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"""Which side is narrow? Per-modality rank against the shared rank.
Truncating a recovering synthetic field to rank four leaves its correlation at
0.90 and its recovery at 6%, so the width of the shared spectrum is a separate
and binding constraint that the correlation does not express. The engineering
question follows immediately: which modality is narrow. Upgrading the vision
encoder is worth its compute only if vision is what limits the shared width,
and if the text side is the narrow one no vision backbone will help.
Three numbers per field pair. Each field's own effective rank, from the
participation ratio of its spectrum, says how many directions the modality
uses at all. The principal angles between the two leading eigenspaces say how
many of those directions the two modalities share. And the per-direction
correlation profile says where along the spectrum agreement dies.
"""
from __future__ import annotations
import argparse
import json
import numpy as np
import torch
from .common import write_json
def parse_args() -> argparse.Namespace:
parser = argparse.ArgumentParser()
parser.add_argument("--fields", nargs="+", required=True)
parser.add_argument("--labels", nargs="+", default=None)
parser.add_argument("--width", type=int, default=32)
parser.add_argument("--output", default="artifacts/vg_5k/shared_rank.json")
return parser.parse_args()
def spectrum(matrix: np.ndarray) -> tuple[np.ndarray, np.ndarray]:
symmetric = (matrix + matrix.T) / 2.0
values, vectors = np.linalg.eigh(symmetric)
order = np.argsort(np.abs(values))[::-1]
return values[order], vectors[:, order]
def effective_rank(values: np.ndarray) -> float:
weights = np.abs(values) / np.abs(values).sum()
weights = weights[weights > 1e-15]
return float(np.exp(-(weights * np.log(weights)).sum()))
def analyse(path: str, label: str, width: int) -> dict:
state = torch.load(path, map_location="cpu", weights_only=False)
visual = state["visual_field"].double().numpy()
text = state["text_field"].double().numpy()
size = len(visual)
mask = ~np.eye(size, dtype=bool)
visual_values, visual_vectors = spectrum(visual)
text_values, text_vectors = spectrum(text)
width = min(width, size)
# principal angles between the two leading eigenspaces: cosines near one
# count as directions the two modalities agree on. The raw count inflates
# with width -- two random w-dimensional subspaces of R^N already overlap
# at cosines near sqrt(w/N) -- so a scene-shuffled null is subtracted and
# every comparison is made at matched width.
overlap = visual_vectors[:, :width].T @ text_vectors[:, :width]
cosines = np.clip(np.linalg.svd(overlap, compute_uv=False), 0.0, 1.0)
shared = int((cosines > 0.7).sum())
null_counts = []
for trial in range(5):
generator = np.random.default_rng(trial)
order = generator.permutation(size)
shuffled = text[np.ix_(order, order)]
_, shuffled_vectors = spectrum(shuffled)
null_overlap = visual_vectors[:, :width].T @ shuffled_vectors[:, :width]
null_cosines = np.clip(np.linalg.svd(null_overlap, compute_uv=False), 0.0, 1.0)
null_counts.append(int((null_cosines > 0.7).sum()))
null = float(np.mean(null_counts))
# where along the spectrum does agreement die
profile = []
for index in range(min(width, 16)):
visual_direction = visual_vectors[:, index]
best = float(np.abs(text_vectors[:, :width].T @ visual_direction).max())
profile.append({"index": index, "best_text_overlap": best})
return {
"label": label,
"size": size,
"correlation": float(np.corrcoef(visual[mask], text[mask])[0, 1]),
"visual_effective_rank": effective_rank(visual_values),
"text_effective_rank": effective_rank(text_values),
"shared_directions_cos_gt_0.7": shared,
"shuffled_null": null,
"shared_above_null": shared - null,
"principal_cosines_top8": [round(float(c), 3) for c in cosines[:8]],
"spectrum_profile": profile,
}
def main() -> None:
args = parse_args()
labels = args.labels or [path.split("/")[-1] for path in args.fields]
rows = [analyse(path, label, args.width)
for path, label in zip(args.fields, labels)]
for row in rows:
print(
f"{row['label']:<26} rho={row['correlation']:.3f} "
f"eff_rank vision={row['visual_effective_rank']:6.1f} "
f"text={row['text_effective_rank']:6.1f} "
f"shared={row['shared_directions_cos_gt_0.7']:3d} "
f"(null {row['shuffled_null']:4.1f}, net {row['shared_above_null']:5.1f})",
flush=True,
)
summary = {
"protocol": (
"Effective rank from the participation ratio of each field's "
"spectrum; shared directions from principal angles between the "
"two leading eigenspaces. No pairing enters beyond the alignment "
"the fields already carry."
),
"width": args.width,
"rows": rows,
"reading": (
"The narrow modality is the one to upgrade. A shared count far "
"below both effective ranks means each side is rich but they are "
"rich about different things."
),
}
write_json(args.output, summary)
if __name__ == "__main__":
main()
|
