Polynomial Density Learning of Spherical Gaussian Mixtures with a Minimum-Weight Bound

We give a polynomial-time improper density learner for mixtures of at most k identity-covariance Gaussians in ℝ^d whose nonzero mixing weights are at least a supplied w_min > 0. The means are unrestricted and need not be separated, linearly independent, or contained in a bounded ball. Training samples, real-arithmetic operations, and the cost of each output sample are polynomial in d, k, w_min⁻¹, ε⁻¹, log(1/δ), where ε is total-variation accuracy and 1−δ is training confidence. In particular, uniformly weighted mixtures admit joint-polynomial density learning without geometric assumptions. The construction extends a positive-atom moment cover to bounded-radius Gaussian location mixtures through compressed radial normalization. Fresh Gaussian split views then yield exact positive localization and globally pruned radius refinement without accumulated noise. At the terminal radius, full routed positive semidefinite moment matrices learn tensor subspaces, while an observable quadratic loss under a common mixture reference learns density coefficients from the original samples. This separation avoids a failure of per-bin Hermite density estimation: a far component of tiny routing mass can have exponentially large local Hermite coefficients. Randomized Hermite evaluation, clipping, and capped rejection sampling yield a valid nonnegative output distribution. The dependence on the minimum original weight is substantive; arbitrary-weight joint-polynomial learning remains outside the theorem.