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Diffstat (limited to 'src/Persistence_representations')
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diff --git a/src/Persistence_representations/example/persistence_weighted_gaussian.cpp b/src/Persistence_representations/example/persistence_weighted_gaussian.cpp new file mode 100644 index 00000000..e95b9445 --- /dev/null +++ b/src/Persistence_representations/example/persistence_weighted_gaussian.cpp @@ -0,0 +1,96 @@ +/* This file is part of the Gudhi Library. The Gudhi library + * (Geometric Understanding in Higher Dimensions) is a generic C++ + * library for computational topology. + * + * Author(s): Mathieu Carriere + * + * Copyright (C) 2018 INRIA (France) + * + * This program is free software: you can redistribute it and/or modify + * it under the terms of the GNU General Public License as published by + * the Free Software Foundation, either version 3 of the License, or + * (at your option) any later version. + * + * This program is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + * GNU General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program. If not, see <http://www.gnu.org/licenses/>. + */ + +#include <gudhi/Persistence_weighted_gaussian.h> + +#include <iostream> +#include <vector> +#include <utility> + +using PD = std::vector<std::pair<double,double> >; +using PWG = Gudhi::Persistence_representations::Persistence_weighted_gaussian; + +int main(int argc, char** argv) { + + std::vector<std::pair<double, double> > persistence1; + std::vector<std::pair<double, double> > persistence2; + + persistence1.push_back(std::make_pair(1, 2)); + persistence1.push_back(std::make_pair(6, 8)); + persistence1.push_back(std::make_pair(0, 4)); + persistence1.push_back(std::make_pair(3, 8)); + + persistence2.push_back(std::make_pair(2, 9)); + persistence2.push_back(std::make_pair(1, 6)); + persistence2.push_back(std::make_pair(3, 5)); + persistence2.push_back(std::make_pair(6, 10)); + + PWG PWG1(persistence1); + PWG PWG2(persistence2); + double sigma = 1; + double tau = 1; + int m = 1000; + + + + // Linear PWG + + std::cout << PWG1.compute_scalar_product (PWG2, sigma, PWG::arctan_weight, m) << std::endl; + std::cout << PWG1.compute_scalar_product (PWG2, sigma, PWG::arctan_weight, -1) << std::endl; + + std::cout << PWG1.distance (PWG2, sigma, PWG::arctan_weight, m) << std::endl; + std::cout << PWG1.distance (PWG2, sigma, PWG::arctan_weight, -1) << std::endl; + + + + + + + + // Gaussian PWG + + std::cout << std::exp( -PWG1.distance (PWG2, sigma, PWG::arctan_weight, m, 2) ) / (2*tau*tau) << std::endl; + std::cout << std::exp( -PWG1.distance (PWG2, sigma, PWG::arctan_weight, -1, 2) ) / (2*tau*tau) << std::endl; + + + + + + + + // PSS + + PD pd1 = persistence1; int numpts = persistence1.size(); for(int i = 0; i < numpts; i++) pd1.emplace_back(persistence1[i].second,persistence1[i].first); + PD pd2 = persistence2; numpts = persistence2.size(); for(int i = 0; i < numpts; i++) pd2.emplace_back(persistence2[i].second,persistence2[i].first); + + PWG pwg1(pd1); + PWG pwg2(pd2); + + std::cout << pwg1.compute_scalar_product (pwg2, 2*std::sqrt(sigma), PWG::pss_weight, m) / (16*pi*sigma) << std::endl; + std::cout << pwg1.compute_scalar_product (pwg2, 2*std::sqrt(sigma), PWG::pss_weight, -1) / (16*pi*sigma) << std::endl; + + std::cout << pwg1.distance (pwg2, 2*std::sqrt(sigma), PWG::pss_weight, m) / (16*pi*sigma) << std::endl; + std::cout << pwg1.distance (pwg2, 2*std::sqrt(sigma), PWG::pss_weight, -1) / (16*pi*sigma) << std::endl; + + + return 0; +} diff --git a/src/Persistence_representations/example/sliced_wasserstein.cpp