New interdisciplinary branch in math physics tackles topological interactions of polymer-like objects.
problem Analyzing topological interactions in fluctuating non-phantom rope-like objects.
method Review of conceptual steps in statistical topology.
result Emergence of statistical topology as a new interdisciplinary field.
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 …
Flexible approach for normal approximations in geometric and topological statistics.
problem Normal approximation for complex statistics not expressible as sums of score functions.
method Flexible add-one cost operator combined with strong stabilization theory.
result Established normal approximation results for geometric and topological statistics.
Geometric surgery theory applied to quantum statistics in spacetime.
problem Constraints of quantum statistics data in spacetime dimensions.
method Geometric-topology surgery theory applied to spacetime manifolds.
result New quantum surgery constraints derived, analogous to Verlinde formula.
New method detects uncertainty in neural networks for out-of-distribution detection.
problem Detecting out-of-distribution inputs to ensure model reliability.
method Predictive topological uncertainty (pTU) based on persistent homology.
result pTU provides a statistical framework for OOD detection.
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.
Quantum surgery formulas link statistics and topology.
problem Formulate constraints for quantum statistics of entangled systems.
method Geometric-topology surgery theory on spacetime manifolds.
result New quantum surgery formulas and constraints derived.
Study shows topological features improve time series classification.
problem Classifying stochastic processes with varying noise and sampling.
method Topological data analysis features compared to statistical and raw features.
result Topological features lead to better classification performance.
We introduce the first unified theory for target tracking using Multiple Hypothesis Tracking, Topological Data Analysis, and machine learning. Our string of innovations are 1) robust topological features are used to encode behavioral information, 2) statistical models are fitted to distributions over these topological …
New method robustifies topological data analysis against outliers.
problem Outliers make topological data analysis unstable.
method Proposed a robust distance function (MoM Dist) for persistent homology.
result MoM Dist sublevel filtrations and weighted filtrations are consistent estimators in adversarial settings.
Introduces generalized almost statistical convergence and its properties.
problem Developing a new convergence concept for sequences.
method Introducing generalized almost statistical convergence and proving its properties.
result Existence of a GAS convergent sequence that is neither statistical nor almost convergent.
Constructing exponential families from statistical manifolds.
problem The central problem of constructing exponential families from statistical manifolds.
method Constructive approach proving every compact statistical manifold admits a foliation of Hessian manifolds.
result Compact orientable leaves are either finite quotients of flat torus or mapping torus with periodic monodromy.
The paper explores statistical and topological properties of sliced probability divergences.
problem Understanding the topological, statistical, and computational consequences of slicing divergences.
method Deriving theoretical properties of sliced probability divergences, including metric axioms preservation and weak continuity.
result Sliced divergences share similar topological properties and have stable sample complexity.
Study on Alexander polynomials in braids, linking number theory and topology.
problem Distribution of Alexander polynomials in specific families of braids.
method Exploration of arithmetic invariants and analogies with number theory.
result New directions in arithmetic topology and statistics.
New methods compare neural network models using geometric and topological summaries.
problem Comparing deep representations of complex networks in models and brains.
method Develops inference methods based on topological data analysis (TDA) and graph-based techniques.
result New statistical methods enable better model comparison and inference.
This chapter covers methods for identifying and inferring graph topologies.
problem Identifying and inferring graph topologies from multidimensional relational data.
method Overview of methods including correlation metrics, covariance selection, kernels, structural equations, and vector autoregressions.
result Supports both batch and online learning with convergence guarantees and leverages high-order statistical information.
Study topological Iwasawa invariants for 3-sphere links, proving density results.
problem Detect and analyze Iwasawa invariants for 3-sphere links.
method Explicit criteria for detecting Iwasawa invariants, statistical analysis of 2-bridge links.
result Density of 2-bridge links with specific Iwasawa invariants.
Quantum system linked to knots and branched coverings.
problem Modeling quantum statistical mechanics for knots and branched coverings.
method Construction of a quantum system using knot theory and arithmetic topology.
result The resulting algebra of observables is a noncommutative Bernoulli crossed product.
Experimental evidence for curve ratios on genus two surfaces.
problem Determining the ratio of topological curve types on surfaces.
method Experimental statistics applied to Mirzakhani's genus two surface results.
result Separating and non-separating curves occur in the ratio 1:48.
Study how knots occupy space using topological methods.
problem Understanding how knots occupy a volume of space.
method Statistical analysis of persistent homology features from Vietoris-Rips complexes.
result Existence of correlations between geometric and topological features of knots.
Data science enhances knot theory by analyzing invariant relations.
problem Understanding the complex relations between knot invariants.
method Topological data analysis applied to knot theory.
result New insights into long-standing conjectures about knots.
Kernel method enhances analysis of topological data.
problem Statistical analysis of persistence diagrams for robust topological properties.
method Kernel embedding and weight factor to control persistence effect.
result Advantages over existing methods in practical data analysis.
Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari-Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. Thi…
The paper connects arithmetic statistics to homological stability in étale cohomology.
problem Understanding the growth of arithmetic statistics over finite fields.
method Importing homological stability from topology to étale cohomology.
result Established subexponential bounds on the growth of unstable cohomology.
We develop a theory of higher-order feature attribution for complex models.
problem Interpreting feature contributions in models with interactions is challenging.
method We extend Integrated Gradients (IG) to higher-order feature attributions.
result We establish natural connections to statistics and topological signal processing.
TDA-based portfolios show better risk-adjusted returns than classical methods.
problem Traditional portfolio selection methods fail to capture complex asset dynamics.
method Topological Data Analysis (TDA) using persistence landscapes to quantify portfolio risk.
result TDA-based portfolios outperform classical models in excess mean return and financial ratios.
