Paper reconstructs compact metric spaces using finite approximations and inverse persistence.
problem Reconstructing the homotopy type of compact metric spaces.
method Using inverse limit of finite approximations and inverse persistence.
result Definition of inverse persistence as a new persistence process.
New approach uses distributed persistence for stable, parallelizable topological analysis of large point clouds.
problem Estimating the full persistence diagram of large point clouds is expensive, unstable, and not a sufficient statistic.
method Proposes distributed persistence as a new invariant, which is perfectly parallelizable, more stable, and has a rich inverse theory.
result The map from point clouds to distributed persistence invariants is a global quasi-isometry, interpolating between purely geometric and topological invariants.
Persistent homology detects curvature from sampled points.
problem Understanding the geometric information encoded in short intervals of persistent homology.
method Persistent homology computations and average persistence landscapes.
result Persistent homology detects curvature of disks from sampled points.
The paper uses polyhedral expansions to capture the shape of compact metric spaces.
problem Capturing the shape of compact metric spaces using finite approximations.
method Inverse sequences of polyhedra based on finite approximations of a compact metric space.
result Proves the General Principle and computes inverse persistent homology groups.
New method tackles incomplete data in RBM inverse Ising problems.
problem Computing data and model expectations in inverse Ising problems with missing observations.
method Combines mean-field approximation, persistent contrastive divergence, and spatial Monte Carlo integration.
result Effective and accurate tuning of model parameters compared to conventional methods.
Regularizes persistent homology gradients for neural network integration.
problem Ill-posed inverse problem in computing gradients of persistent homology.
method Regularization through a grouping term to define gradients for larger entities.
result Ensures gradients are defined with respect to larger entities, not individual points.
The article uses PageRank and persistent homology for scalable graph comparison.
problem Comparing the similarities between complex networks.
method Combines PageRank and persistent homology to compute a scalable graph descriptor.
result Shows the effectiveness of the method on shape mesh datasets.
Improved persistence spheres map measures to functions, stable under partial transport.
problem Representing and comparing measures in topological machine learning.
method Persistence spheres map measures to continuous functions on the sphere, stable under 1-Wasserstein partial transport.
result Persistence spheres provide a stable, parameter-free representation of measures, improving upon existing methods.
In [8] Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Further…
CNN outperforms other methods in gravity inversion.
problem Estimating subsurface density from gravitational field data.
method CNN, VAEs, GANs, iterative solvers (GD, GMRES, LGMRES, ICG).
result CNN provides the most reliable reconstructions.
New method improves neural network robustness to adversarial attacks.
problem Improving adversarial robustness of neural networks.
method Inspired by adaptive control theory, the approach uses persistency of excitation to constrain gradient descent updates.
result Networks trained with the PoE-motivated learning rate schedule are significantly more robust to adversarial attacks.
Hidden semi-Markov models (HSMMs) are latent variable models which allow latent state persistence and can be viewed as a generalization of the popular hidden Markov models (HMMs). In this paper, we introduce a novel spectral algorithm to perform inference in HSMMs. Unlike expectation maximization (EM), our approach cor…
Extends diffusion models to handle exponential family distributions for inverse problems.
problem Intractability of likelihood score for non-Gaussian observations.
method Evidence trick to approximate likelihood score for exponential family distributions.
result Effective Bayesian inference on complex Poisson processes and malaria prevalence prediction.
This study develops a dynamic inverse optimization framework to recover hidden, time-varying preferences from observed allocation trajectories.
problem The gap between classical optimization theory and real-world practice, especially in the presence of drift and shocks.
method Dynamic inverse optimization framework using a drift-aware estimator grounded in convex analysis and online learning theory.
result Sharp static and dynamic regret bounds for the framework, demonstrating its responsiveness to gradual drift and sudden shocks.
In financial markets, the order flow, defined as the process assuming value one for buy market orders and minus one for sell market orders, displays a very slowly decaying autocorrelation function. Since orders impact prices, reconciling the persistence of the order flow with market efficiency is a subtle issue. A poss…
PTOPOFL uses topological descriptors to protect privacy in federated learning.
problem Privacy and data reconstruction attacks in federated learning.
method PTOPOFL replaces gradient communication with persistent homology feature vectors for privacy and topology-guided aggregation.
result PTOPOFL achieves higher AUC and reduces reconstruction risk compared to gradient sharing.
