"""Module for the Simplex Domain."""
import torch
from .domain_interface import DomainInterface
from .cartesian import CartesianDomain
from ..label_tensor import LabelTensor
from ..utils import check_consistency
[docs]
class SimplexDomain(DomainInterface):
"""
Implementation of the simplex domain.
"""
def __init__(self, simplex_matrix, sample_surface=False):
"""
Initialization of the :class:`SimplexDomain` class.
:param list[LabelTensor] simplex_matrix: A matrix representing the
vertices of the simplex.
:param bool sample_surface: A flag to choose the sampling strategy.
If ``True``, samples are taken only from the surface of the simplex.
If ``False``, samples are taken from the interior of the simplex.
Default is ``False``.
:raises ValueError: If the labels of the vertices don't match.
:raises ValueError: If the number of vertices is not equal to the
dimension of the simplex plus one.
.. warning::
Sampling for dimensions greater or equal to 10 could result in a
shrinkage of the simplex, which degrades the quality of the samples.
For dimensions higher than 10, use other sampling algorithms.
:Example:
>>> spatial_domain = SimplexDomain(
[
LabelTensor(torch.tensor([[0, 0]]), labels=["x", "y"]),
LabelTensor(torch.tensor([[1, 1]]), labels=["x", "y"]),
LabelTensor(torch.tensor([[0, 2]]), labels=["x", "y"]),
], sample_surface = True
)
"""
# check consistency of sample_surface as bool
check_consistency(sample_surface, bool)
self._sample_surface = sample_surface
# check consistency of simplex_matrix as list or tuple
check_consistency([simplex_matrix], (list, tuple))
# check everything within simplex_matrix is a LabelTensor
check_consistency(simplex_matrix, LabelTensor)
# check consistency of labels
matrix_labels = simplex_matrix[0].labels
if not all(vertex.labels == matrix_labels for vertex in simplex_matrix):
raise ValueError("Labels don't match.")
# check consistency dimensions
dim_simplex = len(matrix_labels)
if len(simplex_matrix) != dim_simplex + 1:
raise ValueError(
"An n-dimensional simplex is composed by n + 1 tensors of "
"dimension n."
)
# creating vertices matrix
self._vertices_matrix = LabelTensor.vstack(simplex_matrix)
# creating basis vectors for simplex
vert = self._vertices_matrix
self._vectors_shifted = (vert[:-1] - vert[-1]).T
# build cartesian_bound
self._cartesian_bound = self._build_cartesian(self._vertices_matrix)
@property
def sample_modes(self):
"""
List of available sampling modes.
:return: List of available sampling modes.
:rtype: list[str]
"""
return ["random"]
@property
def variables(self):
"""
List of variables of the domain.
:return: List of variables of the domain.
:rtype: list[str]
"""
return sorted(self._vertices_matrix.labels)
def _build_cartesian(self, vertices):
"""
Build the cartesian border for a simplex domain to be used in sampling.
:param list[LabelTensor] vertices: list of vertices defining the domain.
:return: The cartesian border for the simplex domain.
:rtype: CartesianDomain
"""
span_dict = {}
for coord in self.variables:
sorted_vertices = torch.sort(vertices[coord].tensor.squeeze())
# respective coord bounded by the lowest and highest values
span_dict[coord] = [
float(sorted_vertices.values[0]),
float(sorted_vertices.values[-1]),
]
return CartesianDomain(span_dict)
[docs]
def is_inside(self, point, check_border=False):
"""
Check if a point is inside the simplex. It uses an algorithm involving
barycentric coordinates.
:param LabelTensor point: Point to be checked.
:param check_border: If ``True``, the border is considered inside
the simplex. Default is ``False``.
:raises ValueError: If the labels of the point are different from those
passed in the ``__init__`` method.
:return: ``True`` if the point is inside the domain,
``False`` otherwise.
:rtype: bool
"""
if not all(label in self.variables for label in point.labels):
raise ValueError(
"Point labels different from constructor"
f" dictionary labels. Got {point.labels},"
f" expected {self.variables}."
)
point_shift = point - self._vertices_matrix[-1]
point_shift = point_shift.tensor.reshape(-1, 1)
# compute barycentric coordinates
lambda_ = torch.linalg.solve(
self._vectors_shifted * 1.0, point_shift * 1.0
)
lambda_1 = 1.0 - torch.sum(lambda_)
lambdas = torch.vstack([lambda_, lambda_1])
# perform checks
if not check_border:
return all(torch.gt(lambdas, 0.0)) and all(torch.lt(lambdas, 1.0))
return all(torch.ge(lambdas, 0)) and (
any(torch.eq(lambdas, 0)) or any(torch.eq(lambdas, 1))
)
def _sample_interior_randomly(self, n, variables):
"""
Sample at random points from the interior of the simplex. Boundaries are
excluded from this sampling routine.
:param int n: Number of points to sample.
:param list[str] variables: variables to be sampled.
:return: Sampled points.
:rtype: list[torch.Tensor]
"""
# =============== For Developers ================ #
#
# The sampling startegy used is fairly simple.
# First we sample a random vector from the hypercube
# which contains the simplex. Then, if the point
# sampled is inside the simplex, we add it as a valid
# one.
#
# =============================================== #
sampled_points = []
while len(sampled_points) < n:
sampled_point = self._cartesian_bound.sample(
n=1, mode="random", variables=variables
)
if self.is_inside(sampled_point, self._sample_surface):
sampled_points.append(sampled_point)
return torch.cat(sampled_points, dim=0)
def _sample_boundary_randomly(self, n):
"""
Sample at random points from the boundary of the simplex.
:param int n: Number of points to sample.
:return: Sampled points.
:rtype: torch.Tensor
"""
# =============== For Developers ================ #
#
# The sampling startegy used is fairly simple.
# We first sample the lambdas in [0, 1] domain,
# we then set to zero only one lambda, and normalize.
# Finally, we compute the matrix product between the
# lamdas and the vertices matrix.
#
# =============================================== #
sampled_points = []
while len(sampled_points) < n:
# extract number of vertices
number_of_vertices = self._vertices_matrix.shape[0]
# extract idx lambda to set to zero randomly
idx_lambda = torch.randint(
low=0, high=number_of_vertices, size=(1,)
)
# build lambda vector
# 1. sampling [1, 2)
lambdas = torch.rand((number_of_vertices, 1))
# 2. setting lambdas[idx_lambda] to 0
lambdas[idx_lambda] = 0
# 3. normalize
lambdas /= lambdas.sum()
# 4. compute dot product
sampled_points.append(self._vertices_matrix.T @ lambdas)
return torch.cat(sampled_points, dim=1).T
[docs]
def sample(self, n, mode="random", variables="all"):
"""
Sampling routine.
:param int n: Number of points to sample.
:param str mode: Sampling method. Default is ``random``.
Available modes: random sampling, ``random``.
:param list[str] variables: variables to be sampled. Default is ``all``.
:raises NotImplementedError: If the sampling method is not implemented.
:return: Sampled points.
:rtype: LabelTensor
.. warning::
When ``sample_surface=True``, all variables are sampled,
ignoring the ``variables`` parameter.
"""
if variables == "all":
variables = self.variables
elif isinstance(variables, (list, tuple)):
variables = sorted(variables)
if mode in self.sample_modes:
if self._sample_surface:
sample_pts = self._sample_boundary_randomly(n)
else:
sample_pts = self._sample_interior_randomly(n, variables)
else:
raise NotImplementedError(f"mode={mode} is not implemented.")
return LabelTensor(sample_pts, labels=self.variables)
