[jamgochiana]

Fit LatentCovariance with standard latent-variable CovarianceEM, either as a weighted single Gaussian or inside EM with exactly Tied(:F,:D). Mixture components learn shared loadings F and residual variances D, their own full or diagonal A_k, and unrestricted means: Σ_k=F A_k F′+Diagonal(D). PCAEM remains the separate fixed-projection pipeline.

For fixed responsibilities and updated means, a small posterior-moment helper uses latent-space Cholesky solves. The joint M-step pools mass-weighted cross/second moments to update F, updates each A_k, and pools expected squared residuals to update D. variance_floor constrains D. One joint inner report describes this covariance optimization.

The implementation follows the direct matrix-equation style of existing LRD CovarianceEM, reuses _variance and _objective, and shares one core between single-Gaussian and mixture fitting. It uses the existing public workspaces and outer EM dispatch. No custom buffer struct or optimizer framework is introduced. fit! continues from current parameters; docs explain initialization, identifiability, local convergence and the inner sweep budget.

Workspace validation checks F/D ties, latent shape and positive-definite A. Zero responsibilities are omitted before arithmetic, and component arrays remain independently owned. Includes the shared empty-observation mixture prediction fix also in #31. Other partial-sharing patterns remain deferred. No new dependencies or version bump.

Validation

• 816 assertions pass locally, including independent dense posterior/one-sweep references, full/diagonal A, weighted replication, one-component equivalence, continuation, monotonicity, variance floors, zero-responsibility extreme values, and high loading/noise likelihood against a dense BigFloat reference.
• Documentation builds with executed examples; README and fitting guide describe the method and limitations.
• Fresh GitHub Julia 1.11, prerelease and documentation checks pass on b363efd.
• A local simulation of squash-merging Add banded precision Gaussians and exact fitting #31 followed by Fit shared latent covariance models with joint EM #30 combines cleanly: 1,813 assertions pass, and combined docs build.

Performance follow-up

Reusable buffers, blocked accumulation and shared LRD/latent posterior kernels are deferred to #34, which preserves the prototype and its benchmark evidence. This PR retains sample-sized intermediate arrays and centers one component at a time.

A cached MNIST check (10k train / 10k validation, p=784, K=10, rank 30, seed 1, Gaussian dequantization σ=0.01) uses three outer iterations and three inner sweeps, Julia 1.11.5 / BLAS1, and two fresh fits after a small dispatch warm-up. Median fit time is 17.23 s, cumulative allocations 60.69 GB, and mean validation log likelihood 966.2014811920. Covariances, means, weights, training history and per-image validation scores match the original direct implementation exactly. Timings include initialization and in-fit GC; loading, validation scoring and I/O are excluded. Cumulative allocations are not peak RAM; this checks a fixed iteration budget, not convergence. Benchmark assets remain outside the repository.