Source code for psdr.geometry.vertex

from __future__ import print_function

import numpy as np
from scipy.spatial.distance import cdist
import scipy.linalg

#from .domains import Domain

[docs]def voronoi_vertex(domain, Xhat, X0, L = None, randomize = True): r""" Constructs a subset of the Voronoi vertices on a given domain Given a domain :math:`\mathcal{D} \subset \mathbb{R}^m`, a set of points :math:`\lbrace \widehat{\mathbf{x}}_j \rbrace_{j=1}^M`, and a distance metric .. math:: d(\mathbf{x}_1, \mathbf{x}_2) = \|\mathbf{L}(\mathbf{x}_1 - \mathbf{x}_2)\|_2 the vertices of the bounded Voronoi diagram :math:`\mathbf{v}_i \in \mathcal{D}` are those points are those points that satisfy :math:`r` equidistance constraints in the metric :math:`d` .. math:: \| \mathbf{L}(\mathbf{v}_i - \widehat{\mathbf{x}}_{\mathcal{I}_i[1]})\|_2 = \| \mathbf{L}(\mathbf{v}_i - \widehat{\mathbf{x}}_{\mathcal{I}_i[2]})\|_2 = \ldots = \| \mathbf{L}(\mathbf{v}_i - \widehat{\mathbf{x}}_{\mathcal{I}_i[r]})\|_2 and :math:`m-r` constraints from the domain. The bounded Voronoi vertices include the vertices of the domain. The number of domain vertices can grow exponentially in dimension (e.g., consider the cube), but the asymptotic cost of finding these is :math:`\mathcal{O}(mn)` [wikiVEP]_ per vertex where :math:`n` is the number of linear inequality constraints specifying the domain. Hence, the complexity of finding the Voronoi vertices grows exponentially in the dimension of the domain. Thus we use an approach due Lindemann and Cheng [LC05]_ to find a subset of vertices by taking a starting point :math:`\mathbf{x}_0` and sequentially projecting it onto a series of hyperplanes such that after :math:`m` steps we satisfy :math:`m` constraints. Parameters ---------- domain: Domain Domain on which to construct the vertices Xhat: array-like (M, m) M existing points on the domain which define the Voronoi diagram X0: array-like (N, m) Initial points to use to find vertices L: array-like (m, m), optional Weight on distance in 2-norm; defaults to the identity matrix randomize: bool, optional (default: True) If true, when L is rank deficient add a random direction at each step such that we find points on the boundary of the domain satisfying m constraints. Returns ------- X: np.ndarray(N, m) Points satisfying m constraints References ---------- .. [LC05] Iteratively Locating Voronoi Vertices for Dispersion Estimation Stephen R. Lindemann and Peng Cheng Proceedings of the 2005 Interational Conference on Robotics and Automation .. [wikiVEP] Vertex enumeration problem, Wikipedia. https://en.wikipedia.org/wiki/Vertex_enumeration_problem """ # Startup checks #assert isinstance(domain, Domain), "domain must be an instance of the Domain class" assert len(domain.Ls) == 0, "Currently this does not support domains with quadratic constraints" Xhat = np.atleast_2d(np.array(Xhat)) X0 = np.atleast_2d(np.array(X0)).copy() m = len(domain) if L is None: L = np.eye(m) else: L = np.array(L) assert L.shape[1] == m, "Number of columns doesn't match dimension of the space" Lrank = np.linalg.matrix_rank(L) LTL = L.T.dot(L) # Linear inequality constraints for the domain (including lower bound/upper bound box constraints) A = domain.A_aug b = domain.b_aug # This algorithm terminates when each point has m active constraints as we can have no more improvement # hence we substract the number of equality constraints # we also subtract the rank-deficency of L because this results in zero-steps for k in range(m - domain.A_eq.shape[0]): if