Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle, for example, is one-dimensional but its faithful representation requires two real coordinates. In this work, we present a strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise NLDR analysis of a broader range of realistic data sets.
We introduce here a framework to construct coordinates in \emph{finite} Lens spaces for data with nontrivial 1-dimensional $\mathbb{Z}_q$ persistent cohomology, $q\geq 3$. Said coordinates are defined on an open neighborhood of the data, yet constructed with only a small subset of landmarks. We also introduce a dimensionality reduction scheme in $S^{2n-1}/\mathbb{Z}_q$ (Lens-PCA: $\mathsf{LPCA}$), and demonstrate the efficacy of the pipeline $PH^1(\;\cdot\; ; \mathbb{Z}_q)$ class $\Rightarrow$ $S^{2n-1}/\mathbb{Z}_q$ coordinates $\Rightarrow$ $\mathsf{LPCA}$, for nonlinear (topological) dimens…
We present in this paper a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundle…
We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent information. We explain how one can use the existing algorithm for persistent homology to process any of the four modules, and relate it to a recently introduced persistent cohomology algorithm. We present experimental evidence for the practical efficiency of the latter algorithm.
We expand the toolbox of (co)homological methods in computational topology by applying the concept of persistence to sheaf cohomology. Since sheaves (of modules) combine topological information with algebraic information, they allow for variation along an algebraic dimension and along a topological dimension. Consequently, we introduce two different constructions of sheaf cohomology (co)persistence modules. One of them can be viewed as a natural generalization of the construction of simplicial or singular cohomology copersistence modules. We discuss how both constructions relate to each other …
Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and reveal certain interaction patterns in terms of topological invariants. However, in addition to the geometric information, there is a wide variety of non-geometric information of molecular structures, such as element types, atomic partial charges, atomic pairwise interactions, and electrostatic potential function, that is not described by persistent homolog…
This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse function as the authors did in [LNV]. In connection with persistent cohomology, we prove that the rescaled logarithms of these small eigenvalues are asymptotically determined by the lengths of the bar code of the function f. In particular, this proves that these quantities are stable in the C 0 topology on the space of functions. Additionally, our analysis provides a general method for computing the subexponential cor…
We define and study several new interleaving distances for persistent cohomology which take into account the algebraic structures of the cohomology of a space, for instance the cup product or the action of the Steenrod algebra. In particular, we prove that there exists a persistent A-infinity-structure associated to data sets and and we define the associated distance. We prove the stability of these new distances for Cech or Vietoris Rips complexes with respect to the Gromov-Hausdorff distance, and we compare these new distances with each other and the classical one, building some examples whi…
Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the cup-length, which is useful for discriminating spaces. In this paper, we lift the cup-length into the persist…
Northeastern University（东北大学）；The University of Oklahoma（俄克拉荷马大学）；University of Florida（佛罗里达大学）；University of Delaware（特拉华大学）；Colorado State University（科罗拉多州立大学）；The Ohio State University（俄亥俄州立大学）；Khoury College of Computer Sciences（胡里计算机科学学院）
The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a $1$-dimensional hole in the data; the output is a map from the data into the circle that captures this hole, and that is of minimum energy in a suitable sense. However, when applied to several cohomology classes, the output circle-valued maps can be "geometrically correlated" even if the chosen cohomology classes are linearly independent. It is shown in the original work that less correlated maps can be obtained with suitable integer linear com…
We present in this paper an application of the theory of principal bundles to the problem of nonlinear dimensionality reduction in data analysis. More explicitly, we derive, from a 1-dimensional persistent cohomology computation, explicit formulas for circle-valued functions on data with nontrivial underlying topology. We show that the language of principal bundles leads to coordinates defined on an open neighborhood of the data, but computed using only a smaller subset of landmarks. It is in this sense that the coordinates are sparse. Several data examples are presented, as well as theoretica…
Universidad de Buenos Aires（布宜诺斯艾利斯大学）；CONICET（国家科学技术研究委员会）；Université Grenoble I（格勒诺布尔第一大学）
Our objective in this article is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of the (say) simplicial set embedded in a finite dimensional vector space induces a multiplicative filtration (which would not be a so harsh hypothesis in our setting) on the dg algebra given by the complex of simplicial cochains, we may use a result by T. Kadeishvili to get a unique (up to noncanonical equivalence) A_infinity-algebra structure on the complete persistent cohomology of the filtered simplicial (or topological) set. We th…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is mo…
Let $f$ be a Morse function on a smooth compact manifold $M$ with boundary. The path component $\mathrm{PH}_f^{-1}(D)$ containing $f$ of the space of Morse functions giving rise to the same Persistent Homology $D=\mathrm{PH}(f))$ is shown to be the same as the orbit of $f$ under pre-composition $ϕ\mapsto f\circ ϕ$ by diffeomorphisms of $M$ which are isotopic to the identity. Consequently we derive topological properties of the fiber $\mathrm{PH}_f^{-1}(D)$: In particular we compute its homotopy type for many compact surfaces $M$. In the $1$-dimensional settings where $M$ is the unit interval o…
The Ohio State University（俄亥俄州立大学）；Indiana University（印第安纳大学）；Indiana University Network Science Institute（印第安纳大学网络科学研究所）；Inria Saclay（法国国家信息与自动化研究所萨克莱分部）；CUNY College of Staten Island（纽约市立大学史泰登岛学院）；CUNY Graduate Center（纽约市立大学研究生中心）
Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization and dimension reduction. We follow the framework of circular coordinate representation, which allows us to perform dimension reduction and visualization for high-dimensional datasets on a torus using persistent cohomology. In this paper, we propose a method to adapt the circular coordinate framework to take into account the roughness of circular coordinates in change-point and high…
