From a62cf4d2a99b4a7985c61b2a7feb92a82a8218b7 Mon Sep 17 00:00:00 2001 From: Yuren Hao Date: Sat, 1 Aug 2026 14:10:03 -0500 Subject: 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 --- worldalign/energy.py | 137 +++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 137 insertions(+) create mode 100644 worldalign/energy.py (limited to 'worldalign/energy.py') diff --git a/worldalign/energy.py b/worldalign/energy.py new file mode 100644 index 0000000..796c41a --- /dev/null +++ b/worldalign/energy.py @@ -0,0 +1,137 @@ +from __future__ import annotations + +import torch +import torch.nn.functional as F + + +def off_diagonal_mask(size: int, device: torch.device | str) -> torch.Tensor: + return ~torch.eye(size, dtype=torch.bool, device=device) + + +def standardized_relation( + features: torch.Tensor, mask: torch.Tensor | None = None +) -> tuple[torch.Tensor, torch.Tensor]: + """Cosine relation field and its standardized off-diagonal values.""" + features = F.normalize(features.float(), dim=-1) + relation = features @ features.T + if mask is None: + mask = off_diagonal_mask(len(features), features.device) + values = relation[mask] + standardized = (values - values.mean()) / values.std().clamp_min(1e-6) + return relation, standardized + + +def relation_field_energy( + visual_relation: torch.Tensor, + visual_standardized: torch.Tensor, + language_particles: torch.Tensor, + temperatures: tuple[float, ...] = (0.03, 0.07, 0.15), +) -> tuple[torch.Tensor, torch.Tensor]: + """Second-order and multiscale conditional relation energies.""" + language_relation, language_standardized = standardized_relation( + language_particles + ) + mse = F.mse_loss(language_standardized, visual_standardized) + diagonal = torch.eye( + len(language_particles), + dtype=torch.bool, + device=language_particles.device, + ) + conditional_kl = language_particles.new_zeros(()) + for temperature in temperatures: + visual_logits = (visual_relation / temperature).masked_fill( + diagonal, -1e4 + ) + language_logits = (language_relation / temperature).masked_fill( + diagonal, -1e4 + ) + visual_probability = F.softmax(visual_logits, dim=-1) + conditional_kl = conditional_kl + ( + visual_probability + * ( + F.log_softmax(visual_logits, dim=-1) + - F.log_softmax(language_logits, dim=-1) + ) + ).sum(-1).mean() + return mse, conditional_kl + + +def projection_quantile_target( + text_features: torch.Tensor, + particles: int, + projections: int, + generator: torch.Generator, +) -> tuple[torch.Tensor, torch.Tensor]: + """Fixed sliced-distribution target from an unpaired text population.""" + directions = F.normalize( + torch.randn( + text_features.shape[-1], + projections, + generator=generator, + device=text_features.device, + ), + dim=0, + ) + projected = (text_features @ directions).sort(dim=0).values + quantile_indices = ( + torch.linspace( + 0, + len(projected) - 1, + particles, + device=text_features.device, + ) + .round() + .long() + ) + return directions, projected[quantile_indices] + + +def sliced_distribution_energy( + particles: torch.Tensor, + directions: torch.Tensor, + target_quantiles: torch.Tensor, +) -> torch.Tensor: + projected = (F.normalize(particles, dim=-1) @ directions).sort( + dim=0 + ).values + return F.mse_loss(projected, target_quantiles) + + +def prototype_manifold_energy( + particles: torch.Tensor, prototypes: torch.Tensor +) -> torch.Tensor: + particles = F.normalize(particles, dim=-1) + prototypes = F.normalize(prototypes, dim=-1) + return (1 - (particles @ prototypes.T).max(dim=-1).values).mean() + + +def log_sinkhorn( + logits: torch.Tensor, temperature: float, iterations: int = 12 +) -> torch.Tensor: + """Doubly stochastic coupling with differentiable log-domain updates.""" + log_coupling = logits / temperature + for _ in range(iterations): + log_coupling = log_coupling - torch.logsumexp( + log_coupling, dim=1, keepdim=True + ) + log_coupling = log_coupling - torch.logsumexp( + log_coupling, dim=0, keepdim=True + ) + return log_coupling.exp() + + +def retrieval_metrics( + particles: torch.Tensor, paired_text: torch.Tensor +) -> dict[str, float]: + particles = F.normalize(particles.float(), dim=-1) + paired_text = F.normalize(paired_text.float(), dim=-1) + similarity = particles @ paired_text.T + target = similarity.diagonal() + ranks = (similarity > target[:, None]).sum(-1) + 1 + return { + "r@1": float((ranks <= 1).float().mean()), + "r@5": float((ranks <= 5).float().mean()), + "r@10": float((ranks <= 10).float().mean()), + "median_rank": float(ranks.float().median()), + "paired_cosine": float(target.mean()), + } -- cgit v1.2.3