Quasipolynomial-Time Learning of Centered Gaussian Mixtures with Bounded Covariances

We give a sample-only distribution learner for arbitrary-weight mixtures of centered Gaussian distributions with covariance matrices between the identity and twice the identity. The covariances may be noncommuting. For A=2+d+k+epsilon⁻¹+log(1/delta), the number of observations, training operations, stored real words, and operations on each invocation of the learned sampler are bounded by exp(C log^2 A), for a universal constant C. With probability at least 1-delta over training, the sampler's output law has total variation distance at most epsilon from the target. The computational model uses exact real arithmetic and continuous random primitives, with deterministic caps on every loop. The proof iterates a sample-only posterior covariance refinement across logarithmically many Gaussian-noise levels. A structured robust matrix filter gives a Frobenius cover of almost all latent covariance mass at squared radius logarithmic in the number of components and the inverse error budget. Around this cover, a bounded positive-semidefinite dictionary is constructed by enumerating low-degree Hermite polynomials. Root-density approximation and a fixed Renyi moment yield a KL-close feasible comparator without a square-integrable density ratio. Convex likelihood fitting and bounded-factor rejection then give the next approximation and its posterior sampler. The result requires no component labels, covariance oracle, separation, or minimum positive weight. It does not recover component parameters or resolve polynomial-time feature discovery.