Persistence landscapes map diagrams into function spaces for statistical and machine learning applications.
problem Mapping persistence diagrams into function spaces for statistical and machine learning.
method Introducing persistence landscapes, weighted persistence landscapes, and Poisson-weighted persistence landscape kernels.
result Persistence landscapes allow for the application of statistical and machine learning tools, and are stable and invertible.
Approaches for approximating persistent homology for large datasets.
problem Inability to compute persistent homology for large datasets.
method Multiple subsampling framework for statistical approximation of persistent homology.
result Derivation of finite sample convergence rates for empirical means of persistent homology.
Proposes deep graph persistence to address neural persistence issues in deep learning.
problem Variance of weights and lack of spatial structure in deep neural networks impact neural persistence.
method Extends neural persistence to the whole network, considering interactions between layers.
result Deep graph persistence alleviates variance-related issues and captures persistent paths through the network.
Paper proves k-means clustering works on persistence diagrams.
problem Complex geometry of persistence diagram space.
method Proves convergence of k-means on persistence diagram space. result Performance of k-means on persistence diagrams and measures is superior. Formula for interleaving distance of rectangle persistence modules.
problem Calculating distances between rectangle persistence modules.
method Formulas based on rectangle geometry, extended to decomposable modules.
result Closed formulas for interleaving and bottleneck distances.
This paper demonstrates the flaws of co-persistence theory proposed by Bollerslev and Engle (1993) which cause the theory can hardly be applied. With the introduction of the half-life of decay coefficient as the measure of the persistence, and both the weak definition of persistence and co-persistence in variance, this…
Optimizes wavelets for graph classification using spectral wavelet signatures and persistence diagrams.
problem Graph classification with geometric properties encoded in persistence diagrams.
method Optimizes spectral wavelets for graph datasets to capture best-suited features for classification.
result Competitive performance in graph classification problems compared to other persistence-based architectures.
New methods found persistent tangles in knots.
problem Persistent tangles in knot diagrams.
method Non-trivial colorings for tangles.
result Any knot with non-trivial coloring has persistent tangles.
New method for analyzing multiparameter persistence modules from smooth functions.
problem Analyzing multiparameter persistence modules from smooth functions.
method Generalized Morse theory applied to cobordism and Cerf theory.
result Complete description of persistence modules as direct sums of indecomposables.
Develops robust persistence diagrams using kernel methods.
problem Persistence diagrams are sensitive to data perturbations.
method Constructs robust persistence diagrams from superlevel filtrations of robust density estimators using reproducing kernels.
result Robust persistence diagrams are consistent estimators in bottleneck distance.
This paper interprets critical scales in persistent homology for compact metric spaces.
problem Understanding critical scales in persistent homology for general compact metric spaces.
method Analyzing local minima of the distance function and their impact on persistence.
result Each decrease in zero-dimensional persistence and increase in one-dimensional persistence is induced by local minima of the distance function.
Introduces a new space of Radon measures for better understanding persistence diagrams.
problem Lack of optimal transport-based formalism for persistence diagrams.
method Formalizes persistence diagrams as Radon measures on the upper half plane via optimal partial transport.
result Characterizes convergence and barycenters of persistence diagrams.
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.
Study on embedding persistence diagrams into Hilbert spaces, focusing on metric distortion.
problem Understanding metric properties of persistence diagrams in Hilbert spaces.
method Investigate embedding persistence diagrams into separable Hilbert spaces using bi-Lipschitz maps.
result Impossible to find a bi-Lipschitz embedding into finite-dimensional Hilbert spaces.
Kernel for multi-parameter persistent homology connects TDA with ML.
problem Connecting persistent homology with machine learning for multivariate data.
method Integrating a one-parameter kernel weighted along straight lines.
result Stable and efficiently computable kernel for multi-parameter persistence.
