arXiv research

This paper classifies planar-Rips complexes and their unit disk graphs up to homotopy.

Homotopy types of Vietoris-Rips metric thickenings of the circle confirmed.

Contractible Vietoris-Rips complexes for integer n proved using discrete Morse theory.

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 $D$. …

New construction reduces Vietoris-Rips complex construction time.

Generalizes Rips' result on hyperbolic spaces to metric spaces, showing collapses for tree metrics.

This paper interprets critical scales in persistent homology for compact metric spaces.

MuRiT efficiently computes multi-parameter persistence barcodes.

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 $P_n$ 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.

Using ideas of the Dowker duality we prove that the Rips complex at scale $r$ is homotopy equivalent to the nerve of a cover consisting of sets of prescribed diameter. We then develop a functorial version of the Nerve theorem coupled with the Dowker duality, which is presented as a Functorial Dowker-Nerve Diagram. Thes…

This paper investigates the average-case time complexity of certifying RIP matrices.

Paper defines and evaluates DR complex for persistent homology.

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…

We inspect Vietoris-Rips complexes $VR_t(X)$ 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.

The paper connects geometric and topological concepts to bound distances between metric spaces.

Sign-RIP improves robust low-rank matrix recovery by preserving norms even with corrupted measurements.

Researchers decompose Forman-Ricci curvature for efficient computation in VR complexes.

New group constructed from cube complex properties.

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…

Study topological invariants of complexes for Riemannian manifolds.

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…

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.

The study of shadow of Vietoris-Rips complexes and their homotopy properties.

New TDA approach using Finsler metrics.

Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.

We prove 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 …

A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first homology. We call the resulting object persistence. Recent results demonstrate that per…

Develops a new framework for large-scale geometry.

Given a sample of points $X$ in a metric space $M$ and a scale $r>0$, the Vietoris-Rips simplicial complex $\mathrm{VR}(X;r)$ is a standard construction to attempt to recover $M$ from $X$ up to homotopy type. A deficiency of this approach is that $\mathrm{VR}(X;r)$ is not metrizable if it is not locally finite, and thu…

The restricted isometry property (RIP) is a universal tool for data recovery. We explore the implication of the RIP in the framework of generalized sparsity and group measurements introduced in the Part I paper. It turns out that for a given measurement instrument the number of measurements for RIP can be improved by o…

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…

Paper analyzes noisy low-rank matrix optimization, improving RIP bounds and convergence rates.

When the linear measurements of an instance of low-rank matrix recovery satisfy a restricted isometry property (RIP)---i.e. they are approximately norm-preserving---the problem is known to contain no spurious local minima, so exact recovery is guaranteed. In this paper, we show that moderate RIP is not enough to elimin…

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…

Unified pipeline classifies time series using complex networks and persistent homology.

We study isometric actions of finitely presented groups on $\mathbb{R}$-trees. In this paper, we develop a relative version of the Rips machine to study $\textit{pairs}$ of such actions. An important example of a $\textit{pair}$ is a group action on an $\mathbb{R}$-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.

Two groups with same profinite completion have different co-Hopfian properties.

Matrices satisfying the Restricted Isometry Property (RIP) play an important role in the areas of compressed sensing and statistical learning. RIP matrices with optimal parameters are mainly obtained via probabilistic arguments, as explicit constructions seem hard. It is therefore interesting to ask whether a fixed mat…

Study semi-coarse spaces' homotopy and homology, extending coarse geometry.

Nonnegative low-rank matrix recovery can have spurious local minima.

In the present paper we construct a Z^{3}-periodic surface in R^{3} whose almost all plane sections of a certain direction consist of exactly one connected component. This question originates from a problem of Novikov on the semi- classical motion of an electron in strong magnetic field. Our main tool is the Rips machi…