arXiv subjects

Tamal K. Dey

Publications and source records attributed to Tamal K. Dey.

At least 55 records · Page 3Linked to original sources

arXiv2021

Rectangular Approximation and Stability of $2$-parameter Persistence Modules

One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of $1$-parameter persistence modules are stable in the sense that the bottleneck distance between two diagrams equals the interleaving distance between their generating modules. However, in multi-parameter setting this property breaks down in general. A simple special case of persistence modules called rectangle decomposable modules is known to admit a weaker stability property. Using this fact, we derive a stability-like property for $2$-parameter persistence modules. For this, first we consider interval decomposable modules and their optimal approximations with rectangle decomposable modules with respect to the bottleneck distance. We provide a polynomial time algorithm to exactly compute this optimal approximation which, together with the polynomial-time computable bottleneck distance among interval decomposable modules, provides a lower bound on the interleaving distance. Next, we leverage this result to derive a polynomial-time computable distance for general multi-parameter persistence modules which enjoys similar stability-like property. This distance can be viewed as a generalization of the matching distance defined in the literature.

cs.CG↗

arXiv2021

Persistence of Conley-Morse Graphs in Combinatorial Dynamical Systems

Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invariant set. Likewise, considering a Morse decomposition omits much information about the individual Morse sets. In this paper, we propose a method to summarize changes in combinatorial dynamical systems by capturing changes in the so-called Conley-Morse graphs. A Conley-Morse graph contains information about both the structure of a selected Morse decomposition and about the Conley index at each Morse set in the decomposition. Hence, our method summarizes the changing structure of a sequence of dynamical systems at a finer granularity than previous approaches.

math.DS↗

arXiv2021

2D Points Curve Reconstruction Survey and Benchmark

Curve reconstruction from unstructured points in a plane is a fundamental problem with many applications that has generated research interest for decades. Involved aspects like handling open, sharp, multiple and non-manifold outlines, run-time and provability as well as potential extension to 3D for surface reconstruction have led to many different algorithms. We survey the literature on 2D curve reconstruction and then present an open-sourced benchmark for the experimental study. Our unprecedented evaluation on a selected set of planar curve reconstruction algorithms aims to give an overview of both quantitative analysis and qualitative aspects for helping users to select the right algorithm for specific problems in the field. Our benchmark framework is available online to permit reproducing the results, and easy integration of new algorithms.

cs.GR↗

arXiv2021

Computing Zigzag Persistence on Graphs in Near-Linear Time

Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one instrument for analyzing such changing graph data. However, standard persistent homology defined over a growing space cannot always capture such a dynamic process unless shrinking with deletions is also allowed. Hence, zigzag persistence which incorporates both insertions and deletions of simplices is more appropriate in such a setting. Unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general $O(m^ω)$ time complexity are not known, where $ω< 2.37286$ is the matrix multiplication exponent. In this paper, we propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration with $m$ additions and deletions on a graph with $n$ vertices and edges, the algorithm for $0$-dimension runs in $O(m\log^2 n+m\log m)$ time and the algorithm for 1-dimension runs in $O(m\log^4 n)$ time. The algorithm for $0$-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on $2$-manifolds. The algorithm for $1$-dimension pairs a negative edge with the earliest positive edge so that a $1$-cycle containing both edges resides in all intermediate graphs. Both algorithms achieve the claimed time complexity via dynamic graph data structures proposed by Holm et al. In the end, using Alexander duality, we extend the algorithm for $0$-dimension to compute the $(p-1)$-dimensional zigzag persistence for $\mathbb{R}^p$-embedded complexes in $O(m\log^2 n+m\log m+n\log n)$ time.

cs.CG↗

arXiv2020

Persistence of the Conley Index in Combinatorial Dynamical Systems

A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector fields have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to track features in a changing multivector field.

