MoDE: Mixture of Deformation Experts

• MoDE is a dynamic Gaussian Splatting framework that blends outputs from multiple deformation experts, each contributing distinct motion priors.
• It employs a learned temporal gating mechanism with cubic spline interpolation to continuously weight and combine expert predictions.
• Empirical results demonstrate improved dynamic motion expression with modest computational cost, though static scene quality may be impacted.

Mixture of Deformation Experts (MoDE) is a dynamic Gaussian Splatting framework in which multiple canonical deformation experts operate on a shared canonical Gaussian representation and are combined by a learned, temporally continuous gating mechanism (Jin et al., 9 Jul 2026). It was introduced to address a central limitation of dynamic 3D Gaussian Splatting: heterogeneous and spatially varying motion is often poorly matched by any single deformation prior. In MoDE, experts predict candidate deformations for the same canonical Gaussian primitives, and their outputs are blended directly at the Gaussian attribute level before rendering. The formulation is end-to-end and jointly optimized, does not introduce extra training stages, and does not modify the original optimization schedule (Jin et al., 9 Jul 2026).

1. Problem setting and conceptual basis

MoDE is motivated by the observation that existing dynamic Gaussian Splatting methods each encode a strong deformation inductive bias, and no single one dominates consistently across scenes, spatial regions, or time (Jin et al., 9 Jul 2026). The analysis underlying the method identifies three forms of heterogeneity: scene-level, spatial-level, and temporal-level. In that formulation, different scenes favor different deformation formulations, different regions within one scene favor different formulations, and the best deformation formulation may change over time even in one scene (Jin et al., 9 Jul 2026).

The method therefore reinterprets dynamic Gaussian deformation modeling as a Mixture-of-Experts problem. Each deformation formulation is treated as an expert with its own motion prior, and the central task becomes adaptive composition of those priors within one dynamic Gaussian representation (Jin et al., 9 Jul 2026). The paper emphasizes that the relevant failure mode is not insufficient model size, but overcommitment to one deformation parameterization.

A defining feature of MoDE is that expert interaction occurs at the 3D representation level. The experts are inserted directly into the deformable Gaussian Splatting pipeline and are trained jointly on a shared canonical Gaussian representation, rather than being optimized independently and combined only afterward (Jin et al., 9 Jul 2026). This distinguishes MoDE from late ensembling or image-space fusion. The mixture is not an image blend; it is a 3D deformation blend on the same canonical primitive (Jin et al., 9 Jul 2026).

2. Representation and deformation formulation

The shared canonical Gaussian representation is written as

$$\mathcal{G}_i = \{ \mathbf{X}_i, \mathbf{r}_i, \mathbf{s}_i, \sigma_i, \mathbf{C}_i \},$$

where $\mathbf{X}_i$, $\mathbf{r}_i$, and $\mathbf{s}_i$ are the canonical position, rotation, and scale, and $\sigma_i$, $\mathbf{C}_i$ denote opacity/density-type and color/feature attributes as used in the Gaussian representation (Jin et al., 9 Jul 2026).

MoDE restricts experts to methods that share the canonical-deformation pattern. For each Gaussian and time $t$, a model-specific encoding function produces a deformation feature

$\mathbf{f}_i(t) = \Psi(\mathcal{G}_i, t),$

where $\Psi(\cdot)$ depends on the expert type. A common multi-head mapping then predicts attribute offsets: $$\Delta \mathbf{X}_i = \phi_x(\mathbf{f}_i), \quad \Delta \mathbf{r}_i = \phi_r(\mathbf{f}_i), \quad \Delta \mathbf{s}_i = \phi_s(\mathbf{f}_i).$$ The deformed Gaussian attributes are

$\mathbf{X}_i$0

In the MoDE formulation, the deformation outputs discussed are specifically the geometric Gaussian attributes $\mathbf{X}_i$1 (Jin et al., 9 Jul 2026).

