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.
Contractible Vietoris-Rips complexes for integer n proved using discrete Morse theory.
problem Proving contractibility of Vietoris-Rips complexes for Zn. method Used Bestvina-Brady discrete Morse theory to provide a short and improved proof.
result Contractible Vietoris-Rips complexes at large scales for Zn. New construction reduces Vietoris-Rips complex construction time.
problem Efficiently constructing Vietoris-Rips complexes.
method Inductive construction avoiding unnecessary comparisons.
result Significant reduction in computational complexity.
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 has emerged as a novel tool for data analysis in the past two decades. However, there are still very few shapes or even manifolds whose persistent homology barcodes (say of the Vietoris-Rips complex) are fully known. Towards this direction, let Pn be the boundary of a regular polygon in the plane…
We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.
Generalizes Rips' result on hyperbolic spaces to metric spaces, showing collapses for tree metrics.
problem Understanding the contractibility of Vietoris-Rips complexes in metric spaces.
method Extending Rips' result using geodesic defect and apparent pairs gradient.
result Vietoris-Rips complexes collapse to subforests for finite tree metrics.
We inspect Vietoris-Rips complexes VRt(X) of certain metric spaces X using a new generalization of Bestvina-Brady discrete Morse theory. Our main result is a pair of metric criteria on X, called the Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain $VR_t(…
IsUMap improves data visualization of complex geometries.
problem Accurately representing complex, locally distorted metric spaces.
method Integrates UMAP and Isomap with Vietoris-Rips filtrations.
result Significant improvements in data representation quality.
The paper connects geometric and topological concepts to bound distances between metric spaces.
problem Bounding distances between metric spaces using Gromov-Hausdorff distance.
method Using Borsuk-Ulam theorems and Vietoris-Rips complexes, the paper obstructs the existence of certain continuous maps between complexes to bound discontinuities of functions.
result The paper provides new bounds on Gromov-Hausdorff distances between spheres of different dimensions.
Let G be a group acting properly and by isometries on a metric space X; it follows that the quotient or orbit space X/G is also a metric space. We study the Vietoris-Rips and Čech complexes of X/G. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris-Rips or Čech complex go to z…
Researchers decompose Forman-Ricci curvature for efficient computation in VR complexes.
problem Efficiently computing Forman-Ricci curvature in higher-dimensional data.
method Decomposition and set-theoretical proof for local computation of FRC in VR complexes.
result Reveals critical geometric insights overlooked by conventional techniques.
We construct a compact subset K of the four dimensional Euclidean space with the following property: For all values of the parameter in an interval, the Vietoris-Rips complex of K has uncountably generated first homology. This answers a question that arose in work on persistent homology.
We propose the labeled Čech complex, the plain labeled Vietoris-Rips complex, and the locally scaled labeled Vietoris-Rips complex to perform persistent homology inference of decision boundaries in classification tasks. We provide theoretical conditions and analysis for recovering the homology of a decision boundary fr…
This paper classifies planar-Rips complexes and their unit disk graphs up to homotopy.
problem Classifying planar-Rips complexes and their unit disk graphs.
method Simplicial classification, homotopy equivalence, and hereditary properties.
result Classification of planar-Rips complexes and unit disk graphs up to homotopy.
Study topological invariants of complexes for Riemannian manifolds.
problem Understanding topological properties of Riemannian manifolds.
method Analyzing Betti numbers and Euler characteristic of Vietoris-Rips and Čech complexes.
result Betti curve converges to manifold's Betti number within a scale parameter interval.
Homotopy equivalence shown between complex and thickened versions of manifolds.
problem Homotopy equivalence between manifold complexes and thickened versions.
method Natural bijections and homotopy equivalences of Vietoris-Rips and Čech complexes and thickened versions.
result Natural bijections between complexes and thickened versions are homotopy equivalences.
New TDA approach using Finsler metrics.
problem Traditional TDA concepts and methods.
method Introducing Finsler metrics for TDA.
result Relevance of Finsler metrics to TDA.
