Solving the Ho--Nguyen Conjecture for Gaussian Mixtures: Applications to Mixtures of Experts
Over-fitted Gaussian mixtures can contain many parameter configurations with nearly the same density, making the order of density cancellation central to their parameter-learning rates. Ho and Nguyen encoded these cancellations in a polynomial system and conjectured its exact order as a function of cluster size. The key to resolving the conjecture is a common positive-variance shift that turns the coefficient equations into Gaussian moment matching. For every cluster size $\Mhn\ge2$, we prove that the real, nonzero-weight system has order $2\Mhn$: a sharp sign-change bound forces rigidity at that order, while positive Gauss--Hermite quadrature constructs a non-trivial solution through ord...
| Authors | Yuyang Zhang, Qihuang Zhang |
|---|---|
| Primary category | cs.LG |
| Published | 19 October 2026 |
| Last updated at source | 19 October 2026 |
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