s-clausen / cecher

Cecher: efficient computation of Čech persistence barcodes

Copyright © 2023 Sönke Clausen

The Cecher is a C++ software for efficient computation of Čech persistence barcodes outperforming current software (Dionysus, GUDHI). Its main features are:

• a core persistence algorithm adapted from the C++ code of the ultrafast Ripser, featuring its memory-efficient philosophy, and use of a combination of cohomology, clearing, and zero-apparent pairs shortcut.

• a symbolic perturbation of the Čech filtration, allowing to completely skip the construction of many columns of the boundary matrix corresponding to zero-persistence pairs.

• a minimal enclosing sphere algorithm especially efficient in high ambient dimension, outperforming established software like Miniball in the context of a persistence algorithm with a construction using the distance matrix.

• a lazy-exact implementation, where computations are performed primarily with an interval arithmetic type.

• a version of the union-find algorithm in Ripser with compression.

Input & Options

The Cecher supports an input in the form of a decimal point cloud, where a whitespace seperates values (examples). The input is given in a file whose name is passed as a command line argument.

The following preprocessor macros are supported:

• USE_COEFFICIENTS: enable support for coefficients in a prime field Z/pZ (default: p=2).
• EXPLICIT_CENTERS: enable variant of the minimal enclosing sphere algorithm for low ambient dimension.

The following command line arguments are supported:

• --dim k: compute persistent homology up to dimension k (default: k=1).
• --threshold t: compute Čech filtration up to radius t (default: no threshold).
• --modulus p: compute homology with coefficients in the prime field Z/pZ (requires USE_COEFFICIENTS).

Lazy-Exact Implementation

The Cecher falls back on exact computations when flat simplices exist or when minimal circumspheres of simplices have coinciding radii. On most 'real world data' all operations utilize the interval arithmetic (double-precision) type.

Dependencies

The Cecher depends on the library CGAL for its lazy-exact implementation.

How to Build & Run

On a preconfigured system (Windows, macOS or Linux, where CGAL and its dependencies are installed) building and running the Cecher should be as easy as:

cd path\to\cecher
cmake -DUSE_COEFFICIENTS=off -DEXPLICIT_CENTERS=off .
make
.\cecher examples\dragon.txt --dim 2

Adapt preprocessor macros and program arguments to your needs.

Experiments

The following results were obtained on a desktop computer with a 3.4GHz Intel Xeon E3 1231 v3 Server CPU and 16 GB 1,600MHz DDR3 RAM. The number of points is denoted by n, the maximal degree of homology to be computed is denoted by k, and the ambient dimension is denoted by d. Empty entries correspond to RAM usage beyond 12GB.

data	n	d	k	Dionysus	GUDHI	Cecher
random_60	300	60	1	132.6s, 992MB	16.7s, 5.4GB	3.8s, 13MB
torus	800	4	1			11s, 372MB
circle	300	2	1	120.9s, 910MB	5.6s, 523MB	0.4s, 17MB
circle_60	600	60	1			2.9s, 135MB
random_20	100	20	2	101.9s, 902MB	13.6s, 2.2GB	49.7s, 10MB
random	200	3	2		111.6s, 4.9GB	2.5s, 6MB
torus_small	100	4	2	396.8s, 903MB	7.1s, 544MB	1.3s, 8MB
sphere_60	200	60	2		167.9s, 11.4GB	6.9s, 16MB

Credits

The Cecher and its concepts are the result of a thesis advised by Ulrich Bauer and Fabian Roll. The theory can be found here: coming soon.