The paper highlights three canonical-compatible experts. The 4DGaussians expert uses a shared spatiotemporal HexPlane embedding with a deformation MLP and has a bias toward stable, coherent motion, especially in static or slowly varying regions. The Grid4D expert uses higher-capacity spatiotemporal hash encoding and better captures localized, higher-frequency motion and sharper temporal variation. The E-D3DGS expert introduces learnable per-Gaussian embeddings plus temporal embeddings and provides more local adaptivity, especially for structured but dynamic regions and locally coherent higher-frequency motion (Jin et al., 9 Jul 2026).

These experts have the same canonical input/output structure but different $\mathbf{X}_i$2, hence different motion priors. MoDE’s purpose is to compose those priors rather than commit to one (Jin et al., 9 Jul 2026).

3. Temporal routing and optimization

A central design element is the router. Instead of a per-frame MLP router, MoDE uses a per-Gaussian temporal weight spline (Jin et al., 9 Jul 2026). For each Gaussian, the method learns a discrete set of control weights

$\mathbf{X}_i$3

where $\mathbf{X}_i$4 is the number of deformation experts and $\mathbf{X}_i$5 is the number of control points. Given normalized time $\mathbf{X}_i$6, the continuous expert-score vector is obtained by cubic Hermite spline interpolation: $\mathbf{X}_i$7 This gives continuous temporal evolution of expert preferences and first-order temporal continuity (Jin et al., 9 Jul 2026).

MoDE then applies sparse Top-$\mathbf{X}_i$8 soft routing: $\mathbf{X}_i$9 where $\mathbf{r}_i$0 is the set of indices of the Top-$\mathbf{r}_i$1 elements of $\mathbf{r}_i$2. The paper characterizes this as soft because selected experts are blended, and sparse because non-top-$\mathbf{r}_i$3 experts are zeroed (Jin et al., 9 Jul 2026). The rationale given is that Top-$\mathbf{r}_i$4 reduces weak responses and inter-expert interference, while softmax keeps weighting stable among active experts.

Let $\mathbf{r}_i$5 denote expert $\mathbf{r}_i$6’s deformation output. The final Gaussian attributes are

$\mathbf{r}_i$7

This is the core MoDE equation (Jin et al., 9 Jul 2026). Because all experts share canonical Gaussians, expert allocation is local to each Gaussian primitive and evolves continuously in time.

Optimization is carried out inside the original deformable Gaussian Splatting pipeline. The paper does not provide an explicit MoDE loss equation and does not state additional MoDE-specific regularizers beyond the routing and training strategies; the faithful summary is that MoDE uses the standard reconstruction losses of the underlying Gaussian Splatting training pipeline, adds no extra training stage, and preserves the original training schedule and optimization procedure (Jin et al., 9 Jul 2026).

Two stabilization mechanisms are central. First, MoDE uses asymmetric gradient flow: only the baseline deformation expert updates the canonical Gaussian parameters, and gradients from other experts are blocked at the canonical level (Jin et al., 9 Jul 2026). This is intended to stabilize convergence when several experts with different priors share one canonical representation. Second, MoDE uses random gating warm-up: for an initial number of iterations, experts are activated randomly with equal probability, after which spline-based learned gating is enabled (Jin et al., 9 Jul 2026). This is designed to avoid premature expert collapse early in training.

4. Integration constraints and comparison to MoE-GS

The paper’s broader conceptual contribution is an integration-constraints view of multi-deformation modeling in dynamic Gaussian Splatting (Jin et al., 9 Jul 2026). Under this view, MoDE and Mixture of Experts for Dynamic Gaussian Splatting (MoE-GS) represent two distinct strategies that differ in when and how experts interact during training.

In MoDE, experts interact during the core deformation and reconstruction process, share a canonical Gaussian representation, mix outputs at the deformation or Gaussian attribute level, and are optimized jointly without extra training stages (Jin et al., 9 Jul 2026). Because expert interaction happens on the same canonical primitive set, MoDE naturally supports direct Gaussian-level 3D reconstruction.

