World Alignment: unpaired cross-modal correspondence by relational identifiability

Method: scene states are sets of part states; relation fields are built
within each modality and are invariant to how each side labels its own
features; the cross-modal bridge is a coupling searched under an energy
that is a closed-form functional of one matrix; solving is spectral
initialisation followed by exact local refinement.

Evidence: in a procedurally generated closed world, blind recovery of a
hidden image-caption correspondence reaches 95.3% at 256 scenes against
0.39% chance, and the recovered pairs transfer to 200 held-out scenes at
93.0% exact retrieval with random-pair and shuffled-image controls at or
near chance. Cross-modal value correspondence is derived from disjoint
corpora rather than declared. On Visual Genome the field correlation
reaches 0.656 against the 0.9 that polynomial recovery needs, with the
deficit attributed away from segmentation and discretisation.

Protocol: no image-text pair enters any objective, optimiser,
initialisation, or model selection; hidden pairs score orderings only.

Co-Authored-By: Claude <noreply@anthropic.com>

1 files changed, 310 insertions, 0 deletions

diff --git a/worldalign/tier0_pipeline.py b/worldalign/tier0_pipeline.py
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+++ b/worldalign/tier0_pipeline.py
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+"""Tier 0 pipeline: unpaired corpora to aligned relation fields.
+
+Consolidates the components that produced the synthetic world's
+end-to-end result, each of which was developed and measured separately:
+
+- watershed object extraction, which splits touching ring members that
+  connected components merge (exact object count 74.2% to 100%);
+- Gestalt appearance grouping, which assembles members into groups by
+  shared colour, size, and shape rather than by a distance threshold
+  (exact group count 71.9% to 90.2%);
+- size classes from radial extent under one-dimensional k-means per
+  member count, because area confounds size with shape and the classes
+  are gap-separated rather than equally populated (52.7% to 87.9%);
+- text factor families from mutual exclusivity within a group phrase;
+- the cross-modal value correspondence from marginal frequency rank.
+
+Nothing crosses modalities except the frequency ranking, and the two
+corpora it reads are disjoint: no instance appears on both sides.
+"""
+
+from __future__ import annotations
+
+import argparse
+import json
+from collections import Counter, defaultdict
+from pathlib import Path
+
+import numpy as np
+import torch
+import torch.nn.functional as F
+from scipy import ndimage
+from scipy.cluster.hierarchy import fcluster, linkage
+from sklearn.cluster import KMeans
+from tqdm import tqdm
+
+from .common import read_json, seed_everything, write_json
+from .synth_cc_battery import moment_field
+from .synth_set_battery import parse_group_phrases
+from .synth_towers import load_image
+from .tier0_dictionary import partition_text_vocabulary, text_token_statistics
+
+NUMBER_TO_COUNT = {"two": 2, "three": 3, "four": 4}
+SINGULAR = ("a", "an")
+
+
+def parse_args() -> argparse.Namespace:
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--data-dir", default="artifacts/synth_v1")
+    parser.add_argument("--split", choices=["val", "test"], default="test")
+    parser.add_argument("--samples", type=int, default=256)
+    parser.add_argument("--offset", type=int, default=0)
+    parser.add_argument("--fit-scenes", type=int, default=1500)
+    parser.add_argument("--peak-distance", type=int, default=5)
+    parser.add_argument("--group-threshold", type=float, default=0.35)
+    parser.add_argument("--views", type=int, default=0, help="0 uses manifest.")
+    parser.add_argument("--seed", type=int, default=0)
+    parser.add_argument("--output", required=True)
+    parser.add_argument("--states-output", default="")
+    return parser.parse_args()
+
+
+def extract_objects(image: torch.Tensor, peak_distance: int) -> list[dict]:
+    """Foreground objects, splitting touching ones by distance watershed."""
+    array = image.permute(1, 2, 0).numpy()
+    background = np.median(array.reshape(-1, 3), axis=0)
+    foreground = np.abs(array - background).sum(-1) > 0.12
+    distance = ndimage.distance_transform_edt(foreground)
+    window = 2 * peak_distance + 1
+    peaks = (
+        distance >= ndimage.maximum_filter(distance, size=window) - 1e-9
+    ) & (distance > 1.0)
