CHAZAL , Frédéric

Modern data often come as point clouds embedded in high dimensional Euclidean spaces, or possibly more general metric spaces. They are usually not distributed uniformly, but lie around some highly nonlinear geometric structures with nontrivial topology. Topological data analysis (TDA) is an emerging field whose goal is to provide mathematical...

Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane R 2 that can equivalently be seen as discrete measures in R 2. When the data come as a random point cloud, these discrete measures...

Topological Data Analysis (TDA)is a recent and fast growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. This paper is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of TDA for non experts.

Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane $\mathbb{R}^2$ that can equivalently be seen as discrete measures in $\mathbb{R}^2$. When the data is assumed to be random, these...

This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our...

This paper addresses the case where data come as point sets, or more generally as discrete measures. Our motivation is twofold: first we intend to approximate with a compactly supported measure the mean of the measure generating process, that coincides with the intensity measure in the point process framework, or with the expected persistence...

A learning device performs learning by an autoencoder, using waveform data with changes over time that is obtained from an intrinsic movement of an object. The learning device performs persistent homology conversion to calculate a change in the number of connected component according to a threshold change in a value direction for the waveform...

Approximations of Laplace-Beltrami operators on manifolds through graph Lapla-cians have become popular tools in data analysis and machine learning. These discretized operators usually depend on bandwidth parameters whose tuning remains a theoretical and practical problem. In this paper, we address this problem for the unnormalized graph...

Distances to compact sets are widely used in the field of Topological Data Analysis for inferring geometric and topological features from point clouds. In this context, the distance to a probability measure (DTM) has been introduced by Chazal et al. (2011b) as a robust alternative to the distance a compact set. In practice, the DTM can be...

We present a method to construct signatures of periodic-like data. Based on topological considerations, our construction encodes information about the order and values of local extrema. Its main strength is robustness to reparametrisation of the observed signal, so that it depends only on the form of the periodic function. The signature...

In this paper, we provide stability guarantees for two frameworks that are based on the notion of functional maps – the shape difference operators introduced in [?] and the framework of [?] which is used to analyze and visualize the deformations between shapes induced by a functional map. We consider two types of perturbations in our analysis:...

This paper presents an innovative and generic deep learning approach to monitor heart conditions from ECG signals.We focus our attention on both the detection and classification of abnormal heartbeats, known as arrhythmia. We strongly insist on generalization throughout the construction of a deep-learning model that turns out to be effective...

Geometric and topological inference deals with the retrieval of information about a geometric object that is only known through a finite set of possibly noisy sample points. Geometric and topological inference employs many tools from Computational Geometry and Applied Topology. It has connections to Manifold Learning and provides the...

It has been observed since a long time that data are often carrying interesting topological and geometric structures. Characterizing such structures and providing efficient tools to infer and exploit them is a challenging problem that asks for new mathematics and that is motivated by a real need from applications. This paper is an introduction...

The Fermat distance has been recently established as a useful tool for machine learning tasks when a natural distance is not directly available to the practitioner or to improve the results given by Euclidean distances by exploding the geometrical and statistical properties of the dataset. This distance depends on a parameter $α$ that greatly...

In this paper, a stride detector algorithm combined with a technique inspired by zero velocity update (ZUPT) is proposed to reconstruct the trajectory of a pedestrian from an ankle-mounted inertial device. This innovative approach is based on sensor alignment and machine learning. It is able to detect 100% of both normal walking strides and...

In this paper, we propose a novel pooling approach for shape classification and recognition using the bag-of-words pipeline, based on topological persistence, a recent tool from Topological Data Analysis. Our technique extends the standard max-pooling, which summarizes the distribution of a visual feature with a single number, thereby losing...

A new paradigm for point cloud data analysis hasemerged recently, where point clouds are no longertreated as mere compact sets but rather as empiricalmeasures. A notion of distance to such measures hasbeen dened and shown to be stable with respect toperturbations of the measure. This distance can eas-ily be computed pointwise in the case of a...

We consider a signal composed of several periods of a periodic function, of which we observe a noisy reparametrisation. The phase estimation problem consists of finding that reparametrisation, and, in particular, the number of observed periods. Existing methods are well-suited to the setting where the periodic function is known, or at least,...

—In this paper, a strides detection algorithm is proposed using inertial sensors worn on the ankle. This innovative approach based on geometric patterns can detect both normal walking strides and atypical strides such as small steps, side steps and backward walking that existing methods struggle to detect. It is also robust in critical...

Efficient and Robust Persistent Homology for Measures. Computational Geometry, Elsevier, 2016, . HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public...

In this paper, an algorithm for activity recognition is proposed using inertial sensors worn on the ankle. This innovative approach based on geometric patterns uses a stride detector that can detect both normal walking strides and atypical strides such as small steps, side steps and backward walking that existing methods struggle to detect. It...

In this paper, a strides detection algorithm combined with a technique inspired by Zero Velocity Update (ZUPT) is proposed using inertial sensors worn on the ankle. This innovative approach based on a sensors alignment and machine learning can detect both normal walking strides and atypical strides such as small steps, side steps and backward...

BackgroundThis paper exploits recent developments in topological data analysis to present a pipeline for clustering based on Mapper, an algorithm that reduces complex data into a one-dimensional graph.ResultsWe present a pipeline to identify and summarise clusters based on statistically significant topological features from a point cloud using...