A framework for differentiability on persistence barcodes.
problem Differentiability on the space of persistence barcodes.
method Lifts to the space of ordered barcodes to compute derivatives.
result A chain rule enabling gradient descent for objective functions.
We use barcodes to analyze neural networks' loss surfaces, revealing important properties.
problem Understanding the topology of neural networks' loss surfaces.
method Topological data analysis using Morse complexes and barcodes.
result Barcodes of local minima are located in a small part of the loss function's range and decrease with network depth and width.
New metrics and coordinates for barcode space using group theory.
problem Describing and measuring the space of barcodes.
method Geometric group theory applied to barcodes.
result Stratification of barcode space into regions with similar statistical properties.
Entropy measures geodesic flow complexity.
problem Measuring complexity of geodesic flows on manifolds.
method Introduced barcode entropy to measure exponential growth rate of not-too-short bars in Morse-theoretic barcodes.
result Barcode entropy bounds topological entropy and vice versa.
Quantum-enhanced barcode decoding and pattern recognition outperforms classical methods.
problem Improving barcode decoding and pattern recognition using quantum entanglement.
method Quantum hypothesis testing applied to barcode decoding and pattern recognition using entangled quantum sources and measurements.
result Quantum-enhanced methods outperform classical coherent-state strategies for barcode data decoding and classification.
New feature map for topological data analysis improves classification performance.
problem Lack of effective feature maps for topological data analysis.
method Realize barcodes as paths in a vector space, compute path signature, resulting in a feature map.
result Achieves state-of-the-art results on classification benchmarks.
MuRiT efficiently computes multi-parameter persistence barcodes.
problem Efficient computation of multi-parameter persistent homology.
method Vietoris-Rips transformation to reduce multi-parameter to single-parameter computation.
result MuRiT computes pathwise persistence barcodes for multi-filtered flag complexes.
This paper uses Ghrist barcodes to track persistent shapes in video frames.
problem Detecting and tracking persistent shapes in video frames.
method Introduces Ghrist barcodes for persistent Betti numbers derived from vortex nerve complexes in triangulated video frames.
result Persistent Betti numbers of vortex nerves are k+2 for k edges. Paper introduces stable vectorization for multiparameter PH using signed barcodes.
problem Lack of stable vectorization methods for multiparameter persistent homology.
method Signed barcodes as measures for stable vectorization of MPH.
result Stable feature vectors from signed barcodes improve performance in data science.
Introduces TSI, a variance-based measure for persistence barcodes.
problem Capturing structural variability in persistence barcodes.
method Variance-based scalar measure, TSI, and complementary TSigI.
result TSI captures structural variability complementary to entropy.
Study uses barcode theory to bound displacement energy of Legendrian submanifolds.
problem Bounding displacement energy for Legendrian submanifolds.
method Applies barcodes of persistent homology to Chekanov-Eliashberg algebra, linearizing only below a certain action level.
result Shows Legendrians that admit augmentations cannot be C0-approximated by stabilized Legendrians. Homotopy types of Vietoris-Rips metric thickenings of the circle confirmed.
problem Understanding the homotopy types of Vietoris-Rips metric thickenings of the circle.
method Finding quotients of the metric thickenings that preserve homotopy type and showing that the quotient spaces can be described as CW complexes.
result The Vietoris-Rips metric thickenings of the circle are homotopy equivalent to odd-dimensional spheres at the expected scale parameters.
Stable topological summary captures evolving dependency structure in dynamic Bayesian networks.
problem Missing larger-scale patterns in evolving dependency structures in dynamic Bayesian networks.
method Topological approach using Dynamic Bayesian Graphs and persistent homology.
result Stable topological summary (barcodes) captures evolving dependency structure in DBNs.
Paper stabilizes persistent homology rank functions for statistical inference.
problem Stability issues in persistent homology rank functions.
method Derive stability results for rank functions under FDA metrics.
result Rank functions stabilize, improving statistical inference.
