From Gaussian Tokens to Optical Measures: Refinement-Robust Learning on 3DGS

Adaptive refinement makes 3D Gaussian Splatting (3DGS) non-canonical: splitting, merging, and pruning alter primitive count without changing radiance. Yet prevailing set architectures uniformly weight Gaussians, leaving downstream representations vulnerable to refinement. We establish refinement invariance as a core representation symmetry and derive Optical Measure Attention from volume rendering physics. Because optical depth ${\tau }_i=-\log (1-{\alpha }_i)$ is strictly additive under volume compositing, we model scenes as finite measures $\mu ={\sum }_i{\tau }_i{\delta }_{z_i}$, injecting $\log {\tau }_i$ as an attention logit bias and ${\tau }_i$ into normalized moments. This guarantees exact invariance under co-located mass-preserving splits and proves an $\mathcal{O}(r^2)$ error bound under child displacements of radius $r$. Under factor-8 refinement, it preserves 100.00% of Conv predictions versus 94.18% for ordinary attention (median logit drift $15.80\%\to 0.52\%$) and 99.66% vs. 91.44% on query-based pooling; In metric retrieval, it lifts top-10 neighbor retention from $\sim$77% to 96% across six decimation pipelines. Finally, an online streaming kernel executes exact quadrature with memory independent of Gaussian cardinality, reducing peak memory by $18.48\times$. These results establish a principled paradigm: integrate continuous optical measures rather than count discretization primitives.