math.AT↗

arXiv2020

Computing Minimal Persistent Cycles: Polynomial and Hard Cases

Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in a purely topological persistence diagram (also termed as barcode). In our earlier work, we showed that computing minimal 1-dimensional persistent cycles (persistent 1-cycles) for finite intervals is NP-hard while the same for infinite intervals is polynomially tractable. In this paper, we address this problem for general dimensions with $\mathbb{Z}_2$ coefficients. In addition to proving that it is NP-hard to compute minimal persistent d-cycles (d>1) for both types of intervals given arbitrary simplicial complexes, we identify two interesting cases which are polynomially tractable. These two cases assume the complex to be a certain generalization of manifolds which we term as weak pseudomanifolds. For finite intervals from the d-th persistence diagram of a weak (d+1)-pseudomanifold, we utilize the fact that persistent cycles of such intervals are null-homologous and reduce the problem to a minimal cut problem. Since the same problem for infinite intervals is NP-hard, we further assume the weak (d+1)-pseudomanifold to be embedded in $\mathbb{R}^{d+1}$ so that the complex has a natural dual graph structure and the problem reduces to a minimal cut problem. Experiments with both algorithms on scientific data indicate that the minimal persistent cycles capture various significant features of the data.

cs.CG↗

arXiv2020

An efficient algorithm for $1$-dimensional (persistent) path homology

This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their persistence. These concepts are well suited for spaces that are not directed. As a result, one needs a concept of homology that accommodates orientations in input space. Path-homology developed for directed graphs by Grigor'yan, Lin, Muranov and Yau has been effectively adapted for this purpose recently by Chowdhury and Mémoli. They also give an algorithm to compute this path-homology. Our main contribution in this paper is an algorithm that computes this path-homology and its persistence more efficiently for the $1$-dimensional ($H_1$) case. In developing such an algorithm, we discover various structures and their efficient computations that aid computing the $1$-dimensional path-homnology. We implement our algorithm and present some preliminary experimental results.

cs.CG↗

arXiv2020

The Compressed Annotation Matrix: an Efficient Data Structure for Computing Persistent Cohomology

The persistent homology with coefficients in a field F coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors with the simplices. We separate the representation of the simplicial complex from the representation of the cohomology groups, and introduce a new data structure for maintaining the annotation matrix, which is more compact and reduces substancially the amount of matrix operations. In addition, we propose heuristics to simplify further the representation of the cohomology groups and improve both time and space complexities. The paper provides a theoretical analysis, as well as a detailed experimental study of our implementation and comparison with state-of-the-art software for persistent homology and cohomology.

cs.CG↗

arXiv2019

Computing Bottleneck Distance for Multi-parameter Interval Decomposable Persistence Modules

Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For $1$-parameter persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck distance with known efficient algorithms. The question is open for most $n$-parameter persistence modules, $n>1$, because of the well recognized complications of the indecomposables. Here, we consider a reasonably complicated class called {\em $n$-parameter interval decomposable} modules whose indecomposables may have a description of non-constant complexity. We present a polynomial time algorithm to compute the bottleneck distance for these modules from indecomposables, which bounds the interleaving distance from above, and give another algorithm to compute a new distance called {\em dimension distance} that bounds it from below. An earlier version of this paper considered only the $2$-parameter interval decomposable modules~\cite{DeyCheng18}.

cs.CG↗

arXiv2019

Road Network Reconstruction from Satellite Images with Machine Learning Supported by Topological Methods

Automatic Extraction of road network from satellite images is a goal that can benefit and even enable new technologies. Methods that combine machine learning (ML) and computer vision have been proposed in recent years which make the task semi-automatic by requiring the user to provide curated training samples. The process can be fully automatized if training samples can be produced algorithmically. Of course, this requires a robust algorithm that can reconstruct the road networks from satellite images reliably so that the output can be fed as training samples. In this work, we develop such a technique by infusing a persistence-guided discrete Morse based graph reconstruction algorithm into ML framework. We elucidate our contributions in two phases. First, in a semi-automatic framework, we combine a discrete-Morse based graph reconstruction algorithm with an existing CNN framework to segment input satellite images. We show that this leads to reconstructions with better connectivity and less noise. Next, in a fully automatic framework, we leverage the power of the discrete-Morse based graph reconstruction algorithm to train a CNN from a collection of images without labelled data and use the same algorithm to produce the final output from the segmented images created by the trained CNN. We apply the discrete-Morse based graph reconstruction algorithm iteratively to improve the accuracy of the CNN. We show promising experimental results of this new framework on datasets from SpaceNet Challenge.

cs.CV↗

arXiv2019

Computing Height Persistence and Homology Generators in $\mathbb{R}^3$ Efficiently

