Section: New Results

Topological and Geometric Inference

Subsampling Methods for Persistent Homology

Participants : Frédéric Chazal, Bertrand Michel.

In collaboration with B.T. Fasy, F. Lecci, A. Rinaldo and L. 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.

Efficient and Robust Persistent Homology for Measures

Participants : Frédéric Chazal, Steve Oudot.

In collaboration with M. Buchet (Ohio State University) and Donald Sheehy (University of Connecticut).

A new paradigm for point cloud data analysis has emerged recently, where point clouds are no longer treated as mere compact sets but rather as empirical measures. A notion of distance to such measures has been defined and shown to be stable with respect to perturbations of the measure. This distance can eas- ily be computed pointwise in the case of a point cloud, but its sublevel-sets, which carry the geometric infor- mation about the measure, remain hard to compute or approximate. This makes it challenging to adapt many powerful techniques based on the Euclidean distance to a point cloud to the more general setting of the distance to a measure on a metric space. We propose [28] an efficient and reliable scheme to approximate the topological structure of the family of sublevel-sets of the distance to a measure. We obtain an algorithm for approximating the persistent homology of the distance to an empirical measure that works in arbitrary metric spaces. Precise quality and complexity guarantees are given with a discussion on the behavior of our approach in practice.

Topological analysis of scalar fields with outliers

Participants : Frédéric Chazal, Steve Oudot.

In collaboration with M. Buchet, T.K. Dey, F. Fan, Y. Wang (Ohio State University).

Given a real-valued function f defined over a manifold M embedded in Euclidean space, we are interested in recovering structural information about f from the sole information of its values on a finite sample P [27] . Existing methods provide approximation to the persistence diagram of f when the noise is bounded in both the functional and geometric domains. However, they fail in the presence of aberrant values, also called outliers, both in theory and practice. We propose a new algorithm that deals with outliers. We handle aberrant functional values with a method inspired from the k-nearest neighbors regression and the local median filtering, while the geometric outliers are handled using the distance to a measure. Combined with topological results on nested filtrations, our algorithm performs robust topological analysis of scalar fields in a wider range of noise models than handled by current methods. We provide theoretical guarantees on the quality of our approximation and some experimental results illustrating its behavior.

Zigzag Persistence via Reflections and Transpositions

Participants : Clément Maria, Steve Oudot.

We introduce [33] 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.

Stable Topological Signatures for Points on 3D Shapes

Participants : Mathieu Carrière, Steve Oudot, Maksims Ovsjanikovs.

Comparing points on 3D shapes is among the fundamental operations in shape analysis. To facilitate this task, a great number of local point signatures or descriptors have been proposed in the past decades. However, the vast majority of these descriptors concentrate on the local geometry of the shape around the point, and thus are insensitive to its connectivity structure. By contrast, several global signatures have been proposed that successfully capture the overall topology of the shape and thus characterize the shape as a whole. We propose [29] , [43] the first point descriptor that captures the topology structure of the shape as `seen' from a single point, in a multiscale and provably stable way. We also demonstrate how a large class of topological signatures, including ours, can be mapped to vectors, opening the door to many classical analysis and learning methods. We illustrate the performance of this approach on the problems of supervised shape labeling and shape matching. We show that our signatures provide complementary information to existing ones and allow to achieve better performance with less training data in both applications.

Structure and Stability of the 1-Dimensional Mapper

Participants : Mathieu Carrière, Steve Oudot.

Given a continuous function $f:X\to ℝ$ and a cover $I$ of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover $f^{-1}\left(I\right)$. Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of $f$, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. We propose [44] a theoretical framework that relates the structure of the Mapper to the one of the Reeb graph, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of unstability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover $I$ goes to zero.

Persistence Theory: From Quiver Representations to Data Analysis

Participant : Steve Oudot.

Persistence theory emerged in the early 2000s as a new theory in the area of applied and computational topology. This book [35] provides a broad and modern view of the subject, including its algebraic, topological, and algorithmic aspects. It also elaborates on applications in data analysis. The level of detail of the exposition has been set so as to keep a survey style, while providing sufficient insights into the proofs so the reader can understand the mechanisms at work.