+    labels, count = ndimage.label(peaks)
+    if count == 0:
+        labels, count = ndimage.label(foreground)
+    else:
+        while True:
+            grown = ndimage.grey_dilation(labels, size=3)
+            take = (labels == 0) & foreground & (grown > 0)
+            if not take.any():
+                break
+            labels = np.where(take, grown, labels)
+        count = labels.max()
+    objects = []
+    for index in range(1, count + 1):
+        mask = labels == index
+        area = float(mask.sum())
+        if area < 6:
+            continue
+        ys, xs = np.nonzero(mask)
+        centred_y, centred_x = ys - ys.mean(), xs - xs.mean()
+        covariance = np.cov(np.stack([centred_x, centred_y])) + 1e-6 * np.eye(2)
+        eigenvalues = np.linalg.eigvalsh(covariance)
+        eroded = ndimage.binary_erosion(mask)
+        perimeter = max(float((mask & ~eroded).sum()), 1.0)
+        box = (xs.max() - xs.min() + 1) * (ys.max() - ys.min() + 1)
+        objects.append(
+            {
+                "rgb": array[mask].mean(0),
+                "area": area,
+                "extent": float(np.hypot(centred_y, centred_x).max()),
+                "shape": np.array(
+                    [
+                        4 * np.pi * area / perimeter**2,
+                        1 - eigenvalues[0] / eigenvalues[1],
+                        area / max(box, 1),
+                    ]
+                ),
+            }
+        )
+    return objects
+
+
+def appearance(item: dict) -> np.ndarray:
+    return np.concatenate(
+        [item["rgb"] * 3.0, [np.log(item["area"] + 1e-6) * 0.6], item["shape"]]
+    )
+
+
+def group_objects(objects: list[dict], threshold: float) -> list[dict]:
+    """Members of one group share appearance; group by similarity."""
+    if not objects:
+        return []
+    if len(objects) == 1:
+        labels = np.array([0])
+    else:
+        features = np.stack([appearance(item) for item in objects])
+        labels = fcluster(linkage(features, "complete"), threshold, "distance")
+    buckets: dict[int, list[dict]] = {}
+    for item, label in zip(objects, labels):
+        buckets.setdefault(int(label), []).append(item)
+    return [
+        {
+            "rgb": np.mean([item["rgb"] for item in members], axis=0),
+            "members": len(members),
+            "extent": float(np.mean([item["extent"] for item in members])),
+        }
+        for members in buckets.values()
+    ]
+
+
+class VisionCoder:
+    """Colour classes and size classes fitted on the vision corpus alone."""
+
+    def __init__(self, groups: list[dict], classes: int, seed: int) -> None:
+        colours = np.stack([group["rgb"] for group in groups]).astype(np.float64)
+        self.colour_model = KMeans(classes, n_init=10, random_state=seed).fit(colours)
+        frequency = Counter(self.colour_model.labels_.tolist())
+        self.colour_rank = {
+            label: rank for rank, (label, _) in enumerate(frequency.most_common())
+        }
+        by_count: dict[int, list[float]] = defaultdict(list)
+        for group in groups:
+            by_count[min(group["members"], 4)].append(group["extent"])
+        self.size_models = {}
+        for count, extents in by_count.items():
+            model = KMeans(3, n_init=10, random_state=seed).fit(
+                np.asarray(extents, dtype=np.float64)[:, None]
+            )
+            order = np.argsort(model.cluster_centers_[:, 0])
+            self.size_models[count] = (
+                model,
+                {int(label): rank for rank, label in enumerate(order)},
+            )
+
+    def encode(self, groups: list[dict], classes: int) -> torch.Tensor:
+        if not groups:
+            groups = [{"rgb": np.zeros(3), "members": 1, "extent": 1.0}]
+        colours = self.colour_model.predict(
+            np.stack([group["rgb"] for group in groups]).astype(np.float64)
+        )
+        vectors = []
+        for group, colour in zip(groups, colours):
+            model, order = self.size_models[min(group["members"], 4)]
+            size = order[
+                int(model.predict(np.array([[group["extent"]]], dtype=np.float64))[0])
+            ]
+            vectors.append(
+                factor_vector(self.colour_rank[int(colour)], group["members"], size, classes)
+            )
+        return F.normalize(torch.stack(vectors), dim=-1)
+
+
+def factor_vector(colour: int, count: int, size: int, classes: int) -> torch.Tensor:
