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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..e3ed2f6a --- /dev/null +++ b/src/Persistence_representations/include/gudhi/Sliced_Wasserstein.h @@ -0,0 +1,380 @@ +/* This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT. + * See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details. + * Author(s): Mathieu Carriere + * + * Copyright (C) 2018 Inria + * + * Modification(s): + * - YYYY/MM Author: Description of the modification + */ + +#ifndef SLICED_WASSERSTEIN_H_ +#define SLICED_WASSERSTEIN_H_ + +// gudhi include +#include <gudhi/read_persistence_from_file.h> +#include <gudhi/common_persistence_representations.h> +#include <gudhi/Debug_utils.h> + +#include <vector> // for std::vector<> +#include <utility> // for std::pair<>, std::move +#include <algorithm> // for std::sort, std::max, std::merge +#include <cmath> // for std::abs, std::sqrt +#include <stdexcept> // for std::invalid_argument +#include <random> // for std::random_device + +namespace Gudhi { +namespace Persistence_representations { + +/** + * \class Sliced_Wasserstein gudhi/Sliced_Wasserstein.h + * \brief A class implementing the Sliced Wasserstein kernel. + * + * \ingroup Persistence_representations + * + * \details + * In this class, we compute infinite-dimensional representations of persistence diagrams by using the + * Sliced Wasserstein kernel (see \ref sec_persistence_kernels for more details on kernels). We recall that + * infinite-dimensional representations are defined implicitly, so only scalar products and distances are available for + * the representations defined in this class. + * The Sliced Wasserstein kernel is defined as a Gaussian-like kernel between persistence diagrams, where the distance + * used for comparison is the Sliced Wasserstein distance \f$SW\f$ between persistence diagrams, defined as the + * integral of the 1-norm between the sorted projections of the diagrams onto all lines passing through the origin: + * + * \f$ SW(D_1,D_2)=\int_{\theta\in\mathbb{S}}\,\|\pi_\theta(D_1\cup\pi_\Delta(D_2))-\pi_\theta(D_2\cup\pi_\Delta(D_1))\ + * |_1{\rm d}\theta\f$, + * + * where \f$\pi_\theta\f$ is the projection onto the line defined with angle \f$\theta\f$ in the unit circle + * \f$\mathbb{S}\f$, and \f$\pi_\Delta\f$ is the projection onto the diagonal. + * Assuming that the diagrams are in general position (i.e. there is no collinear triple), the integral can be computed + * exactly in \f$O(n^2{\rm log}(n))\f$ time, where \f$n\f$ is the number of points in the diagrams. We provide two + * approximations of the integral: one in which we slightly perturb the diagram points so that they are in general + * position, and another in which we approximate the integral by sampling \f$N\f$ lines in the circle in + * \f$O(Nn{\rm log}(n))\f$ time. The Sliced Wasserstein Kernel is then computed as: + * + * \f$ k(D_1,D_2) = {\rm exp}\left(-\frac{SW(D_1,D_2)}{2\sigma^2}\right).\f$ + * + * The first method is usually much more accurate but also + * much slower. For more details, please see \cite pmlr-v70-carriere17a . + * + **/ + +class Sliced_Wasserstein { + protected: + Persistence_diagram diagram; + int approx; + double sigma; + std::vector<std::vector<double> > projections, projections_diagonal; + + // ********************************** + // Utils. + // ********************************** + + void build_rep() { + if (approx > 0) { + double step = pi / this->approx; + int n = diagram.size(); + + for (int i = 0; i < this->approx; i++) { + std::vector<double> l, l_diag; + for (int j = 0; j < n; j++) { + double px = diagram[j].first; + double py = diagram[j].second; + double proj_diag = (px + py) / 2; + + l.push_back(px * cos(-pi / 2 + i * step) + py * sin(-pi / 2 + i * step)); + l_diag.push_back(proj_diag * cos(-pi / 2 + i * step) + proj_diag * sin(-pi / 2 + i * step)); + } + + std::sort(l.begin(), l.end()); + std::sort(l_diag.begin(), l_diag.end()); + projections.push_back(std::move(l)); + projections_diagonal.push_back(std::move(l_diag)); + } + + diagram.clear(); + } + } + + // Compute the angle formed by two points of a PD + double compute_angle(const Persistence_diagram& diag, int i, int j) const { + if (diag[i].second == diag[j].second) + return pi / 2; + else + return atan((diag[j].first - diag[i].first) / (diag[i].second - diag[j].second)); + } + + // 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(double alpha, double beta) const { + 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(double theta1, double theta2, int p, int q, const Persistence_diagram& diag1, + const Persistence_diagram& diag2) const { + double norm = std::sqrt((diag1[p].first - diag2[q].first) * (diag1[p].first - diag2[q].first) + + (diag1[p].second - diag2[q].second) * (diag1[p].second - diag2[q].second)); + double angle1; + if (diag1[p].first == diag2[q].first) + angle1 = theta1 - pi / 2; + else + angle1 = theta1 - atan((diag1[p].second - diag2[q].second) / (diag1[p].first - diag2[q].first)); + double angle2 = angle1 + theta2 - theta1; + double integral = compute_int_cos(angle1, angle2); + return norm * integral; + } + + // Evaluation of the Sliced Wasserstein Distance between a pair of diagrams. + double compute_sliced_wasserstein_distance(const Sliced_Wasserstein& second) const { + GUDHI_CHECK(this->approx == second.approx, + std::invalid_argument("Error: different approx values for representations")); + + Persistence_diagram diagram1 = this->diagram; + Persistence_diagram diagram2 = second.diagram; + double sw = 0; + + if (this->approx == -1) { + // Add projections onto diagonal. + int n1, n2; + n1 = diagram1.size(); + n2 = diagram2.size(); + double min_ordinate = std::numeric_limits<double>::max(); + double min_abscissa = std::numeric_limits<double>::max(); + double max_ordinate = std::numeric_limits<double>::lowest(); + double max_abscissa = std::numeric_limits<double>::lowest(); + for (int i = 0; i < n2; i++) { + min_ordinate = std::min(min_ordinate, diagram2[i].second); + min_abscissa = std::min(min_abscissa, diagram2[i].first); + max_ordinate = std::max(max_ordinate, diagram2[i].second); + max_abscissa = std::max(max_abscissa, diagram2[i].first); + diagram1.emplace_back((diagram2[i].first + diagram2[i].second) / 2, + (diagram2[i].first + diagram2[i].second) / 2); + } + for (int i = 0; i < n1; i++) { + min_ordinate = std::min(min_ordinate, diagram1[i].second); + min_abscissa = std::min(min_abscissa, diagram1[i].first); + max_ordinate = std::max(max_ordinate, diagram1[i].second); + max_abscissa = std::max(max_abscissa, diagram1[i].first); + 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. + double epsilon = 0.0001; + double thresh_y = (max_ordinate - min_ordinate) * epsilon; + double thresh_x = (max_abscissa - min_abscissa) * epsilon; + std::random_device rd; + std::default_random_engine re(rd()); + std::uniform_real_distribution<double> uni(-1, 1); + for (int i = 0; i < num_pts_dgm; i++) { + double u = uni(re); + diagram1[i].first += u * thresh_x; + diagram1[i].second += u * thresh_y; + diagram2[i].first += u * thresh_x; + diagram2[i].second += u * thresh_y; + } + + // 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(), + [](const 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(), + [](const 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++) { + order1[orderp1[i]] = i; + order2[orderp2[i]] = i; + } + + // 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 / this->approx; + std::vector<double> v1, v2; + for (int i = 0; i < this->approx; i++) { + v1.clear(); + v2.clear(); + std::merge(this->projections[i].begin(), this->projections[i].end(), second.projections_diagonal[i].begin(), + second.projections_diagonal[i].end(), std::back_inserter(v1)); + std::merge(second.projections[i].begin(), second.projections[i].end(), this->projections_diagonal[i].begin(), + this->projections_diagonal[i].end(), std::back_inserter(v2)); + + int n = v1.size(); + double f = 0; + for (int j = 0; j < n; j++) f += std::abs(v1[j] - v2[j]); + sw += f * step; + } + } + + return sw / pi; + } + + public: + /** \brief Sliced Wasserstein kernel constructor. + * \implements Topological_data_with_distances, Real_valued_topological_data, Topological_data_with_scalar_product + * \ingroup Sliced_Wasserstein + * + * @param[in] _diagram persistence diagram. + * @param[in] _sigma bandwidth parameter. + * @param[in] _approx number of directions used to approximate the integral in the Sliced Wasserstein distance, set + * to -1 for random perturbation. If positive, then projections of the diagram points on all + * directions are stored in memory to reduce computation time. + * + */ + Sliced_Wasserstein(const Persistence_diagram& _diagram, double _sigma = 1.0, int _approx = 10) + : diagram(_diagram), approx(_approx), sigma(_sigma) { + build_rep(); + } + + /** \brief Evaluation of the kernel on a pair of diagrams. + * \ingroup Sliced_Wasserstein + * + * @pre approx and sigma attributes need to be the same for both instances. + * @param[in] second other instance of class Sliced_Wasserstein. + * + */ + double compute_scalar_product(const Sliced_Wasserstein& second) const { + GUDHI_CHECK(this->sigma == second.sigma, + std::invalid_argument("Error: different sigma values for representations")); + return std::exp(-compute_sliced_wasserstein_distance(second) / (2 * this->sigma * this->sigma)); + } + + /** \brief Evaluation of the distance between images of diagrams in the Hilbert space of the kernel. + * \ingroup Sliced_Wasserstein + * + * @pre approx and sigma attributes need to be the same for both instances. + * @param[in] second other instance of class Sliced_Wasserstein. + * + */ + double distance(const Sliced_Wasserstein& second) const { + GUDHI_CHECK(this->sigma == second.sigma, + std::invalid_argument("Error: different sigma values for representations")); + return std::sqrt(this->compute_scalar_product(*this) + second.compute_scalar_product(second) - + 2 * this->compute_scalar_product(second)); + } + +}; // class Sliced_Wasserstein +} // namespace Persistence_representations +} // namespace Gudhi + +#endif // SLICED_WASSERSTEIN_H_ |