# -*- coding: utf-8 -*-
"""Implementation of KG2E."""
import math
from typing import Any, ClassVar, Mapping, Optional
import torch
import torch.autograd
from ..base import EntityRelationEmbeddingModel
from ...constants import DEFAULT_EMBEDDING_HPO_EMBEDDING_DIM_RANGE
from ...losses import Loss
from ...nn import Embedding
from ...regularizers import Regularizer
from ...triples import TriplesFactory
from ...typing import DeviceHint
from ...utils import clamp_norm
__all__ = [
'KG2E',
]
_LOG_2_PI = math.log(2. * math.pi)
[docs]class KG2E(EntityRelationEmbeddingModel):
r"""An implementation of KG2E from [he2015]_.
KG2E aims to explicitly model (un)certainties in entities and relations (e.g. influenced by the number of triples
observed for these entities and relations). Therefore, entities and relations are represented by probability
distributions, in particular by multi-variate Gaussian distributions $\mathcal{N}_i(\mu_i,\Sigma_i)$
where the mean $\mu_i \in \mathbb{R}^d$ denotes the position in the vector space and the diagonal variance
$\Sigma_i \in \mathbb{R}^{d \times d}$ models the uncertainty.
Inspired by the :class:`pykeen.models.TransE` model, relations are modeled as transformations from head to tail
entities: $\mathcal{H} - \mathcal{T} \approx \mathcal{R}$ where
$\mathcal{H} \sim \mathcal{N}_h(\mu_h,\Sigma_h)$, $\mathcal{H} \sim \mathcal{N}_t(\mu_t,\Sigma_t)$,
$\mathcal{R} \sim \mathcal{P}_r = \mathcal{N}_r(\mu_r,\Sigma_r)$, and
$\mathcal{H} - \mathcal{T} \sim \mathcal{P}_e = \mathcal{N}_{h-t}(\mu_h - \mu_t,\Sigma_h + \Sigma_t)$
(since head and tail entities are considered to be independent with regards to the relations).
The interaction model measures the similarity between $\mathcal{P}_e$ and $\mathcal{P}_r$ by
means of the Kullback-Liebler Divergence (:meth:`KG2E.kullback_leibler_similarity`).
.. math::
f(h,r,t) = \mathcal{D_{KL}}(\mathcal{P}_e, \mathcal{P}_r)
Besides the asymmetric KL divergence, the authors propose a symmetric variant which uses the expected
likelihood (:meth:`KG2E.expected_likelihood`)
.. math::
f(h,r,t) = \mathcal{D_{EL}}(\mathcal{P}_e, \mathcal{P}_r)
"""
#: The default strategy for optimizing the model's hyper-parameters
hpo_default: ClassVar[Mapping[str, Any]] = dict(
embedding_dim=DEFAULT_EMBEDDING_HPO_EMBEDDING_DIM_RANGE,
c_min=dict(type=float, low=0.01, high=0.1, scale='log'),
c_max=dict(type=float, low=1.0, high=10.0),
)
def __init__(
self,
triples_factory: TriplesFactory,
embedding_dim: int = 50,
loss: Optional[Loss] = None,
preferred_device: DeviceHint = None,
random_seed: Optional[int] = None,
dist_similarity: Optional[str] = None,
c_min: float = 0.05,
c_max: float = 5.,
regularizer: Optional[Regularizer] = None,
) -> None:
r"""Initialize KG2E.
:param embedding_dim: The entity embedding dimension $d$. Is usually $d \in [50, 350]$.
:param dist_similarity: Either 'KL' for kullback-liebler or 'EL' for expected liklihood. Defaults to KL.
:param c_min:
:param c_max:
"""
super().__init__(
triples_factory=triples_factory,
embedding_dim=embedding_dim,
loss=loss,
preferred_device=preferred_device,
random_seed=random_seed,
regularizer=regularizer,
entity_constrainer=clamp_norm,
entity_constrainer_kwargs=dict(maxnorm=1., p=2, dim=-1),
relation_constrainer=clamp_norm,
relation_constrainer_kwargs=dict(maxnorm=1., p=2, dim=-1),
)
# Similarity function used for distributions
if dist_similarity is None or dist_similarity.upper() == 'KL':
self.similarity = self.kullback_leibler_similarity
elif dist_similarity.upper() == 'EL':
self.similarity = self.expected_likelihood
else:
raise ValueError(f'Unknown distribution similarity: "{dist_similarity}".')
