HomologyInferenceWithWeakFeatureSize.jl

Version

1.2.0

Copyright (C) 2024 Parker Edwards

Installation

For global installation from Julia in package mode (press ] to enter package mode, backspace to exit): add https://github.com/P-Edwards/HomologyInferenceWithWeakFeatureSize.jl.git

To install as a standalone local package: Clone this repository to a directory. Then, from Julia in package mode: activate <path/to/cloned/copy/of/root/directory> followed by instantiate.

Basic usage

To use multiple cores, run export JULIA_NUM_THREADS=<number_to_use> before starting Julia. Examples are available at https://github.com/P-Edwards/wfs-and-reach-examples.

This package primarily exposes two functions:

compute_weak_feature_size(F;maximum_bottleneck_order=nothing,threshold=1e-8,solve_with_monodromy=false,system_variables=variables(F),multi_system=false)

Where

• F is a list of polynomials
• For maximum_bottleneck_order >= 2 the function will stop at the indicated order. Default is the number of variables for F + 1. This may become very expensive for order > 3.
• threshold is a numerical threshold to determine the value at which quantities are considered indistinguishable from 0.
• solve_with_monodromy is a flag that toggles between a polyhedral homotopy solving method and a monodromy based one. There are currently no correctness guarantees with the monodromy method, but it is faster and has worked experimentally.
• system_variables is a vector of variables to use for the original system. Useful for requiring the use of variables that do not occur in the functions in F.
• multi_system is a flag which switches between single system mode (the default), and multi-system mode (described below).

The output is a number, which is the best lower bound on the weak feature size computed using geometric bottlenecks up to the provided order.

Example: Compute the weak feature size of an ellipsoid, which occurs at a 2-bottleneck.

using HomologyInferenceWithWeakFeatureSize, HomotopyContinuation
@var x y z
F = [x^2 + y^2 + z^2/2 + x*z/7 - 1]
compute_weak_feature_size(F;maximum_bottleneck_order=2)

Output: 0.9950352569779831

The second function is

homology_inference(F;wfs=nothing,subsampling_coefficient=1.0,homology_up_to_degree=1,maximum_bottleneck_order=nothing,delta=1e-10,threshold=1e-7,return_sample=false)

This computes a dense sample of the space according to the weak feature size, then finds the Betti numbers of the corresponding algebraic manifold V_R(F), provided V(F) is a smooth complete intersection of codimension d and F contains d equations. Inputs are:

• F a list of polynomials
• wfs in the default case the weak feature size will be computed using the above function up to the provided maximum_bottleneck_order. The user can instead provide a precomputed number.
• subsampling_coefficient controls the trade-off between sample density and subsampling after the initial sampling. The default is reasonable.
• homology_up_to_degree controls the maximum homology degree for homology inference. Homology in degree 2 usually requires substantial memory. Homology in degree 3 is unlikely to complete.
• delta and threshold are technical inputs. The defaults are reasonable.
• If return_sample is set to true, the output will include the dense sample of the algebraic manifold that was computed.

The output is a list with entry:

• output["betti_numbers"] a list of the computed Betti numbers in ascending order of degree
• output["wfs"] the computed weak feature size
• output["proportion_of_wfs"] multiplying wfs by this entry gives the initial density of the sample that was computed, before subsampling
• output["diagrams"] is a list of persistence diagrams in the format of PersistenceDiagrams.jl. The diagrams are in ascending order of degree.

Multi-system mode

The package also supports weak feature size type computations between multiple varieties, which is useful for computing, e.g., the distance between two varieties V(G) and V(H)

compute_weak_feature_size(array_of_systems;maximum_bottleneck_order=nothing,threshold=1e-8,solve_with_monodromy=false,system_variables=variables(F[1]),multi_system=true)

Where

• array_of_systems is a vector with maximum_bottleneck_order elements that are lists of polynomials

Depedencies

This package uses both sampling_varieties by Oliver Gafvert and Ripserer.jl by Matija Čufar.

License

This repository is distributed under GPLv3.