## Homological Illusions of Persistence and Stability (2008)

Citations: | 12 - 3 self |

### BibTeX

@TECHREPORT{Morozov08homologicalillusions,

author = {Dmitriy Morozov},

title = {Homological Illusions of Persistence and Stability},

institution = {},

year = {2008}

}

### OpenURL

### Abstract

In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of points in the extended plane called persistence diagram. We start with the question of ridding the function of topological noise as suggested by its persistence diagram. We give an algorithm for hierarchically finding such ε-simplifications on 2-manifolds as well as answer the question of when it is impossible to simplify a function in higher dimensions. We continue by examining time-varying functions. The original algorithm computes the persistence pairing from an ordering of the simplices in a triangulation and takes worstcase time cubic in the number of simplices. We describe how to maintain the pairing in linear time per transposition of consecutive simplices. A side effect of the update algorithm is an elementary proof of the stability of persistence diagrams. We introduce a parametrized family of persistence diagrams called persistence vineyards and illustrate the concept with a vineyard describing a folding of a small peptide. We also base a simple algorithm to compute the rank invariant of a collection of functions on the update procedure.

### Citations

1688 | A Global Geometric Framework for Nonlinear Dimensionality - Tenenbaum, Silva, et al. |

818 |
Algebraic Topology
- Hatcher
- 2002
(Show Context)
Citation Context ...le topological invariant called homology. 2.1 Complexes and homology We review simplicial complexes, homology groups, and briefly mention concepts from homotopy theory. We refer the reader to Hatcher =-=[52]-=- or Munkres [64] for a thorough study of these subjects. Complexes. We distinguish between abstract and geometric simplicial complexes. Given a collection of sets, we say that a subset of any p + 1 of... |

734 | Laplacian Eigenmaps for Dimensionality Reduction and
- Belkin
(Show Context)
Citation Context ...rovides proofs of correctness under assumptions of dense sampling [3, 20]. Machine learning uses statistical methods to study high-dimensional data that describes relatively low-dimensional manifolds =-=[10, 65]-=-. Finally, topological data analysis relies on algebraic methods to reveal the topological structure of high-dimensional data [76]. In this chapter we combine characteristics found in computational ge... |

647 | Surface reconstruction from unorganized points." Computer Graphics 26(2): 7178
- Hoppe, DeRose
- 1992
(Show Context)
Citation Context ...wn emphasis on the subject. Computer graphics and visualization stresses fast algorithms inspired by work in numerical analysis and image processing and focuses on data that describes surfaces in R 3 =-=[54, 80]-=-. Computational geometry favors combinatorial algorithms based on Delaunay triangulations [33] and provides proofs of correctness under assumptions of dense sampling [3, 20]. Machine learning uses sta... |

451 |
Elements of Algebraic Topology
- Munkres
- 1993
(Show Context)
Citation Context ...nvariant called homology. 2.1 Complexes and homology We review simplicial complexes, homology groups, and briefly mention concepts from homotopy theory. We refer the reader to Hatcher [52] or Munkres =-=[64]-=- for a thorough study of these subjects. Complexes. We distinguish between abstract and geometric simplicial complexes. Given a collection of sets, we say that a subset of any p + 1 of them is an abst... |

451 | The entropy formula for the Ricci flow and its geometric applications
- Perelman
(Show Context)
Citation Context ...connected, famously restating his conjecture to require the manifold to have a trivial fundamental group [71]. The problem remained open for a century until Grigori Perelman recently provided a proof =-=[67, 68, 69]-=-. See [63] for a historical account. Poincaré homology sphere ˆ S 3 presents us with an example of a manifold on which not all ε-simplifications exist. Given any function f : ˆ S 3 → R, its 0- and 3-d... |

434 | Three-dimensional alpha shapes
- Edelsbrunner, Mucke
- 1994
(Show Context)
Citation Context ...nal skeleton of the Čech complex whose size is ( ) |U| . n+1 Alpha shapes. A more efficient combinatorial representation of the distance function is the alpha shape complex, or alpha shapes for short =-=[34]-=-. To introduce it we need to recall 20some background from computational geometry. Given a set of points U, the Voronoi cell of u ∈ U is the set of points that are closer to u than to any other point... |