Explains how geometry and statistics intertwine, focusing on information geometry.
problem Understanding the interplay between geometry and statistics.
method Introduces differential topology, geometry, probability, and (pre-)Frobenius manifolds.
result Discovers connections between geometry and statistics, particularly in information geometry.
New algebraic geometry and statistical manifold connections proven.
problem Understanding the structure of statistical manifolds and their algebraic properties.
method Developed relations between algebraic geometry, information theory, and Topological Field Theory.
result Statistical pre-Frobenius manifolds form algebraic varieties and have hexagonal, isoclinic webs.
A reinforcement learning framework evolves persistence diagrams on PD space for stochastic dynamics.
problem Stochastic modeling and evolution of persistence diagrams (PDs) for topological analysis.
method Reinforcement learning framework for PD space, evolving diagrams through topology-aware local edit operations.
result Established conditions for geometric ergodicity of Markov chains on PD space, yielding unique stationary laws.
Discovering topological quantum field theories in 2+1 and 3+1 dimensions.
problem Exploring topological orders in condensed matter lattice models.
method Calculating braiding statistics and link invariants of anyon excitations.
result Identifying new spin topological quantum field theories with specific knot/link invariants.
Experimental results on veering triangulations of 3-manifolds.
problem Understanding the combinatorial structure of veering triangulations.
method Algorithmic construction and experimental analysis.
result Experimental insights into the structure of veering triangulations and their relation to topological invariants.
Researchers use topological summaries in regression and classification tasks.
problem Lack of positive semi-definite property in topological summaries.
method Defined a topological exponential kernel and showed its effectiveness in regression and classification.
result Topological summaries can be successfully used in supervised learning tasks.
We present two paradigms relating algebraic, topological and quantum computational statistics for the topological model for quantum computation. In particular we suggest correspondences between the computational power of topological quantum computers, computational complexity of link invariants and images of braid grou…
Survey of techniques for diagnosing pediatric sleep apnea from inexpensive data.
problem Diagnosing pediatric sleep apnea from limited and variable data.
method Exploratory data analysis using correlation networks, Mapper, SVD; supervised and unsupervised learning techniques.
result Analysis of various learning techniques applied to pediatric sleep apnea data.
A new method detects interactions in neural networks using topological analysis.
problem Detecting interactions between input features in neural networks.
method Topological analysis of neural network connectivity to quantify interaction strength.
result The PID algorithm outperforms state-of-the-art baselines in interaction detection tasks.
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.
Improved modeling of persistence diagrams for data analysis.
problem Determining significant outliers in persistence diagrams.
method Modification of the RST (Replicating Statistical Topology) model using MCMC Metropolis-Hastings algorithm.
result The modified RST model improves the goodness of fit in persistence diagram analysis.
Artificial neural networks map quantum phases of disordered topological superconductors.
problem Classifying quantum phases of disordered topological superconductors.
method Supervised artificial neural network trained on ensemble averages of quasiparticle distributions.
result Artificial neural networks can classify quantum phases with high confidence, identifying unknown phases.
The paper explores the topology of polygonal meshes and their properties.
problem Understanding the topological properties of polygonal meshes.
method Overview of topological concepts, definitions of intrinsic and extrinsic topology, proofs of Euler and Euler-Poincaré formulas, and discussion on cutting meshes.
result Detailed understanding and definitions of polygonal mesh topology, including intrinsic and extrinsic properties.
TopoFisher learns topological summaries by maximizing Fisher information, improving parameter efficiency and inference quality.
problem Simulation-based inference misses key information in low-order statistics, especially for non-Gaussian fields.
method TopoFisher uses a differentiable persistent-homology pipeline that learns topological summaries by maximizing local Gaussian Fisher information.
result TopoFisher recovers much of the available information and outperforms fixed topological vectorizations in weak gravitational lensing.
The paper addresses statistical inference for cluster trees, providing methods to quantify uncertainty.
problem Quantifying uncertainty in cluster tree estimates.
method Developed metrics for comparing cluster trees, proposed methods for constructing and summarizing confidence sets, introduced a partial ordering for pruning insignificant features.
result Proposed methods for constructing and summarizing confidence sets for the unknown true cluster tree.
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.
Canonical quantization of abelian BF-type topological field theory coupled to extended sources on generic d-dimensional manifolds and with curved line bundles is studied. Sheaf cohomology is used to construct the appropriate topological extension of the action and the topological flux quantization conditions, in terms …
A faster, more stable method for optimizing topological functions.
problem Optimizing topological functions is computationally expensive and unstable.
method Introduces a novel backpropagation scheme for faster and more robust optimization.
result Produces more robust optima and stable visualizations.
Study detects P-type bifurcations in single system realizations using unreliable kernel density estimates.
problem Detecting P-type bifurcations in signals with unreliable kernel density estimates.
method Create persistence diagrams from single system realization, statistically analyze resulting set, compare point process modeling methods.
result Subsampling outperforms other point process modeling methods in predicting P-type bifurcations.
The main aim of this paper is to extend Bochner's technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical structures. It deals, in particular, with Hodge's theory, Bochner-Weitzenbock and Simon's type formulas. Moreover, a few global and local theorem…
A new Riemannian framework for efficiently comparing topological persistence diagrams.
problem Efficiently comparing topological persistence diagrams without point-to-point matching.
method Developed a Riemannian geometry framework to represent and compare persistence diagrams.
result The new framework reduces computational complexity and enables statistical analysis.
Unified geometric approach to quantum indeterminacy.
problem Quantum indeterminacy and uncertainty principles.
method Geometric formulation using convex geometry and symplectic topology.
result Robertson-Schrodinger inequalities emerge as geometric principles.