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.
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. The Rips complex at scale r is homotopy equivalent to the nerve of a cover of diameter r.
problem Reconstructing spaces using Rips complexes and covers.
method Functorial Dowker-Nerve Diagram, homotopy equivalence, cover of diameter r.
result General framework for reconstructing spaces by Rips complexes.
Fix a finite set of points in Euclidean n-space $\euc^n$, thought of as a point-cloud sampling of a certain domain $D\subset\euc^n$. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. …
Morse theory on complexes for CAT(0) groups.
problem Finding universal spaces for proper actions of CAT(0) groups.
method Equivariant discrete Morse theory on Vietoris-Rips complexes.
result Exhibited finite universal spaces for proper actions of all asymptotically CAT(0) groups.
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. Study of metric spaces and group actions using Vietoris-Rips and Čech complexes.
problem Understanding the homotopy type of quotient spaces under group actions.
method Intermediate scale parameters for Vietoris-Rips and Čech complexes.
result First scale parameter where homotopy type of projective spaces changes.
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.
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.
New Morse theory applied to Vietoris-Rips complexes for topological data analysis and geometric group theory.
problem Understanding homotopy types of Vietoris-Rips complexes for metric spaces.
method Generalization of Bestvina-Brady discrete Morse theory applied to Vietoris-Rips complexes.
result Metric criteria (Morse and Link) to deduce homotopy types of VRt(X). 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.
This paper investigates the average-case time complexity of certifying RIP matrices.
problem Certifying the restricted isometry property (RIP) for large sparsity levels in random Gaussian matrices.
method Analysis of the low-degree likelihood ratio to determine the average-case time complexity.
result Subexponential runtime of NildeΩ(s2/M) is required for certifying RIP matrices. Given a set of points that sample a shape, the Rips complex of the data points is often used in machine-learning to provide an approximation of the shape easily-computed. It has been proved recently that the Rips complex captures the homotopy type of the shape assuming the vertices of the complex meet some mild samplin…
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.
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.
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.
Sign-RIP improves robust low-rank matrix recovery by preserving norms even with corrupted measurements.
problem Robust low-rank matrix recovery in the presence of corrupted measurements.
method Proposed Sign-RIP, a robust restricted isometry property.
result Sign-RIP guarantees uniform convergence of subdifferentials in robust low-rank matrix recovery.
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.
New group constructed from cube complex properties.
problem Creating a new group from cube complex properties.
method Cubical Rips construction for finitely presented groups.
result New group surjects onto a given group with specific properties.
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.
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.
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.
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.
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. 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.
We analyze decision boundaries using topological data analysis.
problem Quantifying deep neural network complexity for model selection.
method We use labeled Čech complex, plain labeled Vietoris-Rips complex, and locally scaled labeled Vietoris-Rips complex to infer persistent homology of decision boundaries.
result We provide theoretical conditions and analysis for recovering the homology of a decision boundary from samples.
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.
Metric thickenings help recover manifold homology from samples.
problem Recovering manifold homology from finite samples embedded in Euclidean space.
method Introducing metric thickenings of Vietoris--Rips and Čech complexes.
result Metric thickenings are homotopy equivalent to the manifold for scale parameters less than the reach.
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.
In "Rips complexes and covers in the uniform category" \cite{Rips} the authors define, following James \cite{J}, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given…
The restricted isometry property (RIP) for design matrices gives guarantees for optimal recovery in sparse linear models. It is of high interest in compressed sensing and statistical learning. This property is particularly important for computationally efficient recovery methods. As a consequence, even though it is in …
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…
Develops a new framework for large-scale geometry.
problem Characterizing large-scale models of metric spaces.
method Categorical framework for metric Rips filtration and universal quasigeodesic cones.
result Establishes universal properties and adjointness of the Rips colimit.
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.
Paper analyzes noisy low-rank matrix optimization, improving RIP bounds and convergence rates.
problem Noisy low-rank matrix optimization with general objective functions.
method Develops new mathematical framework and proves convergence rate under RIP condition.
result Any spurious local solution is close to ground truth when RIP constant is less than 1/3.
Improves data recovery with optimized measurements and generalized sparsity models.
problem Data recovery with optimized measurements and generalized sparsity models.
method Optimizing over families of Banach spaces, investigating preservation of difference of sparse vectors, extending RIP to group structured measurements, and extending Fourier measurement concepts to infinite dimensions.
result Optimal scaling of number of measurements for group structured measurements and improved RIP in infinite dimensions.
Paper shows moderate RIP is insufficient for avoiding spurious local minima in matrix recovery.
problem The need for moderate RIP to avoid spurious local minima in matrix recovery.
method Analyzes the necessity of RIP constants and provides counterexamples.
result Counterexamples show spurious local minima exist even with moderate RIP.
Paper proves sufficient conditions for tensor recovery using t-RIP with random measurements.
problem Establish robust recovery guarantees for low-tubal-rank tensors.
method Probabilistic arguments and random sub-Gaussian distributions to ensure t-RIP conditions.
result Minimal number of linear measurements nearly optimal for tensor recovery.
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.
New matrices satisfy RIP with correlated entries for various applications.
problem Constructing RIP matrices with dependent entries.
method Introduced a new ensemble of random matrices XR where X is a fixed matrix and R is a random matrix from various models. result The constructed matrices XR satisfy the RIP with high probability. We study isometric actions of finitely presented groups on R-trees. In this paper, we develop a relative version of the Rips machine to study pairs of such actions. An important example of a pair is a group action on an R-tree and a subgroup action on its minimal invariant su…
Recently, Rips produced an example of a double of two free groups which has unsolvable generalized word problem. In this paper, we show that Rips's example fits into a large class of doubles of groups, each member of which contains F_2 x F_2 and therefore has unsolvable generalized word problem and is incoherent.
The paper analyzes conditions for solving low-rank matrix recovery problems with noisy measurements.
problem Low-rank matrix recovery with corrupted measurements.
method Analysis of the restricted isometry property (RIP) and local search methods.
result Sharp bounds on the maximum distance between local minimizers and the ground truth.
Two groups with same profinite completion have different co-Hopfian properties.
problem Understanding co-Hopfian properties in residually finite groups.
method Using a specific construction involving a finitely presented acyclic group with trivial profinite completion.
result Found two groups with same profinite completion but different co-Hopfian properties.
Unified framework for generalized sparsity and RIP analysis.
problem Analyzing inverse problems with sparsity models.
method Proposed generalized notions of sparsity and a unified RIP framework.
result Extends RIP analysis to broader contexts including tensor products.
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.