342 | Surface reconstruction by Voronoi filtering
- Amenta, Bern
- 1999
(Show Context)
Citation Context ...o the reconstruction problem have been explored by the machine learning community, most of the above homology learning methods as well most surface reconstruction techniques in computational geometry =-=[3, 33, 36]-=- reduce to studying the distance function dU : Rn → R which is defined on the ambient space of the data by measuring the distance at every point to the closest point in the data set. The resulting fun... |

301 |
Topology matching for fully automatic similarity estimation of 3D shapes
- Hilaga, Shinagawa, et al.
- 2001
(Show Context)
Citation Context ...o far we miss this luxury in higher dimensions. The simplification of continuous functions is a central problem in visualization. It may be used to clean up Morse-Smale complexes [38] and Reeb graphs =-=[53, 72]-=-, which are powerful tools in the study and visualization of continuous data in scientific computing. This strengthens the importance of extending our results to three- and higher-dimensional spheres ... |

231 | Data Structures for Mobile Data
- Basch, Guibas, et al.
- 1997
(Show Context)
Citation Context ... can be viewed as performing Pσ(0) Pσ(1) 0 1 Figure 4.1: Sketch of the arrangement formed by piecewise rational curves representing trajectories of the simplices. a kinetic sort of the simplices. See =-=[8, 9]-=- for discussion of the framework of kinetic data structures. To maintain persistence diagrams we are interested in updating R = DV decomposition from Section 2.2 which gives us the pairing after two c... |

228 | Ricci flow with surgery on three-manifolds. http://arxiv
- Perelman
- 2003
(Show Context)
Citation Context ...connected, famously restating his conjecture to require the manifold to have a trivial fundamental group [71]. The problem remained open for a century until Grigori Perelman recently provided a proof =-=[67, 68, 69]-=-. See [63] for a historical account. Poincaré homology sphere ˆ S 3 presents us with an example of a manifold on which not all ε-simplifications exist. Given any function f : ˆ S 3 → R, its 0- and 3-d... |

194 |
Stratified Morse Theory
- Goresky, MacPherson
- 1988
(Show Context)
Citation Context ...is thesis, we lay the foundations by parametrizing all our results in terms of scale parameters. Results and prior work. Stratified spaces have been studied extensively in the mathematical literature =-=[48, 81]-=-. Particularly relevant to the line of work presented in this chapter is the development of intersection homology [47] and of persistence for intersection homology [12]. There is a striking paucity in... |

171 |
Learning mixtures of Gaussians
- Dasgupta
- 1999
(Show Context)
Citation Context ...irable to be able to work with a point set drawn from a distribution that is uniformly random on a topological space and Gaussian in normal directions. This line of research was initiated by Dasgupta =-=[30, 31]-=- for the restricted case of 0-manifolds, i.e. the mixture of Gaussians. Carlsson et al. use various input preprocessing heuristics aimed to deal with outlier sensitivity of distance functions [17]. Mo... |

120 |
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
- Perelman
(Show Context)
Citation Context ...connected, famously restating his conjecture to require the manifold to have a trivial fundamental group [71]. The problem remained open for a century until Grigori Perelman recently provided a proof =-=[67, 68, 69]-=-. See [63] for a historical account. Poincaré homology sphere ˆ S 3 presents us with an example of a manifold on which not all ε-simplifications exist. Given any function f : ˆ S 3 → R, its 0- and 3-d... |

116 |
Stability of persistence diagrams
- Cohen-Steiner, Edelsbrunner, et al.
(Show Context)
Citation Context ...invariant called homology. It keeps track of components, tunnels, 1voids of the space as well as their high-dimensional counter-parts. A number of techniques have been proposed for learning homology =-=[5, 17, 22, 23, 25, 76, 65]-=-. The best example of practical advantage of knowing homology of the space is the work of Carlsson et al. [17] where the authors discover that at a certain scale the space describing the distribution ... |

