Computing Conley-Morse Persistence Barcode Efficiently by Updating Matrix Decompositions

What the math gives to ML

The paper's transferable asset is an incremental persistence computation scheme: instead of recomputing zigzag persistence after every change, it maintains homological decompositions while blocks split, merge, or change direction, using isomorphisms implemented by elementary transpositions. This suggests an online topological monitor or regularizer for neural representations whose activation complexes evolve during training, where recomputing persistent homology at every checkpoint is otherwise prohibitively expensive. The most practical first use is not end-to-end differentiation, but maintaining a barcode-derived stability statistic for activations, routing graphs, or token-neighborhood complexes and testing whether it predicts or improves training stability.