Mixture-of-Experts as Soft Clustering: A Dual Jacobian-PCA Spectral Geometry Perspective

Mixture-of-Experts (MoE) architectures are widely used for efficiency and conditional computation, but their effect on the geometry of learned functions and representations remains poorly understood. We study MoEs through a geometric lens, interpreting routing as soft partitioning into overlapping expert-local charts. We introduce a Dual Jacobian-PCA spectral probe that analyzes local function geometry via Jacobian singular value spectra and representation geometry via weighted PCA of routed hidden states. Using a controlled MLP-MoE setting with exact Jacobian computation, we compare dense, Top-k, and fully soft routing under matched capacity. Across random seeds, MoE routing consistently reduces local sensitivity: expert-local Jacobians show smaller leading singular values and faster spectral decay than dense baselines. Weighted PCA reveals that expert-local representations distribute variance across more principal directions, indicating higher effective rank. We further observe low alignment among expert Jacobians, suggesting decomposition into low-overlap expert-specific transformations. Routing sharpness modulates these effects: Top-k routing yields more concentrated, lower-rank expert structure, while fully soft routing produces broader, higher-rank representations. Experiments on a 3-layer transformer with WikiText confirm curvature reduction on natural language and show lower cross-expert alignment for Top-k routing. These findings support interpreting MoEs as soft partitionings of function space that flatten local curvature while redistributing representation variance, yielding testable predictions for expert scaling, hallucination reduction, and ensemble diversity.