114 | Finding the Homology of Submanifolds with High Confidence from Random Samples. Discrete Computational Geometry
- Niyogi, Smale, et al.
- 2008
(Show Context)
Citation Context ...invariant called homology. It keeps track of components, tunnels, 1voids of the space as well as their high-dimensional counter-parts. A number of techniques have been proposed for learning homology =-=[5, 17, 22, 23, 25, 76, 65]-=-. The best example of practical advantage of knowing homology of the space is the work of Carlsson et al. [17] where the authors discover that at a certain scale the space describing the distribution ... |

97 |
Triangulating topological spaces
- Edelsbrunner, Shah
- 1997
(Show Context)
Citation Context ...words, Del(U|X) is the nerve of the collection of sets X ∩ Vi. For X = R n we get the usual notion of Delaunay triangulation and for X ⊂ R n we get the restricted Delaunay triangulation as defined in =-=[42]-=-. Generically, Del(U|R n ) is a simplicial complex geometrically realized in R n ; see Figure 5.7. We are also interested in the restricFigure 5.7: Delaunay triangulations dual to the Voronoi decompos... |

85 |
Topological estimation using witness complexes
- Silva, Carlsson
- 2004
(Show Context)
Citation Context ...invariant called homology. It keeps track of components, tunnels, 1voids of the space as well as their high-dimensional counter-parts. A number of techniques have been proposed for learning homology =-=[5, 17, 22, 23, 25, 76, 65]-=-. The best example of practical advantage of knowing homology of the space is the work of Carlsson et al. [17] where the authors discover that at a certain scale the space describing the distribution ... |

84 |
Critical points and curvature for embedded polyhedra
- Banchoff
- 1967
(Show Context)
Citation Context ...am. • Vineyards trace critical values and do not require any notion of critical points. However, when critical points are available, such as for smooth and for piecewise linear functions on manifolds =-=[7, 62]-=-, we can use the update algorithm to maintain their association with the points in the persistence diagram. Can we exploit this ability to gain a better understanding of the stability or instability o... |

80 | Nonlinear manifold learning for visual speech recognition
- Bregler, S
- 1995
(Show Context)
Citation Context ...esses that depend on a large number of independent parameters. The problem of reconstructing this subset (or one such subset from the class of possibilities) is often referred to as manifold learning =-=[13]-=-. The name betrays the tacit assumption that the subset is thought to have the topological structure of a manifold. In other words, it is locally homeomorphic to R m , or possibly to the m-dimensional... |

70 | Fully automatic registration of multiple 3d data sets
- Huber, Hebert
- 2003
(Show Context)
Citation Context ...ty or instability of critical points? In particular, can this ability be developed into a global alignment algorithm for shapes that is more general and more reliable than what is currently available =-=[46, 55]-=-? Finally, we would like to suggest that vineyards should not be limited to homotopies but rather considered an analysis and visualization tool for parametrized families of functions. A point in case ... |

57 | A topological hierarchy for functions on triangulated surfaces
- Bremer, Edelsbrunner, et al.
- 2004
(Show Context)
Citation Context ...e complexes initiated in [38]. Such complexes capture information about the gradient vector field by partitioning the domain into regions of uniform flow. While the simplification algorithms given in =-=[14, 38, 49]-=- follow the persistence order, they only simplify the complex and not the function itself. The use of the simplified complex together with the original data may be tolerable for visualization purposes... |

56 |
les points singuliers d’une forme de pfaff complètement intégrable ou d’une fonction numérique
- Sur
- 1946
(Show Context)
Citation Context ...o far we miss this luxury in higher dimensions. The simplification of continuous functions is a central problem in visualization. It may be used to clean up Morse-Smale complexes [38] and Reeb graphs =-=[53, 72]-=-, which are powerful tools in the study and visualization of continuous data in scientific computing. This strengthens the importance of extending our results to three- and higher-dimensional spheres ... |

