diff options
Diffstat (limited to 'src/Persistence_representations/include/gudhi')
-rw-r--r-- | src/Persistence_representations/include/gudhi/Persistence_weighted_gaussian.h | 72 | ||||
-rw-r--r-- | src/Persistence_representations/include/gudhi/Sliced_Wasserstein.h | 58 |
2 files changed, 118 insertions, 12 deletions
diff --git a/src/Persistence_representations/include/gudhi/Persistence_weighted_gaussian.h b/src/Persistence_representations/include/gudhi/Persistence_weighted_gaussian.h index a6efa72d..f824225a 100644 --- a/src/Persistence_representations/include/gudhi/Persistence_weighted_gaussian.h +++ b/src/Persistence_representations/include/gudhi/Persistence_weighted_gaussian.h @@ -45,7 +45,40 @@ using Weight = std::function<double (std::pair<double,double>) >; namespace Gudhi { namespace Persistence_representations { - +/** + * \class Persistence_weighted_gaussian gudhi/Persistence_weighted_gaussian.h + * \brief A class implementing the Persistence Weighted Gaussian Kernel and a specific case of it called the Persistence Scale Space Kernel. + * + * \ingroup Persistence_representations + * + * \details + * The Persistence Weighted Gaussian Kernel is built with Gaussian Kernel Mean Embedding, meaning that each persistence diagram is first + * sent to the Hilbert space of a Gaussian kernel with bandwidth parameter \f$\sigma >0\f$ using a weighted mean embedding \f$\Phi\f$: + * + * \f$ \Phi\,:\,D\,\rightarrow\,\sum_{p\in D}\,w(p)\,{\rm exp}\left(-\frac{\|p-\cdot\|_2^2}{2\sigma^2}\right) \f$, + * + * Usually, the weight function is chosen to be an arctan function of the distance of the point to the diagonal: + * \f$w(p) = {\rm arctan}(C\,|y-x|^\alpha)\f$, for some parameters \f$C,\alpha >0\f$. + * Then, their scalar product in this space is computed: + * + * \f$ k(D_1,D_2)=\langle\Phi(D_1),\Phi(D_2)\rangle + * \,=\,\sum_{p\in D_1}\,\sum_{q\in D_2}\,w(p)\,w(q)\,{\rm exp}\left(-\frac{\|p-q\|_2^2}{2\sigma^2}\right).\f$ + * + * Note that one may apply a second Gaussian kernel to their distance in this space and still get a kernel. + * + * It follows that the computation time is \f$O(n^2)\f$ where \f$n\f$ is the number of points + * in the diagrams. This time can be improved by computing approximations of the kernel + * with \f$m\f$ Fourier features \cite Rahimi07randomfeatures. In that case, the computation time becomes \f$O(mn)\f$. + * + * The Persistence Scale Space Kernel is a Persistence Weighted Gaussian Kernel between modified diagrams: + * the symmetric of each point with respect to the diagonal is first added in each diagram, and then the weight function + * is set to be +1 if the point is above the diagonal and -1 otherwise. + * + * For more details, please consult <i>Persistence Weighted Kernel for Topological Data Analysis</i>\cite Kusano_Fukumizu_Hiraoka_PWGK + * and <i>A Stable Multi-Scale Kernel for Topological Machine Learning</i>\cite Reininghaus_Huber_ALL_PSSK . + * It implements the following concepts: Topological_data_with_distances, Topological_data_with_scalar_product. + * +**/ class Persistence_weighted_gaussian{ protected: @@ -56,8 +89,17 @@ class Persistence_weighted_gaussian{ public: - Persistence_weighted_gaussian(PD _diagram){diagram = _diagram; sigma = 1.0; approx = 1000; weight = arctan_weight;} - Persistence_weighted_gaussian(PD _diagram, double _sigma, int _approx, Weight _weight){diagram = _diagram; sigma = _sigma; approx = _approx; weight = _weight;} + /** \brief Persistence Weighted Gaussian Kernel constructor. + * \ingroup Persistence_weighted_gaussian + * + * @param[in] _diagram persistence diagram. + * @param[in] _sigma bandwidth parameter of the Gaussian Kernel used for the Kernel Mean Embedding of the diagrams. + * @param[in] _approx number of random Fourier features in case of approximate computation, set to -1 for exact computation. + * @param[in] _weight weight function for the points in the diagrams. + * + */ + Persistence_weighted_gaussian(PD _diagram, double _sigma = 1.0, int _approx = 1000, Weight _weight = arctan_weight){diagram = _diagram; sigma = _sigma; approx = _approx; weight = _weight;} + PD get_diagram(){return this->diagram;} double get_sigma(){return this->sigma;} int get_approx(){return this->approx;} @@ -68,7 +110,12 @@ class Persistence_weighted_gaussian{ // Utils. // ********************************** - + /** \brief Specific weight of Persistence Scale Space Kernel. + * \ingroup Persistence_weighted_gaussian + * + * @param[in] p point in 2D. + * + */ static double pss_weight(std::pair<double,double> p){ if(p.second > p.first) return 1; else return -1; @@ -108,7 +155,12 @@ class Persistence_weighted_gaussian{ // Scalar product + distance. // ********************************** - + /** \brief Evaluation of the kernel on a pair of diagrams. + * \ingroup Persistence_weighted_gaussian + * + * @param[in] second other instance of class