arXiv subjects

Yusu Wang

Publications and source records attributed to Yusu Wang.

At least 91 records · Page 5Linked to original sources

arXiv2017

A Complete Characterization of the 1-Dimensional Intrinsic Cech Persistence Diagrams for Metric Graphs

Metric graphs are special types of metric spaces used to model and represent simple, ubiquitous, geometric relations in data such as biological networks, social networks, and road networks. We are interested in giving a qualitative description of metric graphs using topological summaries. In particular, we provide a complete characterization of the 1-dimensional intrinsic Cech persistence diagrams for metric graphs using persistent homology. Together with complementary results by Adamaszek et. al, which imply results on intrinsic Cech persistence diagrams in all dimensions for a single cycle, our results constitute important steps toward characterizing intrinsic Cech persistence diagrams for arbitrary metric graphs across all dimensions.

math.AT↗

arXiv2017

Unperturbed: spectral analysis beyond Davis-Kahan

Classical matrix perturbation results, such as Weyl's theorem for eigenvalues and the Davis-Kahan theorem for eigenvectors, are general purpose. These classical bounds are tight in the worst case, but in many settings sub-optimal in the typical case. In this paper, we present perturbation bounds which consider the nature of the perturbation and its interaction with the unperturbed structure in order to obtain significant improvements over the classical theory in many scenarios, such as when the perturbation is random. We demonstrate the utility of these new results by analyzing perturbations in the stochastic blockmodel where we derive much tighter bounds than provided by the classical theory. We use our new perturbation theory to show that a very simple and natural clustering algorithm -- whose analysis was difficult using the classical tools -- nevertheless recovers the communities of the blockmodel exactly even in very sparse graphs.

stat.ML↗

arXiv2017

Computing the Gromov-Hausdorff Distance for Metric Trees

The Gromov-Hausdorff (GH) distance is a natural way to measure distance between two metric spaces. We prove that it is $\mathrm{NP}$-hard to approximate the Gromov-Hausdorff distance better than a factor of $3$ for geodesic metrics on a pair of trees. We complement this result by providing a polynomial time $O(\min\{n, \sqrt{rn}\})$-approximation algorithm for computing the GH distance between a pair of metric trees, where $r$ is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an $O(\sqrt{n})$-approximation algorithm.

cs.CG↗

arXiv2017

Graphons, mergeons, and so on!

In this work we develop a theory of hierarchical clustering for graphs. Our modeling assumption is that graphs are sampled from a graphon, which is a powerful and general model for generating graphs and analyzing large networks. Graphons are a far richer class of graph models than stochastic blockmodels, the primary setting for recent progress in the statistical theory of graph clustering. We define what it means for an algorithm to produce the "correct" clustering, give sufficient conditions in which a method is statistically consistent, and provide an explicit algorithm satisfying these properties.

stat.ML↗

arXiv2017

Declutter and Resample: Towards parameter free denoising

In many data analysis applications the following scenario is commonplace: we are given a point set that is supposed to sample a hidden ground truth $K$ in a metric space, but it got corrupted with noise so that some of the data points lie far away from $K$ creating outliers also termed as {\em ambient noise}. One of the main goals of denoising algorithms is to eliminate such noise so that the curated data lie within a bounded Hausdorff distance of $K$. Popular denoising approaches such as deconvolution and thresholding often require the user to set several parameters and/or to choose an appropriate noise model while guaranteeing only asymptotic convergence. Our goal is to lighten this burden as much as possible while ensuring theoretical guarantees in all cases. Specifically, first, we propose a simple denoising algorithm that requires only a single parameter but provides a theoretical guarantee on the quality of the output on general input points. We argue that this single parameter cannot be avoided. We next present a simple algorithm that avoids even this parameter by paying for it with a slight strengthening of the sampling condition on the input points which is not unrealistic. We also provide some preliminary empirical evidence that our algorithms are effective in practice.

