Geometry-Selective Distribution Comparison via Normal Log-Density Taylor Signatures

MMD and KSD quantify distributional discrepancy but do not explicitly distinguish changes along local tangent and normal directions. We propose the Normal Log-Density Taylor Signature (NLTS), which uses a reference distribution $P$ to define local normal directions and compares reference log-density responses across samples from $P$ and $Q$. Under a local manifold model with small normal noise, we show that the negative-log-density Hessian identifies the local normal subspace, while, for spectrally isolated normal directions, projected scores provide the corresponding first-order Taylor coefficients. Synthetic experiments verify the intended normal–tangent selectivity. On CIFAR-10-C, NLTS remains effective with learned scores, while in single-cell perturbations its power is associated with geometric changes characterized separately by local PCA. Together, these results support the use of reference-defined local density geometry for selective high-dimensional distribution comparison. Code will be made publicly available.