Vietoris
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RedZeD: Computing persistent homology by Reduction to Zero Differentials
Computer Science > Computational Geometry [Submitted on 4 Jun 2026] Title:RedZeD: Computing persistent homology by Reduction to Zero Differentials View PDF HTML (experimental)Abstract:We introduce a new algorithm for computing persistent homology of Vietoris--Rips filtrations, which in many cases offers a considerable speedup over the existing implementation of the persistence pairing algorithm. The key innovation, called active enumeration, is made possible by a new theoretical framework of...
Learning Coherent Representations: A Topological Approach to Interpretability
arXiv:2606.02841v1 Announce Type: new Abstract: Deep neural networks learn representations where individual features often lack interpretable meaning; a single neuron may activate for scattered, unrelated inputs. We introduce coherence, a geometric property inspired by neural coding in the brain, where neurons like grid cells and head direction cells respond to contiguous regions of state space. A non-negative matrix is coherent if each row (sample) attends to geometrically clustered columns...
Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry
arXiv:2606.04227v2 Announce Type: replace Abstract: We present an algorithmic framework for incremental maintenance of first sheaf cohomology $H^1(X; \mathcal{F})$ on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of $H^1$ via factorization of the coboundary matrix requires $O(n^3)$ time; when the complex evolves with a stream of $m$ edits, full recomputation after each edit costs $O(mn^3)$. Under a bounded...
$p$-adic Bi-Filtrations for Topological Machine Learning on Genomic Sequences
arXiv:2606.06117v1 Announce Type: cross Abstract: We introduce pVR, a topological machine learning framework for alignment-free genomic sequence classification that combines $p$-adic numbers with topological data analysis. Each DNA sequence is encoded along two complementary axes: a $p$-adic distance on $k$-mer prefixes, which captures hierarchical positional structure, and a compositional $L_1$ distance on $k$-mer frequencies, which captures local sequence content. The two distances jointly...
Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry
new Abstract: We present an algorithmic framework for incremental maintenance of first sheaf cohomology $H^1(X; \mathcal{F})$ on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of $H^1$ via factorization of the coboundary matrix requires $O(n^3)$ time; when the complex evolves with a stream of $m$ edits, full recomputation after each edit costs $O(mn^3)$. Under a bounded local geometry assumption -- bounded...