cs.CG↗

arXiv2017

Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers

Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called \emph{nerve} of a cover of the domain. In this paper, we aim to analyze the topological information encoded in these structures in order to provide better understanding of these structures and facilitate their practical usage. More specifically, we show that the one-dimensional homology of the nerve complex $N(\mathcal{U})$ of a path-connected cover $\mathcal{U}$ of a domain $X$ cannot be richer than that of the domain $X$ itself. Intuitively, this result means that no new $H_1$-homology class can be "created" under a natural map from $X$ to the nerve complex $N(\mathcal{U})$. Equipping $X$ with a pseudometric $d$, we further refine this result and characterize the classes of $H_1(X)$ that may survive in the nerve complex using the notion of \emph{size} of the covering elements in $\mathcal{U}$. These fundamental results about nerve complexes then lead to an analysis of the $H_1$-homology of Reeb spaces, mappers and multiscale mappers. The analysis of $H_1$-homology groups unfortunately does not extend to higher dimensions. Nevertheless, by using a map-induced metric, establishing a Gromov-Hausdorff convergence result between mappers and the domain, and interleaving relevant modules, we can still analyze the persistent homology groups of (multiscale) mappers to establish a connection to Reeb spaces.

cs.CG↗

arXiv2017

Analyzing and Visualizing Scalar Fields on Graphs

The value proposition of a dataset often resides in the implicit interconnections or explicit relationships (patterns) among individual entities, and is often modeled as a graph. Effective visualization of such graphs can lead to key insights uncovering such value. In this article we propose a visualization method to explore graphs with numerical attributes associated with nodes (or edges) -- referred to as scalar graphs. Such numerical attributes can represent raw content information, similarities, or derived information reflecting important network measures such as triangle density and centrality. The proposed visualization strategy seeks to simultaneously uncover the relationship between attribute values and graph topology, and relies on transforming the network to generate a terrain map. A key objective here is to ensure that the terrain map reveals the overall distribution of components-of-interest (e.g. dense subgraphs, k-cores) and the relationships among them while being sensitive to the attribute values over the graph. We also design extensions that can capture the relationship across multiple numerical attributes (scalars). We demonstrate the efficacy of our method on several real-world data science tasks while scaling to large graphs with millions of nodes.

cs.DB↗

arXiv2016

Measuring Distance between Reeb Graphs

One of the prevailing ideas in geometric and topological data analysis is to provide descriptors that encode useful information about hidden objects from observed data. The Reeb graph is one such descriptor for a given scalar function. The Reeb graph provides a simple yet meaningful abstraction of the input domain, and can also be computed efficiently. Given the popularity of the Reeb graph in applications, it is important to understand its stability and robustness with respect to changes in the input function, as well as to be able to compare the Reeb graphs resulting from different functions. In this paper, we propose a metric for Reeb graphs, called the functional distortion distance. Under this distance measure, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by (and thus more discriminative than) the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions. As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs.

cs.CG↗

arXiv2016

SimBa: An Efficient Tool for Approximating Rips-filtration Persistence via Simplicial Batch-collapse

In topological data analysis, a point cloud data P extracted from a metric space is often analyzed by computing the persistence diagram or barcodes of a sequence of Rips complexes built on $P$ indexed by a scale parameter. Unfortunately, even for input of moderate size, the size of the Rips complex may become prohibitively large as the scale parameter increases. Starting with the Sparse Rips filtration introduced by Sheehy, some existing methods aim to reduce the size of the complex so as to improve the time efficiency as well. However, as we demonstrate, existing approaches still fall short of scaling well, especially for high dimensional data. In this paper, we investigate the advantages and limitations of existing approaches. Based on insights gained from the experiments, we propose an efficient new algorithm, called SimBa, for approximating the persistent homology of Rips filtrations with quality guarantees. Our new algorithm leverages a batch collapse strategy as well as a new sparse Rips-like filtration. We experiment on a variety of low and high dimensional data sets. We show that our strategy presents a significant size reduction, and our algorithm for approximating Rips filtration persistence is order of magnitude faster than existing methods in practice.

cs.CG↗

arXiv2016

Mutiscale Mapper: A Framework for Topological Summarization of Data and Maps

Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. \textsf{Mapper}, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map defined on the data, and produces a summary of the data by using a cover of the codomain of the map. This cover, via a pullback operation to the domain, produces a simplicial complex connecting the data points. The resulting view of the data through a cover of the codomain offers flexibility in analyzing the data. However, it offers only a view at a fixed scale at which the cover is constructed. Inspired by the concept, we explore a notion of a tower of covers which induces a tower of simplicial complexes connected by simplicial maps, which we call {\em multiscale mapper}. We study the resulting structure, its stability, and design practical algorithms to compute its associated persistence diagrams efficiently. Specifically, when the domain is a simplicial complex and the map is a real-valued piecewise-linear function, the algorithm can compute the exact persistence diagram only from the 1-skeleton of the input complex. For general maps, we present a combinatorial version of the algorithm that acts only on \emph{vertex sets} connected by the 1-skeleton graph, and this algorithm approximates the exact persistence diagram thanks to a stability result that we show to hold. We also relate the multiscale mapper with the Čech complexes arising from a natural pullback pseudometric defined on the input domain.

