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, 142 insertions, 0 deletions

diff --git a/worldalign/common.py b/worldalign/common.py
new file mode 100644
index 0000000..aa7b174
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+++ b/worldalign/common.py
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+from __future__ import annotations
+
+import json
+import math
+import os
+import random
+from pathlib import Path
+from typing import Any
+
+import numpy as np
+import torch
+import torch.nn.functional as F
+
+
+DATASET_NAME = "nlphuji/flickr30k"
+DATASET_SPLIT = "test"
+
+
+def seed_everything(seed: int) -> None:
+    random.seed(seed)
+    np.random.seed(seed)
+    torch.manual_seed(seed)
+    if torch.cuda.is_available():
+        torch.cuda.manual_seed_all(seed)
+
+
+def read_json(path: str | os.PathLike[str]) -> dict[str, Any]:
+    with open(path, encoding="utf-8") as f:
+        return json.load(f)
+
+
+def write_json(path: str | os.PathLike[str], value: Any) -> None:
+    path = Path(path)
+    path.parent.mkdir(parents=True, exist_ok=True)
+    with open(path, "w", encoding="utf-8") as f:
+        json.dump(value, f, indent=2, ensure_ascii=False)
+
+
+def normalized(x: torch.Tensor, eps: float = 1e-8) -> torch.Tensor:
+    return F.normalize(x.float(), dim=-1, eps=eps)
+
+
+def pairwise_cosine_distance(x: np.ndarray) -> np.ndarray:
+    x = x.astype(np.float64, copy=False)
+    x /= np.linalg.norm(x, axis=1, keepdims=True).clip(min=1e-12)
+    d = 1.0 - x @ x.T
+    np.fill_diagonal(d, 0.0)
+    scale = np.median(d[d > 0])
+    return d / max(float(scale), 1e-12)
+
+
+def linear_cka(x: torch.Tensor, y: torch.Tensor) -> float:
+    x = x.float() - x.float().mean(0, keepdim=True)
+    y = y.float() - y.float().mean(0, keepdim=True)
+    xty = x.T @ y
+    numerator = (xty * xty).sum()
+    xx = x.T @ x
+    yy = y.T @ y
+    denominator = torch.sqrt((xx * xx).sum() * (yy * yy).sum())
+    return float((numerator / denominator.clamp_min(1e-12)).item())
+
+
+def retrieval_metrics(
+    image_features: torch.Tensor,
+    text_features: torch.Tensor,
+    ks: tuple[int, ...] = (1, 5, 10),
+) -> dict[str, float]:
+    image_features = normalized(image_features)
+    text_features = normalized(text_features)
+    similarities = image_features @ text_features.T
+    n = similarities.shape[0]
+    truth = torch.arange(n, device=similarities.device)
+
+    i2t_order = similarities.argsort(dim=1, descending=True)
+    t2i_order = similarities.T.argsort(dim=1, descending=True)
+    i2t_rank = (i2t_order == truth[:, None]).nonzero()[:, 1]
+    t2i_rank = (t2i_order == truth[:, None]).nonzero()[:, 1]
+
+    result: dict[str, float] = {}
+    for k in ks:
+        result[f"i2t_r@{k}"] = float((i2t_rank < k).float().mean().item())
+        result[f"t2i_r@{k}"] = float((t2i_rank < k).float().mean().item())
+    result["i2t_median_rank"] = float(i2t_rank.float().median().item() + 1)
+    result["t2i_median_rank"] = float(t2i_rank.float().median().item() + 1)
+    result["chance_r@1"] = 1.0 / max(n, 1)
+    return result
+
+
+def batch_indices(n: int, batch_size: int, shuffle: bool = False, seed: int = 0):
+    order = np.arange(n)
+    if shuffle:
+        rng = np.random.default_rng(seed)
+        rng.shuffle(order)
+    for start in range(0, n, batch_size):
+        yield order[start : start + batch_size]
+
+
+def sliced_wasserstein(
+    x: torch.Tensor,
+    y: torch.Tensor,
+    num_projections: int = 64,
+) -> torch.Tensor:
+    """Differentiable empirical sliced W2 for equal-sized minibatches."""
+    n = min(x.shape[0], y.shape[0])
+    x = x[:n]
+    y = y[:n]
+    directions = torch.randn(
+        x.shape[-1], num_projections, device=x.device, dtype=x.dtype
+    )
+    directions = F.normalize(directions, dim=0)
+    x_proj = (x @ directions).sort(dim=0).values
+    y_proj = (y @ directions).sort(dim=0).values
+    return (x_proj - y_proj).square().mean()
+
+
+def cosine_isometry_loss(source: torch.Tensor, mapped: torch.Tensor) -> torch.Tensor:
+    source = normalized(source)
+    mapped = normalized(mapped)
+    source_gram = source @ source.T
+    mapped_gram = mapped @ mapped.T
+    mask = ~torch.eye(source.shape[0], dtype=torch.bool, device=source.device)
+    return (source_gram[mask] - mapped_gram[mask]).square().mean()
+
+
+def cosine_loss(x: torch.Tensor, y: torch.Tensor) -> torch.Tensor:
+    return 1.0 - F.cosine_similarity(x.float(), y.float(), dim=-1).mean()
+
+
+def dtype_for_device(device: str) -> torch.dtype:
+    return torch.bfloat16 if device.startswith("cuda") else torch.float32
+
+
+def parameter_count(module: torch.nn.Module) -> int:
+    return sum(p.numel() for p in module.parameters())
+
+
+def cosine_schedule(step: int, steps: int, warmup: int) -> float:
+    if step < warmup:
+        return (step + 1) / max(1, warmup)
+    progress = (step - warmup) / max(1, steps - warmup)
+    return 0.5 * (1.0 + math.cos(math.pi * progress))
+