MoE-GS, by contrast, trains experts independently first, does not require a shared canonical representation, and combines experts only in a separate routing stage after expert optimization (Jin et al., 9 Jul 2026). In that setting, mixture occurs primarily in image space via a volume-aware pixel router, and because experts live in non-canonical, heterogeneous Gaussian spaces, there is no direct Gaussian correspondence across experts (Jin et al., 9 Jul 2026).

The paper’s Table 1 summarizes the trade-offs as follows: Direct Gaussian-level 3D reconstruction, MoDE ○ and MoE-GS ×; compatibility with non-canonical deformations, MoDE × and MoE-GS ○; training stability, MoDE △ and MoE-GS ○; training efficiency, MoDE ○ and MoE-GS △; expert specialization flexibility, MoDE △ and MoE-GS ○ (Jin et al., 9 Jul 2026). This comparison clarifies the defining structural constraint of MoDE: expert composition occurs at representation level, but only for experts that are compatible with a shared canonical space.

A common misunderstanding is to treat MoDE as merely an ensemble over rendered images. The paper explicitly rejects that view: expert outputs are mixed directly at the Gaussian attribute level, so MoDE performs weighted deformation composition before rendering (Jin et al., 9 Jul 2026). Another common misunderstanding is to assume that MoDE is a general-purpose wrapper for arbitrary dynamic Gaussian methods. In fact, canonical compatibility is a hard design constraint.

5. Empirical behavior, strengths, and limitations

The paper evaluates MoDE on the N3V dataset using canonical experts 4DGaussians, Grid4D, and E-D3DGS (Jin et al., 9 Jul 2026). Compared to single-deformation baselines, MoDE yields more expressive motion in dynamic regions, better reconstruction around articulated or rapidly moving structures, and visibly different motion behavior due to deformation-level composition (Jin et al., 9 Jul 2026). Figure 1 is described as showing that MoDE can increase per-Gaussian motion flexibility compared with the more rigid motion of a single expert such as 4DGaussians.

The main MoDE PSNR results are reported as follows. For the 4DGaussians baseline, PSNR is $\mathbf{r}_i$8; with E-D3DGS expert it is $\mathbf{r}_i$9, and with Grid4D expert it is $\mathbf{s}_i$0. For the Grid4D baseline, PSNR is $\mathbf{s}_i$1; with 4DGaussians expert it is $\mathbf{s}_i$2, and with E-D3DGS expert it is $\mathbf{s}_i$3. For the E-D3DGS baseline, PSNR is $\mathbf{s}_i$4; with 4DGaussians expert it is $\mathbf{s}_i$5, and with Grid4D expert it is $\mathbf{s}_i$6 (Jin et al., 9 Jul 2026). The direct implication is that MoDE improves the average PSNR for 4DGaussians- and Grid4D-based systems, but not consistently for E-D3DGS-based ones.

The paper’s explanation is that MoDE helps when the baseline deformation prior is too narrow (Jin et al., 9 Jul 2026). This is especially visible for 4DGaussians, whose HexPlane-induced smoothness can be too rigid; MoDE with E-D3DGS adds more local per-Gaussian flexibility, improving dynamic motion expressiveness (Jin et al., 9 Jul 2026). The analysis is more cautious for E-D3DGS: the baseline already has strong local deformation modeling, and adding another expert under a shared canonical representation can overconstrain optimization rather than help (Jin et al., 9 Jul 2026).

The dynamic/static trade-off is one of the method’s most important practical lessons. The ablation in Figure 2 is described as showing that dynamic regions improve, while static regions may degrade (Jin et al., 9 Jul 2026). Because N3V contains large static areas and PSNR is computed over the full image, static-region degradation can offset dynamic-region gains. This explains why improved motion expressiveness in dynamic regions does not always produce a whole-image metric improvement.

