Softmax-GS: Overlap-Aware 3D Gaussian Splatting

• The paper introduces Softmax-GS, a novel method using softmax-based overlap competition and per-Gaussian learnable parameters to address view inconsistency and diffuse boundaries.
• It enforces order invariance and preserves transmittance, ensuring stable compositional rendering even with overlapping Gaussians.
• Benchmark evaluations show that Softmax-GS improves PSNR, reduces artifacts, and achieves efficient rendering speeds compared to vanilla 3D GS.

Softmax-GS is a 3D Gaussian Splatting method for novel view synthesis that addresses two failure modes of vanilla 3D GS: view inconsistency caused by overlapping Gaussians and diffuse object boundaries caused by long Gaussian tails. It does so by enforcing a softmax-based competition in overlapping regions between two Gaussians, while learning per-Gaussian parameters that span a continuous spectrum from smooth color blending to crisp, well-defined boundaries. Its formulation is constructed to preserve order invariance for any two overlapping Gaussians and to keep the output transmittance unchanged irrespective of the extent of overlapping, so that overlap handling does not introduce undesirable discontinuities in the rendered output (Ziwen et al., 30 Apr 2026).

1. Problem setting and motivation

Vanilla 3D Gaussian Splatting assumes that trimmed Gaussians have disjoint support, so they can be sorted and composited via

$$I(\mathbf x)=\sum_{k=1}^K c_k\,a_k\prod_{j<k}(1-a_j), \qquad a_k=\int_0^L o_k(\mathbf x,l)\,dl.$$

In practice, Gaussians overlap. The reported consequences are twofold: overlaps break the no-overlap assumption and cause sudden “popping” as viewpoint changes, while long Gaussian tails produce overly smooth object boundaries and therefore require many small Gaussians to approximate sharp edges.

Softmax-GS directly relaxes the no-overlap assumption. Rather than treating overlap as an anomaly outside the compositing model, it introduces a learned competition rule in overlap regions and a separate mechanism for per-Gaussian boundary sharpening. This is significant because the two issues are coupled: overlap handling alone does not resolve diffuse edges, and edge sharpening alone does not resolve arbitrary layering artifacts. The ablations reported for simple geometries make that division explicit (Ziwen et al., 30 Apr 2026).

2. Softmax competition in overlapping regions

For any two overlapping Gaussians $j,k$ with Gaussian-exponent maps $p_j,p_k$, where higher $p$ means stronger local contribution, Softmax-GS defines per-sample softmax weights

$$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$

Here $\beta$ is learnable and controls competition strength. The limiting cases are stated explicitly: $\beta=0$ gives equal blending $w_j=w_k=1/2$, whereas large $\beta$ yields winner-take-all behavior.

If each Gaussian projects to a 2D absorbance $a_i$, the softmax-weighted absorbances are

$j,k$0

These modified absorbances replace the original $j,k$1 in the splatting integrator: $j,k$2

The role of this construction is not merely to reweight colors. It enforces competition at the opacity level, so the overlap region becomes a learned compromise between mutual blending and mutual exclusion. This is the central mechanism by which Softmax-GS converts ambiguous overlap into a differentiable, controllable interaction (Ziwen et al., 30 Apr 2026).

3. Learnable controls for blending, boundaries, and depth separation

Softmax-GS introduces three per-Gaussian learnable scalars: $j,k$3, $j,k$4, and $j,k$5.

The parameter $j,k$6 controls boundary sharpness through the Generalized Exponential Function. For a standard 1D Gaussian along a ray, the exponent is $j,k$7, but Softmax-GS uses

$j,k$8

Larger $j,k$9 yields sharper falloff. This mechanism is independent of overlap competition and targets the diffuse-boundary problem directly.

The parameter $p_j,p_k$0 controls the strength of the softmax competition. Increasing $p_j,p_k$1 makes the competition crisper and thus sharpens boundaries specifically where Gaussians overlap.

The parameter $p_j,p_k$2 controls how overlap influence decays as Gaussians separate in depth. For two Gaussians at depths $p_j,p_k$3, the decay factor is

$p_j,p_k$4

and the final interpolated absorbance is

$p_j,p_k$5

Thus, distant Gaussians revert to standard non-overlap behavior.

Taken together, the three parameters define the method’s reported “blend or bound” spectrum. $p_j,p_k$6 sharpens individual Gaussian boundaries, $p_j,p_k$7 regulates competition strength in shared support, and $p_j,p_k$8 suppresses competition when overlap is not physically significant because of depth separation (Ziwen et al., 30 Apr 2026).

4. Order invariance and transmittance preservation

A key design goal is that, for any small set of overlapping Gaussians, permuting their sort order or changing their overlap extent should not change the final transmittance

$p_j,p_k$9

Softmax-GS imposes two constraints on each pairwise update $p$0:

1. preserve the ratio induced by softmax competition,

$p$1

2. maintain the original transmittance

$p$2

Solving these equations yields closed-form corrected absorbances. After depth-based interpolation, a global scaling $p$3 is then applied so that

$p$4

The paper describes this as a two-step correction guaranteeing both invariance to processing order and exact matching of the non-overlap transmittance. This aspect distinguishes Softmax-GS from a naive overlap-aware blending rule: the goal is not only sharper overlap handling, but also compositional stability under reordering and varying overlap extent (Ziwen et al., 30 Apr 2026).

5. Rendering pipeline, optimization, and complexity

Initialization seeds Gaussians from SfM points or densifies from a coarse surface. The initial settings are $p$5, $p$6, and $p$7. During forward rendering, for each pixel or ray, the method selects the nearest $p$8 Gaussians, with $p$9 given as an example, computes 2D splat projections $$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$0, and applies the sequential Softmax-GS update from nearest to farthest.

