saloon: Householder QR and least squares; maroon: linear and ridge regression (Hoon)

[sigilante]

Step 4 of the foundation work list in #86: QR and the first regression models. Hoon only, no jet hints, off main (the #87/#88/#90 stack is merged, so nothing is stacked).

Saloon — %linalg

• +qr — the thin Householder factorization A = Q*R for rows >= cols. Q has orthonormal columns; R is upper triangular with an exactly-zero subdiagonal rather than rounding residue. A zero column needs no reflection and is skipped.
• +trsv-r — back substitution against R, read as it is (the companion of +qr, as +trsv-lo/+trsv-up are of +chol).
• +lstsq — min |A*x - v| as R^-1 * Q^T*v. Solving through QR avoids the squared condition number of the normal equations. Needs full column rank.
• +gram (x^T*x) and +matvec-t (m^T*x), neither materializing a transpose.
• +house, +reflect, +sum-sq — the pieces +qr is built from.

Sign convention. +house uses LAPACK's choice, alpha = -sign(x0)*|x|, so v0 = x0 - alpha never cancels. A side effect is that the factors match numpy.linalg.qr including signs — negative diagonal entries in R and all — which the oracle verifies rather than assumes.

Maroon — linear models

• +linreg — ordinary least squares through Saloon's +lstsq.
• +ridge — solves (Xc^T*Xc + alpha*I) coef = Xc^T*yc with +chol-solve. Any alpha > 0 makes that positive definite even when x is rank-deficient, which is half the point of ridge; alpha = 0 reduces to OLS.
• Both centre the data and recover the intercept afterwards (intercept = mean(y) - mean(x).coef), as scikit-learn does with fit_intercept=True — which also keeps the intercept out of the ridge penalty. Both return [coef intercept].
• +predict, +mse, +r2 — r2 crashes on a constant target, where it is 0/0.

No transpose jet anywhere

The lagoon transpose jet crashes on every runtime older than urbit/vere#1057, which is most deployed ships today. So +lstsq uses +matvec-t and +ridge uses +gram instead of mmul of a transpose — and this PR also moves Maroon's existing +cov onto +gram, the one place #90 used transpose. +gram is exactly symmetric, since entries (i,j) and (j,i) multiply the same pairs.

Tests: 36 arms

saloon-qr.hoon (19) and maroon-regression.hoon (17).

Exact, compared as whole rays — inputs whose column norms are powers of two, so every Householder vector, scale and reflection is representable:

• diag(2,2) → Q = -I, R = -2I (the LAPACK signs); the same on a tall 3×2
• R exactly -5 for [3 4]^T; a zero column skipped, not divided by
• lstsq = [2 3] on a tall system whose third row is unreachable, and on a square one
• y = 3x + 1 on x = [0 0 2 2]: the centred feature [-1 -1 1 1] has norm exactly 2, giving coef 3, intercept 1, mse 0, r2 1
• ridge with alpha = 12 makes the system [[16]], so coef 0.75, intercept 3.25, mse 5.0625 and r2 0.4375 are all exact; alpha = 0 equals OLS

Approximate, within a tolerance: R and Q against numpy.linalg.qr (signs included), Q*R ≈ A, Q^T*Q ≈ I (via +gram), lstsq against numpy.linalg.lstsq and against the normal equations, and two-feature fits against sklearn's LinearRegression and Ridge(alpha=1.0), plus the fit's R².

Oracles: saloon/tools/linalg_check.py (56 checks, now with a Householder mirror in exact rationals) and maroon/tools/ml_check.py (35 checks, with sklearn).

Verification

Fresh fakeship at hoon-135/zuse-408 on the vere #1057 build (with the @rq fix, since merged upstream as urbit/vere#1112):

python3 saloon/tools/linalg_check.py        # 56 checks, 0 failures
python3 maroon/tools/ml_check.py            # 35 checks, 0 failures
-test /=num=/tests/lib/saloon-qr            # 19, ok=%.y
-test /=num=/tests/lib/maroon-regression    # 17, ok=%.y
-test /=num=/tests/lib                      # 438 OK / 4 failed  (was 402 / 4)

+36 OK and the failures unchanged; maroon-pca (19) and saloon-linalg (44) stay green after the +cov change. The 4 are the pre-existing saloon-unum posit transcendentals, which fail on every runtime.

Not in this PR

LU with pivoting (not needed yet — Cholesky and QR cover the solves the models use), jets, and the next #86 step: gradient-descent optimizers, then logistic regression.

Refs #86.

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https://claude.ai/code/session_01TgBnKsUPYzkoPZonjePnaq