8b7b61b3bd
* Adding Equations, solving typos * improve _code.rst * the team rst and restuctore index.rst * fixing errors --------- Co-authored-by: Dario Coscia <dariocoscia@dhcp-015.eduroam.sissa.it>
239 lines
9.1 KiB
Python
239 lines
9.1 KiB
Python
import torch
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from .location import Location
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from pina.geometry import CartesianDomain
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from pina import LabelTensor
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from ..utils import check_consistency
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class SimplexDomain(Location):
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"""PINA implementation of a Simplex."""
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def __init__(self, simplex_matrix, sample_surface=False):
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"""
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:param simplex_matrix: A matrix of LabelTensor objects representing
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a vertex of the simplex (a tensor), and the coordinates of the
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point (a list of labels).
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:type simplex_matrix: list[LabelTensor]
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:param sample_surface: A variable for choosing sample strategies. If
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``sample_surface=True`` only samples on the Simplex surface
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frontier are taken. If ``sample_surface=False``, no such criteria
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is followed.
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:type sample_surface: bool
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.. warning::
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Sampling for dimensions greater or equal to 10 could result
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in a shrinking of the simplex, which degrades the quality
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of the samples. For dimensions higher than 10, other algorithms
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for sampling should be used.
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:Example:
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>>> spatial_domain = SimplexDomain(
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[
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LabelTensor(torch.tensor([[0, 0]]), labels=["x", "y"]),
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LabelTensor(torch.tensor([[1, 1]]), labels=["x", "y"]),
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LabelTensor(torch.tensor([[0, 2]]), labels=["x", "y"]),
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], sample_surface = True
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)
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"""
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# check consistency of sample_surface as bool
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check_consistency(sample_surface, bool)
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self._sample_surface = sample_surface
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# check consistency of simplex_matrix as list or tuple
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check_consistency([simplex_matrix], (list, tuple))
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# check everything within simplex_matrix is a LabelTensor
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check_consistency(simplex_matrix, LabelTensor)
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# check consistency of labels
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matrix_labels = simplex_matrix[0].labels
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if not all(vertex.labels == matrix_labels for vertex in simplex_matrix):
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raise ValueError(f"Labels don't match.")
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# check consistency dimensions
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dim_simplex = len(matrix_labels)
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if len(simplex_matrix) != dim_simplex + 1:
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raise ValueError(
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"An n-dimensional simplex is composed by n + 1 tensors of dimension n."
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)
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# creating vertices matrix
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self._vertices_matrix = LabelTensor.vstack(simplex_matrix)
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# creating basis vectors for simplex
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# self._vectors_shifted = (
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# (self._vertices_matrix.T)[:, :-1] - (self._vertices_matrix.T)[:, None, -1]
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# ) ### TODO: Remove after checking
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vert = self._vertices_matrix
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self._vectors_shifted = (vert[:-1] - vert[-1]).T
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# build cartesian_bound
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self._cartesian_bound = self._build_cartesian(self._vertices_matrix)
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@property
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def variables(self):
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return self._vertices_matrix.labels
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def _build_cartesian(self, vertices):
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"""
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Build Cartesian border for Simplex domain to be used in sampling.
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:param vertex_matrix: matrix of vertices
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:type vertices: list[list]
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:return: Cartesian border for triangular domain
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:rtype: CartesianDomain
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"""
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span_dict = {}
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for i, coord in enumerate(self.variables):
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sorted_vertices = sorted(vertices, key=lambda vertex: vertex[i])
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# respective coord bounded by the lowest and highest values
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span_dict[coord] = [
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float(sorted_vertices[0][i]),
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float(sorted_vertices[-1][i])
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]
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return CartesianDomain(span_dict)
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def is_inside(self, point, check_border=False):
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"""
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Check if a point is inside the simplex.
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Uses the algorithm described involving barycentric coordinates:
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https://en.wikipedia.org/wiki/Barycentric_coordinate_system.
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:param point: Point to be checked.
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:type point: LabelTensor
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:param check_border: Check if the point is also on the frontier
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of the simplex, default ``False``.
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:type check_border: bool
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:return: Returning ``True`` if the point is inside, ``False`` otherwise.
