Section: New Results

Topological and Geometric Inference

Only distances are required to reconstruct submanifolds

Participants : Jean-Daniel Boissonnat, Steve Oudot.

In collaboration with Ramsay Dyer (Johann Bernouilli Institute, University of Groningen, Pays Bas) and Arijit Ghosh (Max-Planck-Institut für Informatik, Saarbrücken, Germany).

In [45] , we give the first algorithm that outputs a faithful reconstruction of a submanifold of Euclidean space without maintaining or even constructing complicated data structures such as Voronoi diagrams or Delaunay complexes. Our algorithm uses the witness complex and relies on the stability of power protection, a notion introduced in this paper. The complexity of the algorithm depends exponentially on the intrinsic dimension of the manifold, rather than the dimension of ambient space, and linearly on the dimension of the ambient space. Another interesting feature of this work is that no explicit coordinates of the points in the point sample is needed. The algorithm only needs the distance matrix as input, i.e., only distance between points in the point sample as input.

Computing Persistent Homology with Various Coefficient Fields in a Single Pass

Participants : Jean-Daniel Boissonnat, Clément Maria.

In [32] , we introduce an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent homological features in the diagrams for the different coefficient fields. This computation allows us to infer the prime divisors of the torsion coefficients of the integral homology groups of the topological space at any scale, hence furnishing a more informative description of topology than persistence in a single coefficient field. We provide theoretical complexity analysis as well as detailed experimental results.

Recognizing shrinkable complexes is NP-complete

Participants : Olivier Devillers, Marc Glisse.

In collaboration with Dominique Attali (Gipsa-lab, Grenoble), Sylvain Lazard (Inria Nancy - Grand Est)

We say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove [31] that it is NP-complete to decide whether a (three-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes which are not shrinkable.

Zigzag Zoology: Rips Zigzags for Homology Inference

Participant : Steve Oudot.

In collaboration with Donald Sheehy (University of Connecticut)

For points sampled near a compact set X, the persistence barcode of the Rips filtration built from the sample contains information about the homology of X as long as X satisfies some geometric assumptions. The Rips filtration is prohibitively large, however zigzag persistence can be used to keep the size linear. We present [28] several species of Rips-like zigzags and compare them with respect to the signal-to-noise ratio, a measure of how well the underlying homology is represented in the persistence barcode relative to the noise in the barcode at the relevant scales. Some of these Rips-like zigzags have been available as part of the Dionysus library for several years while others are new. Interestingly, we show that some species of Rips zigzags will exhibit less noise than the (non-zigzag) Rips filtration itself. Thus, the Rips zigzag can offer improvements in both size complexity and signal-to-noise ratio. Along the way, we develop new techniques for manipulating and comparing persistence barcodes from zigzag modules. We give methods for reversing arrows and removing spaces from a zigzag. We also discuss factoring zigzags and a kind of interleaving of two zigzags that allows their barcodes to be compared. These techniques were developed to provide our theoretical analysis of the signal-to-noise ratio of Rips-like zigzags, but they are of independent interest as they apply to zigzag modules generally.

Zigzag Persistence via Reflections and Transpositions

Participants : Clément Maria, Steve Oudot.

We introduce [40] a simple algorithm for computing zigzag persistence, designed in the same spirit as the standard persistence algorithm. Our algorithm reduces a single matrix, maintains an explicit set of chains encoding the persistent homology of the current zigzag, and updates it under simplex insertions and removals. The total worst-case running time matches the usual cubic bound. A noticeable difference with the standard persistence algorithm is that we do not insert or remove new simplices "at the end" of the zigzag, but rather "in the middle". To do so, we use arrow reflections and transpositions, in the same spirit as reflection functors in quiver theory. Our analysis introduces a new kind of reflection called the "weak-diamond", for which we are able to predict the changes in the interval decomposition and associated compatible bases. Arrow transpositions have been studied previously in the context of standard persistent homology, and we extend the study to the context of zigzag persistence. For both types of transformations, we provide simple procedures to update the interval decomposition and associated compatible homology basis.