b/src/Persistence_representations/example/sliced_wasserstein.cpp new file mode 100644 index 00000000..673d8474 --- /dev/null +++ b/src/Persistence_representations/example/sliced_wasserstein.cpp @@ -0,0 +1,55 @@ +/* This file is part of the Gudhi Library. The Gudhi library + * (Geometric Understanding in Higher Dimensions) is a generic C++ + * library for computational topology. + * + * Author(s): Mathieu Carriere + * + * Copyright (C) 2018 INRIA (France) + * + * This program is free software: you can redistribute it and/or modify + * it under the terms of the GNU General Public License as published by + * the Free Software Foundation, either version 3 of the License, or + * (at your option) any later version. + * + * This program is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + * GNU General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program. If not, see <http://www.gnu.org/licenses/>. + */ + +#include <gudhi/Sliced_Wasserstein.h> + +#include <iostream> +#include <vector> +#include <utility> + +using SW = Gudhi::Persistence_representations::Sliced_Wasserstein; + +int main(int argc, char** argv) { + + std::vector<std::pair<double, double> > persistence1; + std::vector<std::pair<double, double> > persistence2; + + persistence1.push_back(std::make_pair(1, 2)); + persistence1.push_back(std::make_pair(6, 8)); + persistence1.push_back(std::make_pair(0, 4)); + persistence1.push_back(std::make_pair(3, 8)); + + persistence2.push_back(std::make_pair(2, 9)); + persistence2.push_back(std::make_pair(1, 6)); + persistence2.push_back(std::make_pair(3, 5)); + persistence2.push_back(std::make_pair(6, 10)); + + SW SW1(persistence1); + SW SW2(persistence2); + + std::cout << SW1.compute_sliced_wasserstein_distance(SW2,100) << std::endl; + std::cout << SW1.compute_sliced_wasserstein_distance(SW2,-1) << std::endl; + std::cout << SW1.compute_scalar_product(SW2,1,100) << std::endl; + std::cout << SW1.distance(SW2,1,100,1) << std::endl; + + return 0; +} diff --git a/src/Persistence_representations/include/gudhi/Persistence_weighted_gaussian.h b/src/Persistence_representations/include/gudhi/Persistence_weighted_gaussian.h new file mode 100644 index 00000000..2884885c --- /dev/null +++ b/src/Persistence_representations/include/gudhi/Persistence_weighted_gaussian.h @@ -0,0 +1,143 @@ +/* This file is part of the Gudhi Library. The Gudhi library + * (Geometric Understanding in Higher Dimensions) is a generic C++ + * library for computational topology. + * + * Author(s): Mathieu Carriere + * + * Copyright (C) 2018 INRIA (France) + * + * This program is free software: you can redistribute it and/or modify + * it under the terms of the GNU General Public License as published by + * the Free Software Foundation, either version 3 of the License, or + * (at your option) any later version. + * + * This program is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + * GNU General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program. If not, see <http://www.gnu.org/licenses/>. + */ + +#ifndef PERSISTENCE_WEIGHTED_GAUSSIAN_H_ +#define PERSISTENCE_WEIGHTED_GAUSSIAN_H_ + +#ifdef GUDHI_USE_TBB +#include <tbb/parallel_for.h> +#endif + +// gudhi include +#include <gudhi/read_persistence_from_file.h> + +// standard include +#include <cmath> +#include <iostream> +#include <vector> +#include <limits> +#include <fstream> +#include <sstream> +#include <algorithm> +#include <string> +#include <utility> +#include <functional> +#include <boost/math/constants/constants.hpp> + +double pi = boost::math::constants::pi<double>(); +using PD = std::vector<std::pair<double,double> >; + +namespace Gudhi { +namespace Persistence_representations { + +class Persistence_weighted_gaussian{ + + protected: + PD diagram; + + public: + + Persistence_weighted_gaussian(PD _diagram){diagram = _diagram;} + PD get_diagram(){return this->diagram;} + + + // ********************************** + // Utils. + // ********************************** + + + static double pss_weight(std::pair<double,double> P){ + if(P.second > P.first) return 1; + else return -1; + } + + static double arctan_weight(std::pair<double,double> P){ + return atan(P.second - P.first); + } + + template<class Weight = std::function<double (std::pair<double,double>) > > + std::vector<std::pair<double,double> > Fourier_feat(PD D, std::vector<std::pair<double,double> > Z, Weight weight = arctan_weight){ + int m = D.size(); std::vector<std::pair<double,double> > B; int M = Z.size(); + for(int i = 0; i < M; i++){ + double d1 = 0; double d2 = 0; double zx = Z[i].first; double zy = Z[i].second; + for(int j = 0; j < m; j++){ + double x = D[j].first; double y = D[j].second; + d1 += weight(D[j])*cos(x*zx + y*zy); + d2 += weight(D[j])*sin(x*zx + y*zy); + } + B.emplace_back(d1,d2); + } + return B; + } + + std::vector<std::pair<double,double> > random_Fourier(double sigma, int M = 1000){ + std::normal_distribution<double> distrib(0,1); std::vector<std::pair<double,double> > Z; std::random_device rd; + for(int i = 0; i < M; i++){ + std::mt19937 e1(rd()); std::mt19937 e2(rd()); + double zx = distrib(e1); double zy = distrib(e2); + Z.emplace_back(zx/sigma,zy/sigma); + } + return Z; + } + + + + // ********************************** + // Scalar product + distance. + // ********************************** + + + template<class Weight = std::function<double (std::pair<double,double>) > > + double compute_scalar_product(Persistence_weighted_gaussian second, double sigma, Weight weight = arctan_weight, int m = 1000){ + + PD diagram1 = this->diagram; PD diagram2 = second.diagram; + + if(m == -1){ + int num_pts1 = diagram1.size(); int num_pts2 = diagram2.size(); double k = 0; + for(int i = 0; i < num_pts1; i++) + for(int j = 0; j < num_pts2; j++) + k += weight(diagram1[i])*weight(diagram2[j])*exp(-((diagram1[i].first - diagram2[j].first) * (diagram1[i].first - diagram2[j].first) + + (diagram1[i].second - diagram2[j].second) * (diagram1[i].second - diagram2[j].second)) + /(2*sigma*sigma)); + return k; + } + else{ + std::vector<std::pair<double,double> > z = random_Fourier(sigma, m); + std::vector<std::pair<double,double> > b1 = Fourier_feat(diagram1,z,weight); + std::vector<std::pair<double,double> > b2 = Fourier_feat(diagram2,z,weight); + double d = 0; for(int i = 0; i < m; i++) d += b1[i].first*b2[i].first + b1[i].second*b2[i].second; + return d/m; + } + } + + template<class Weight = std::function<double (std::pair<double,double>) > > + double distance(Persistence_weighted_gaussian second, double sigma, Weight weight = arctan_weight, int m = 1000, double power = 1) { + return std::pow(this->compute_scalar_product(*this, sigma, weight, m) + second.compute_scalar_product(second, sigma, weight, m)-2*this->compute_scalar_product(second, sigma, weight, m), power/2.0); + } + + +}; + +} // namespace Persistence_weighted_gaussian +} // namespace Gudhi + +#endif // PERSISTENCE_WEIGHTED_GAUSSIAN_H_ diff --git a/src/Persistence_representations/include/gudhi/Sliced_Wasserstein.h b/src/Persistence_representations/include/gudhi/Sliced_Wasserstein.h new file mode 100644 index 00000000..4fa6151f --- /dev/null +++ b/src/Persistence_representations/include/gudhi/Sliced_Wasserstein.h @@ -0,0 +1,285 @@ +/* This file is part of the Gudhi Library. The Gudhi library + * (Geometric Understanding in Higher