53 | An Introduction to Morse Theory - Matsumoto - 1997 |

49 | Algebraic topology - Maunder - 1970 |

44 |
Surface reconstruction by wrapping finite point sets in space. Discrete and Computational Geometry 32
- EDELSBRUNNER
(Show Context)
Citation Context ...o the reconstruction problem have been explored by the machine learning community, most of the above homology learning methods as well most surface reconstruction techniques in computational geometry =-=[3, 33, 36]-=- reduce to studying the distance function dU : Rn → R which is defined on the ambient space of the data by measuring the distance at every point to the closest point in the data set. The resulting fun... |

44 |
User’s guide to spectral sequences, Publish or Perish
- McCleary
(Show Context)
Citation Context ...r very mild assumptions on the conditioning of the space. We explain this observation in detail in Section 2.4. The concept of persistence can be seen embedded within the theory of spectral sequences =-=[61]-=- but has not been treated as a concept in its own right until [39]. The latter paper also describes a fast algorithm for modulo 2 homology and demonstrates that persistence is relevant to applications... |

35 | Topology-based simplification for feature extraction from 3D scalar fields
- Gyulassy, Natarajan, et al.
- 2005
(Show Context)
Citation Context ...e complexes initiated in [38]. Such complexes capture information about the gradient vector field by partitioning the domain into regions of uniform flow. While the simplification algorithms given in =-=[14, 38, 49]-=- follow the persistence order, they only simplify the complex and not the function itself. The use of the simplified complex together with the original data may be tolerable for visualization purposes... |

32 | Towards persistence-based reconstruction in Euclidean spaces
- Chazal, Oudot
- 2008
(Show Context)
Citation Context |

32 |
The Topological Classification of Stratified Spaces
- Weinberger
- 1994
(Show Context)
Citation Context ...is thesis, we lay the foundations by parametrizing all our results in terms of scale parameters. Results and prior work. Stratified spaces have been studied extensively in the mathematical literature =-=[48, 81]-=-. Particularly relevant to the line of work presented in this chapter is the development of intersection homology [47] and of persistence for intersection homology [12]. There is a striking paucity in... |

30 | Kinetic Data Structures
- Basch
- 1999
(Show Context)
Citation Context ... can be viewed as performing Pσ(0) Pσ(1) 0 1 Figure 4.1: Sketch of the arrangement formed by piecewise rational curves representing trajectories of the simplices. a kinetic sort of the simplices. See =-=[8, 9]-=- for discussion of the framework of kinetic data structures. To maintain persistence diagrams we are interested in updating R = DV decomposition from Section 2.2 which gives us the pairing after two c... |

30 |
Extending persistence using Poincaré and Lefschetz duality
- Cohen-Steiner, Edelsbrunner, et al.
(Show Context)
Citation Context ...imension and we write Dgm(f) for the infinite series of diagrams; we simplify language by ignoring the difference between a single diagram and an entire series. Cohen-Steiner, Edelsbrunner, and Harer =-=[26]-=- use relative homology to augment the above sequence of homomorphisms giving rise to persistent homology. Denoting the superlevel set of a function f at threshold a by X a = f[a, ∞), equal to the set ... |

30 |
Computing persistent homology. Discrete Comput. Geom., 33(2):249–274, 2005. imsart-coll ver. 2009/08/13 file: larry.tex date: February 23, 2010 Homology for Random Fields and Complexes 21 Robert J. Adler Electrical Engineering Technion
- Zomorodian, Carlsson
- 2010
(Show Context)
Citation Context ... modulo 2 homology and demonstrates that persistence is relevant to applications, including the study of protein structure. The concept and the algorithm have been extended to homology over fields in =-=[82]-=-. The stability of persistence diagrams has been established in [25], opening the concept up to additional applications, including the inference of homology from point clouds, see also [76], the compa... |

29 |
Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen
- Vietoris
- 1927
(Show Context)
Citation Context ...be seen as approximating distance functions for homology inference, especially as distilled in [19, 23]. Adopting the techniques for assessing local homology presented in this thesis to Vietoris-Rips =-=[78]-=- and Witness complexes is a practically important and promising course of study. Finally, while theoretically simple and easy to analyze, the Hausdorff model of noise allowed by the notion of ε-approx... |

