Learning Arbitrary Finite Gaussian Location Mixtures

We give a sample-only improper total-variation learner for finite Gaussian location mixtures with arbitrary means and weights and known common covariance. The exact-real sample, training, real-register representation and per-output costs are fixed polynomials in dimension, component count, inverse accuracy and logarithmic inverse failure probability. The construction combines a finite full-jet randomized gate, independent Gaussian views obtained from fresh original observations, rank-k compression of the actual accepted latent measures, a square-root PSD density dictionary, directional likelihood fitting with linear omitted-mass loss, and a robust posterior mass-cover refinement. An explicit noncircular noise schedule starts from raw observations and removes the need for a supplied mean radius. Every rejection loop is capped. Standard Gaussian analysis, convex optimization and robust mean estimation are identified explicitly; the last ingredient is implemented by a fully specified finite weighted filter, with a cited high-probability statistical subset theorem.