1 files changed, 50 insertions, 3 deletions
diff --git a/src/Kernels/doc/Intro_kernels.h b/src/Kernels/doc/Intro_kernels.h
index be97a6cf..163690b1 100644
--- a/src/Kernels/doc/Intro_kernels.h
+++ b/src/Kernels/doc/Intro_kernels.h
@@ -37,17 +37,64 @@ namespace kernel {
  * 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,
- * the Persistence Weighted Gaussian kernel and the Sliced Wasserstein kernel.
+ * 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
+ * \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