28 |
Insolubility of the problem of homeomorphy
- MARKOV
- 1960
(Show Context)
Citation Context ...t notion of equivalence that we use is that of homeomorphism which is a continuous bijection between two spaces whose inverse is also continuous. However, homeomorphism type of a space is undecidable =-=[58]-=-, while we are interested in efficient algorithms to study topological spaces. We therefore restrict our attention to a much weaker, but easily computable topological invariant called homology. 2.1 Co... |

27 | Towards the Poincaré conjecture and the classification of 3-manifolds
- Milnor
- 2003
(Show Context)
Citation Context ...y restating his conjecture to require the manifold to have a trivial fundamental group [71]. The problem remained open for a century until Grigori Perelman recently provided a proof [67, 68, 69]. See =-=[63]-=- for a historical account. Poincaré homology sphere ˆ S 3 presents us with an example of a manifold on which not all ε-simplifications exist. Given any function f : ˆ S 3 → R, its 0- and 3-dimensional... |

24 |
complément à l’analysis situs
- Poincaré, Second
- 1900
(Show Context)
Citation Context ...at we now know as Poincaré homology sphere which has homology of a 3-sphere, but is not simply-connected, famously restating his conjecture to require the manifold to have a trivial fundamental group =-=[71]-=-. The problem remained open for a century until Grigori Perelman recently provided a proof [67, 68, 69]. See [63] for a historical account. Poincaré homology sphere ˆ S 3 presents us with an example o... |

21 | Persistence-sensitive simplification functions on 2manifolds
- Edelsbrunner, Morozov, et al.
- 2006
(Show Context)
Citation Context ...implification. It is natural to try to manipulate the values at vertices of the function f to obtain g while maintaining the correct pairing in the lower-star filtration. This approach is explored in =-=[40]-=-. However, in the next two sections we pursue a different strategy. We start with a lower-star filtration given by f and assign to each simplex the maximum of the values of its vertices, thus obtainin... |

18 | I/O-efficient batched union-find and its applications to terrain analysis
- Agarwal, Arge, et al.
- 2006
(Show Context)
Citation Context ...and statements from homotopy theory, and make the former statement precise. Given two functions f0 : X → Y and f1 : X → Y, we say that they are homotopic if there exists a continuous function F : X × =-=[0, 1]-=- → Y such that F (x, 0) = f0(x) and F (x, 1) = f1(x). We denote this by f ≃ g. Two topological spaces X and Y are said to be homotopy equivalent if there exist functions f : X → Y and g : Y → X such t... |

18 | Fast smallest-enclosing-ball computation in high dimensions
- Fischer, Gartner
(Show Context)
Citation Context ...s by dimension, we obtain a filtration which can be used with the algorithm described in Section 2.2. Gärtner et al. give an algorithm to compute the value of č(σ) for any simplex of the Čech complex =-=[43, 45]-=-. The size of the Čech complex with parameter infinity is exponential in the number of points, card Č(∞) = 2|U| . We can minimize the waste by observing that the homology of dimension greater than (n ... |

16 | Curve and surface reconstruction
- Dey
- 1997
(Show Context)
Citation Context ...o the reconstruction problem have been explored by the machine learning community, most of the above homology learning methods as well most surface reconstruction techniques in computational geometry =-=[3, 33, 36]-=- reduce to studying the distance function dU : Rn → R which is defined on the ambient space of the data by measuring the distance at every point to the closest point in the data set. The resulting fun... |

15 |
Intersection cohomology of cs-spaces and Zeeman’s filtration
- Habegger, Saper
- 1991
(Show Context)
Citation Context ... the same local structure in X. 56Connection with local homology. We note that the filtration in the definition of the stratified space is not unique. However, there is a natural coarsest filtration =-=[50]-=- which consists of the components in the partition of X defined by calling points x and y equivalent if there exist neighborhoods of x and y and a homeomorphism between these neighborhoods that maps x... |