The reported cost-performance comparison further reflects the method’s intended niche. The paper gives the following values: 4DGaussians, $\mathbf{s}_i$7 PSNR and $\mathbf{s}_i$8 h; 4DGaussians w/ E-D3DGS, $\mathbf{s}_i$9 PSNR and $\sigma_i$0 h; 4DGaussians w/ Grid4D, $\sigma_i$1 PSNR and $\sigma_i$2 h; MoE-GS $\sigma_i$3, $\sigma_i$4 PSNR and $\sigma_i$5 h (Jin et al., 9 Jul 2026). MoDE therefore gives modest gains at modest extra cost, whereas MoE-GS gives larger gains at much larger cost (Jin et al., 9 Jul 2026).

The limitations discussed in the paper are explicit. First, MoDE only works for experts that share a canonical Gaussian formulation; non-canonical methods like trajectory-based or keyframe-based models cannot be directly inserted (Jin et al., 9 Jul 2026). Second, because experts are jointly trained on one canonical representation, they are not free to specialize completely, and their interaction can constrain complementary behavior (Jin et al., 9 Jul 2026). Third, shared optimization can be unstable unless carefully managed, which motivates blocked gradients to canonical Gaussians from non-baseline experts and random gating warm-up (Jin et al., 9 Jul 2026). Fourth, improved dynamic modeling can come at the cost of static-region quality, depending on the baseline and scene composition (Jin et al., 9 Jul 2026).

6. Broader interpretations, analogues, and disambiguation

Although the direct named method is the dynamic Gaussian Splatting framework above, the phrase “Mixture of Deformation Experts” also serves as a useful interpretive lens for several adjacent lines of work. In mechanical metamaterials, “Harvesting Deformation Modes for Micromorphic Homogenization from Experiments on Mechanical Metamaterials” does not use the term Mixture of Deformation Experts, but presents a deterministic kinematic superposition in which a smooth mean displacement field, one or more long-range correlated fluctuation modes, and a residual local microfluctuation field are separated; in that synthesis, the patterned fluctuation modes act like a small set of deformation “experts,” while their spatial amplitudes act like gating or mixing fields (Maraghechi et al., 2024). This suggests a physics-based analogue of MoDE grounded in micromorphic computational homogenization rather than probabilistic routing.

In face alignment, “Robust Face Alignment Using a Mixture of Invariant Experts” is described as highly relevant to a Mixture of Deformation Experts interpretation because each expert is specialized to a different subset of the joint space of pose and expressions, the current shape is softly assigned to prototype-conditioned experts, and outputs are aggregated by expert weights (Tuzel et al., 2015). The terminology there is “Mixture of Invariant Experts,” not Mixture of Deformation Experts, but structurally it is a mixture model over deformation regimes induced by pose and expression.

In deformable image registration, SHMoAReg is described as a close variant of a Mixture of Deformation Experts because its decoder contains a Spatial Heterogeneous Mixture of Experts module in which multiple expert branches directly predict deformation-field components, routing is voxel-level, and prediction is direction-specific for $\sigma_i$6 (Zheng et al., 24 Sep 2025). The strongest MoDE-like element is that the experts output deformation predictions rather than generic latent features.

The acronym itself is also overloaded. “MoDE: Effective Multi-task Parameter Efficient Fine-Tuning with a Mixture of Dyadic Experts” uses the same abbreviation for Mixture of Dyadic Experts, a LoRA-based PEFT method rather than a deformation model (Ning et al., 2024). That naming collision is purely terminological. In the literature summarized here, the direct usage of Mixture of Deformation Experts denotes a representation-level answer to multi-deformation modeling in dynamic 3D Gaussian Splatting (Jin et al., 9 Jul 2026).

Across these uses and analogues, a recurring design pattern is visible: a shared representation, multiple specialized deformation priors or templates, and a routing or mixing mechanism that varies across tokens, Gaussians, voxels, or space-time. In the direct MoDE formulation, however, that pattern is instantiated specifically as efficient, representation-level, canonical-space deformation composition under the structural constraints of dynamic Gaussian Splatting (Jin et al., 9 Jul 2026).