New invariants study Morse functions' equivalence classes in persistent homology.
problem Understanding equivalence classes of Morse functions on spheres for persistent homology.
method Graph-equivalent and height-equivalent Morse functions, with fundamental moves.
result Established new invariants to discern Morse functions more effectively.
Symplectic homology matches dual capacities for convex domains.
problem Understanding symplectic capacities and Reeb flows on convex domains.
method Isomorphic filtered symplectic homology to dual singular homology.
result Gutt-Hutchings capacities match spectral invariants for convex domains.
Floer theory constructs filtrations on quantum cohomology for symplectic manifolds.
problem Quantum cohomology of symplectic manifolds with C∗-actions. method Floer theory applied to C∗-actions on symplectic manifolds. result Constructs a family of filtrations on quantum cohomology for Conical Symplectic Resolutions.
Detection of rare variants by resequencing is important for the identification of individuals carrying disease variants. Rapid sequencing by new technologies enables low-cost resequencing of target regions, although it is still prohibitive to test more than a few individuals. In order to improve cost trade-offs, it has…
The fingerprint classification problem is to sort fingerprints into pre-determined groups, such as arch, loop, and whorl. It was asserted in the literature that minutiae points, which are commonly used for fingerprint matching, are not useful for classification. We show that, to the contrary, near state-of-the-art clas…
A new method clusters complex networks using topological and geometric structure.
problem Clustering complex networks with intricate topology.
method Centroid-based clustering strategy using Wasserstein distance and barycenter for persistence barcodes.
result Demonstrated effectiveness on simulated and real-world networks.
A new algorithm reduces the size of datasets for TDA.
problem Processing large datasets with high dimensions in TDA is computationally infeasible.
method Introduced Characteristic Lattice Algorithm (CLA) for data reduction.
result CLA reduces dataset size while preserving geometric and topological features.
Paper compares dimension reduction methods using topological analysis on EEG data.
problem Comparing dimension reduction methods on EEG data.
method Topological data analysis, including persistent homology, Wasserstein distance, and hypothesis tests.
result Different dimension reduction methods show significant qualitative differences across topological homologies.
A deep probabilistic model analyzes DNA-encoded library data for efficient screening.
problem Complex data from DNA-encoded library experiments mask underlying signals.
method Compositional deep probabilistic model of DEL data, modeling latent reactions between synthons.
result DEL-Compose model demonstrates strong performance and valuable insights.
Topology-GS improves 3D GS for better structural and feature integrity.
problem Compromised pixel-level and feature-level integrity in 3D GS.
method Incorporates Local Persistent Voronoi Interpolation (LPVI) and PersLoss based on persistent homology.
result Topology-GS outperforms existing methods in PSNR, SSIM, and LPIPS metrics.
Enhanced Bruhat decomposition studies Morse theory and Reidemeister torsion.
problem Investigating the Bruhat numbers associated with Morse functions.
method Using a variation of the classical Bruhat decomposition for GL(F). result The product of Bruhat numbers is independent of the Morse function and interpretable as Reidemeister torsion.
Persistent entropy detects phase transitions in complex systems.
problem Detecting phase transitions in complex systems.
method Established a general theorem for persistent entropy to reliably detect phase transitions, introduced operational framework for finite-time computations.
result Persistent entropy exhibits an asymptotically non-vanishing gap across phases, robust numerical signatures across experiments.
Unified toolkit for comparing neural representations using SRTD and NTS.
problem Heuristic asymmetry and unbounded scores in existing divergences.
method Developed SRTD and NTS to address these issues.
result Unified, robust, and scale-invariant metric for comparing neural representations.
Study on Vietoris-Rips complexes of regular polygons, revealing complex homotopy types.
problem Understanding the homotopy types and persistent homology of Vietoris-Rips complexes of regular polygons.
method Use of persistent homology, cyclic graphs, and winding fractions.
result Characterization of homotopy types and persistent homology of Vietoris-Rips complexes of Pn up to a scale parameter.