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:rtype: bool
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.. note::
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When ``sample_surface`` in the ``__init()__``
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is set to ``True``, then the method only checks
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points on the surface, and not inside the domain.
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"""
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if not all(label in self.variables for label in point.labels):
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raise ValueError("Point labels different from constructor"
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f" dictionary labels. Got {point.labels},"
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f" expected {self.variables}.")
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point_shift = point - self._vertices_matrix[-1]
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point_shift = point_shift.tensor.reshape(-1, 1)
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# compute barycentric coordinates
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lambda_ = torch.linalg.solve(self._vectors_shifted * 1.0,
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point_shift * 1.0)
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lambda_1 = 1.0 - torch.sum(lambda_)
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lambdas = torch.vstack([lambda_, lambda_1])
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# perform checks
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if not check_border:
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return all(torch.gt(lambdas, 0.0)) and all(torch.lt(lambdas, 1.0))
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return all(torch.ge(lambdas, 0)) and (any(torch.eq(lambdas, 0))
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or any(torch.eq(lambdas, 1)))
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def _sample_interior_randomly(self, n, variables):
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"""
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Randomly sample points inside a simplex of arbitrary
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dimension, without the boundary.
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:param int n: Number of points to sample in the shape.
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:param variables: pinn variable to be sampled, defaults to ``all``.
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:type variables: str or list[str], optional
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:return: Returns tensor of n sampled points.
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:rtype: torch.Tensor
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"""
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# =============== For Developers ================ #
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#
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# The sampling startegy used is fairly simple.
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# First we sample a random vector from the hypercube
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# which contains the simplex. Then, if the point
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# sampled is inside the simplex, we add it as a valid
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# one.
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#
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# =============================================== #
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sampled_points = []
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while len(sampled_points) < n:
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sampled_point = self._cartesian_bound.sample(n=1,
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mode="random",
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variables=variables)
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if self.is_inside(sampled_point, self._sample_surface):
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sampled_points.append(sampled_point)
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return torch.cat(sampled_points, dim=0)
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def _sample_boundary_randomly(self, n):
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"""
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Randomly sample points on the boundary of a simplex
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of arbitrary dimensions.
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:param int n: Number of points to sample in the shape.
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:return: Returns tensor of n sampled points
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:rtype: torch.Tensor
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"""
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# =============== For Developers ================ #
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#
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# The sampling startegy used is fairly simple.
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# We first sample the lambdas in [0, 1] domain,
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# we then set to zero only one lambda, and normalize.
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# Finally, we compute the matrix product between the
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# lamdas and the vertices matrix.
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#
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# =============================================== #
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sampled_points = []
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while len(sampled_points) < n:
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# extract number of vertices
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number_of_vertices = self._vertices_matrix.shape[0]
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# extract idx lambda to set to zero randomly
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idx_lambda = torch.randint(low=0,
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high=number_of_vertices,
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size=(1, ))
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# build lambda vector
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# 1. sampling [1, 2)
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lambdas = torch.rand((number_of_vertices, 1))
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# 2. setting lambdas[idx_lambda] to 0
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lambdas[idx_lambda] = 0
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# 3. normalize
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lambdas /= lambdas.sum()
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# 4. compute dot product
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sampled_points.append(self._vertices_matrix.T @ lambdas)
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return torch.cat(sampled_points, dim=1).T
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def sample(self, n, mode="random", variables="all"):
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"""
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Sample n points from Simplex domain.
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:param int n: Number of points to sample in the shape.
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:param str mode: Mode for sampling, defaults to ``random``. Available modes include: ``random``.
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:param variables: Variables to be sampled, defaults to ``all``.
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:type variables: str | list[str]
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:return: Returns ``LabelTensor`` of n sampled points.
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:rtype: LabelTensor
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.. warning::
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When ``sample_surface = True`` in the initialization, all
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the variables are sampled, despite passing different once
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in ``variables``.
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"""
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if mode in ["random"]:
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if self._sample_surface:
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sample_pts = self._sample_boundary_randomly(n)
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else:
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sample_pts = self._sample_interior_randomly(n, variables)
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else:
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raise NotImplementedError(f"mode={mode} is not implemented.")
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return LabelTensor(sample_pts, labels=self.variables) |