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Minimal Cycle Representatives in Persistent Homology Using Linear Programming: An Empirical Study With User's Guide.

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  • 1Mathematics, Statistics, and Computer Science Department, Macalester College, Saint Paul, MN, United States.

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Summary
This summary is machine-generated.

Optimizing cycle representatives in persistent homology reduces ambiguity and data interpretation challenges. Computational cost varies, but integer programming offers similar efficiency to linear programming for cycle basis optimization.

Keywords:
computational persistent homologygeneratorsl1 and l0 minimizationlinear programmingminimal cycle representativestopological data analysis

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Area of Science:

  • Computational Topology
  • Data Analysis
  • Applied Mathematics

Background:

  • Persistent homology characterizes topological features in data using cycle representatives.
  • Non-unique representatives lead to ambiguity in data interpretation.
  • Optimization offers a method to select meaningful representatives.

Purpose of the Study:

  • To evaluate the effectiveness and computational cost of various optimization procedures for constructing homological cycle bases in persistent homology.
  • To compare different loss algorithms (edge-loss, triangle-loss) and weighting schemes (uniform, length, area).
  • To analyze the impact of linear programming methods and solvers on optimization performance.

Main Methods:

  • Investigated dimension one persistent homology with rational coefficients.
  • Employed minimization optimization procedures using linear programming.
  • Applied uniform-weighted and length-weighted edge-loss algorithms.
  • Utilized uniform-weighted and area-weighted triangle-loss algorithms.
  • Optimized over column bases of simplicial boundary matrices using general-purpose solvers.

Main Results:

  • Optimization effectively reduces the size of cycle representatives, with varying degrees based on data characteristics.
  • The computational cost of optimization generally exceeds the cost of basis computation.
  • Linear solver choice significantly impacts cycle optimization time.
  • Integer programming yields comparable computation times to linear programming with the Gurobi solver.
  • Integer and non-integer solutions often yield the same cost and are valid for restricted optimization.
  • Distinct results were observed for Erdős-Rényi random clique complexes compared to real-world and synthetic point cloud data.

Conclusions:

  • Optimization is a valuable technique for refining cycle representatives in persistent homology, though computational costs must be considered.
  • The choice of optimization method, weighting scheme, and linear solver influences both effectiveness and efficiency.
  • Further research into specific data types, like random clique complexes, is warranted for nuanced topological analysis.