+    vector = torch.zeros(classes + 4 + 3)
+    vector[colour] = 1.0
+    vector[classes + min(count - 1, 3)] = 1.0
+    vector[classes + 4 + size] = 1.0
+    return vector
+
+
+def encode_caption(caption: str, colour_rank: dict[str, int], classes: int) -> torch.Tensor:
+    vectors = []
+    for phrase in parse_group_phrases(caption):
+        tokens = phrase.split()
+        colour = next((colour_rank[t] for t in tokens if t in colour_rank), 0)
+        count = (
+            1
+            if any(token in SINGULAR for token in tokens)
+            else next(
+                (NUMBER_TO_COUNT[t] for t in tokens if t in NUMBER_TO_COUNT), 1
+            )
+        )
+        size = 0 if "small" in tokens else (2 if "large" in tokens else 1)
+        vectors.append(factor_vector(colour, count, size, classes))
+    if not vectors:
+        vectors = [factor_vector(0, 1, 1, classes)]
+    return F.normalize(torch.stack(vectors), dim=-1)
+
+
+def moment_state(states: torch.Tensor) -> torch.Tensor:
+    first = states.mean(0)
+    second = (states[:, :, None] * states[:, None, :]).mean(0).flatten()
+    return torch.cat([first, second])
+
+
+def main() -> None:
+    args = parse_args()
+    seed_everything(args.seed)
+    manifest = read_json(Path(args.data_dir, "manifest.json"))
+    captions = read_json(Path(args.data_dir, "captions.json"))["captions"]
+    image_dir = Path(manifest["image_dir"])
+    views = args.views or manifest["visual_views"]
+    rows = manifest[args.split][args.offset : args.offset + args.samples]
+
+    families = partition_text_vocabulary(
+        text_token_statistics(captions, manifest["text_only_train"])
+    )
+    colour_words = families["colour_words"]
+    classes = len(colour_words)
+    colour_rank = {word: rank for rank, word in enumerate(colour_words)}
+
+    fit_groups = [
+        group
+        for row in tqdm(
+            manifest["vision_only_train"][: args.fit_scenes], desc="fit vision"
+        )
+        for group in group_objects(
+            extract_objects(
+                load_image(image_dir / f"scene{row:06d}_v0.png"), args.peak_distance
+            ),
+            args.group_threshold,
+        )
+    ]
+    coder = VisionCoder(fit_groups, classes, args.seed)
+
+    per_view_fields = []
+    view_states = []
+    for view in range(views):
+        sets = [
+            coder.encode(
+                group_objects(
+                    extract_objects(
+                        load_image(image_dir / f"scene{row:06d}_v{view}.png"),
+                        args.peak_distance,
+                    ),
+                    args.group_threshold,
+                ),
+                classes,
+            )
+            for row in tqdm(rows, desc=f"encode view {view}")
+        ]
+        per_view_fields.append(moment_field(sets))
+        if view == 0:
+            view_states = [moment_state(item) for item in sets]
+    visual_field = torch.stack(per_view_fields).mean(0)
+
+    text_sets = [encode_caption(captions[row][0], colour_rank, classes) for row in rows]
+    text_field = moment_field(text_sets)
+
+    mask = ~np.eye(len(rows), dtype=bool)
+    correlation = float(
+        np.corrcoef(
+            visual_field.double().numpy()[mask], text_field.double().numpy()[mask]
+        )[0, 1]
+    )
+    torch.save(
+        {"visual_field": visual_field, "text_field": text_field, "rows": rows},
+        args.output,
+    )
+    if args.states_output:
+        torch.save(
+            {
+                "vision_states": torch.stack(view_states),
+                "text_states": torch.stack([moment_state(item) for item in text_sets]),
+                "rows": rows,
+            },
+            args.states_output,
+        )
+    summary = {
+        "data_dir": args.data_dir,
+        "split": args.split,
+        "samples": len(rows),
+        "colour_classes": classes,
+        "text_families": {
+            key: families[key] for key in ("count_words", "colour_words", "size_words")
+        },
+        "field_correlation_at_truth": correlation,
+        "note": (
+            "The dictionary is derived from disjoint corpora; the "
+            "correlation is a diagnostic computed with hidden pairs and "
+            "never used by the pipeline."
+        ),
+    }
+    print(json.dumps({"field_correlation_at_truth": correlation}))
+    write_json(args.output.replace(".pt", ".json"), summary)
+    print(f"Wrote {args.output}")
+
+
+if __name__ == "__main__":
+    main()