Over-fitted Gaussian mixtures can contain many parameter configurations with nearly identical densities, and the order of that density cancellation controls how quickly parameters can be learned. Ho and Nguyen encoded these cancellations in a polynomial system and conjectured its exact order as a function of cluster size. Zhang and Zhang now prove the conjecture for every cluster size M ≥ 2.
The proof hinges on a common positive-variance shift that turns the coefficient equations into Gaussian moment matching. The authors show that the real, nonzero-weight system has order 2M: a sharp sign-change bound forces rigidity at that order, while positive Gauss–Hermite quadrature constructs a nontrivial solution through order 2M−1. Numerical checks at 110-digit working precision recover the predicted leading term for two to six components, and finite multistart searches recover the same observed powers for two to five components.
The exact order yields an explicit pointwise maximum-likelihood exponent and an exact local minimax order on specified shrinking neighbourhoods of a fixed known mixing measure. The same coefficient viewpoint separates high-order cancellation from exact parameter symmetries in mixtures of experts, showing why statistical rate arguments must account for exact parameter symmetries.