import torch from .location import Location from pina.geometry import CartesianDomain from pina import LabelTensor from ..utils import check_consistency class SimplexDomain(Location): """PINA implementation of a Simplex.""" def __init__(self, simplex_matrix, sample_surface=False): """ :param simplex_matrix: A matrix of LabelTensor objects representing a vertex of the simplex (a tensor), and the coordinates of the point (a list of labels). :type simplex_matrix: list[LabelTensor] :param sample_surface: A variable for choosing sample strategies. If `sample_surface=True` only samples on the Simplex surface frontier are taken. If `sample_surface=False`, no such criteria is followed. :type sample_surface: bool .. warning:: Sampling for dimensions greater or equal to 10 could result in a shrinking of the simplex, which degrades the quality of the samples. For dimensions higher than 10, other algorithms for sampling should be used. :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 self._coordinates = simplex_matrix[0].labels if not all(vertex.labels == self._coordinates for vertex in simplex_matrix): raise ValueError(f"Labels don't match.") # creating vertices matrix self._vertices_matrix = torch.cat(simplex_matrix) self._vertices_matrix.labels = self._coordinates # creating basis vectors for simplex self._vectors_shifted = ( (self._vertices_matrix.T)[:, :-1] - (self._vertices_matrix.T)[:, None, -1] ) # build cartesian_bound self._cartesian_bound = self._build_cartesian(self.variables) @property def coordinates(self): return self._coordinates @property def variables(self): return self._vertices_matrix def _build_cartesian(self, vertices): """ Build Cartesian border for Simplex domain to be used in sampling. :param vertex_matrix: matrix of vertices :type vertices: list[list] :return: Cartesian border for triangular domain :rtype: CartesianDomain """ span_dict = {} for i, coord in enumerate(self._coordinates): sorted_vertices = sorted(vertices, key=lambda vertex: vertex[i]) # respective coord bounded by the lowest and highest values span_dict[coord] = [sorted_vertices[0][i], sorted_vertices[-1][i]] return CartesianDomain(span_dict) def is_inside(self, point, check_border=False): """ Check if a point is inside the simplex. Uses the algorithm described involving barycentric coordinates: https://en.wikipedia.org/wiki/Barycentric_coordinate_system. .. note:: When ```'sample_surface'``` in the ```'__init()__'``` is set to ```'True'```, then the method only checks points on the surface, and not inside the domain. :param point: Point to be checked. :type point: LabelTensor :param check_border: Check if the point is also on the frontier of the simplex, default False. :type check_border: bool :return: Returning True if the point is inside, False otherwise. :rtype: bool """ if not all([label in self.coordinates for label in point.labels]): raise ValueError( "Point labels different from constructor" f" dictionary labels. Got {point.labels}," f" expected {self.variables}." ) # shift point point_shift = point.T - (self._vertices_matrix.T)[:, None, -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): """ Randomly sample points inside a simplex of arbitrary dimension, without the boundary. :param int n: Number of points to sample in the shape. :param variables: pinn variable to be sampled, defaults to 'all'. :type variables: str or list[str], optional :return: Returns tensor of n sampled points. :rtype: 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): """ Randomly sample points on the boundary of a simplex of arbitrary dimensions. :param int n: Number of points to sample in the shape. :return: Returns tensor of n 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[1] # 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 @ lambdas) return torch.cat(sampled_points, dim=1).T def sample(self, n, mode="random", variables="all"): """ Sample n points from Simplex domain. :param int n: Number of points to sample in the shape. :param str mode: Mode for sampling, defaults to 'random'. Available modes include: 'random'. :param variables: pinn variable to be sampled, defaults to 'all'. :type variables: str or list[str], optional :return: Returns LabelTensor of n sampled points :rtype: LabelTensor(tensor) .. warning:: When ``sample_surface = True`` in the initialization, all the variables are sampled, despite passing different once in ``variables``. """ if mode in ["random"]: 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)