Local Equivalence and Intrinsic Metrics between Reeb Graphs

As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as {\em functional distortion} or {\em interleaving} that are provably more discriminative than the so-called {\em bottleneck distance}, being true metrics whereas the latter is only a pseudo-metric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. Here we take the opposite view on the problem and show that the bottleneck distance is in fact good enough {\em locally}, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the {\em intrinsic metrics} induced by these distances, which turn out to be all {\em globally} equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them

Sliced Wasserstein Kernel for Persistence Diagrams

Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe topological properties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in various learning contexts. They do not, however, live in a space naturally endowed with a Hilbert structure and are usually compared with specific distances, such as the bottleneck distance. To incorporate PDs in a learning pipeline, several kernels have been proposed for PDs with a strong emphasis on the stability of the RKHS distance w.r.t. perturbations of the PDs. In this article, we use the Sliced Wasserstein approximation SW of the Wasserstein distance to define a new kernel for PDs, which is not only provably stable but also provably discriminative (depending on the number of points in the PDs) w.r.t. the Wasserstein distance $d_1$ between PDs. We also demonstrate its practicality, by developing an approximation technique to reduce kernel computation time, and show that our proposal compares favorably to existing kernels for PDs on several benchmarks.Comment: Minor modification

Statistical Analysis and Parameter Selection for Mapper

In this article, we study the question of the statistical convergence of the 1-dimensional Mapper to its continuous analogue, the Reeb graph. We show that the Mapper is an optimal estimator of the Reeb graph, which gives, as a byproduct, a method to automatically tune its parameters and compute confidence regions on its topological features, such as its loops and flares. This allows to circumvent the issue of testing a large grid of parameters and keeping the most stable ones in the brute-force setting, which is widely used in visualization, clustering and feature selection with the Mapper.Comment: Minor modification

Statistical analysis of Mapper for stochastic and multivariate filters

Reeb spaces, as well as their discretized versions called Mappers, are common descriptors used in Topological Data Analysis, with plenty of applications in various fields of science, such as computational biology and data visualization, among others. The stability and quantification of the rate of convergence of the Mapper to the Reeb space has been studied a lot in recent works [BBMW19, CO17, CMO18, MW16], focusing on the case where a scalar-valued filter is used for the computation of Mapper. On the other hand, much less is known in the multivariate case, when the codomain of the filter is $\mathbb{R}^p$, and in the general case, when it is a general metric space $(Z, d_Z)$, instead of $\mathbb{R}$. The few results that are available in this setting [DMW17, MW16] can only handle continuous topological spaces and cannot be used as is for finite metric spaces representing data, such as point clouds and distance matrices. In this article, we introduce a slight modification of the usual Mapper construction and we give risk bounds for estimating the Reeb space using this estimator. Our approach applies in particular to the setting where the filter function used to compute Mapper is also estimated from data, such as the eigenfunctions of PCA. Our results are given with respect to the Gromov-Hausdorff distance, computed with specific filter-based pseudometrics for Mappers and Reeb spaces defined in [DMW17]. We finally provide applications of this setting in statistics and machine learning for different kinds of target filters, as well as numerical experiments that demonstrate the relevance of our approac

PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures

Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of persistence diagrams is not Hilbert, they end up being difficult inputs for most Machine Learning techniques. To address this concern, several vectorization methods have been put forward that embed persistence diagrams into either finite-dimensional Euclidean space or (implicit) infinite dimensional Hilbert space with kernels. In this work, we focus on persistence diagrams built on top of graphs. Relying on extended persistence theory and the so-called heat kernel signature, we show how graphs can be encoded by (extended) persistence diagrams in a provably stable way. We then propose a general and versatile framework for learning vectorizations of persistence diagrams, which encompasses most of the vectorization techniques used in the literature. We finally showcase the experimental strength of our setup by achieving competitive scores on classification tasks on real-life graph datasets

Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.Comment: 23 pages, 3 figures, 8 table

Lifetime exposure to ambient ultraviolet radiation and the risk for cataract extraction and age-related macular degeneration : the Alienor Study