Persistence_weighted_gaussian. Warning: sigma, approx and weight parameters need to be the same for both instances!!! + * + */ double compute_scalar_product(Persistence_weighted_gaussian second){ PD diagram1 = this->diagram; PD diagram2 = second.diagram; @@ -131,11 +183,17 @@ class Persistence_weighted_gaussian{ } } - double distance(Persistence_weighted_gaussian second, double power = 1) { + /** \brief Evaluation of the distance between images of diagrams in the Hilbert space of the kernel. + * \ingroup Persistence_weighted_gaussian + * + * @param[in] second other instance of class Persistence_weighted_gaussian. Warning: sigma, approx and weight parameters need to be the same for both instances!!! + * + */ + double distance(Persistence_weighted_gaussian second) { if(this->sigma != second.get_sigma() || this->approx != second.get_approx()){ std::cout << "Error: different representations!" << std::endl; return 0; } - else return std::pow(this->compute_scalar_product(*this) + second.compute_scalar_product(second)-2*this->compute_scalar_product(second), power/2.0); + else return std::pow(this->compute_scalar_product(*this) + second.compute_scalar_product(second)-2*this->compute_scalar_product(second), 0.5); } diff --git a/src/Persistence_representations/include/gudhi/Sliced_Wasserstein.h b/src/Persistence_representations/include/gudhi/Sliced_Wasserstein.h index f2ec56b7..bfb77384 100644 --- a/src/Persistence_representations/include/gudhi/Sliced_Wasserstein.h +++ b/src/Persistence_representations/include/gudhi/Sliced_Wasserstein.h @@ -45,6 +45,30 @@ using PD = std::vector<std::pair<double,double> >; namespace Gudhi { namespace Persistence_representations { +/** + * \class Sliced_Wasserstein gudhi/Sliced_Wasserstein.h + * \brief A class implementing the Sliced Wasserstein Kernel. + * + * \ingroup Persistence_representations + * + * \details + * 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. + * The integral can be either computed exactly in \f$O(n^2{\rm log}(n))\f$ time, where \f$n\f$ is the number of points + * in the diagrams, or approximated 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$ + * + * For more details, please consult <i>Sliced Wasserstein Kernel for Persistence Diagrams</i>\cite pmlr-v70-carriere17a . + * It implements the following concepts: Topological_data_with_distances, Topological_data_with_scalar_product. + * +**/ class Sliced_Wasserstein { protected: @@ -83,8 +107,15 @@ class Sliced_Wasserstein { } - Sliced_Wasserstein(PD _diagram){diagram = _diagram; approx = 100; sigma = 0.001; build_rep();} - Sliced_Wasserstein(PD _diagram, double _sigma, int _approx){diagram = _diagram; approx = _approx; sigma = _sigma; build_rep();} + /** \brief Sliced Wasserstein Kernel constructor. + * \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 exact computation. + * + */ + Sliced_Wasserstein(PD _diagram, double _sigma = 1.0, int _approx = 100){diagram = _diagram; approx = _approx; sigma = _sigma; build_rep();} PD get_diagram(){return this->diagram;} int get_approx(){return this->approx;} @@ -163,6 +194,12 @@ class Sliced_Wasserstein { // Scalar product + distance. // ********************************** + /** \brief Evaluation of the Sliced Wasserstein Distance between a pair of diagrams. + * \ingroup Sliced_Wasserstein + * + * @param[in] second other instance of class Sliced_Wasserstein. Warning: approx parameter needs to be the same for both instances!!! + * + */ double compute_sliced_wasserstein_distance(Sliced_Wasserstein second) { PD diagram1 = this->diagram; PD diagram2 = second.diagram; double sw = 0; @@ -277,14 +314,25 @@ class Sliced_Wasserstein { return sw/pi; } - + /** \brief Evaluation of the kernel on a pair of diagrams. + * \ingroup Sliced_Wasserstein + * + * @param[in] second other instance of class Sliced_Wasserstein. Warning: sigma and approx parameters need to be the same for both instances!!! + * + */ double compute_scalar_product(Sliced_Wasserstein second){ return std::exp(-compute_sliced_wasserstein_distance(second)/(2*this->sigma*this->sigma)); } - double distance(Sliced_Wasserstein second, double power = 1) { + /** \brief Evaluation of the distance between images of diagrams in the Hilbert space of the kernel. + * \ingroup Sliced_Wasserstein + * + * @param[in] second other instance of class Sliced_Wasserstein. Warning: sigma and approx parameters need to be the same for both instances!!! + * + */ + double distance(Sliced_Wasserstein second) { if(this->sigma != second.sigma || this->approx != second.approx){std::cout << "Error: different representations!" << std::endl; return 0;} - else return std::pow(this->compute_scalar_product(*this) + second.compute_scalar_product(second)-2*this->compute_scalar_product(second), power/2.0); + else return std::pow(this->compute_scalar_product(*this) + second.compute_scalar_product(second)-2*this->compute_scalar_product(second), 0.5); } |