Dimensions) is a generic C++ + * library for computational topology. + * + * Author(s): Mathieu Carriere + * + * Copyright (C) 2018 INRIA (France) + * + * This program is free software: you can redistribute it and/or modify + * it under the terms of the GNU General Public License as published by + * the Free Software Foundation, either version 3 of the License, or + * (at your option) any later version. + * + * This program is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + * GNU General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program. If not, see <http://www.gnu.org/licenses/>. + */ + +#ifndef SLICED_WASSERSTEIN_H_ +#define SLICED_WASSERSTEIN_H_ + +#ifdef GUDHI_USE_TBB +#include <tbb/parallel_for.h> +#endif + +// gudhi include +#include <gudhi/read_persistence_from_file.h> + +// standard include +#include <cmath> +#include <iostream> +#include <vector> +#include <limits> +#include <fstream> +#include <sstream> +#include <algorithm> +#include <string> +#include <utility> +#include <functional> +#include <boost/math/constants/constants.hpp> + +double pi = boost::math::constants::pi<double>(); +using PD = std::vector<std::pair<double,double> >; + +namespace Gudhi { +namespace Persistence_representations { + +class Sliced_Wasserstein { + + protected: + PD diagram; + + public: + + Sliced_Wasserstein(PD _diagram){diagram = _diagram;} + PD get_diagram(){return this->diagram;} + + + // ********************************** + // Utils. + // ********************************** + + // Compute the angle formed by two points of a PD + double compute_angle(PD diag, int i, int j){ + std::pair<double,double> vect; double x1,y1, x2,y2; + x1 = diag[i].first; y1 = diag[i].second; + x2 = diag[j].first; y2 = diag[j].second; + if (y1 - y2 > 0){ + vect.first = y1 - y2; + vect.second = x2 - x1;} + else{ + if(y1 - y2 < 0){ + vect.first = y2 - y1; + vect.second = x1 - x2; + } + else{ + vect.first = 0; + vect.second = abs(x1 - x2);} + } + double norm = std::sqrt(vect.first*vect.first + vect.second*vect.second); + return asin(vect.second/norm); + } + + // Compute the integral of |cos()| between alpha and beta, valid only if alpha is in [-pi,pi] and beta-alpha is in [0,pi] + double compute_int_cos(const double & alpha, const double & beta){ + double res = 0; + if (alpha >= 0 && alpha <= pi){ + if (cos(alpha) >= 0){ + if(pi/2 <= beta){res = 2-sin(alpha)-sin(beta);} + else{res = sin(beta)-sin(alpha);} + } + else{ + if(1.5*pi <= beta){res = 2+sin(alpha)+sin(beta);} + else{res = sin(alpha)-sin(beta);} + } + } + if (alpha >= -pi && alpha <= 0){ + if (cos(alpha) <= 0){ + if(-pi/2 <= beta){res = 2+sin(alpha)+sin(beta);} + else{res = sin(alpha)-sin(beta);} + } + else{ + if(pi/2 <= beta){res = 2-sin(alpha)-sin(beta);} + else{res = sin(beta)-sin(alpha);} + } + } + return res; + } + + double compute_int(const double & theta1, const double & theta2, const int & p, const int & q, const PD & PD1, const PD & PD2){ + double norm = std::sqrt( (PD1[p].first-PD2[q].first)*(PD1[p].first-PD2[q].first) + (PD1[p].second-PD2[q].second)*(PD1[p].second-PD2[q].second) ); + double angle1; + if (PD1[p].first > PD2[q].first) + angle1 = theta1 - asin( (PD1[p].second-PD2[q].second)/norm ); + else + angle1 = theta1 - asin( (PD2[q].second-PD1[p].second)/norm ); + double angle2 = angle1 + theta2 - theta1; + double integral = compute_int_cos(angle1,angle2); + return norm*integral; + } + + + + + // ********************************** + // Scalar product + distance. + // ********************************** + + double compute_sliced_wasserstein_distance(Sliced_Wasserstein second, int approx) { + + PD diagram1 = this->diagram; PD diagram2 = second.diagram; double sw = 0; + + if(approx == -1){ + + // Add projections onto diagonal. + int n1, n2; n1 = diagram1.size(); n2 = diagram2.size(); double max_ordinate = std::numeric_limits<double>::lowest(); + for (int i = 0; i < n2; i++){ + max_ordinate = std::max(max_ordinate, diagram2[i].second); + diagram1.emplace_back( (diagram2[i].first+diagram2[i].second)/2, (diagram2[i].first+diagram2[i].second)/2 ); + } + for (int i = 0; i < n1; i++){ + max_ordinate = std::max(max_ordinate, diagram1[i].second); + diagram2.emplace_back( (diagram1[i].first+diagram1[i].second)/2, (diagram1[i].first+diagram1[i].second)/2 ); + } + int num_pts_dgm = diagram1.size(); + + // Slightly perturb the points so that the PDs are in generic positions. + int mag = 0; while(max_ordinate > 10){mag++; max_ordinate/=10;} + double thresh = pow(10,-5+mag); + srand(time(NULL)); + for (int i = 0; i < num_pts_dgm; i++){ + diagram1[i].first += thresh*(1.0-2.0*rand()/RAND_MAX); diagram1[i].second += thresh*(1.0-2.0*rand()/RAND_MAX); + diagram2[i].first += thresh*(1.0-2.0*rand()/RAND_MAX); diagram2[i].second += thresh*(1.0-2.0*rand()/RAND_MAX); + } + + // Compute all angles in both PDs. + std::vector<std::pair<double, std::pair<int,int> > > angles1, angles2; + for (int i = 0; i < num_pts_dgm; i++){ + for (int j = i+1; j < num_pts_dgm; j++){ + double theta1 = compute_angle(diagram1,i,j); double theta2 = compute_angle(diagram2,i,j); + angles1.emplace_back(theta1, std::pair<int,int>(i,j)); + angles2.emplace_back(theta2, std::pair<int,int>(i,j)); + } + } + + // Sort angles. + std::sort(angles1.begin(), angles1.end(), [=](std::pair<double, std::pair<int,int> >& p1, const std::pair<double, std::pair<int,int> >& p2){return (p1.first < p2.first);}); + std::sort(angles2.begin(), angles2.end(), [=](std::pair<double, std::pair<int,int> >& p1, const std::pair<double, std::pair<int,int> >& p2){return (p1.first < p2.first);}); + + // Initialize orders of the points of both PDs (given by ordinates when theta = -pi/2). + std::vector<int> orderp1, orderp2; + for (int i = 0; i < num_pts_dgm; i++){ orderp1.push_back(i); orderp2.push_back(i); } + std::sort( orderp1.begin(), orderp1.end(), [=](int i, int j){ if(diagram1[i].second != diagram1[j].second) return (diagram1[i].second < diagram1[j].second); else return (diagram1[i].first > diagram1[j].first); } ); + std::sort( orderp2.begin(), orderp2.end(), [=](int i, int j){ if(diagram2[i].second != diagram2[j].second) return (diagram2[i].second < diagram2[j].second); else return (diagram2[i].first > diagram2[j].first); } ); + + // Find the inverses of the orders. + std::vector<int> order1(num_pts_dgm); std::vector<int> order2(num_pts_dgm); + for(int i = 0; i < num_pts_dgm; i++) for (int j = 0; j < num_pts_dgm; j++) if(orderp1[j] == i){ order1[i] = j; break; } + for(int i = 0; i < num_pts_dgm; i++) for (int j = 0; j < num_pts_dgm; j++) if(orderp2[j] == i){ order2[i] = j; break; } + + // Record all inversions of points in the orders as theta varies along the positive half-disk. + std::vector<std::vector<std::pair<int,double> > > anglePerm1(num_pts_dgm); + std::vector<std::vector<std::pair<int,double> > > anglePerm2(num_pts_dgm); + + int m1 = angles1.size(); + for (int i = 0; i < m1; i++){ + double theta = angles1[i].first; int p = angles1[i].second.first; int q = angles1[i].second.second; + anglePerm1[order1[p]].emplace_back(p,theta); + anglePerm1[order1[q]].emplace_back(q,theta); + int a = order1[p]; int b = order1[q]; order1[p] = b; order1[q] = a; + } + + int m2 = angles2.size(); + for (int i = 0; i < m2; i++){ + double theta = angles2[i].first; int p = angles2[i].second.first; int q = angles2[i].second.second; + anglePerm2[order2[p]].emplace_back(p,theta); + anglePerm2[order2[q]].emplace_back(q,theta); + int a = order2[p]; int b = order2[q]; order2[p] = b; order2[q] = a; + } + + for (int i = 0; i < num_pts_dgm; i++){ + anglePerm1[order1[i]].emplace_back(i,pi/2); + anglePerm2[order2[i]].emplace_back(i,pi/2); + } + + // Compute the SW distance with the list of inversions. + for (int i = 0; i < num_pts_dgm; i++){ + std::vector<std::pair<int,double> > u,v; u = anglePerm1[i]; v = anglePerm2[i]; + double theta1, theta2; theta1 = -pi/2; + unsigned int ku, kv; ku = 0; kv = 0; theta2 = std::min(u[ku].second,v[kv].second); + while(theta1 != pi/2){ + if(diagram1[u[ku].first].first != diagram2[v[kv].first].first || diagram1[u[ku].first].second != diagram2[v[kv].first].second) + if(theta1 != theta2) + sw += compute_int(theta1, theta2, u[ku].first, v[kv].first, diagram1, diagram2); + theta1 = theta2; + if ( (theta2 == u[ku].second) && ku < u.size()-1 ) ku++; + if ( (theta2 == v[kv].second) && kv < v.size()-1 ) kv++; + theta2 = std::min(u[ku].second, v[kv].second); + } + } + } + + + else{ + double step = pi/approx; + + // Add projections onto diagonal. + int n1, n2; n1 = diagram1.size(); n2 = diagram2.size(); + for (int i = 0; i < n2; i++) + diagram1.emplace_back( (diagram2[i].first + diagram2[i].second)/2, (diagram2[i].first + diagram2[i].second)/2 ); + for (int i = 0; i < n1; i++) + diagram2.emplace_back( (diagram1[i].first + diagram1[i].second)/2, (diagram1[i].first + diagram1[i].second)/2 ); + int n = diagram1.size(); + + // Sort and compare all projections. + #ifdef GUDHI_USE_TBB + tbb::parallel_for(0, approx, [&](int i){ + std::vector<std::pair<int,double> > l1, l2; + for (int j = 0; j < n; j++){ + l1.emplace_back( j, diagram1[j].first*cos(-pi/2+i*step) + diagram1[j].second*sin(-pi/2+i*step) ); + l2.emplace_back( j, diagram2[j].first*cos(-pi/2+i*step) + diagram2[j].second*sin(-pi/2+i*step) ); + } + std::sort(l1.begin(),l1.end(), [=](const std::pair<int,double> & p1, const std::pair<int,double> & p2){return p1.second < p2.second;}); + std::sort(l2.begin(),l2.end(), [=](const std::pair<int,double> & p1, const std::pair<int,double> & p2){return p1.second < p2.second;}); + double f = 0; for (int j = 0; j < n; j++) f += std::abs(l1[j].second - l2[j].second); + sw += f*step; + }); + #else + for (int i = 0; i < approx; i++){ + std::vector<std::pair<int,double> > l1, l2; + for (int j = 0; j < n; j++){ + l1.emplace_back( j, diagram1[j].first*cos(-pi/2+i*step) + diagram1[j].second*sin(-pi/2+i*step) ); + l2.emplace_back( j, diagram2[j].first*cos(-pi/2+i*step) + diagram2[j].second*sin(-pi/2+i*step) ); + } + std::sort(l1.begin(),l1.end(), [=](const std::pair<int,double> & p1, const std::pair<int,double> & p2){return p1.second < p2.second;}); + std::sort(l2.begin(),l2.end(), [=](const std::pair<int,double> & p1, const std::pair<int,double> & p2){return p1.second < p2.second;}); + double f = 0; for (int j = 0; j < n; j++) f += std::abs(l1[j].second - l2[j].second); + sw += f*step; + } + #endif + } + + return sw/pi; + } + + + double compute_scalar_product(Sliced_Wasserstein second, double sigma, int approx = 100) { + return std::exp(-compute_sliced_wasserstein_distance(second, approx)/(2*sigma*sigma)); + } + + double distance(Sliced_Wasserstein second, double sigma, int approx = 100, double power = 1) { + return std::pow(this->compute_scalar_product(*this, sigma, approx) + second.compute_scalar_product(second, sigma, approx)-2*this->compute_scalar_product(second, sigma, approx), power/2.0); + } + + +}; + +} // namespace Sliced_Wasserstein +} // namespace Gudhi + +#endif // SLICED_WASSERSTEIN_H_ |