"""Affinity matrices with normalizations using nearest neighbor distances."""
# Author: Hugues Van Assel <vanasselhugues@gmail.com>
# Cédric Vincent-Cuaz <cedric.vincent-cuaz@inria.fr>
#
# License: BSD 3-Clause License
from typing import Tuple, Union, Optional
import torch
from torchdr.affinity.base import Affinity, LogAffinity
from torchdr.utils import batch_transpose, kmin, logsumexp_red, sum_red, wrap_vectors
@wrap_vectors
def _log_SelfTuning(C, sigma):
r"""Return the self-tuning affinity matrix with sample-wise bandwidth.
The bandwidth is determined by the distance from a point
to its K-th neirest neighbor in log domain.
Parameters
----------
C : torch.Tensor or pykeops.torch.LazyTensor of shape (n, n)
Pairwise distance matrix.
sigma : torch.Tensor of shape (n,)
Sample-wise bandwidth parameter.
Returns
-------
log_P : torch.Tensor or pykeops.torch.LazyTensor
"""
sigma_t = batch_transpose(sigma)
return -C / (sigma * sigma_t)
@wrap_vectors
def _log_MAGIC(C, sigma):
r"""Return the MAGIC affinity matrix with sample-wise bandwidth.
The bandwidth is determined by the distance from a point
to its K-th neirest neighbor in log domain.
Parameters
----------
C : torch.Tensor or pykeops.torch.LazyTensor of shape (n, n)
Pairwise distance matrix.
sigma : torch.Tensor of shape (n,)
Sample-wise bandwidth parameter.
Returns
-------
log_P : torch.Tensor or pykeops.torch.LazyTensor
"""
return -C / sigma
[docs]
class SelfTuningAffinity(LogAffinity):
r"""Self-tuning affinity introduced in :cite:`zelnik2004self`.
The affinity has a sample-wise bandwidth :math:`\mathbf{\sigma} \in \mathbb{R}^n`.
.. math::
\exp \left( - \frac{C_{ij}}{\sigma_i \sigma_j} \right)
In the above, :math:`\mathbf{C}` is the pairwise distance matrix and
:math:`\sigma_i` is the distance from the K'th nearest neighbor of data point
:math:`\mathbf{x}_i`.
Parameters
----------
K : int, optional
K-th neirest neighbor .
normalization_dim : int or Tuple[int], optional
Dimension along which to normalize the affinity matrix.
metric : str, optional
Metric to use for pairwise distances computation.
zero_diag : bool, optional
Whether to set the diagonal of the affinity matrix to zero.
device : str, optional
Device to use for computations.
backend : {"keops", "faiss", None}, optional
Which backend to use for handling sparsity and memory efficiency.
Default is None.
verbose : bool, optional
Verbosity. Default is False.
"""
def __init__(
self,
K: int = 7,
normalization_dim: Union[int, Tuple[int]] = (0, 1),
metric: str = "sqeuclidean",
zero_diag: bool = True,
device: Optional[str] = None,
backend: Optional[str] = None,
verbose: bool = False,
):
super().__init__(
metric=metric,
zero_diag=zero_diag,
device=device,
backend=backend,
verbose=verbose,
)
self.K = K
self.normalization_dim = normalization_dim
def _compute_log_affinity(self, X: torch.Tensor):
r"""Fit the self-tuning affinity model to the provided data.
Parameters
----------
X : torch.Tensor
Input data.
Returns
-------
log_affinity_matrix : torch.Tensor or pykeops.torch.LazyTensor
The computed affinity matrix in log domain.
"""
C, _ = self._distance_matrix(X)
minK_values, minK_indices = kmin(C, k=self.K, dim=1)
self.sigma_ = minK_values[:, -1]
log_affinity_matrix = _log_SelfTuning(C, self.sigma_)
if self.normalization_dim is not None:
self.log_normalization_ = logsumexp_red(
log_affinity_matrix, self.normalization_dim
)
log_affinity_matrix = log_affinity_matrix - self.log_normalization_
return log_affinity_matrix
[docs]
class MAGICAffinity(Affinity):
r"""Compute the MAGIC affinity introduced in :cite:`van2018recovering`.
The construction is as follows. First, it computes a Gaussian
kernel with sample-wise bandwidth :math:`\mathbf{\sigma} \in \mathbb{R}^n`.
.. math::
P_{ij} \leftarrow \exp \left( - \frac{C_{ij}}{\sigma_i} \right)
In the above, :math:`\mathbf{C}` is the pairwise distance matrix and
:math:`\sigma_i` is the distance from the K'th nearest neighbor of data point
:math:`\mathbf{x}_i`.
Then it averages the affinity matrix with its transpose:
.. math::
P_{ij} \leftarrow \frac{P_{ij} + P_{ji}}{2} \:.
Finally, it normalizes the affinity matrix along each row:
.. math::
P_{ij} \leftarrow \frac{P_{ij}}{\sum_{t} P_{it}} \:.
Parameters
----------
K : int, optional
K-th neirest neighbor .
metric : str, optional
Metric to use for pairwise distances computation.
zero_diag : bool, optional
Whether to set the diagonal of the affinity matrix to zero.
device : str, optional
Device to use for computations.
backend : {"keops", "faiss", None}, optional
Which backend to use for handling sparsity and memory efficiency.
Default is None.
verbose : bool, optional
Verbosity. Default is False.
"""
def __init__(
self,
K: int = 7,
metric: str = "sqeuclidean",
zero_diag: bool = True,
device: Optional[str] = None,
backend: Optional[str] = None,
verbose: bool = False,
):
super().__init__(
metric=metric,
zero_diag=zero_diag,
device=device,
backend=backend,
verbose=verbose,
)
self.K = K
def _compute_affinity(self, X: torch.Tensor):
r"""Fit the MAGIC affinity model to the provided data.
Parameters
----------
X : torch.Tensor
Input data.
Returns
-------
affinity_matrix : torch.Tensor or pykeops.torch.LazyTensor
The computed affinity matrix.
"""
C, _ = self._distance_matrix(X)
minK_values, minK_indices = kmin(C, k=self.K, dim=1)
self.sigma_ = minK_values[:, -1]
affinity_matrix = _log_MAGIC(C, self.sigma_).exp()
affinity_matrix = (affinity_matrix + batch_transpose(affinity_matrix)) / 2
self.normalization_ = sum_red(affinity_matrix, 1)
affinity_matrix = affinity_matrix / self.normalization_
return affinity_matrix