Persistent homology can recognize knotting in curves.
problem Recognizing knotting in curves
method Compute one-dimensional persistent homology, extract cycle representatives, and assign a hypergraph curvature-based score.
result Systematic differences between knotted and unknotted structures are revealed.
This article analyzes the relationship between co-persistence and hedging which indicates co-persistence ratio is just the long-term hedging ratio. The new method of exhaustive search algorithm for deriving co-persistence ratio is derived in the article. And we also develop a new hedging strategy of combining co-persis…
The paper studies geometric properties of geodesic spaces using Rips and Čech filtrations.
problem Understanding geometric properties of geodesic spaces.
method Applying fundamental group and homology groups to Rips or Čech filtrations.
result Rips critical points correspond to circles of specific lengths and persistence encodes space properties.
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a …
A new approach to reinforcement learning improves policy performance by adjusting control frequency.
problem Improving reinforcement learning performance by optimizing control frequency.
method Introducing action persistence and a novel algorithm, PFQI, to learn optimal value function at a given persistence.
result PFQI effectively learns optimal value function with action persistence, improving reinforcement learning performance.
A new method compares persistent cycles in topological data.
problem Comparing persistent homology representations of two spaces.
method Direct comparison of individual persistent cycles based on persistence intervals and spatial placement.
result Demonstrated the effectiveness of the method in topological inference.
New model predicts persistent weather patterns better than persistence model.
problem Improving weather forecasting accuracy for persistent patterns.
method Mixture of experts model using ordinal neural network.
result The model outperforms persistence and other models for longer time horizon predictions.
Embeds persistence diagrams into Hilbert spaces to use kernel methods.
problem No inner product structure on persistence diagrams.
method Shows non-embeddability of persistence diagrams into Hilbert spaces.
result Persistence diagrams with bottleneck distance do not coarse embed into Hilbert spaces.
We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, …
Paper approximates geodesic space persistence with finite samples.
problem Geodesic spaces have uncountable Rips complexes, making persistence analysis difficult.
method Develops finite samples to approximate geodesic space persistence and proves stability.
result Persistence of a geodesic space can be obtained from finite samples, and stability holds.
Study cosmic structures using Topological Data Analysis and Persistence Energy.
problem Investigate cosmic web evolution in ΛCDM cosmologies. method Apply LITE method to embed persistence diagrams into vector spaces and analyze cosmic structures.
result Discover a correlation between Persistence Energy and redshift values.
New lattice path method for statistical inference of persistent diagrams.
problem Statistical inference on persistent diagrams.
method Lattice path representation and combinatorial enumerations.
result Topological changes observed in spike proteins of COVID-19 virus.
The paper examines how long-memory dynamics, rough-volatility, and persistence affect equity volatility forecasting.
problem The study investigates how long-memory dynamics, rough-volatility, and persistence impact equity volatility forecasting.
method The paper combines semiparametric long-memory estimation, rough-volatility diagnostics, and structured forecasting regressions.
result Persistence measures improve out-of-sample volatility forecasts, particularly during periods of elevated market volatility and in volatility-managed portfolio applications.
This review explores TDA and TDL beyond persistent homology.
problem Limitations of persistent homology in capturing topological invariants and homotopic evolution.
method Spectral representations, sheaf theory, Mayer topology, interaction topology, differential topology, geometric topology.
result Review of topological tools for various data types.
Revises SWK for persistence diagrams using Figalli-Gigli distance.
problem Efficiently embedding persistence diagrams in a Hilbert space.
method Directly use Figalli-Gigli distance to build a positive definite kernel.
result SFGK shares properties with SWK and performs similarly on benchmarks.
Study examines persistence diagrams in machine learning, proposing permutation tests.
problem Understanding the power and limitations of persistence diagrams in machine learning.
method Carried out experiments on graph and shape data, proposed permutation tests for persistence diagrams.
result Persistence pairing shows significant improvement in various tasks, but the most critical values are most discriminative.
New method converts complex PDs into stable vectors for ML.
problem Complex structure of persistence diagrams makes them hard to use in ML.
method Persistence bag-of-words (BoW) for vectorizing PDs.
result Achieves state-of-the-art performance and speed in ML.