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diff --git a/src/Kernels/doc/Intro_kernels.h b/src/Kernels/doc/Intro_kernels.h deleted file mode 100644 index 163690b1..00000000 --- a/src/Kernels/doc/Intro_kernels.h +++ /dev/null @@ -1,108 +0,0 @@ -/* 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) 2017 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 <http://www.gnu.org/licenses/>. - */ - -#ifndef DOC_KERNEL_INTRO_KERNEL_H_ -#define DOC_KERNEL_INTRO_KERNEL_H_ - -namespace Gudhi { - -namespace kernel { - -/** \defgroup kernel Kernels - * - * \author Mathieu Carrière - * - * @{ - * - * Kernels are generalized scalar products. They take the form of functions whose evaluations on pairs of persistence diagrams are equal - * to the scalar products of the images of the diagrams under some feature map into a (generally unknown and infinite dimensional) - * Hilbert space. Kernels are - * very useful to handle any type of data for algorithms that require at least a Hilbert structure, such as Principal Component Analysis - * or Support Vector Machines. In this package, we implement three kernels for persistence diagrams: - * the Persistence Scale Space Kernel (PSSK)---see \cite Reininghaus_Huber_ALL_PSSK, - * the Persistence Weighted Gaussian Kernel (PWGK)---see \cite Kusano_Fukumizu_Hiraoka_PWGK, - * and the Sliced Wasserstein Kernel (SWK)---see \cite pmlr-v70-carriere17a. - * - * \section pwg Persistence Weighted Gaussian Kernel and Persistence Scale Space Kernel - * - * The PWGK 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$. - * Then, either their scalar product in this space is - * computed (Linear Persistence Weighted Gaussian Kernel): - * - * \f$ LPWGK(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$, - * - * or a second Gaussian kernel with bandwidth parameter \f$\tau >0\f$ is applied to their distance in this space - * (Gaussian Persistence Weighted Gaussian Kernel): - * - * \f$ GPWGK(D_1,D_2)={\rm exp}\left(-\frac{\|\Phi(D_1)-\Phi(D_2)\|^2}{2\tau^2} \right)\f$, - * where \f$\|\Phi(D_1)-\Phi(D_2)\|^2 = \langle\Phi(D_1)-\Phi(D_2),\Phi(D_1)-\Phi(D_2)\rangle\f$. - * - * It follows that the computation time is \f$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 PSSK is a Linear 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. - * - * \section sw Sliced Wasserstein Kernel - * - * 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$m\f$ lines in the circle in \f$O(mn{\rm log}(n))\f$ time. The SWK is then computed as: - * - * \f$ SWK(D_1,D_2) = {\rm exp}\left(-\frac{SW(D_1,D_2)}{2\sigma^2}\right).\f$ - * - * When launching: - * - * \code $> ./BasicEx ../../../../data/persistence_diagram/PD1 ../../../../data/persistence_diagram/PD2 - * \endcode - * - * the program output is: - * - * \include Kernels/kernel.txt - * - * - * \copyright GNU General Public License v3. - * \verbatim Contact: gudhi-users@lists.gforge.inria.fr \endverbatim - */ -/** @} */ // end defgroup kernel - -} // namespace kernel - -} // namespace Gudhi - -#endif // DOC_KERNEL_INTRO_KERNEL_H_ |