Unified framework connects deformation theory and derived categories for multiparameter persistence.
problem Algebraic complexity of multiparameter persistence modules hinders classification, stability, and interpretability.
method Combines deformation theory and derived categories to study multiparameter persistence geometrically.
result Unified conjecture relating interleaving distance to derived convolution metrics established.
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.
Stable density-based clustering via multiparameter persistence.
problem Density-based clustering stability to data perturbations.
method Degree-Rips construction, correspondence-interleaving distance, multiparameter stability analysis.
result Persistable pipeline yields stable, consistent density-based clustering.
Extends Morse-Forman theory to vector-valued functions for multiparameter persistence.
problem Computing multiparameter persistence with new tools and methods.
method Adapting Forman's theory to vectorial setting and using combinatorial topological dynamics.
result Established more general result for sublevel sets and found a way to induce Morse decomposition.
CuMPerLay vectorizes CMP for deep learning, improving image analysis.
problem Complex multifiltration structures hinder using CMP in deep learning.
method Introduces a new algorithm for vectorizing MP homologies of cubical complexes.
result Differentiable vectorization enables robust topological feature vectors for deep learning.
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.
Develops 2-categorical methods for multi-parameter persistence.
problem Fundamental limitations of traditional persistence modules.
method 2-categorical structures to capture hierarchical interactions.
result New invariants effectively characterize multidimensional topological features.
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.
We establish multiparameter resolvent trace expansions for elliptic boundary value problems, polyhomogeneous both in the resolvent and the auxiliary parameter. The present analysis is rooted in the joint project with Matthias Lesch on multiparameter resolvent trace expansions on revolution surfaces with applications to…
The paper uses MDM theory to analyze multifiltering functions on simplicial complexes.
problem Understanding multifiltering functions through discrete Morse theory.
method Applying multiparameter discrete Morse theory to vector-valued multifiltering functions.
result Any multifiltering function can be approximated by a compatible MDM function.
We prove the existence of multiparameter isospectral deformations of metrics on SO(n) (n=9 and n≥11), SU(n) (n≥8), and Sp(n) (n≥4). For these examples, we follow a metric construction developed by Schueth who had given one-parameter families of isospectral metrics on orthogonal and unitary group…
The central problem of strip theory is the calculation of potential flowaround 2D sections. One particular method of solutions to this problem is conformal mapping of the body section to the unit circle over which a solution of potential flow is available. Here, a new multiparameter conformal mapping method is presente…
We present Natural Gradient Boosting (NGBoost), an algorithm for generic probabilistic prediction via gradient boosting. Typical regression models return a point estimate, conditional on covariates, but probabilistic regression models output a full probability distribution over the outcome space, conditional on the cov…
Random complexes can be embedded linearly if certain conditions on parameters are met.
problem Embedding random simplicial complexes linearly in Euclidean space.
method Established strict inequalities on parameters for linear embedding into R^(2d).
result Necessary and sufficient conditions for linear embedding of random complexes.
This study explores how choosing noninformative priors affects Thompson Sampling in multiparameter bandit models.
problem The optimality of Thompson Sampling (TS) in multiparameter bandit models depends on the choice of priors, especially when models are complex.
method The study extends regret analysis to uniform distributions and proposes a modified TS policy, TS-T, to achieve asymptotic optimality.
result Changing noninformative priors can significantly affect the expected regret in multiparameter bandit models.
We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the cla…
Given a Coxeter system (W,S) and a positive real multiparameter $\bq$, we study the "weighted L2-cohomology groups," of a certain simplicial complex Σ associated to (W,S). These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to (W,S) and the multiparameter q. The…
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.
Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine lea…
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.
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.
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.
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…
Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come. In this article, by considering the space of persistence diagrams as a space of d…
Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {\em diagram distances}, most of the recent attempts at using persistence diagrams in machine learning have been done through kernel methods, i.e., embeddings of …
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 …
The choice of the control frequency of a system has a relevant impact on the ability of reinforcement learning algorithms to learn a highly performing policy. In this paper, we introduce the notion of action persistence that consists in the repetition of an action for a fixed number of decision steps, having the effect…
Given a compact geodesic space X we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of X to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their preci…
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.
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.
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, …
This article addresses persistent tangles. These are tangles whose presence in a knot diagram forces that diagram to be knotted. We provide new methods for constructing persistent tangles. Our techniques rely mainly on the existence of non-trivial colorings for the tangles in question. Our main result in this article i…
Study introduces a new method for multiple parameter regularization in polynomial functional regression.
problem Handling varying regularization parameters in polynomial functional regression.
method Developed a theoretically grounded algorithm for multiple parameter regularization and model aggregation.
result Promising results from evaluations on synthetic and real-world data.
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.
New financial models use tempered stable subordination for better correlation dynamics.
problem Building financial models with better correlation dynamics.
method Introducing tempered stable Sato subordinators and additive inhomogeneous processes.
result The new process has time-dependent correlation, improving fit for financial data.
New model predicts energy prices volatility by smoothing time variation and persistence.
problem Separate study of volatility's time variation and persistence.
method Dynamic persistence model that allows shocks with heterogeneous persistence to vary smoothly over time.
result Significantly improves volatility forecasts over state-of-the-art models.
Persistent homology reveals geometric features of metric spaces, especially geodesic circles.
problem Detecting geometric features in metric spaces using persistent homology.
method Analyzing algebraic elements (footprints) in persistent homology of metric spaces and subspace.
result Higher-dimensional persistent homology captures lower-dimensional geometric features.
Persistent Legendrian contact homology distinguishes knots using height functional.
problem Distinguishing Legendrian knots in R3. method Persistent homology applied to Chekanov-Eliashberg DGA, with height functional.
result Strong Morse inequalities for persistent Legendrian contact homology.