The study of shadow of Vietoris-Rips complexes and their homotopy properties.
problem Understanding the geometric/topological behavior of the shadow projection map p. method Inverse system techniques from shape theory to study systems of shadow complexes.
result The limit map limp exhibits favorable homotopy-theoretic properties when X is an ANR. Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.
problem Applying Milnor-Schwarz lemma to inverse monoids.
method Two proofs provided: elementary and using Vietoris-Rips complex.
result Generalization of Milnor-Schwarz lemma to inverse monoids.
We prove contractibility of VR complexes for integer lattices up to dimension 5.
problem Contractibility of Vietoris-Rips complexes for integer lattices.
method Analyzing the homotopy type and contractibility of VR complexes for integer lattices.
result Contractibility of VR complexes for integer lattices up to dimension 5.
Given a sample of points X in a metric space M and a scale r>0, the Vietoris-Rips simplicial complex VR(X;r) is a standard construction to attempt to recover M from X up to homotopy type. A deficiency of this approach is that VR(X;r) is not metrizable if it is not locally finite, and thu…
Given a sample Y from an unknown manifold X embedded in Euclidean space, it is possible to recover the homology groups of X by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set Y. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular…
Unified pipeline classifies time series using complex networks and persistent homology.
problem Classifying univariate time series using various graph constructions and metrics.
method Time series to graph, graph to dissimilarity matrix, filtration to persistence diagrams, vectorization to features.
result Persistence-based features are robust to noise and optimal graph type depends on signal structure.
Study semi-coarse spaces' homotopy and homology, extending coarse geometry.
problem Extend homotopy and homology concepts to semi-coarse spaces.
method Analyze homotopy and construct homology groups invariant under semi-coarse homotopy equivalence.
result Show semi-coarse homology is isomorphic to Vietoris-Rips homology for graphs.
A new method tracks index using topological data analysis for sparse portfolios.
problem Sparse index tracking with robust risk management.
method Topological learning via Vietoris-Rips filtration for sparse regularization.
result The method outperforms state-of-the-art techniques in various market conditions.
Paper defines and evaluates DR complex for persistent homology.
problem Computing persistent homology of Euclidean point cloud data.
method Delaunay-Rips complex construction for speed and stability.
result DR produces stable persistence diagrams under point cloud perturbations.
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.
Paper uses TDA to assess cryptocurrency risk by measuring phase space instability.
problem Traditional risk measures fail to capture market dynamics' geometric structure.
method Applied Takens' Delay Embedding Theorem to generate point cloud, computed persistent homology groups, defined Topological Persistence Norm.
result Proposed leverage calibration heuristic based on persistence of 1-dimensional cycles.
Unified probabilistic foundation for fuzzy simplicial sets in dimensionality reduction.
problem Lack of clear probabilistic interpretation in fuzzy simplicial sets.
method Introducing a probabilistic framework explaining fuzzy simplicial sets as marginals of probability measures on simplicial sets.
result Unified probabilistic theoretical foundation for fuzzy simplicial sets.
Efficiently sparsifies simplicial complexes using local densities of states.
problem Prohibitive computational requirements for dense simplicial complexes.
method Probabilistic sparsification using local densities of states and kernel-ignoring decomposition.
result Approximates the spectrum of the original SC with a sparser surrogate SC.
Proposes Topology Distance for evaluating GANs.
problem Challenges in evaluating GANs' goodness.
method Builds Vietoris-Rips complex on image features and defines TD based on latent manifold comparisons.
result Demonstrates TD's superiority over existing metrics.
This paper introduces persistent equivariant cohomology and applies it to circle actions.
problem Understanding the cohomology of filtered spaces with group actions.
method Persistent Borel equivariant cohomology, Serre spectral sequence, Gysin homomorphism.
result Explicit description and cohomology computation for circle actions.
Study reveals how dengue spread patterns vary across different years in Recife, Brazil.
problem Understanding spatial organization of dengue transmission in urban areas.
method Spatial analysis of dengue cases using topological data analysis and Vietoris-Rips filtrations.
result Critical percolation thresholds define distinct geometric regimes of dengue spread.
