The machinery of topological data analysis becomes increasingly popular in a broad range of machine learning tasks, ranging from anomaly detection and manifold learning to graph classification. Persistent homology is one of the key approaches here, allowing us to systematically assess the evolution of various hidden patterns in the data as we vary a scale parameter. The extracted patterns, or homological features, along with information on how long such features persist throughout the considered filtration of a scale parameter, convey a critical insight into salient data characteristics and data organization.
In this work, we introduce two new and easily interpretable topological summaries for single and multi-parameter persistence, namely, saw functions and multi-persistence grid functions, respectively. Compared to the existing topological summaries which tend to assess the numbers of topological features and/or their lifespans at a given filtration step, our proposed saw and multi-persistence grid functions allow us to explicitly account for essential complementary information such as the numbers of births and deaths at each filtration step.
These new topological summaries can be regarded as the complexity measures of the evolving subspaces determined by the filtration and are of particular utility for applications of persistent homology on graphs. We derive theoretical guarantees on the stability of the new saw and multi-persistence grid functions and illustrate their applicability for graph classification tasks.
Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures
DataShape, Centre Inria d’Université Côte d’Azur（DataShape，蔚蓝海岸大学Inria中心）；Mathematics, Vrije Universiteit Amsterdam（阿姆斯特丹自由大学数学学院）；GeomeriX, Inria Saclay and École polytechnique（GeomeriX，法国国家信息与自动化研究所萨克雷中心、巴黎综合理工学院）；Mathematics, University of Oxford（牛津大学数学学院）
University of Pennsylvania（宾夕法尼亚大学）；BlueLight AI（蓝光大话人工智能）
Multiparameter persistent homology has been largely neglected as an input to machine learning algorithms. We consider the use of lattice-based convolutional neural network layers as a tool for the analysis of features arising from multiparameter persistence modules. We find that these show promise as an alternative to convolutions for the classification of multidimensional persistence modules.
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 persistenc…
As the field of Topological Data Analysis continues to show success in theory and in applications, there has been increasing interest in using tools from this field with methods for machine learning. Using persistent homology, specifically persistence diagrams, as inputs to machine learning techniques requires some mathematical creativity. The space of persistence diagrams does not have the desirable properties for machine learning, thus methods such as kernel methods and vectorization methods have been developed. One such featurization of persistence diagrams by Perea, Munch and Khasawneh use…
Novartis（诺华）；University of Texas at Dallas（德克萨斯大学达拉斯分校）；Jet Propulsion Laboratory, Caltech（加州理工学院喷气推进实验室）；Temple University（天普大学）
In computer-aided drug discovery (CADD), virtual screening (VS) is used for identifying the drug candidates that are most likely to bind to a molecular target in a large library of compounds. Most VS methods to date have focused on using canonical compound representations (e.g., SMILES strings, Morgan fingerprints) or generating alternative fingerprints of the compounds by training progressively more complex variational autoencoders (VAEs) and graph neural networks (GNNs). Although VAEs and GNNs led to significant improvements in VS performance, these methods suffer from reduced performance wh…
University at Buffalo, SUNY（纽约州立大学布法罗分校）；POSTECH（浦项科技大学）
Persistent Homology (PH) is a useful tool to study the underlying structure of a data set. Persistence Diagrams (PDs), which are 2D multisets of points, are a concise summary of the information found by studying the PH of a data set. However, PDs are difficult to incorporate into a typical machine learning workflow. To that end, two main methods for representing PDs have been developed: kernel methods and vectorization methods. In this paper we propose a new finite-dimensional vector, called the interconnectivity vector, representation of a PD adapted from Bag-of-Words (BoW). This new represen…
Université Côte d’Azur（蔚蓝海岸大学）；Inria（法国国家信息与自动化研究所）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；LIGM（几何与模型化实验室）；Université Gustave Eiffel（古斯塔夫·埃菲尔大学）；The University of Tokyo（东京大学）；Fujitsu Limited（富士通有限公司）
The use of topological descriptors in modern machine learning applications, such as Persistence Diagrams (PDs) arising from Topological Data Analysis (TDA), has shown great potential in various domains. However, their practical use in applications is often hindered by two major limitations: the computational complexity required to compute such descriptors exactly, and their sensitivity to even low-level proportions of outliers. In this work, we propose to bypass these two burdens in a data-driven setting by entrusting the estimation of (vectorization of) PDs built on top of point clouds to a n…
A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science
a Department of Mathematics, Colorado State University, Fort Collins, CO b Dept.\ of Electrical；Computer Engineering, Colorado State University, Fort Collins, CO c Cooperative Institute for Research in the Atmosphere (CIRA), Colorado State University, Fort Collins, CO
