Split Then Select: Moment-Preserving Density Control for Generalized Primitive Splatting

Primitive splatting has become a powerful representation for efficient differentiable rendering, but existing density-control strategies remain largely heuristic and often suffer from uncontrolled primitive growth. In this work, we present Moment-Preserving Density Control (MPDC), a general theoretical framework for densification in primitive splatting. Unlike conventional density-control strategies that select first and split later, MPDC follows a split-then-select paradigm. Our key insight is that a principled split should first preserve the current rendering, rather than immediately perturb the image. By matching local moments, MPDC decomposes a primitive into multiple offspring while keeping the rendered output, and hence the loss, unchanged up to higher-order error. Although such a split does not directly reduce the loss, it changes the parameter space: a saddle point in the original parameterization may no longer remain a saddle point after splitting. This provides a distinct saddle-escaping mechanism from prior approaches that move primitives along negative-curvature directions. Building on this view, we derive closed-form splitting rules for different primitive parameterizations and then select only primitives whose moment-preserving splits reduce an upper bound of the loss through a splitting-matrix criterion. Experiments on three diverse primitive splatting methods show that MPDC substantially reduces primitive counts without sacrificing rendering quality, while improving memory efficiency and rendering speed. Code will be released upon publication.