4. Computation Pipeline¶
The following figure illustrates RIVET’s pipeline for working with the 2-parameter persistent homology of data.
We now explain this pipeline:
RIVET can accept as input a data set, a bifiltration, or an FIRep. RIVET always works with a single homology degree at a time; when giving data or a bifiltration as input, one specifies the degree of homology to consider.
RIVET accepts data in the form of either a point cloud in \(\mathbb R^n\), or a finite metric space (represented as distance matrix). Optionally, a function on the points can also be given by the user or precomputed by RIVET. If a function is specified, then RIVET computes a function-Rips bifiltration. If no function is specified, then RIVET computes the degree-Rips bifiltration.
Given a bifiltration \(F\), RIVET constructs an FIRep for \(H_j(F)\) in the specified degree \(j\). If \(F\) is multi-critical, RIVET uses the trick of Chacholski et al. LINK to obtain the FIRep.
Given an FIRep, RIVET computes a minimal presentation of its homology module. This computation also yields the Hilbert function of the module with almost no extra work. The 0th and 1st bigraded Betti numbers of a bipersistence module \(M\) can be read directly off of the minimal presentation. Given these and the Hilbert function, a simple formula yields the 2nd bigraded Betti numbers as well.
RIVET uses the minimal presentation of \(M\) to compute the augmented arrangement of \(M\). This is a line arrangement in the right half plane, together with a barcode at each face of the line arrangement. The augmented arrangement is used to perform fast queries of the fibered barcode of \(M\).