Multiparameter Persistence Images for Topological Machine Learning

International audienceIn the last decade, there has been increasing interest in topological data analysis, a new methodology for using geometric structures in data for inference and learning. A central theme in the area is the idea of persistence, which in its most basic form studies how measures of shape change as a scale parameter varies. There are now a number of frameworks that support statistics and machine learning in this context. However, in many applications there are several different parameters one might wish to vary: for example, scale and density. In contrast to the one-parameter setting, techniques for applying statistics and machine learning in the setting of multiparameter persistence are not well understood due to the lack of a concise representation of the results. We introduce a new descriptor for multiparameter persistence, which we call the Multiparameter Persistence Image, that is suitable for machine learning and statistical frameworks, is robust to perturbations in the data, has finer resolution than existing descriptors based on slicing, and can be efficiently computed on data sets of realistic size. Moreover, we demonstrate its efficacy by comparing its performance to other multiparameter descriptors on several classification tasks

Efficient Approximation of Multiparameter Persistence Modules

Topological Data Analysis is a growing area of data science, which aims at computing and characterizing the geometry and topology of data sets, in order to produce useful descriptors for subsequent statistical and machine learning tasks. Its main computational tool is persistent homology, which amounts to track the topological changes in growing families of subsets of the data set itself, called ltrations, and encode them in an algebraic object, called persistence module. Even though algorithms and theoretical properties of modules are now well-known in the single-parameter case, that is, when there is only one ltration to study, much less is known in the multi-parameter case, where several ltrations are given at once. ough more complicated, the resulting persistence modules are usually richer and encode more information, making them be er descriptors for data science. In this article, we present the rst approximation scheme, which is based on bered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence, for computing and decomposing general multi-parameter persistence modules. Our algorithm has controlled complexity and running time, and works in arbitrary dimension, i.e., with an arbitrary number of ltrations. Moreover, when restricting to speci c classes of multi-parameter persistence modules, namely the ones that can be decomposed into intervals, we establish theoretical results about the approximation error between our estimate and the true module in terms of interleaving distance. Finally, we present empirical evidence validating output quality and speed-up on several data sets

Local Signatures using Persistence Diagrams

In this article, we address the problem of devising signatures using the framework of persistent homology.Considering a compact length space with curvature bounded above, we build, either for every point or for the shape itself, a topological signature that is provably stable to perturbations of the space in the Gromov-Hausdorff distance. This signature has been used in 3D shape analysis tasks, such as shape segmentation and matching. Here, we provide general statements and formal proofs of stability for this signature

Topology identifies emerging adaptive mutations in SARS-CoV-2

The COVID-19 pandemic has lead to a worldwide effort to characterize its evolution through the mapping of mutations in the genome of the coronavirus SARS-CoV-2. Ideally, one would like to quickly identify new mutations that could confer adaptive advantages (e.g. higher infectivity or immune evasion) by leveraging the large number of genomes. One way of identifying adaptive mutations is by looking at convergent mutations, mutations in the same genomic position that occur independently. However, the large number of currently available genomes precludes the efficient use of phylogeny-based techniques. Here, we establish a fast and scalable Topological Data Analysis approach for the early warning and surveillance of emerging adaptive mutations based on persistent homology. It identifies convergent events merely by their topological footprint and thus overcomes limitations of current phylogenetic inference techniques. This allows for an unbiased and rapid analysis of large viral datasets. We introduce a new topological measure for convergent evolution and apply it to the GISAID dataset as of February 2021, comprising 303,651 high-quality SARS-CoV-2 isolates collected since the beginning of the pandemic. We find that topologically salient mutations on the receptor-binding domain appear in several variants of concern and are linked with an increase in infectivity and immune escape, and for many adaptive mutations the topological signal precedes an increase in prevalence. We show that our method effectively identifies emerging adaptive mutations at an early stage. By localizing topological signals in the dataset, we extract geo-temporal information about the early occurrence of emerging adaptive mutations. The identification of these mutations can help to develop an alert system to monitor mutations of concern and guide experimentalists to focus the study of specific circulating variants

Optimizing persistent homology based functions

Solving optimization tasks based on functions and losses with a topological flavor is a very active,growing field of research in data science and Topological Data Analysis, with applications in non-convexoptimization, statistics and machine learning. However, the approaches proposed in the literatureare usually anchored to a specific application and/or topological construction, and do not come withtheoretical guarantees. To address this issue, we study the differentiability of a general map associatedwith the most common topological construction, that is, the persistence map. Building on real analyticgeometry arguments, we propose a general framework that allows us to define and compute gradientsfor persistence-based functions in a very simple way. We also provide a simple, explicit and sufficientcondition for convergence of stochastic subgradient methods for such functions. This result encompassesall the constructions and applications of topological optimization in the literature. Finally, we provideassociated code, that is easy to handle and to mix with other non-topological methods and constraints, aswell as some experiments showcasing the versatility of our approach

Persistent homology based characterization of the breast cancer immune microenvironment: a feasibility study

International audiencePersistent homology is a powerful tool in topological data analysis. The main output, persistence diagrams, encode the geometry and topology of given datasets. We present a novel application of persistent homology to characterize the biological environment surrounding breast cancers, known as the tumor microenvironment. Specifically, we will characterize the spatial arrangement of immune and malignant epithelial (tumor) cells within the breast cancer immune microenvironment. Quantitative and robust characterizations are built by computing persistence diagrams from quantitative multiplex immunofluorescence, which is a technology which allows us to obtain spatial coordinates and protein intensities on individual cells. The resulting persistence diagrams are evaluated as characteristic biomarkers predictive of cancer subtype and prognostic of overall survival. For a cohort of approximately 700 breast cancer patients with median 8.5-year clinical follow-up, we show that these persistence diagrams outperform and complement the usual descriptors which capture spatial relationships with nearest neighbor analysis. Our results thus suggest new methods which can be used to build topology-based biomarkers which are characteristic and predictive of cancer subtype and response to therapy as well as prognostic of overall survival

Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphs

International audienceAlthough neural networks are capable of reaching astonishing performances on a wide variety of contexts, properly training networks on complicated tasks requires expertise and can be expensive from a computational perspective. In industrial applications, data coming from an open-world setting might widely differ from the benchmark datasets on which a network was trained. Being able to monitor the presence of such variations without retraining the network is of crucial importance. In this article, we develop a method to monitor trained neural networks based on the topological properties of their activation graphs. To each new observation, we assign a Topological Uncertainty, a score that aims to assess the reliability of the predictions by investigating the whole network instead of its final layer only, as typically done by practitioners. Our approach entirely works at a post-training level and does not require any assumption on the network architecture, optimization scheme, nor the use of data augmentation or auxiliary datasets; and can be faithfully applied on a large range of network architectures and data types. We showcase experimentally the potential of Topological Uncertainty in the context of trained network selection, Out-Of-Distribution detection, and shift-detection, both on synthetic and real datasets of images and graphs