# element-wise covariance bounds
self.c_min = c_min
self.c_max = c_max
# Additional covariance embeddings
self.entity_covariances = Embedding.init_with_device(
num_embeddings=triples_factory.num_entities,
embedding_dim=embedding_dim,
device=self.device,
# Ensure positive definite covariances matrices and appropriate size by clamping
constrainer=torch.clamp,
constrainer_kwargs=dict(min=self.c_min, max=self.c_max),
)
self.relation_covariances = Embedding.init_with_device(
num_embeddings=triples_factory.num_relations,
embedding_dim=embedding_dim,
device=self.device,
# Ensure positive definite covariances matrices and appropriate size by clamping
constrainer=torch.clamp,
constrainer_kwargs=dict(min=self.c_min, max=self.c_max),
)
def _reset_parameters_(self): # noqa: D102
# Constraints are applied through post_parameter_update
super()._reset_parameters_()
for emb in [
self.entity_covariances,
self.relation_covariances,
]:
emb.reset_parameters()
[docs] def post_parameter_update(self) -> None: # noqa: D102
super().post_parameter_update()
for cov in (
self.entity_covariances,
self.relation_covariances,
):
cov.post_parameter_update()
def _score(
self,
h_indices: Optional[torch.LongTensor] = None,
r_indices: Optional[torch.LongTensor] = None,
t_indices: Optional[torch.LongTensor] = None,
) -> torch.FloatTensor:
"""
Compute scores for NTN.
:param h_indices: shape: (batch_size,)
:param r_indices: shape: (batch_size,)
:param t_indices: shape: (batch_size,)
:return: shape: (batch_size, num_entities)
"""
# Get embeddings
mu_h = self.entity_embeddings.get_in_canonical_shape(indices=h_indices)
mu_r = self.relation_embeddings.get_in_canonical_shape(indices=r_indices)
mu_t = self.entity_embeddings.get_in_canonical_shape(indices=t_indices)
sigma_h = self.entity_covariances.get_in_canonical_shape(indices=h_indices)
sigma_r = self.relation_covariances.get_in_canonical_shape(indices=r_indices)
sigma_t = self.entity_covariances.get_in_canonical_shape(indices=t_indices)
# Compute entity distribution
mu_e = mu_h - mu_t
sigma_e = sigma_h + sigma_t
return self.similarity(mu_e=mu_e, mu_r=mu_r, sigma_e=sigma_e, sigma_r=sigma_r)
[docs] def score_hrt(self, hrt_batch: torch.LongTensor) -> torch.FloatTensor: # noqa: D102
return self._score(h_indices=hrt_batch[:, 0], r_indices=hrt_batch[:, 1], t_indices=hrt_batch[:, 2]).view(-1, 1)
[docs] def score_t(self, hr_batch: torch.LongTensor) -> torch.FloatTensor: # noqa: D102
return self._score(h_indices=hr_batch[:, 0], r_indices=hr_batch[:, 1])
[docs] def score_h(self, rt_batch: torch.LongTensor) -> torch.FloatTensor: # noqa: D102
return self._score(r_indices=rt_batch[:, 0], t_indices=rt_batch[:, 1])
[docs] @staticmethod
def expected_likelihood(
mu_e: torch.FloatTensor,
mu_r: torch.FloatTensor,
sigma_e: torch.FloatTensor,
sigma_r: torch.FloatTensor,
epsilon: float = 1.0e-10,
) -> torch.FloatTensor:
r"""Compute the similarity based on expected likelihood.
