Correct the held-out slice scale, and hold out half of each mode

[fishidaho]

PARAFAC2 models slice k as X_k ≈ P_k (B diag(a_k)) Cᵗ, where P_k has orthonormal columns (P_kᵗ P_k = I). Because P_k is orthonormal irrespective of how many cells the slice holds, the reconstruction's norm

‖P_k (B diag(a_k)) Cᵗ‖_F  =  ‖(B diag(a_k)) Cᵗ‖_F

is independent of n_k. All slice energy lives in a_k.

a_k is fitted against n_train_k cells, then applied unchanged to the n_test_k held-out cells. Since real slice energy grows like sqrt(n_k), the held-out block's magnitude is overstated by sqrt(n_train_k / n_test_k) — 2× at the default held_out_cell_frac=0.2.

Noiseless, exactly-rank-4 data, fit at the true rank 4:

metric	value
Train Block R2X (in-sample)	0.9996
BiCV R2X (held-out)	−0.034
‖prediction‖ / ‖actual‖	1.99
R² after one global rescale	0.970

The direction is right; only the magnitude is wrong. And the factor tracks the cell-count ratio exactly — noiseless rank-4 data, varying only held_out_cell_frac:

held_out_frac	n_train/cond	n_test/cond	sqrt(ratio)	observed ratio	R2X as-is
0.10	180	20	3.0000	3.0551	−3.3154
0.20	160	40	2.0000	2.0060	−0.0702
0.35	130	70	1.3628	1.3890	0.8247
0.50	100	100	1.0000	1.0095	0.9900
0.75	50	150	0.5774	0.5781	0.8139

The score is correct only at held_out_cell_frac = 0.5, where the ratio is 1.

The fix

Simple 6 line fix to multiply each condition's held-out loadings by sqrt(n_test_k / n_train_k). Keeps the orthonormal convention and corrects the known factor. Assumes per-cell energy is homogeneous within a condition which should be true under BiCV's random split.

It is also important to note that both held-out fractions now default to one half which follows the [Owen and Perry] (https://arxiv.org/abs/0908.2062) paper this whole method is inspired by. They reported even in their abstract that "in simulated examples we find that a method leaving out half the rows and half the columns performs well". That is also the one split at which the old score happened to be unbiased, so it is the setting under which older numbers remain comparable, and it recovers the true rank in every simulated scenario tested -- which the correction alone did not at 0.2.

Measured after the change: noiseless rank-4 data at its own rank scores 0.97-0.99 at every fraction (was -0.034); the spread across fractions 0.1-0.7 falls from 3.75 to 0.02; argmax lands on the true rank for (4, noise 0.3), (6, noise 0.3) and (4, noise 1.0); pure noise still scores negative. Row-order invariance from the alignment fix underneath is undisturbed, agreeing to 8.9e-16 between contiguous and fully interleaved conditions.

[aarmey]

Very nice catch!