For backpropagation, intermediate $$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$1 values over the $$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$2 entries are cached to enable linear-time backprop. Trainable parameters per Gaussian are the position $$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$3, shape $$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$4, color $$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$5, base opacity $$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$6, and the three Softmax-GS scalars $$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$7. Optimization uses Adam or AdamW for 30K steps, and 10K steps for toy patterns.

The reconstruction loss is

$$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$8

A regularizer on $$w_k \;=\;\frac{\exp\bigl(\beta\,p_k\bigr)}{\exp\bigl(\beta\,p_j\bigr)\;+\;\exp\bigl(\beta\,p_k\bigr)}, \qquad w_j \;=\;1 - w_k.$$9 and $\beta$0 is added along each ray: $\beta$1 with the variance weighted by inverse distance to the front-most Gaussian and $\beta$2. Optional small $\beta$3 or size-pruning regularizers on $\beta$4 are also noted.

The computational claim is that Softmax-GS keeps linear-time complexity per ray. On the “train” scene it runs at approximately $\beta$5 of 3D GS speed, and a “mini” version with approximately $\beta$6 fewer Gaussians out-speeds vanilla 3D GS at 150 FPS (Ziwen et al., 30 Apr 2026).

6. Ablations and benchmark behavior

The ablation study on simple 2D toy patterns—circle4, rect4, and oval4—uses only 4 Gaussians optimized for 10K steps. Four settings are reported.

Setting	Reported PSNR	Reported behavior
3D GS	$\beta$7–20 dB	poor edges
Sharp edge only ($\beta$8 only)	$\beta$9–31 dB	controls individual blur but still enforces arbitrary layering
Softmax only ($\beta=0$0 only)	$\beta=0$1–21 dB	blends but boundaries still diffuse
Softmax-GS ($\beta=0$2)	$\beta=0$3–29 dB	best of both worlds

These results indicate that neither boundary sharpening alone nor overlap competition alone is sufficient. Varying the Gaussian count from $\beta=0$4 to $\beta=0$5 shows that Softmax-GS maintains a consistent advantage over vanilla 3D GS and the sharp-only variant across all budgets.

On real-world benchmarks—Mip-NeRF360, Tanks & Temples, and Deep Blending—the paper reports three operating regimes. Under a sparse budget with $\beta=0$6 of the original Gaussians, 3D GS$\beta=0$7 achieves PSNR $\beta=0$8, SSIM $\beta=0$9, LPIPS $w_j=w_k=1/2$0, with 104 K splats, whereas Softmax-GS$w_j=w_k=1/2$1 achieves PSNR $w_j=w_k=1/2$2, SSIM $w_j=w_k=1/2$3, LPIPS $w_j=w_k=1/2$4, with 105 K splats, corresponding to approximately $w_j=w_k=1/2$5 dB. Among sparse-scene specialized methods, GES and 3DCS use approximately $w_j=w_k=1/2$6–$w_j=w_k=1/2$7 M Gaussians and reach PSNR approximately $w_j=w_k=1/2$8–$w_j=w_k=1/2$9, while Softmax-GS$\beta$0, at approximately $\beta$1 M Gaussians, reaches PSNR $\beta$2, SSIM $\beta$3, LPIPS $\beta$4. With full Gaussian counts of $\beta$5–$\beta$6 M splats, 3D GS achieves PSNR approximately $\beta$7, SSIM approximately $\beta$8, LPIPS approximately $\beta$9, whereas Softmax-GS achieves PSNR $a_i$0, SSIM $a_i$1, LPIPS $a_i$2, with the same rendering speed within $a_i$3. A GS-MCMC refined optimization combined with Softmax reaches PSNR $a_i$4, SSIM $a_i$5, LPIPS $a_i$6 (Ziwen et al., 30 Apr 2026).

The view-consistency evaluation uses RAFT flow on rendered frame pairs. Softmax-GS reduces both MSE and flow-warp error $a_i$7 compared with 3D GS and StopThePop. The reported interpretation is that the method improves both reconstruction quality and parameter efficiency while specifically addressing popping artifacts caused by overlap.

7. Terminological scope and relation to other “GS” formulations

The name “Softmax-GS” in current usage refers specifically to the 3D Gaussian Splatting method described above, not to the broader Gumbel-Softmax family and not to softmax gating in mixture-of-experts models. That distinction matters because several nearby literatures use similar abbreviations for different technical objects.

In discrete optimization and latent-variable modeling, “GS” typically denotes Gumbel-Softmax or Straight-Through Gumbel-Softmax. Examples include the Gumbel-softmax Optimization framework for combinatorial optimization on graphs (Liu et al., 2019), the Straight-Through Gumbel-Softmax estimator and the decoupled forward/backward-temperature variant (Shah et al., 2024), visual referential games trained with ST-GS (Denamganaï et al., 2020), integer user association with a GS-based GNN (Lyu et al., 5 Jun 2025), and low-precision scalar quantization for LLMs via Gumbel-Softmax sampling in GSQ (Dadgarnia et al., 20 Apr 2026). In a separate theoretical line, “softmax gating” refers to gating probabilities in multinomial logistic or Gaussian mixture-of-experts models rather than to overlap handling in Gaussian splatting (Nguyen et al., 2023, Nguyen et al., 2023).

A common source of confusion is therefore purely terminological: Softmax-GS is not a synonym for Gumbel-Softmax, ST-GS, or softmax gating. Its technical contribution lies in overlap-aware compositing and boundary control for 3D Gaussian Splatting, with learnable $a_i$8, $a_i$9, and $j,k$00 parameters, order-invariant pairwise updates, and explicit transmittance preservation (Ziwen et al., 30 Apr 2026).