
Umatch factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology
Persistent homology is a leading tool in topological data analysis (TDA)...
read it

Homological Scaffold via Minimal Homology Bases
The homological scaffold leverages persistent homology to construct a to...
read it

Volume Optimal Cycle: Tightest representative cycle of a generator on persistent homology
This paper shows a mathematical formalization, algorithms and computatio...
read it

ContinuousTime Systems for Solving 01 Integer Linear Programming Feasibility Problems
The 01 integer linear programming feasibility problem is an important N...
read it

Constructing classification trees using column generation
This paper explores the use of Column Generation (CG) techniques in cons...
read it

Graver Bases via Quantum Annealing with Application to NonLinear Integer Programs
We propose a novel hybrid quantumclassical approach to calculate Graver...
read it

Cost of selfishness in the allocation of cities in the Multiple Travelling Salesmen Problem
The decision to centralise or decentralise human organisations requires ...
read it
Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guide
Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the nonuniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. In this work, we provide a study of the effectiveness and computational cost of several ℓ_1minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniformweighted and lengthweighted edgeloss algorithms as well as uniformweighted and areaweighted triangleloss algorithms. We conduct these optimizations via standard linear programming methods, applying generalpurpose solvers to optimize over column bases of simplicial boundary matrices. Our key findings are: (i) optimization is effective in reducing the size of cycle representatives, (ii) the computational cost of optimizing a basis of cycle representatives exceeds the cost of computing such a basis in most data sets we consider, (iii) the choice of linear solvers matters a lot to the computation time of optimizing cycles, (iv) the computation time of solving an integer program is not significantly longer than the computation time of solving a linear program for most of the cycle representatives, using the Gurobi linear solver, (v) strikingly, whether requiring integer solutions or not, we almost always obtain a solution with the same cost and almost all solutions found have entries in 1, 0, 1 and therefore, are also solutions to a restricted ℓ_0 optimization problem, and (vi) we obtain qualitatively different results for generators in ErdősRényi random clique complexes.
READ FULL TEXT
Comments
There are no comments yet.