Recently it has been shown that computing the dimension of the first homology group $H_1(K)$ of a simplicial $2$-complex $K$ embedded linearly in $\mathbb{R}^4$ is as hard as computing the rank of a sparse $0-1$ matrix. This puts a major roadblock to computing persistence and a homology basis (generators) for complexes embedded in $\mathbb{R}^4$ and beyond in less than quadratic or even near-quadratic time. But, what about dimension three? It is known that persistence for piecewise linear functions on a complex $K$ with $n$ simplices can be computed in $O(n\log n)$ time and a set of generators of total size $k$ can be computed in $O(n+k)$ time when $K$ is a graph or a surface linearly embedded in $\mathbb{R}^3$. But, the question for general simplicial complexes $K$ linearly embedded in $\mathbb{R}^3$ is not completely settled. No algorithm with a complexity better than that of the matrix multiplication is known for this important case. We show that the persistence for {\em height functions} on such complexes, hence called {\em height persistence}, can be computed in $O(n\log n)$ time. This allows us to compute a basis (generators) of $H_i(K)$, $i=1,2$, in $O(n\log n+k)$ time where $k$ is the size of the output. This improves significantly the current best bound of $O(n^ω)$, $ω$ being the matrix multiplication exponent. We achieve these improved bounds by leveraging recent results on zigzag persistence in computational topology, new observations about Reeb graphs, and some efficient geometric data structures.

cs.CG↗

arXiv2018

Persistent 1-Cycles: Definition, Computation, and Its Application

Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of intervals included in a persistence diagram, some applications need to find representative cycles for the intervals. In this paper, we address the problem of computing these representative cycles, termed as persistent 1-cycles, for $\text{H}_1$-persistent homology with $\mathbb{Z}_2$ coefficients. The definition of persistent cycles is based on the interval module decomposition of persistence modules, which reveals the structure of persistent homology. After showing that the computation of the optimal persistent 1-cycles is NP-hard, we propose an alternative set of meaningful persistent 1-cycles that can be computed with an efficient polynomial time algorithm. We also inspect the stability issues of the optimal persistent 1-cycles and the persistent 1-cycles computed by our algorithm with the observation that the perturbations of both cannot be properly bounded. We design a software which applies our algorithm to various datasets. Experiments on 3D point clouds, mineral structures, and images show the effectiveness of our algorithm in practice.

cs.CG↗

arXiv2018

Filtration Simplification for Persistent Homology via Edge Contraction

Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex $K$. These topological changes are summarized in persistence diagrams. We propose two contraction operators which when applied to $K$ and its associated filtration, bound the perturbation in the persistence diagrams. The first assumes that the underlying space of $K$ is a $2$-manifold and ensures that simplices are paired with the same simplices in the contracted complex as they are in the original. The second is for arbitrary $d$-complexes, and bounds the bottleneck distance between the initial and contracted $p$-dimensional persistence diagrams. This is accomplished by defining interleaving maps between persistence modules which arise from chain maps defined over the filtrations. In addition, we show how the second operator can efficiently compose across multiple contractions. We conclude with experiments demonstrating the second operator's utility on manifolds.

cs.CG↗

arXiv2018

Spectral concentration and greedy k-clustering

A popular graph clustering method is to consider the embedding of an input graph into R^k induced by the first k eigenvectors of its Laplacian, and to partition the graph via geometric manipulations on the resulting metric space. Despite the practical success of this methodology, there is limited understanding of several heuristics that follow this framework. We provide theoretical justification for one such natural and computationally efficient variant. Our result can be summarized as follows. A partition of a graph is called strong if each cluster has small external conductance, and large internal conductance. We present a simple greedy spectral clustering algorithm which returns a partition that is provably close to a suitably strong partition, provided that such a partition exists. A recent result shows that strong partitions exist for graphs with a sufficiently large spectral gap between the k-th and (k+1)-st eigenvalues. Taking this together with our main theorem gives a spectral algorithm which finds a partition close to a strong one for graphs with large enough spectral gap. We also show how this simple greedy algorithm can be implemented in near-linear time for any fixed k and error guarantee. Finally, we evaluate our algorithm on some real-world and synthetic inputs.

