ComplexPolynomial¶

class gtda.diagrams.ComplexPolynomial(n_coefficients=10, polynomial_type='R', n_jobs=None)[source]¶

Coefficients of complex polynomials whose roots are obtained from points in persistence diagrams.

Given a persistence diagram consisting of birth-death-dimension triples [b, d, q], subdiagrams corresponding to distinct homology dimensions are first considered separately. For each subdiagram, the polynomial whose roots are complex numbers obtained from its birth-death pairs is computed, and its n_coefficients_ highest-degree complex coefficients excluding the top one are stored into a single real vector by concatenating the vector of all real parts with the vector of all imaginary parts 1 (if not enough coefficients are available to form a vector of the required length, padding with zeros is performed). Finally, all such vectors coming from different subdiagrams are concatenated to yield a single vector for the diagram.

There are three possibilities for mapping birth-death pairs \((b, d)\) to complex polynomial roots. They are:

\begin{gather*} R(b, d) = b + \mathrm{i} d, \\ S(b, d) = \frac{d - b}{\sqrt{2} r} (b + \mathrm{i} d), \\ T(b, d) = \frac{d - b}{2} [\cos{r} - \sin{r} + \mathrm{i}(\cos{r} + \sin{r})], \end{gather*}

where \(r = \sqrt{b^2 + d^2}\).

Important note:

Input collections of persistence diagrams for this transformer must satisfy certain requirements, see e.g. fit.

Parameters

polynomial_type ('R' | 'S' | 'T', optional, default: 'R') – Type of complex polynomial to compute.
n_coefficients (list, int or None, optional, default: 10) – Number of complex coefficients per homology dimension. If an int then the number of coefficients will be equal to that value for each homology dimension. If None then, for each homology dimension in the collection of persistence diagrams seen in fit, the number of complex coefficients is defined to be the largest number of off-diagonal points seen among all subdiagrams in that homology dimension, minus one.
n_jobs (int or None, optional, default: None) – The number of jobs to use for the computation. None means 1 unless in a joblib.parallel_backend context. -1 means using all processors.

homology_dimensions_¶

Homology dimensions seen in fit, sorted in ascending order.

Type: list

n_coefficients_¶

Effective number of complex coefficients per homology dimension. Set in fit.

Type: list