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46
Finding the homology of submanifolds with high confidence from random samples
, 2004
"... Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn ” the hom ..."
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Cited by 177 (6 self)
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Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn ” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learningtheoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to selfintersection of the submanifold. We are also able to treat the situation where the data is “noisy ” and lies near rather than on the submanifold in question.
Computing Persistent Homology
 Discrete Comput. Geom
"... We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enabl ..."
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Cited by 148 (20 self)
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We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This results generalizes and extends the previously known algorithm that was restricted to subcomplexes of S and Z2 coefficients. Finally, our study implies the lack of a simple classification over nonfields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.
Statistical ranking and combinatorial Hodge theory
 Mathematical Programming
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A topological view of unsupervised learning from noisy data
 SIAM Journal of Computing
, 2011
"... Abstract. In this paper, we take a topological view of unsupervised learning. From this point of view, clustering may be interpreted as trying to find the number of connected components of any underlying geometrically structured probability distribution in a certain sense that we will make precise. ..."
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Cited by 35 (1 self)
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Abstract. In this paper, we take a topological view of unsupervised learning. From this point of view, clustering may be interpreted as trying to find the number of connected components of any underlying geometrically structured probability distribution in a certain sense that we will make precise. We construct a geometrically structured probability distribution that seems appropriate for modeling data in very high dimensions. A special case of our construction is the mixture of Gaussians where there is Gaussian noise concentrated around a finite set of points (the means). More generally we consider Gaussian noise concentrated around a low dimensional manifold and discuss how to recover the homology of this underlying geometric core from data that do not lie on it. We show that if the variance of the Gaussian noise is small in a certain sense, then the homology can be learned with high confidence by an algorithm that has a weak (linear) dependence on the ambient dimension. Our algorithm has a natural interpretation as a spectral learning algorithm using a combinatorial Laplacian of a suitable dataderived simplicial complex.
On the Betti numbers of chessboard complexes
 J. ALGEBRAIC COMBIN
, 1998
"... In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0th Betti numb ..."
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Cited by 31 (3 self)
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In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovasz, Vrecica, and Zivaljevic in [BLVZ94]. Our formula also shows that all eigenvalues of the Laplacians of the simplicial complexes are integers, and it gives a formula (involving partitions) for the multiplicities of the eigenvalues.
Shifted simplicial complexes are Laplacian integral
 Trans. Amer. Math. Soc
, 2002
"... Abstract. We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also ..."
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Cited by 30 (7 self)
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Abstract. We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs. 1.
Coreduction homology algorithm
 Discrete & Computational Geometry
"... Abstract. A new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes, particularly strong for low dimensional sets embedded in high dimensions, is presented. The algorithm runs in linear time. The paper presents the theoretical background of the algorithm, ..."
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Cited by 29 (9 self)
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Abstract. A new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes, particularly strong for low dimensional sets embedded in high dimensions, is presented. The algorithm runs in linear time. The paper presents the theoretical background of the algorithm, the algorithm itself, experimental results based on an implementation for cubical sets as well as some theoretical complexity estimates. 1.
Homology algorithm based on acyclic subspace
"... We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace ma ..."
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Cited by 22 (8 self)
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We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be processed in the algebraic way, and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms.
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 20 (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.