.. math::
D((\mu_e, \Sigma_e), (\mu_r, \Sigma_r)))
= \frac{1}{2} \left(
(\mu_e - \mu_r)^T(\Sigma_e + \Sigma_r)^{-1}(\mu_e - \mu_r)
+ \log \det (\Sigma_e + \Sigma_r) + d \log (2 \pi)
\right)
= \frac{1}{2} \left(
\mu^T\Sigma^{-1}\mu
+ \log \det \Sigma + d \log (2 \pi)
\right)
:param mu_e: torch.Tensor, shape: (s_1, ..., s_k, d)
The mean of the first Gaussian.
:param mu_r: torch.Tensor, shape: (s_1, ..., s_k, d)
The mean of the second Gaussian.
:param sigma_e: torch.Tensor, shape: (s_1, ..., s_k, d)
The diagonal covariance matrix of the first Gaussian.
:param sigma_r: torch.Tensor, shape: (s_1, ..., s_k, d)
The diagonal covariance matrix of the second Gaussian.
:param epsilon: float (default=1.0)
Small constant used to avoid numerical issues when dividing.
:return: torch.Tensor, shape: (s_1, ..., s_k)
The similarity.
"""
d = sigma_e.shape[-1]
sigma = sigma_r + sigma_e
mu = mu_e - mu_r
#: a = \mu^T\Sigma^{-1}\mu
safe_sigma = torch.clamp_min(sigma, min=epsilon)
sigma_inv = torch.reciprocal(safe_sigma)
a = torch.sum(sigma_inv * mu ** 2, dim=-1)
#: b = \log \det \Sigma
b = safe_sigma.log().sum(dim=-1)
return a + b + d * _LOG_2_PI
[docs] @staticmethod
def kullback_leibler_similarity(
mu_e: torch.FloatTensor,
mu_r: torch.FloatTensor,
sigma_e: torch.FloatTensor,
sigma_r: torch.FloatTensor,
epsilon: float = 1.0e-10,
) -> torch.FloatTensor:
r"""Compute the similarity based on KL divergence.
This is done between two Gaussian distributions given by mean mu_* and diagonal covariance matrix sigma_*.
.. math::
D((\mu_e, \Sigma_e), (\mu_r, \Sigma_r)))
= \frac{1}{2} \left(
tr(\Sigma_r^{-1}\Sigma_e)
+ (\mu_r - \mu_e)^T\Sigma_r^{-1}(\mu_r - \mu_e)
- \log \frac{det(\Sigma_e)}{det(\Sigma_r)} - k_e
\right)
Note: The sign of the function has been flipped as opposed to the description in the paper, as the
Kullback Leibler divergence is large if the distributions are dissimilar.
:param mu_e: torch.Tensor, shape: (s_1, ..., s_k, d)
The mean of the first Gaussian.
:param mu_r: torch.Tensor, shape: (s_1, ..., s_k, d)
The mean of the second Gaussian.
:param sigma_e: torch.Tensor, shape: (s_1, ..., s_k, d)
The diagonal covariance matrix of the first Gaussian.
:param sigma_r: torch.Tensor, shape: (s_1, ..., s_k, d)
The diagonal covariance matrix of the second Gaussian.
:param epsilon: float (default=1.0)
Small constant used to avoid numerical issues when dividing.
:return: torch.Tensor, shape: (s_1, ..., s_k)
The similarity.
"""
d = mu_e.shape[-1]
safe_sigma_r = torch.clamp_min(sigma_r, min=epsilon)
sigma_r_inv = torch.reciprocal(safe_sigma_r)
#: a = tr(\Sigma_r^{-1}\Sigma_e)
a = torch.sum(sigma_e * sigma_r_inv, dim=-1)
#: b = (\mu_r - \mu_e)^T\Sigma_r^{-1}(\mu_r - \mu_e)
mu = mu_r - mu_e
b = torch.sum(sigma_r_inv * mu ** 2, dim=-1)
#: c = \log \frac{det(\Sigma_e)}{det(\Sigma_r)}
# = sum log (sigma_e)_i - sum log (sigma_r)_i
c = sigma_e.clamp_min(min=epsilon).log().sum(dim=-1) - safe_sigma_r.log().sum(dim=-1)
return -0.5 * (a + b - c - d)