While exposure to ultraviolet radiation (UVR) is a recognized risk factor for cataract, its association is more controversial with age-related macular degeneration (AMD). We report the associations of lifetime exposure to ambient UVR with cataract extraction and AMD. The Alienor Study is a population-based study of 963 residents of Bordeaux (France), aged 73 years or more. Lifetime exposure to ambient UVR was estimated from residential history and Eurosun satellite-based estimations of ground UVR. It was divided in three groups (lower quartile, intermediate quartiles, upper quartile), using the intermediate quartiles as the reference. Early and late AMD was classified from retinal color photographs. Cataract extraction was defined as absence of the natural lens at slit-lamp. After multivariate adjustment, subjects in the upper quartile of lifetime ambient UVR exposure were at increased risk for cataract extraction (odds ratio [OR] = 1.53; 95% confidence interval [CI], 1.04-2.26; P = 0.03) and for early AMD (OR = 1.59; 95% CI, 1.04-2.44; P = 0.03), by comparison with subjects in the intermediate quartiles. Subjects in the lower quartile of UVR exposure also were at increased risk for early AMD (OR = 1.69; 95% CI, 1.06-2.69; P = 0.03), by comparison with those with medium exposure. Associations of late AMD with UVR exposure was not statistically significant. This study further confirms the increased risk for cataract extraction in subjects exposed to high ambient UVR. Moreover, it suggests that risk for early AMD is increased in subjects exposed to high UVR, but also to low UVR, by comparison with medium exposures

A Deletion in Exon 9 of the LIPH Gene Is Responsible for the Rex Hair Coat Phenotype in Rabbits (Oryctolagus cuniculus)

The fur of common rabbits is constituted of 3 types of hair differing in length and diameter while that of rex animals is essentially made up of amazingly soft down-hair. Rex short hair coat phenotypes in rabbits were shown to be controlled by three distinct loci. We focused on the “r1” mutation which segregates at a simple autosomal-recessive locus in our rabbit strains. A positional candidate gene approach was used to identify the rex gene and the corresponding mutation. The gene was primo-localized within a 40 cM region on rabbit chromosome 14 by genome scanning families of 187 rabbits in an experimental mating scheme. Then, fine mapping refined the region to 0.5 cM (Z = 78) by genotyping an additional 359 offspring for 94 microsatellites present or newly generated within the first defined interval. Comparative mapping pointed out a candidate gene in this 700 kb region, namely LIPH (Lipase Member H). In humans, several mutations in this major gene cause alopecia, hair loss phenotypes. The rabbit gene structure was established and a deletion of a single nucleotide was found in LIPH exon 9 of rex rabbits (1362delA). This mutation results in a frameshift and introduces a premature stop codon potentially shortening the protein by 19 amino acids. The association between this deletion and the rex phenotype was complete, as determined by its presence in our rabbit families and among a panel of 60 rex and its absence in all 60 non-rex rabbits. This strongly suggests that this deletion, in a homozygous state, is responsible for the rex phenotype in rabbits

A Deletion in Exon 9 of the LIPH Gene Is Responsible for the Rex Hair Coat Phenotype in Rabbits (Oryctolagus cuniculus)

The fur of common rabbits is constituted of 3 types of hair differing in length and diameter while that of rex animals is essentially made up of amazingly soft down-hair. Rex short hair coat phenotypes in rabbits were shown to be controlled by three distinct loci. We focused on the “r1” mutation which segregates at a simple autosomal-recessive locus in our rabbit strains. A positional candidate gene approach was used to identify the rex gene and the corresponding mutation. The gene was primo-localized within a 40 cM region on rabbit chromosome 14 by genome scanning families of 187 rabbits in an experimental mating scheme. Then, fine mapping refined the region to 0.5 cM (Z = 78) by genotyping an additional 359 offspring for 94 microsatellites present or newly generated within the first defined interval. Comparative mapping pointed out a candidate gene in this 700 kb region, namely LIPH (Lipase Member H). In humans, several mutations in this major gene cause alopecia, hair loss phenotypes. The rabbit gene structure was established and a deletion of a single nucleotide was found in LIPH exon 9 of rex rabbits (1362delA). This mutation results in a frameshift and introduces a premature stop codon potentially shortening the protein by 19 amino acids. The association between this deletion and the rex phenotype was complete, as determined by its presence in our rabbit families and among a panel of 60 rex and its absence in all 60 non-rex rabbits. This strongly suggests that this deletion, in a homozygous state, is responsible for the rex phenotype in rabbits