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17
Proximity of persistence modules and their diagrams
, 2008
"... Topological persistence has proven to be a key concept for the study of realvalued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case o ..."
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Cited by 82 (15 self)
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Topological persistence has proven to be a key concept for the study of realvalued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case of continuous functions defined over triangulable spaces. In this paper, we present new stability results that do not suffer from the above restrictions. Furthermore, by working at an algebraic level directly, we make it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence. Along the way, we extend the definition of persistence diagram to a larger setting, introduce the notions of discretization of a persistence module and associated pixelization map, define a proximity measure between persistence modules, and show how to interpolate between persistence modules, thereby lending a more analytic character to this otherwise algebraic setting. We believe these new theoretical concepts and tools shed new light on the theory of persistence, in addition to simplifying proofs and enabling new applications.
Persistence Diagrams of Cortical Surface Data
 Information Processing in Medical Imaging, LNCS
"... Abstract. We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persis ..."
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Cited by 24 (7 self)
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Abstract. We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persistence diagrams. These are scatter plots of paired local critical values of the signal. One of these diagrams visually shows how the number of connected components of the sublevel sets of the signal changes. The use of local critical values of a function differs from the usual statistical parametric mapping framework, which mainly uses the mean signal in quantifying imaging data. Our proposed method uses all the local critical values in characterizing the signal and by doing so offers a completely new data reduction and analysis framework for quantifying the signal. As an illustration, we apply this method to a 1D simulated signal and 2D cortical thickness data. 1
Zigzag Persistent Homology in Matrix Multiplication Time
 IN: PROCEEDINGS OF THE TWENTYSEVENTH ANNUAL SYMPOSIUM ON COMPUTATIONAL GEOMETRY, 2011
, 2011
"... We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n matrices in M(n) time, our al ..."
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Cited by 21 (0 self)
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We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n matrices in M(n) time, our algorithm runs in O(M(n) + n 2 log 2 n) time for a sequence of n additions and deletions. In particular, the running time is O(n 2.376), by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes O(n 3) time in the worst case.
Optimal topological simplification of discrete functions on surfaces
 Discrete & Computational Geometry
"... We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance δ from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homol ..."
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Cited by 19 (6 self)
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We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance δ from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2δ, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed. 1
Persistent homology for kernels, images, and cokernels
 Proc. Nineteenth Annual ACM SIAM Symposium on Discrete Algorithms
, 2009
"... Motivated by the measurement of local homology and of functions on noisy domains, we extend the notion of persistent homology to sequences of kernels, images, and cokernels of maps induced by inclusions in a filtration of pairs of spaces. Specifically, we note that persistence in this context is wel ..."
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Cited by 16 (5 self)
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Motivated by the measurement of local homology and of functions on noisy domains, we extend the notion of persistent homology to sequences of kernels, images, and cokernels of maps induced by inclusions in a filtration of pairs of spaces. Specifically, we note that persistence in this context is well defined, we prove that the persistence diagrams are stable, and we explain how to compute them.
Persistencesensitive simplification of functions on surfaces in linear time
, 2009
"... Persistence provides a way of grading the importance of homological features in the sublevel sets of a realvalued function. Following the de nition given by Edelsbrunner, Morozov and Pascucci, an εsimpli cation of a function f is a function g in which the homological noise of persistence less than ..."
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Cited by 14 (6 self)
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Persistence provides a way of grading the importance of homological features in the sublevel sets of a realvalued function. Following the de nition given by Edelsbrunner, Morozov and Pascucci, an εsimpli cation of a function f is a function g in which the homological noise of persistence less than ε has been removed. In this paper, we give an algorithm for constructing an εsimpli cation of a function de ned on a triangulated surface in linear time. Our algorithm is very simple, easy to implement and follows directly from the study of the εsimpli cation of a function on a tree. We also show that the computation of persistence de ned on a graph can be performed in linear time in a RAM model. This gives an overall algorithm in linear time for both computing and simplifying the homological noise of a function f on a surface.
Comparison of persistent homologies for vector functions: from continuous to discrete and back
 Computers and Mathematics with Applications
, 2013
"... ar ..."
Separating Features from Noise with Persistence and Statistics
"... In this thesis, we explore techniques in statistics and persistent homology, which detect features among data sets such as graphs, triangulations and point cloud. We accompany our theorems with algorithms and experiments, to demonstrate their effectiveness in practice. We start with the derivation o ..."
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In this thesis, we explore techniques in statistics and persistent homology, which detect features among data sets such as graphs, triangulations and point cloud. We accompany our theorems with algorithms and experiments, to demonstrate their effectiveness in practice. We start with the derivation of graph scan statistics, a measure useful to assess the statistical significance of a subgraph in terms of edge density. We cluster graphs into denselyconnected subgraphs based on this measure. We give algorithms for finding such clusterings and experiment on realworld data. We next study statistics on persistence, for piecewiselinear functions defined on the triangulations of topological spaces. We derive persistence pairing probabilities among vertices in the triangulation. We also provide upper bounds for total persistence in expectation. We continue by examining the elevation function defined on the triangulation of a surface. Its local maxima obtained by persistence pairing are useful in describing features of the triangulations of protein surfaces. We describe an algorithm to compute these local maxima, with a runtime tenthousand times faster in practice than previous method. We connect such improvement with the total Gaussian curvature of the surfaces.
unknown title
"... Abstract. We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persis ..."
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Abstract. We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persistence diagrams. These diagrams visually show how the number of connected components of the sublevel sets of the signal changes. The use of local critical values of a function differs from the usual statistical parametric mapping framework, which mainly uses the mean signal in quantifying imaging data. Our proposed method uses all the local critical values in characterizing the signal and by doing so offers a completely new data reduction and analysis framework for quantifying the signal. As an illustration, we apply this method to a 1D simulated signal and 2D cortical thickness data. In case of the latter, extra homological structures are evident in an control group over the autistic group. 1