cs.CG↗

arXiv2015

Metric embedding with outliers

We initiate the study of metric embeddings with \emph{outliers}. Given some metric space $(X,ρ)$ we wish to find a small set of outlier points $K \subset X$ and either an isometric or a low-distortion embedding of $(X\setminus K,ρ)$ into some target metric space. This is a natural problem that captures scenarios where a small fraction of points in the input corresponds to noise. For the case of isometric embeddings we derive polynomial-time approximation algorithms for minimizing the number of outliers when the target space is an ultrametric, a tree metric, or constant-dimensional Euclidean space. The approximation factors are 3, 4 and 2, respectively. For the case of embedding into an ultrametric or tree metric, we further improve the running time to $O(n^2)$ for an $n$-point input metric space, which is optimal. We complement these upper bounds by showing that outlier embedding into ultrametrics, trees, and $d$-dimensional Euclidean space for any $d\geq 2$ are all NP-hard, as well as NP-hard to approximate within a factor better than 2 assuming the Unique Game Conjecture. For the case of non-isometries we consider embeddings with small $\ell_{\infty}$ distortion. We present polynomial-time \emph{bi-criteria} approximation algorithms. Specifically, given some $ε> 0$, let $k_ε$ denote the minimum number of outliers required to obtain an embedding with distortion $ε$. For the case of embedding into ultrametrics we obtain a polynomial-time algorithm which computes a set of at most $3k_ε$ outliers and an embedding of the remaining points into an ultrametric with distortion $O(ε\log n)$. For embedding a metric of unit diameter into constant-dimensional Euclidean space we present a polynomial-time algorithm which computes a set of at most $2k_ε$ outliers and an embedding of the remaining points with distortion $O(\sqrtε)$.

cs.DS↗

arXiv2015

Beyond Hartigan Consistency: Merge Distortion Metric for Hierarchical Clustering

Hierarchical clustering is a popular method for analyzing data which associates a tree to a dataset. Hartigan consistency has been used extensively as a framework to analyze such clustering algorithms from a statistical point of view. Still, as we show in the paper, a tree which is Hartigan consistent with a given density can look very different than the correct limit tree. Specifically, Hartigan consistency permits two types of undesirable configurations which we term over-segmentation and improper nesting. Moreover, Hartigan consistency is a limit property and does not directly quantify difference between trees. In this paper we identify two limit properties, separation and minimality, which address both over-segmentation and improper nesting and together imply (but are not implied by) Hartigan consistency. We proceed to introduce a merge distortion metric between hierarchical clusterings and show that convergence in our distance implies both separation and minimality. We also prove that uniform separation and minimality imply convergence in the merge distortion metric. Furthermore, we show that our merge distortion metric is stable under perturbations of the density. Finally, we demonstrate applicability of these concepts by proving convergence results for two clustering algorithms. First, we show convergence (and hence separation and minimality) of the recent robust single linkage algorithm of Chaudhuri and Dasgupta (2010). Second, we provide convergence results on manifolds for topological split tree clustering.

stat.ML↗

arXiv2015

Parameter-free Topology Inference and Sparsification for Data on Manifolds

In topology inference from data, current approaches face two major problems. One concerns the selection of a correct parameter to build an appropriate complex on top of the data points; the other involves with the typical `large' size of this complex. We address these two issues in the context of inferring homology from sample points of a smooth manifold of known dimension sitting in an Euclidean space $\mathbb{R}^k$. We show that, for a sample size of $n$ points, we can identify a set of $O(n^2)$ points (as opposed to $O(n^{\lceil \frac{k}{2}\rceil})$ Voronoi vertices) approximating a subset of the medial axis that suffices to compute a distance sandwiched between the well known local feature size and the local weak feature size (in fact, the approximating set can be further reduced in size to $O(n)$). This distance, called the lean feature size, helps pruning the input set at least to the level of local feature size while making the data locally uniform. The local uniformity in turn helps in building a complex for homology inference on top of the sparsified data without requiring any user-supplied distance threshold. Unlike most topology inference results, ours does not require that the input is dense relative to a {\em global} feature such as {\em reach} or {\em weak feature size}; instead it can be adaptive with respect to the local feature size. We present some empirical evidence in support of our theoretical claims.

