Indeed, an easier way to generate a persistence landscape is to associate a triangle function to each persistence diagram point and sample this function to get vectors.
Then the k-th persistence landscape is nothing but the k-th largest value of each coordinate of these vectors!PersLay parameters for the generation of persistence landscapes (source: https://arxiv.
org/abs/1904.
09378).
A simple architecture for persistence diagramsThis opens the use of persistence diagrams in a wide variety of tasks that was not accessible before!.Indeed, even for thousands of persistence diagrams, it is way too expensive (in terms of running time) to cross validate over all possible feature maps with traditional classifiers such as SVM or random forests.
On the other hand, it can be done with a few lines of code with PersLay :-DMore precisely, the output of PersLay can be used as input for any subsequent neural network.
In the article associated to PersLay, we study a simple architecture, in which several persistence diagrams are generated from each data point, and each of these diagrams is treated separately by a specific PersLay channel.
The ouputs of all channels are then concatenated (with some additional features), and we use a single fully-connected layer to produce a result with the right dimension.
A simple architecture using PersLay.
For those of you who would like to know more about this, there is a tutorial on PersLay and this architecture, with some cool applications in graph classification.
I really hope that you are also excited about this new combination of neural networks and Topological Data Analysis.
There are many other applications beside classification that can be performed with neural networks, so stay tuned for other upcoming posts!.