amygdala: select top-k eigenvectors AFTER PCA, not per-story truncation

Kent: 'full rank is going to give you everything — you still have to
select down, but you can do that /after/ PCA'.

Previously I was discarding per-story via k=20 truncation of SVD.
That destroyed per-head discriminability before we ever saw the
eigenvalue spectrum. Then the alternative 'keep full rank' run
accumulated too many shared directions, making the top-1 eigenvector
arbitrary within a flat spectrum.

Correct approach: keep per-story subspaces at full rank (no info
loss) and select k eigenvectors of M = M_pos - M_base at the final
step, weighted sum by eigenvalue. This captures the multi-dimensional
shared subspace when the spectrum is flat (common case), and reduces
to the top-1 behavior when the spectrum has a clear gap.

New --subspace-eigen-k flag (default 5). Clamps negative weights to 0
so wrong-sign directions don't contribute.

Co-Authored-By: Proof of Concept <poc@bcachefs.org>

training/amygdala_training/train_steering_vectors.py

@@ -282,16 +282,22 @@ def _subspace_concept_direction(
     pos_V: list[torch.Tensor],           # list of [hidden, k_i] per story
     base_V: list[torch.Tensor],
     hidden: int,
+    *,
+    top_k: int = 5,
 ) -> tuple[torch.Tensor, torch.Tensor]:
     """Subspace-common-direction CAA alternative.
 
     Builds M_pos = (1/n_pos) Σ V_i V_i^T over positive stories and M_base the
-    same over baselines. Returns the top eigenvector of (M_pos - M_base) —
-    the direction most-common to positives after subtracting what's generic
-    across baselines — plus its eigenvalue spectrum (for diagnostics).
+    same over baselines. Returns a weighted sum of the top-k eigenvectors of
+    (M_pos - M_base), weights = eigenvalues (so stronger common directions
+    contribute more), unit-normed. Returns the full eigenvalue spectrum for
+    diagnostics.
 
-    The top eigenvalue approaches 1 if the concept appears in every positive
-    story's subspace with unit weight and is absent from the baseline.
+    top_k=1 recovers the previous behavior (top eigenvector only). top_k>1
+    captures richer structure when the concept lives in a multi-dimensional
+    shared subspace — which the flat eigenvalue spectrum observed in
+    practice suggests is the common case. Selection happens AFTER the
+    eigendecomposition so nothing is lost up to that point.
     """
     device = pos_V[0].device if pos_V else torch.device("cpu")
     dtype = torch.float32
@@ -310,13 +316,18 @@ def _subspace_concept_direction(
     M_base = acc(base_V)
     M = M_pos - M_base
 
-    # Symmetric eigendecomposition — top eigenvalue/vector.
+    # Symmetric eigendecomposition.
     eigvals, eigvecs = torch.linalg.eigh(M)
-    # eigh returns ascending; top is the last column.
-    top_vec = eigvecs[:, -1]
-    # Unit-norm (eigvecs are unit already, but defensively).
-    top_vec = top_vec / top_vec.norm().clamp_min(1e-6)
-    return top_vec, eigvals
+    # eigh returns ascending; top-k are the last k columns.
+    k = max(1, min(top_k, eigvecs.shape[1]))
+    top_vals = eigvals[-k:]                 # [k], ascending within top-k
+    top_vecs = eigvecs[:, -k:]              # [hidden, k]
+    # Weighted sum of top-k eigenvectors, weights = eigenvalues. Clamp
+    # negative weights to 0 (wrong-sign directions shouldn't contribute).
+    w = top_vals.clamp_min(0.0)
+    combined = top_vecs @ w                 # [hidden]
+    combined = combined / combined.norm().clamp_min(1e-6)
+    return combined, eigvals
 
 
 def _load_corpus(stories_dir: Path, paired_dir: Path | None) -> tuple[
@@ -858,6 +869,17 @@ def main() -> None:
              "residual and ~500-token stories, that's ~500 vectors per story. "
              "Memory is fine: 112 × 5120 × 500 × 4 bytes ≈ 1.1 GB.",
     )
+    ap.add_argument(
+        "--subspace-eigen-k",
+        type=int,
+        default=5,
+        help="Number of top eigenvectors of M_pos - M_base to combine into "
+             "the concept direction. Weighted sum by eigenvalue (so strongest "
+             "common directions contribute most). eigen_k=1 recovers "
+             "single-eigenvector behavior. Higher values (5-10) capture "
+             "richer structure when the concept's shared-subspace spectrum "
+             "is flat (which it tends to be in practice).",
+    )
     ap.add_argument(
         "--quality-report",
         action="store_true",
@@ -1045,6 +1067,7 @@ def main() -> None:
                 base_V += [bs[target_l] for bs in (base_subspaces or [])]
                 top_vec, eigvals = _subspace_concept_direction(
                     pos_V, base_V, hidden=hidden_dim,
+                    top_k=args.subspace_eigen_k,
                 )
                 per_layer_vectors[l_idx, e_idx] = top_vec
                 # Keep the top-20 eigenvalues for quality-report diagnostics.