Status: accepted · Date: 2026-09-13 · Applies: 0095, 0116

Decision 0116 read both .NET PCA incumbents and found one gap: below net8.0, nothing reports how much variance a principal component explains. ML.NET projects and exposes no eigenvalue; NumFlat has EigenValues and ships net8.0 only. 0116 scoped the lot, left it for a caller under decision 0095's rule, and left one question open on purpose: which package it goes in.

#701 is that caller, and it framed the question on a premise:

Lodestar.Decomposition's stated subject is decompositions over a CsrMatrix, so a dense-only member would be the first thing in it that does not take one.

That premise is false, and the exported surface says so. Lodestar.Decomposition 0.2.0 already publishes QrDecomposition.Householder, which takes a ReadOnlySpan<double> and two dimensions: dense, row-major, no CsrMatrix anywhere in the signature. Decision 0095 published it because Lodestar.Stats.Regression needed it. The package's subject was already "the decompositions this repository writes by hand", with sparse input as the reason most of them exist.

PrincipalComponentVariance.Compute ships in Lodestar.Decomposition 0.3.0, taking a row-major span and its two dimensions. It returns the explained variance, its ratio, the cumulative curve and the total variance, one entry per component up to min(n, p), at scikit-learn's PCA parity.

• Not a type named Pca. It has no components, no mean and no Transform; 0116 delegated all three, and a type called PCA would promise them.
• The eigenvalues of the centred block's Gram matrix, XᵀX or XXᵀ whichever is smaller, solved by the one-sided Jacobi kernel this package already carries. Measured on an AMD Ryzen 7 8700G with a Stopwatch, 2,000 × 50: Jacobi over the whole centred block took 21.9 ms against NumFlat's 1.6 ms; reducing to R first, 7.5 ms; the Gram route, 4.9 ms; the Gram route with a row-major Gram and tracked column norms, 1.9 ms. The BenchmarkDotNet numbers are in docs/guides/performance.md.
• Two refusals scikit-learn does not make. Fewer than two rows throws, since the variance divides by n − 1. A matrix with no variance at all throws, where scikit-learn divides by the zero total and returns NaN for every ratio. A NaN or an infinity in the input throws, as scikit-learn's input validation does.

• Lodestar.Preprocessing. The pipeline argument is real: a scaler usually comes before a PCA, and that package already fits on row-major spans. But it has no linear algebra, so it would need either a second copy of the Jacobi kernel or an eleventh inter-package edge, to Lodestar.Decomposition, for one member. Both cost more than the adjacency is worth, and scikit-learn itself files PCA under sklearn.decomposition.
• Lodestar.Stats. Covariance is statistics. But decision 0095 publishes that package's numerical layer member by member, and it holds tails and quantiles, not matrices. The kernel this needs is in Lodestar.Decomposition.
• scikit-learn's svd_solver="full" computation as written, the singular values of the centred block. It keeps the condition number unsquared, and it lost on cost: 21.9 ms, or 7.5 ms through R, against 1.9 ms. Squaring the condition costs absolute accuracy near ε · λ₁, which no ratio can show, and scikit-learn's own covariance_eigh solver makes the same trade for tall blocks. The frozen corpus pins svd_solver="full" and agrees at 1e-9.
• A projection alongside it. Refused by 0116 and not reopened: both incumbents project, and what they lack is this number.

• Lodestar.Decomposition goes to 0.3.0: a new public type, nothing removed.
• docs/migration/sklearn.md's explained_variance_ratio_ row stops reading gap and points here; the projection row is unchanged.
• JacobiSvd gains a values-only entry point that tracks column norms. The public QrDecomposition, the TruncatedSvd path and their corpora do not move.
• NumFlat 1.3.4 joins bench/Lodestar.Text.Benchmarks as the incumbent. It is MIT-licensed and ships nothing from there.