Morse theoretic signal compression and reconstruction on chain complexes

Area of Science:

• Computational topology
• Applied algebraic topology
• Topological Data Analysis (TDA)

Background:

• Cellular signal processing integrates Topological Data Analysis (TDA) and machine learning.
• Current methods utilize combinatorial Laplacians and Hodge decomposition for signal processing.
• Discrete Morse theory is employed for computational efficiency by reducing complex sizes while maintaining topology.

Purpose of the Study:

• To develop a signal compression and reconstruction method for chain complexes using algebraic discrete Morse theory.
• To reduce and reconstruct based chain complexes and their associated signals via deformation retracts.
• To preserve the global topological structure of both the complex and the signals during compression and reconstruction.

Main Methods:

• Leveraging algebraic discrete Morse theory for signal processing on chain complexes.
• Utilizing deformation retracts to reduce and reconstruct based chain complexes and signals.
• Proving the equivalence of deformation retracts and Morse matchings for finite-dimensional chain complexes.

Main Results:

• Demonstrated that deformation retracts on real degree-wise finite-dimensional based chain complexes are equivalent to Morse matchings.
• Analyzed signal changes under Morse matchings, showing trivial reconstruction error on specific Hodge decomposition components.
• Developed an algorithm for computing Morse matchings with minimized reconstruction error.

Conclusions:

• The proposed method effectively compresses and reconstructs signals on chain complexes while preserving essential topological features.
• The findings offer a computationally efficient approach to signal processing in TDA and related fields.
• The developed algorithm provides a practical tool for minimizing signal reconstruction error in complex topological structures.