Topological analysis of scalar fields with outliers

Participants : Mickaël Buchet, Frédéric Chazal, Steve Oudot.

In collaboration with Tamal K. Dey (University of Ohio) Fengtao Fan (University of Ohio) Yusu Wang (University of Ohio)

We extend [57] the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to approximate the sub-level sets of this function by a union of metric balls and extend previous results on sparse Rips filtrations to this setting. This robust and efficient approach to topological data analysis is illustrated with several examples from an implementation.

Efficient and Robust Persistent Homology for Measures.

Participants : Mickaël Buchet, Frédéric Chazal, Steve Oudot.

In collaboration with Donald Sheehy (University of Connecticut)

In  [34] , we extend the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to approximate the sub-level sets of this function by a union of metric balls and extend previous results on sparse Rips filtrations to this setting. This robust and efficient approach to topological data analysis is illustrated with several examples from an implementation.

Persistence-based Structural Recognition

Participants : Frédéric Chazal, Maksims Ovsjanikovs.

In collaboration with Chunyuan Li (former intern in Saclay in 2013)

In [39] we present a framework for object recognition using topological persistence. In particular, we show that the so-called persistence diagrams built from functions defined on the objects can serve as compact and informative descriptors for images and shapes. Complementary to the bag-of-features representation, which captures the distribution of values of a given function, persistence diagrams can be used to characterize its structural properties, reflecting spatial information in an invariant way. In practice, the choice of function is simple: each dimension of the feature vector can be viewed as a function. The proposed method is general: it can work on various multimedia data, including 2D shapes, textures and triangle meshes. Extensive experiments on 3D shape retrieval, hand gesture recognition and texture classification demonstrate the performance of the proposed method in comparison with state-of-the-art methods. Additionally, our approach yields higher recognition accuracy when used in conjunction with the bag-offeatures.

Convergence rates for persistence diagram estimation in Topological Data Analysis

Participants : Frédéric Chazal, Marc Glisse, Bertrand Michel.

In collaboration with Catherine Labruère (University of Burgundy)

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In [36] , we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results.

Stochastic Convergence of Persistence Landscapes and Silhouettes

Participant : Frédéric Chazal.

In collaboration with Brittany Fasy (Tulane University) Fabrizio Lecci (Carnegie Mellon University) Alessandro Rinaldo (Carnegie Mellon University) Larry Wasserman (Carnegie Mellon University)

Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we can summarize the persistent homology with the persistence landscape, introduced by Bubenik, which converts a diagram into a well-behaved real-valued function. In [35] , we investigate the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence of the bootstrap. In addition, we introduce an alternate functional summary of persistent homology, which we call the silhouette, and derive an analogous statistical theory.

Subsampling Methods for Persistent Homology

Participants : Frédéric Chazal, Bertrand Michel.

In collaboration with Brittany Fasy (Tulane University) Fabrizio Lecci (Carnegie Mellon University) Alessandro Rinaldo (Carnegie Mellon University) Larry Wasserman (Carnegie Mellon University)

Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the persistent homology is prohibitive due to the combinatorial nature of the existing algorithms. We propose to compute the persistent homology of several subsamples of the data and then combine the resulting estimates. We study the risk of two estimators and we prove that the subsampling approach carries stable topological information while achieving a great reduction in computational complexity.

The observable structure of persistence modules

Participant : Frédéric Chazal.

In collaboration with Vin de Silva (Pomona College) William Crawley-Boevey (University of Leeds)

In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is de- finable. However, unlike persistence modules indexed over a totally ordered finite set or the natural numbers, such diagrams do not provide a complete invariant of q-tame modules. The purpose of [59] is to show that the category of persistence modules can be adjusted to overcome this issue. We introduce the observable category of persis- tence modules: a localization of the usual category, in which the classical properties of q-tame modules still hold but where the persistence diagram is a complete isomorphism invariant and all q-tame modules admit an interval decomposition.