steenroder.barcodes

steenroder.barcodes(k, filtration, absolute=False, filtration_values=None, return_filtration_values=False, maxdim=None, n_jobs=1, verbose=False)[source]

Given a filtration, compute ordinary persistent (relative or absolute) cohomology barcodes and relative Steenrod barcodes.

Parameters

  • k (int) – Positive integer defining the cohomology operation Sq^k to be performed.

  • filtration (sequence of list-like of int) – Represents a simplex-wise filtration. Entry i is a list/tuple/set containing the integer indices of the vertices defining the i-th simplex in the filtration.

  • absolute (bool, optional, default: False) – If True, return the ordinary persistent absolute homology barcode, and move inessential relative Steenrod bars to one degree lower while keeping essential bars in their degree. If False, return the ordinary persistent relative cohomology barcode and relative Steenrod barcode.

  • filtration_values (ndarray or None, optional, default: None) – Optionally, a single 1D array of filtration values for each simplex in the filtration. Ordinary and Steenrod bars with equal birth and death filtration values are discarded by the computation.

  • return_filtration_values (bool, optional, default: False) – If True, birth and deaths will be expressed as filtration values instead of filtration indices. Ignored if filtration_values is None.

  • maxdim (int or None, optional, default: None) – Maximum simplex dimension to be included. None means that all simplices are included.

  • n_jobs (int, optional, default: 1) – [Experimental] Controls the number of threads to be used during parallel computation of the Steenrod squares. -1 means using all available physical cores.

  • verbose (bool, optional, default: False) – Whether to print timings for the intermediate steps in the computation.

Returns

  • barcode (list of ndarray) – For each dimension d, a 2D int or float array of shape (n_bars, 2) containing the births and deaths of persistent relative cohomology classes in degree d. If absolute is False, the birth of a bar is in entry 1 and the death in entry 0; otherwise, the positions are reversed. Births and death are expressed either as global filtration indices or as filtration values depending on filtration_values and return_filtration_values. If they are expressed as indices, essential bars have death equal to -1; otherwise, essential bars have death equal to numpy.inf.

  • steenrod_barcode (list of ndarray) – The (relative) Sq^k-barcode. For each dimension d, a 2D int or float array of shape (n_bars, 2) containing the birth (entry 1) and death (entry 0) indices of Steenrod bars. The same conventions as for barcode are used for birth and death values.