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The rigid-alignment mechanism is based on the Kabsch formulation used by
AlphaFold 3 implementations. The implementation is maintained locally and
does not import an upstream runtime package.
"""
from __future__ import annotations
import warnings
import torch
import torch.nn.functional as functional
from einops import einsum
def _weighted_centroid(
coordinates: torch.Tensor,
weights: torch.Tensor,
) -> torch.Tensor:
# coordinates: (..., n, 3); weights: (..., n, 1).
return (coordinates * weights).sum(dim=-2, keepdim=True) / weights.sum(
dim=-2,
keepdim=True,
) # (..., 1, 3)
def _warn_if_alignment_is_ambiguous(
mask: torch.Tensor,
singular_values: torch.Tensor,
*,
num_points: int,
coordinate_dim: int,
) -> None:
if torch.any(mask.sum(dim=-1) < coordinate_dim + 1):
warnings.warn(
"The size of one of the point clouds is <= dim+1. "
"`WeightedRigidAlign` cannot return a unique rotation.",
RuntimeWarning,
stacklevel=3,
)
if (singular_values.abs() <= 1e-15).any() and num_points >= coordinate_dim + 1:
warnings.warn(
"Excessively low rank of cross-correlation between aligned "
"point clouds. `WeightedRigidAlign` cannot return a unique rotation.",
RuntimeWarning,
stacklevel=3,
)
def weighted_rigid_align(
true_coords: torch.Tensor,
pred_coords: torch.Tensor,
weights: torch.Tensor,
mask: torch.Tensor,
) -> torch.Tensor:
"""Align true coordinates to predicted coordinates with weighted Kabsch.
``true_coords`` and ``pred_coords`` have shape ``(..., n, 3)``. The
returned tensor is detached because alignment defines a fixed target for
the diffusion loss.
"""
output_shape = torch.broadcast_shapes(true_coords.shape, pred_coords.shape)
*batch_shape, num_points, coordinate_dim = output_shape
point_weights = (mask * weights).unsqueeze(-1) # (..., n, 1)
true_centroid = _weighted_centroid(true_coords, point_weights) # (..., 1, 3)
pred_centroid = _weighted_centroid(pred_coords, point_weights) # (..., 1, 3)
true_centered = true_coords - true_centroid # (..., n, 3)
pred_centered = pred_coords - pred_centroid # (..., n, 3)
covariance = einsum(
point_weights * pred_centered,
true_centered,
"... n i, ... n j -> ... i j",
) # (..., 3, 3)
original_dtype = covariance.dtype
covariance_fp32 = covariance.to(torch.float32) # (..., 3, 3)
left_vectors, singular_values, right_vectors_h = torch.linalg.svd(
covariance_fp32,
driver="gesvd" if covariance_fp32.is_cuda else None,
) # left/right: (..., 3, 3); singular_values: (..., 3)
right_vectors = right_vectors_h.mH # (..., 3, 3)
_warn_if_alignment_is_ambiguous(
mask,
singular_values,
num_points=num_points,
coordinate_dim=coordinate_dim,
)
preliminary_rotation = torch.einsum(
"... i j, ... k j -> ... i k",
left_vectors,
right_vectors,
).to(torch.float32) # (..., 3, 3)
orientation = torch.eye(
coordinate_dim,
dtype=covariance_fp32.dtype,
device=covariance.device,
)[None].repeat(*batch_shape, 1, 1) # (..., 3, 3)
orientation[..., -1, -1] = torch.det(preliminary_rotation) # (...)
rotation = einsum(
left_vectors,
orientation,
right_vectors,
"... i j, ... j k, ... l k -> ... i l",
).to(original_dtype) # (..., 3, 3)
aligned = (
einsum(true_centered, rotation, "... n i, ... j i -> ... n j") + pred_centroid
) # (..., n, 3)
aligned.detach_()
return aligned # (..., n, 3)
def _smooth_lddt_for_example(
pred_coords: torch.Tensor,
true_coords: torch.Tensor,
is_nucleotide: torch.Tensor,
coords_mask: torch.Tensor,
*,
nucleic_acid_cutoff: float,
other_cutoff: float,
) -> torch.Tensor:
# pred_coords/true_coords: (n, 3); is_nucleotide/coords_mask: (n,).
true_distances = torch.cdist(true_coords, true_coords) # (n, n)
nucleotide_rows = is_nucleotide.bool().unsqueeze(-1).expand_as(
true_distances
) # (n, n)
pair_mask = torch.where(
nucleotide_rows,
true_distances < nucleic_acid_cutoff,
true_distances < other_cutoff,
) # (n, n)
pair_mask &= ~torch.eye(
pred_coords.shape[0],
dtype=torch.bool,
device=pred_coords.device,
) # (n, n)
coordinate_rows = coords_mask.bool() # (n,)
pair_mask &= coordinate_rows.unsqueeze(-1) # (n, n)
pair_mask &= coordinate_rows.unsqueeze(-2) # (n, n)
pair_indices = pair_mask.nonzero() # (n_pair, 2)
true_pair_distances = true_distances[
pair_indices[:, 0], pair_indices[:, 1]
] # (n_pair,)
pred_pair_distances = functional.pairwise_distance(
pred_coords[pair_indices[:, 0]],
pred_coords[pair_indices[:, 1]],
) # (n_pair,)
distance_error = torch.abs(true_pair_distances - pred_pair_distances) # (n_pair,)
smooth_agreement = (
sum(torch.sigmoid(threshold - distance_error) for threshold in (0.5, 1.0, 2.0, 4.0)) / 4.0
) # (n_pair,)
return smooth_agreement.sum() / (pair_indices.shape[0] + 1e-5) # ()
def smooth_lddt_loss(
pred_coords: torch.Tensor,
true_coords: torch.Tensor,
is_nucleotide: torch.Tensor,
coords_mask: torch.Tensor,
nucleic_acid_cutoff: float = 30.0,
other_cutoff: float = 15.0,
multiplicity: int = 1,
) -> torch.Tensor:
"""Return one minus the smooth local-distance agreement.
Coordinate tensors have shape ``(b, n, 3)``. Sequence-level masks may
be shared across repeated diffusion samples through ``multiplicity``.
"""
agreements = [
_smooth_lddt_for_example(
pred_coords[index],
true_coords[index],
is_nucleotide[index // multiplicity],
coords_mask[index // multiplicity],
nucleic_acid_cutoff=nucleic_acid_cutoff,
other_cutoff=other_cutoff,
)
for index in range(true_coords.shape[0])
] # each: ()
return 1.0 - torch.stack(agreements).mean(dim=0) # ()
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