A class implementing the Sliced Wasserstein kernel.
In this class, we compute infinite-dimensional representations of persistence diagrams by using the Sliced Wasserstein kernel (see Kernels on persistence diagrams 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 SW 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:
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,
where \pi_\theta is the projection onto the line defined with angle \theta in the unit circle \mathbb{S}, and \pi_\Delta 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 O(n^2{\rm log}(n)) time, where n 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 N lines in the circle in O(Nn{\rm log}(n)) time. The Sliced Wasserstein Kernel is then computed as:
k(D_1,D_2) = {\rm exp}\left(-\frac{SW(D_1,D_2)}{2\sigma^2}\right).
The first method is usually much more accurate but also much slower. For more details, please see [16] .