Quantum method detects financial stress regimes from market data.
problem Detecting financial stress regimes from market data.
method Adapted Pauli Correlation Encoding to quantum topological data analysis.
result Quantum method can recover Betti numbers exactly at every scale.
Study on knot types using thickness and length constraints.
problem Understanding the ideal stratum and deformation persistence of knot types.
method Ropelength-filtered spaces and admissible deformations.
result The first birth level of admissible components corresponds to the ropelength of the knot.
New method predicts Alzheimer's risk with individual uncertainty estimates.
problem Predicting conversion from mild cognitive impairment to Alzheimer's disease.
method Persistent homology of clinical trajectories combined with stacking ensemble.
result Pipeline achieves high accuracy and individual-level uncertainty quantification.
New condition prevents hyperbolic spaces from matching curve complexes.
problem Identifying when hyperbolic spaces cannot match curve complexes.
method Analyzing specific hyperbolic complexes and identifying a condition.
result Identified a condition preventing quasi-isometry between hyperbolic spaces and curve complexes.
Study on complex line fields on almost-complex manifolds, proving existence conditions.
problem Existence of linearly independent complex line fields on almost-complex manifolds.
method Prove necessary and sufficient conditions for the existence of one, two, or three fields over certain manifolds.
result Necessary and sufficient condition for the existence of complex line fields over certain manifolds.
Homotopy types of curve and arc complexes are studied.
problem Understanding the homotopy types of curve and arc complexes.
method Proving homotopy equivalence and contractibility of complexes.
result Fine curve complex is homotopy equivalent to curve complex, fine arc complex is contractible.
This research explores complex-valued neural networks and their implementation.
problem The challenges of implementing complex-valued neural networks and their potential for non-complex data.
method Detailed theory and implementation of CVNN, including Wirtinger calculus, complex backpropagation, and modules like complex layers and activation functions. Python implementation using cvnn toolbox.
result Demonstrates the potential of CVNN for non-complex data through simulations.
Paper introduces fat CW complexes including all closed manifolds.
problem No specific problem stated, focuses on introducing new CW complexes.
method Introduces a new smooth version of CW complexes called fat CW complexes.
result Fat CW complexes include all closed manifolds and have desirable properties.
The paper discusses q-deformations of the Aomoto complex.
problem Deformation of cochain complexes associated with hyperplane arrangements.
method Replaces entries of coboundary maps with q-analogues and analyzes the resulting structures. result The q-deformation can be a cochain complex under certain conditions and yields local system cohomology groups. Study calculates global sections on complex curves.
problem Global sections of chiral de Rham complexes on complex curves.
method Calculation on closed complex curves with genus g ≥ 2.
result Space of global sections determined.
The paper studies lifts of complex structures on a manifold.
problem Understanding higher-order lifts of extended almost complex structures.
method Proved theorems on Nijenhuis tensor and introduced a new tensor field.
result Basic results on almost analytic complex vectors are investigated.
In this paper, we first provide an updated survey of the geometry of complex Cartan spaces. New characterizations for some particular classes of complex Cartan spaces are pointed out, e.g. Landsberg-Cartan, strongly Berwald-Cartan and others. We introduce the Cartan-Randers spaces which offer examples of Berwald-Cartan…
Study L2 Hilbert complexes on complex manifolds.
problem Analyse L2 Hilbert complexes on complex manifolds. method Define and study L2 Aeppli-Bott-Chern Hilbert complex; examine properties on various manifolds; use self-adjoint extensions of differential operators. result Kernels of operators on compact Hermitian manifolds are isomorphic to Aeppli or Bott-Chern cohomology.
The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.
problem Existence and uniqueness of almost complex blow-ups on almost complex manifolds.
method Definition and construction of almost complex blow-ups, proving their existence and uniqueness.
result Existence and uniqueness of almost complex blow-ups on 4D almost complex manifolds.