k >= Lrank and not randomize: break # Find the nearest neighbors # As we intend to do this in high dimensions, # we don't use a tree-based distance approach # as these don't scale well D = cdist(L.dot(X0.T).T, L.dot(Xhat.T).T) I = np.argsort(D, axis = 1) # set the search direction to move away from the closest point h = X0 - Xhat[I[:,0]] h = LTL.dot(h.T).T # which inequality constraints on the domain are active active = np.isclose(A.dot(X0.T).T , b) for i in range(X0.shape[0]): # For each point, project onto the feasible directions nullspace = [domain.A_eq.T] # active inequality constraints for j in range(A.shape[0]): if active[i,j] : nullspace += [A[j].reshape(-1,1)] # constraints from other points in the domain for j in range(1, min(k+1, len(Xhat))): if np.isclose(D[i,I[i,0]], D[i,I[i,j]]): nullspace += [LTL.dot(Xhat[I[i,0]] - Xhat[I[i,j]]).reshape(-1,1)] #h[i] += (X0[i] - Xhat[I[i,j]]) nullspace = np.hstack(nullspace) # If there are no active constraints, don't do anything if nullspace.shape[1]>0: Q, R = scipy.linalg.qr(nullspace, overwrite_a = True, mode = 'economic') h[i] -= Q.dot(Q.T.dot(h[i])) if np.isclose(np.linalg.norm(h[i]), 0) and randomize: # If L is low-rank we can have situations where h[i] # as constructed above is in the nullspace of constraints # and hence is approximately zero. When randomize=True # we choose a random search direction so we can still make # progress towards satisfying m constraints. h[i] = np.random.randn(*h[i].shape) h[i] -= Q.dot(Q.T.dot(h[i])) #print("k=%d, constraints %d, norm %g" % (k, Q.shape[1], np.linalg.norm(h[i]))) # Now we find the furthest we can step along this direction before either hitting a # (1) separating hyperplane separating x0 and points in Xhat or # (2) the boundary of the domain alpha = np.inf*np.ones(X0.shape[0]) # (1) Take a step until we run into a separating hyperplane for xhat in Xhat: # we setup a hyperplane (p - p0)^* n = 0 # where p0 is a point on the hyperplane # separating the closest point Xhat[:,I[:,0]] and xhat p0 = 0.5*(L.dot( (xhat + Xhat[I[:,0]]).T).T) n = L.dot( (xhat - Xhat[I[:,0]]).T).T # Inner product of normal n with search direction h nh = np.sum(n*L.dot(h.T).T, axis = 1) # Inner product for the numerator (x0 - p0)^* n numerator = -np.sum( (L.dot(X0.T).T - p0)*n, axis = 1) alpha_c = np.inf*np.ones(alpha.shape) act = ~np.isclose(np.abs(nh), 0) alpha_c[act] = numerator[act]/nh[act] # When xhat is the closest point, we get a divide by zero error # We cannot move backwards, so these are also set to infinity alpha_c[alpha_c < 0 ] = np.inf alpha = np.minimum(alpha, alpha_c) # (2) Find intersection with the domain # cf., Domain._extent_ineq AX0 = A.dot(X0.T).T Ah = A.dot(h.T).T with np.errstate(divide = 'ignore', invalid = 'ignore'): alpha_c = (b - AX0)/Ah alpha_c[~np.isfinite(alpha_c)] = np.inf alpha_c[alpha_c <= 0] = np.inf # When the step size is near zero, the above formula is inaccurate # causing points to emerge outside of the domain. Hence, we zero the # step size for these points. alpha_c[ np.max(np.abs(h), axis = 1) < 1e-10] = 0. alpha_c = np.min(alpha_c, axis = 1) alpha = np.minimum(alpha, alpha_c) # There are some cases above where both alpha and alpha_c will yield an infinite step # in this case we zero these just to be safe alpha[~np.isfinite(alpha)] = 0. # Now finally take the step X0 += alpha.reshape(-1,1)*h # Keep only those points that are inside the domain # (all points should be, but sometimes numerical issues push outside) #X0 = X0[domain.isinside(X0)] return X0