/* 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
*
* 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 .
*/
#ifndef SLICED_WASSERSTEIN_H_
#define SLICED_WASSERSTEIN_H_
// gudhi include
#include
#include
#include
#include // for std::vector<>
#include // for std::pair<>, std::move
#include // for std::sort, std::max, std::merge
#include // for std::abs, std::sqrt
#include // for std::invalid_argument
#include // 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 > 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 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::max();
double min_abscissa = std::numeric_limits::max();
double max_ordinate = std::numeric_limits::lowest();
double max_abscissa = std::numeric_limits::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 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 > > 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(i, j));
angles2.emplace_back(theta2, std::pair(i, j));
}
}
// Sort angles.
std::sort(angles1.begin(), angles1.end(),
[](const std::pair >& p1,
const std::pair >& p2) { return (p1.first < p2.first); });
std::sort(angles2.begin(), angles2.end(),
[](const std::pair >& p1,
const std::pair >& p2) { return (p1.first < p2.first); });
// Initialize orders of the points of both PDs (given by ordinates when theta = -pi/2).
std::vector 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 order1(num_pts_dgm);
std::vector 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 > > anglePerm1(num_pts_dgm);
std::vector > > 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 > 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 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_