15 | Coarse and reliable geometric alignment for protein docking
- Wang, Agarwal, et al.
- 2005
(Show Context)
Citation Context ... but rather considered an analysis and visualization tool for parametrized families of functions. A point in case is the elevation function [2] whose maxima have been useful in coarse protein docking =-=[79]-=-. For a surface M in space, this function is based on the sphere of height functions, which provide a homotopy F : M × S 2 → R. The elevation function can be constructed from the S 2 -parametrized vin... |

14 | A topological view of unsupervised learning from noisy data. Unpublished technical report
- Niyogi, Smale, et al.
- 2008
(Show Context)
Citation Context ....e. the mixture of Gaussians. Carlsson et al. use various input preprocessing heuristics aimed to deal with outlier sensitivity of distance functions [17]. Most recently Niyogi, Smale, and Weinberger =-=[66]-=- propose a clean-up algorithm for making the homology inference problem in the Gaussian noise setting manageable with the distance function approach. Development of persistence-based omniscalar method... |

13 | Reconstructing manifold and nonmanifold surfaces from point clouds
- Wang, Oliveira, et al.
- 2005
(Show Context)
Citation Context ...wn emphasis on the subject. Computer graphics and visualization stresses fast algorithms inspired by work in numerical analysis and image processing and focuses on data that describes surfaces in R 3 =-=[54, 80]-=-. Computational geometry favors combinatorial algorithms based on Delaunay triangulations [33] and provides proofs of correctness under assumptions of dense sampling [3, 20]. Machine learning uses sta... |

11 |
ZOMORODIAN A.: Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds
- EDELSBRUNNER, HARER
(Show Context)
Citation Context ...inuous functions has been studied before, in many different areas and from many different angles. The work related most directly to ours is on the simplification of Morse-Smale complexes initiated in =-=[38]-=-. Such complexes capture information about the gradient vector field by partitioning the domain into regions of uniform flow. While the simplification algorithms given in [14, 38, 49] follow the persi... |

10 |
Discrete computation of size functions
- Frosini
- 1992
(Show Context)
Citation Context ...ve provided insight into periodic gene expression [32]. Independently, the same ideas were developed from a somewhat different angle and restricted to zero-homology by a group of researchers in Italy =-=[15, 44]-=-. Using different terminology, they introduced persistence diagrams and proved stability, albeit only for the evolution of components in the sublevel sets [4]. The ideas of persistence also emerged in... |

9 |
Edelsbrunner, Dmitriy Morozov, Vines and vineyards by updating persistence in linear time
- Cohen-Steiner, Herbert
(Show Context)
Citation Context ...for computing the persistence pairing of the sequence of homology groups of a filtration in worst-case time cubic in the number of simplices. We now recall their algorithm using its interpretation in =-=[29]-=-. Computation. To compute persistent homology for the sequence of complexes Ki we let D be the m-by-m incidence matrix. We reduce D using left-to-right modulo-2 column additions until the lowest one o... |

8 |
Inequalities for the curvature of curves and surfaces
- Cohen-Steiner, Edelsbrunner
- 2007
(Show Context)
Citation Context ...pt up to additional applications, including the inference of homology from point clouds, see also [76], the comparison of shapes, see also [16], and the analysis of discrete curvature measures, see 3=-=[24]-=-. Persistence has been applied to the analysis of image patch data [17]. Recent results on the Lp-stability of persistence diagrams of Lipschitz functions [27] have provided insight into periodic gene... |

6 |
Optimal matching between reduced size functions
- d’Amico, Frosini, et al.
- 2003
(Show Context)
Citation Context ...y by a group of researchers in Italy [15, 44]. Using different terminology, they introduced persistence diagrams and proved stability, albeit only for the evolution of components in the sublevel sets =-=[4]-=-. The ideas of persistence also emerged in the work of Robins [74]. Contributions. In this thesis we explore several applications of persistence, and in doing so extend the theory. Several of the defi... |