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15
Barcodes: The persistent topology of data
, 2007
"... Abstract. This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in highdimensional data. The primary mathematical tool considered is a homology theory for pointcloud data sets—persis ..."
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Cited by 43 (2 self)
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Abstract. This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in highdimensional data. The primary mathematical tool considered is a homology theory for pointcloud data sets—persistent homology—and a novel representation of this algebraic characterization— barcodes. We sketch an application of these techniques to the classification of natural images. 1. The shape of data When a topologist is asked, “How do you visualize a fourdimensional object?” the appropriate response is a Socratic rejoinder: “How do you visualize a threedimensional object? ” We do not see in three spatial dimensions directly, but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended. We spend a significant portion of our first year of life learning how to infer threedimensional spatial data from paired planar projections. Years of practice have tuned a remarkable ability to extract global structure from representations
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the spec ..."
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Cited by 20 (7 self)
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We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.
QUANTIFYING HOMOLOGY CLASSES
, 2008
"... We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements’ size have the minimal sum. We provide a greedy ..."
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Cited by 15 (4 self)
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We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements’ size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in O(β^4 n³ log² n) time, where n is the size of the simplicial complex and β is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results.
Quantifying homology classes II: Localization and stability
, 2007
"... Abstract. In the companion paper [7], we measured homology classes and computed the optimal homology basis. This paper addresses two related problems, namely, localization and stability. We localize a class with the cycle minimizing a certain objective function. We explore three different objective ..."
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Cited by 11 (3 self)
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Abstract. In the companion paper [7], we measured homology classes and computed the optimal homology basis. This paper addresses two related problems, namely, localization and stability. We localize a class with the cycle minimizing a certain objective function. We explore three different objective functions, namely, volume, diameter and radius. We show that it is NPhard to compute the smallest cycle using the former two. We also prove that the measurement defined in [7] is stable with regard to small changes of the geometry of the concerned space. 1.
Hardness results for homology localization
 In SODA ’10: Proc. 21st Ann. ACMSIAM Sympos. Discrete Algorithms (2010
"... We address the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure. We focus on the volume measure, that is, the 1norm of a cycle. Two main results are presented. First, we prove the problem is NPhard to approximate w ..."
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Cited by 10 (1 self)
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We address the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure. We focus on the volume measure, that is, the 1norm of a cycle. Two main results are presented. First, we prove the problem is NPhard to approximate within any constant factor. Second, we prove that for homology of dimension two or higher, the problem is NPhard to approximate even when the Betti number is O(1). A side effect is the inapproximability of the problem of computing the nonbounding cycle with the smallest volume, and computing cycles representing a homology basis with the minimal total volume. We also discuss other geometric measures (diameter and radius) and show their disadvantages in homology localization. Our work is restricted to homology over the Z2 field. 1
Minimum Cuts and Shortest NonSeparating Cycles via Homology Covers
 SYMPOSIUM ON DISCRETE ALGORITHMS
, 2011
"... Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two ap ..."
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Cited by 10 (4 self)
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Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two applications of our algorithm. The first is an algorithm to compute the shortest nonseparating directed cycle in G in 2 O(g) n log n time, improving the recent algorithm of Cabello et al. [SOCG 2010] for all g = o(log n). The second is a combinatorial algorithm to compute minimum (s, t)cuts in undirected surface graphs in 2 O(g) n log n time, improving an algorithm of Chambers et al. [SOCG 2009] for all positive g. Unlike earlier algorithms for surface graphs that construct and search finite portions of the universal cover, our algorithms use another canonical covering space, called the Z 2homology cover.
Approximating loops in a shortest homology basis from point data
 arXiv:0909.5654v2[cs.CG] (2009), Online. URL http://arxiv.org/abs/0909.5654
"... Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of loops from a point data that presumably sample a smooth manif ..."
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Cited by 8 (4 self)
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Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of loops from a point data that presumably sample a smooth manifold M ⊂ R d. These loops approximate a shortest basis of the one dimensional homology group H1(M) over coefficients in finite field Z2. Previous results addressed the issue of computing the rank of the homology groups from point data, but there is no result on approximating the shortest basis of a manifold from its point sample. In arriving our result, we also present a polynomial time algorithm for computing a shortest basis of H1(K) for any finite simplicial complex K whose edges have nonnegative weights.
The tidy set: A minimal simplicial set for computing homology of clique complexes
 In Proc. ACM Symposium of Computational Geometry
, 2010
"... We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the VietorisRips complex and the weak witness complex, methods that are pop ..."
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Cited by 7 (1 self)
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We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the VietorisRips complex and the weak witness complex, methods that are popular in topological data analysis. The key feature of our approach is that it skips constructing the clique complex. We give algorithms for constructing tidy sets, implement them, and present experiments. Our preliminary results show that tidy sets are orders of magnitude smaller than clique complexes, giving us a homology engine with small memory requirements.
Measuring and localizing homology classes
 The Computing Research Repository (CoRR
, 2007
"... We develop a method for measuring and localizing homology classes. This involves two problems. First, we define relevant notions of size for both a homology class and a homology group basis, using ideas from relative homology. Second, we propose an algorithm to compute the optimal homology basis, us ..."
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Cited by 4 (0 self)
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We develop a method for measuring and localizing homology classes. This involves two problems. First, we define relevant notions of size for both a homology class and a homology group basis, using ideas from relative homology. Second, we propose an algorithm to compute the optimal homology basis, using techniques from persistent homology and finite field algebra. Classes of the computed optimal basis are localized with cycles conveying their sizes. The algorithm runs in O(β 4 n 3 log 2 n) time, where n is the size of the simplicial complex and β is the Betti number of the homology group. 1
Computational topology
 Algorithms and Theory of Computation Handbook
, 2010
"... According to the Oxford English Dictionary, the word topology is derived of topos ( � ) meaning place, andlogy ( ���), a variant of the verb ´��� � , meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in t ..."
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Cited by 3 (2 self)
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According to the Oxford English Dictionary, the word topology is derived of topos ( � ) meaning place, andlogy ( ���), a variant of the verb ´��� � , meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in turn, undertakes the challenge of studying topology using a computer.