Direct Conditional Parameterization for N-Dimensional Splatting

N-dimensional splatting extends 3D Gaussian Splatting with conditioning variables such as view direction and time to model view-dependent and dynamic effects. Each primitive is lifted into a joint distribution over 3D position and conditioning variables, and at render time is conditionally sliced at the query to recover a 3D primitive whose mean, opacity, and covariance vary continuously with view or time. N-DGS (covering 6DGS and 7DGS) and UBS together represent the leading N-dimensional splatting formulations across Gaussian and Beta kernels, but share the same conditional-slicing step: all three effects are derived from the joint covariance matrix per primitive in every forward pass, requiring matrix inversions and regression-matrix multiplications that scale with primitive count. The conditional-slicing step is kernel-agnostic, so a single improvement carries across families. We introduce direct conditional parameterization, replacing the covariance-derived form with explicit per-effect parameters: a Cholesky precision factor for opacity and a displacement matrix with learnable per-dimension coupling for position. Applied to both kernels, the parameterization yields dGS (Gaussian) and dBS (Beta). Across five static and dynamic benchmarks, dGS and dBS match or exceed their covariance-derived counterparts on quality (up to +1.26 dB in the main MCMC evaluation) while running 6.9-7.7x faster at the slicing step and up to 2.66x faster end-to-end rendering. Direct conditioning is a kernel-agnostic drop-in replacement for covariance-derived conditioning in N-dimensional splatting.