Barcodes of Towers and a Streaming Algorithm for Persistent Homology.

Area of Science:

• Computational Topology
• Applied Mathematics
• Computer Science

Background:

• Towers of simplicial complexes are used in topological data analysis.
• Computing persistent barcodes for towers is computationally challenging.
• Existing methods may not be efficient for large-scale data.

Purpose of the Study:

• To develop an efficient method for computing a filtration from a tower of simplicial complexes.
• To ensure the computed filtration retains the same persistent barcode as the original tower.
• To enable practical computation using streaming algorithms.

Main Methods:

• Utilizing a coning strategy adapted from Dey et al.
• Developing a streaming algorithm for filtration computation.
• Combining the filtration approach with matrix reduction for barcode computation.

Main Results:

• A filtration is generated with the same persistent barcode as the tower.
• The filtration size is asymptotically close to the tower size.
• The streaming algorithm is efficient in theory and practice.
• Space complexity is independent of tower length, depending only on subcomplex size.
• The method successfully handles towers with billions of complexes.

Conclusions:

• The proposed method provides an efficient and scalable approach to compute persistent barcodes from towers of simplicial complexes.
• The streaming algorithm is practical for large datasets in computational topology.
• This work advances the applicability of topological data analysis to massive data.