cs.CG↗

arXiv2015

Fréchet Distance for Curves, Revisited

$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\eps}{\varepsilon} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)}$ We revisit the problem of computing Fréchet distance between polygonal curves under $L_1$, $L_2$, and $L_\infty$ norms, focusing on discrete Fréchet distance, where only distance between vertices is considered. We develop efficient algorithms for two natural classes of curves. In particular, given two polygonal curves of $n$ vertices each, a $\eps$-approximation of their discrete Fréchet distance can be computed in roughly $O(nκ^3\log n/\eps^3)$ time in three dimensions, if one of the curves is \emph{$κ$-bounded}. Previously, only a $κ$-approximation algorithm was known. If both curves are the so-called \emph{\backbone~curves}, which are widely used to model protein backbones in molecular biology, we can $\eps$-approximate their Fréchet distance in near linear time in two dimensions, and in roughly $O(n^{4/3}\log nm)$ time in three dimensions. In the second part, we propose a pseudo--output-sensitive algorithm for computing Fréchet distance exactly. The complexity of the algorithm is a function of a quantity we call the \emph{\bwnumber{}}, which is quadratic in the worst case, but tends to be much smaller in practice.

cs.CG↗

arXiv2015

Topological analysis of scalar fields with outliers

Given a real-valued function $f$ defined over a manifold $M$ embedded in $\mathbb{R}^d$, we are interested in recovering structural information about $f$ from the sole information of its values on a finite sample $P$. Existing methods provide approximation to the persistence diagram of $f$ when geometric noise and functional noise are bounded. However, they fail in the presence of aberrant values, also called outliers, both in theory and practice. We propose a new algorithm that deals with outliers. We handle aberrant functional values with a method inspired from the k-nearest neighbors regression and the local median filtering, while the geometric outliers are handled using the distance to a measure. Combined with topological results on nested filtrations, our algorithm performs robust topological analysis of scalar fields in a wider range of noise models than handled by current methods. We provide theoretical guarantees and experimental results on the quality of our approximation of the sampled scalar field.

cs.CG↗

arXiv2014

Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs

The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has widely been used in applications, however its use on real data means that it is desirable and increasingly necessary to have methods for comparison of Reeb graphs. Recently, several methods to define metrics on the space of Reeb graphs have been presented. In this paper, we focus on two: the functional distortion distance and the interleaving distance. The former is based on the Gromov--Hausdorff distance, while the latter utilizes the equivalence between Reeb graphs and a particular class of cosheaves. However, both are defined by constructing a near-isomorphism between the two graphs of study. In this paper, we show that the two metrics are strongly equivalent on the space of Reeb graphs. In particular, this gives an immediate proof of bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.

math.AT↗

arXiv2014

Dimension Detection with Local Homology

Detecting the dimension of a hidden manifold from a point sample has become an important problem in the current data-driven era. Indeed, estimating the shape dimension is often the first step in studying the processes or phenomena associated to the data. Among the many dimension detection algorithms proposed in various fields, a few can provide theoretical guarantee on the correctness of the estimated dimension. However, the correctness usually requires certain regularity of the input: the input points are either uniformly randomly sampled in a statistical setting, or they form the so-called $(\varepsilon,δ)$-sample which can be neither too dense nor too sparse. Here, we propose a purely topological technique to detect dimensions. Our algorithm is provably correct and works under a more relaxed sampling condition: we do not require uniformity, and we also allow Hausdorff noise. Our approach detects dimension by determining local homology. The computation of this topological structure is much less sensitive to the local distribution of points, which leads to the relaxation of the sampling conditions. Furthermore, by leveraging various developments in computational topology, we show that this local homology at a point $z$ can be computed \emph{exactly} for manifolds using Vietoris-Rips complexes whose vertices are confined within a local neighborhood of $z$. We implement our algorithm and demonstrate the accuracy and robustness of our method using both synthetic and real data sets.

cs.CG↗

arXiv2014

Computing Topological Persistence for Simplicial Maps

Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under $\mathbb{Z}_2$ coefficients for a sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis. First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provides an algorithm for computing the persistence induced by simplicial maps. Our main result is that the above simple minded approach can be improved for a sequence of simplicial maps given in a monotone direction. A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses--two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally. Finally, we exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.

cs.CG↗