cs.DS↗

arXiv2018

Persistent Homology of Morse Decompositions in Combinatorial Dynamics

We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics directly from the sample. We study the homological persistence of {\em Morse decompositions} of such systems, an important descriptor of the dynamics, as a tool for validating the reconstruction. Our framework can be viewed as a step toward extending the classical persistence theory to "vector cloud" data. We present experimental results on two numerical examples.

math.AT↗

arXiv2018

Graph Reconstruction by Discrete Morse Theory

Recovering hidden graph-like structures from potentially noisy data is a fundamental task in modern data analysis. Recently, a persistence-guided discrete Morse-based framework to extract a geometric graph from low-dimensional data has become popular. However, to date, there is very limited theoretical understanding of this framework in terms of graph reconstruction. This paper makes a first step towards closing this gap. Specifically, first, leveraging existing theoretical understanding of persistence-guided discrete Morse cancellation, we provide a simplified version of the existing discrete Morse-based graph reconstruction algorithm. We then introduce a simple and natural noise model and show that the aforementioned framework can correctly reconstruct a graph under this noise model, in the sense that it has the same loop structure as the hidden ground-truth graph, and is also geometrically close. We also provide some experimental results for our simplified graph-reconstruction algorithm.

cs.CG↗

arXiv2018

Efficient algorithms for computing a minimal homology basis

Efficient computation of shortest cycles which form a homology basis under $\mathbb{Z}_2$-additions in a given simplicial complex $\mathcal{K}$ has been researched actively in recent years. When the complex $\mathcal{K}$ is a weighted graph with $n$ vertices and $m$ edges, the problem of computing a shortest (homology) cycle basis is known to be solvable in $O(m^2n/\log n+ n^2m)$-time. Several works \cite{borradaile2017minimum, greedy} have addressed the case when the complex $\mathcal{K}$ is a $2$-manifold. The complexity of these algorithms depends on the rank $g$ of the one-dimensional homology group of $\mathcal{K}$. This rank $g$ has a lower bound of $Θ(n)$, where $n$ denotes the number of simplices in $\mathcal{K}$, giving an $O(n^4)$ worst-case time complexity for the algorithms in \cite{borradaile2017minimum,greedy}. This worst-case complexity is improved in \cite{annotation} to $O(n^ω+ n^2g^{ω-1})$ for general simplicial complexes where $ω< 2.3728639$ \cite{le2014powers} is the matrix multiplication exponent. Taking $g=Θ(n)$, this provides an $O(n^{ω+1})$ worst-case algorithm. In this paper, we improve this time complexity. Combining the divide and conquer technique from \cite{DivideConquer} with the use of annotations from \cite{annotation}, we present an algorithm that runs in $O(n^ω+n^2g)$ time giving the first $O(n^3)$ worst-case algorithm for general complexes. If instead of minimal basis, we settle for an approximate basis, we can improve the running time even further. We show that a $2$-approximate minimal homology basis can be computed in $O(n^ω\sqrt{n \log n})$ expected time. We also study more general measures for defining the minimal basis and identify reasonable conditions on these measures that allow computing a minimal basis efficiently.

math.AT↗

arXiv2017

Comparing Graphs via Persistence Distortion

Metric graphs are ubiquitous in science and engineering. For example, many data are drawn from hidden spaces that are graph-like, such as the cosmic web. A metric graph offers one of the simplest yet still meaningful ways to represent the non-linear structure hidden behind the data. In this paper, we propose a new distance between two finite metric graphs, called the persistence-distortion distance, which draws upon a topological idea. This topological perspective along with the metric space viewpoint provide a new angle to the graph matching problem. Our persistence-distortion distance has two properties not shared by previous methods: First, it is stable against the perturbations of the input graph metrics. Second, it is a continuous distance measure, in the sense that it is defined on an alignment of the underlying spaces of input graphs, instead of merely their nodes. This makes our persistence-distortion distance robust against, for example, different discretizations of the same underlying graph. Despite considering the input graphs as continuous spaces, that is, taking all points into account, we show that we can compute the persistence-distortion distance in polynomial time. The time complexity for the discrete case where only graph nodes are considered is much faster. We also provide some preliminary experimental results to demonstrate the use of the new distance measure.

cs.CG↗