSS model. Results demonstrate a clear progression: "By only replacing Gaussians with Student's t distributions (SGD + t-distribution), it already outperforms Mip-NeRF, 3DGS, and GES, demonstrating improved expressivity." "Further with SGHMC, it is already the best method, showing the advantage of the proposed sampling." On the Tanks & Temples dataset: SGD + positive t-distribution achieves 23.80 PSNR; SGHMC + positive t-distribution achieves 24.53 PSNR; the full model achieves 24.87 PSNR. The authors conclude that "while individual techniques alone might provide merely small improvements, SSS as a whole is the state-of-the-art."

The authors proposed Student Splatting and Scooping (SSS), a new non-monotonic mixture model, consisting of positive and negative Student's t distributions, learned by a principled SGHMC sampling. SSS contains a simple yet strong and non-trivial generalization of 3DGS and its variants. SSS outperforms existing methods in rendering quality, and shows high parameter efficiency, e.g. achieving comparable quality with less than one-third of components. The work demonstrates that questioning the fundamental assumptions of a successful framework — the choice of distribution, the sign of weights, and the optimization method — can yield substantial improvements.

SSS has acknowledged limitations. "Its primitives are restricted to symmetric and smooth t-distributions, limiting its representation." The "sampling also needs hyperparameter tuning such as the percentage of negative components." In the future, the authors plan to combine other distribution families (e.g. Laplace) with t-distribution to further enhance expressivity, and to make the SGHMC self-adaptive to achieve better balances between positive and negative components.