The Mapper algorithm is an essential tool for visualizing complex, high dimensional data in topology data analysis (TDA) and has been widely used in biomedical research. It outputs a combinatorial graph whose structure implies the shape of the data. However,the need for manual parameter tuning and fixed intervals, along with fixed overlapping ratios may impede the performance of the standard Mapper algorithm. Variants of the standard Mapper algorithms have been developed to address these limitations, yet most of them still require manual tuning of parameters. Additionally, many of these variants, including the standard version found in the literature, were built within a deterministic framework and overlooked the uncertainty inherent in the data. To relax these limitations, in this work, we introduce a novel framework that implicitly represents intervals through a hidden assignment matrix, enabling automatic parameter optimization via stochastic gradient descent. In this work, we develop a soft Mapper framework based on a Gaussian mixture model(GMM) for flexible and implicit interval construction. We further illustrate the robustness of the soft Mapper algorithm by introducing the Mapper graph mode as a point estimation for the output graph. Moreover, a stochastic gradient descent algorithm with a specific topological loss function is proposed for optimizing parameters in the model. Both simulation and application studies demonstrate its effectiveness in capturing the underlying topological structures. In addition, the application to an RNA expression dataset obtained from the Mount Sinai/JJ Peters VA Medical Center Brain Bank (MSBB) successfully identifies a distinct subgroup of Alzheimer's Disease.

Motivation: The Mapper algorithm is an essential tool to explore shape of data in topology data analysis. With a dataset as an input, the Mapper algorithm outputs a graph representing the topological features of the whole dataset. This graph is often regarded as an approximation of a reeb graph of data. The classic Mapper algorithm uses fixed interval lengths and overlapping ratios, which might fail to reveal subtle features of data, especially when the underlying structure is complex. Results: In this work, we introduce a distribution guided Mapper algorithm named D-Mapper, that utilizes the property of the probability model and data intrinsic characteristics to generate density guided covers and provides enhanced topological features. Our proposed algorithm is a probabilistic model-based approach, which could serve as an alternative to non-prababilistic ones. Moreover, we introduce a metric accounting for both the quality of overlap clustering and extended persistence homology to measure the performance of Mapper type algorithm. Our numerical experiments indicate that the D-Mapper outperforms the classical Mapper algorithm in various scenarios. We also apply the D-Mapper to a SARS-COV-2 coronavirus RNA sequences dataset to explore the topological structure of different virus variants. The results indicate that the D-Mapper algorithm can reveal both vertical and horizontal evolution processes of the viruses. Availability: Our package is available at https://github.com/ShufeiGe/D-Mapper.

Understanding and predicting rock mechanical properties, such as elastic moduli, plays a crucial role across a range of geoscience and engineering fields including energy resources engineering (Sone and Zoback, 2013; Madhubabu et al., 2016), geotechnical engineering (Zhang et al., 2021), and seismology (Byerlee and Brace, 1968). These properties govern how rocks react to in-situ stresses, shaping the macroscopic behaviors of geological structures. Accurate characterization of these properties is therefore instrumental in predicting phenomena such as seismic wave propagation, reservoir behavior, and slope stability. Fundamentally, these macroscopic behaviors originate from the intricate features at the microscopic scale. Specifically, the effective elastic moduli of rocks are influenced by three main factors: (1) the composition of pores and minerals, (2) the properties of these constituents, and (3) the geometric details of the rock's microstructure (Mavko et al., 2020). While the first two factors can be relatively straightforwardly measured and characterized, capturing the complexity of the geometric details often presents a substantial challenge. Historically, traditional micromechanics homogenizations and rock physics relied on empirical relationships or theoretical models based on idealized microstructures to estimate rock properties (Li and Wang, 2008; Han et al., 1986; Mindlin, 1949; Hill, 1952).

Topological Data Analysis (TDA) is a mathematical method using techniques from topology for the analysis of complex, multi-dimensional data that has been widely and successfully applied in several fields such as medicine, material science, biology, and others. This survey summarizes the state of the art of TDA in yet another application area: industrial manufacturing and production in the context of Industry 4.0. We perform a rigorous and reproducible literature search of applications of TDA on the setting of industrial production and manufacturing. The resulting works are clustered and analyzed based on their application area within the manufacturing process and their input data type. We highlight the key benefits of TDA and their tools in this area and describe its challenges, as well as future potential. Finally, we discuss which TDA methods are underutilized in (the specific area of) industry and the identified types of application, with the goal of prompting more research in this profitable area of application.

Visual object recognition in unseen and cluttered indoor environments is a challenging problem for mobile robots. Toward this goal, we extend our previous work to propose the TOPS2 descriptor, and an accompanying recognition framework, THOR2, inspired by a human reasoning mechanism known as object unity. We interleave color embeddings obtained using the Mapper algorithm for topological soft clustering with the shape-based TOPS descriptor to obtain the TOPS2 descriptor. THOR2, trained using synthetic data, achieves substantially higher recognition accuracy than the shape-based THOR framework and outperforms RGB-D ViT on two real-world datasets: the benchmark OCID dataset and the UW-IS Occluded dataset. Therefore, THOR2 is a promising step toward achieving robust recognition in low-cost robots.

With the advent of Big Data, algorithms that try to extract information from them are ubiquitous. Clustering algorithms are a subcategory of Machine Learning algorithms with a wide range of applications. Notions like closeness, distance and shape are central to clustering. It is then natural to try to use ideas and techniques from topology to improve clustering algorithms. This chapter examines some ways on how this could happen. In Section 2 a brief introduction to the clustering task is presented. In Subsection 2.1 a definition is presented along with notation.

This work brings methods from topological data analysis to knot theory and develops new data analysis tools inspired by this application. We explore a vast collection of knot invariants and relations between then using Mapper and Ball Mapper algorithms. In particular, we develop versions of the Ball Mapper algorithm that incorporate symmetries and other relations within the data, and provide ways to compare data arising from different descriptors, such as knot invariants. Additionally, we extend the Mapper construction to the case where the range of the lens function is high dimensional rather than a 1-dimensional space, that also provides ways of visualizing functions between high-dimensional spaces. We illustrate the use of these techniques on knot theory data and draw attention to potential implications of our findings in knot theory.

The emerging subject of deep learning mathematical analysis [1] has been tasked with answering some "mysterious" facts that appear to be inexplicable using traditional mathematical methodologies. They are attempting to comprehend what a neural network actually does. Deep Neural Networks (DNN) transform data at each layer, producing a new representation as output. DNN attempts to divide data in a classification problem, enhancing this action layer by layer until it reaches an output layer when DNN provides its best possible result. Under the manifold hypothesis (natural data creates lower-dimensional manifolds in its embedding space), this task can be viewed as the separation of lower-dimensional manifolds in a data space. DNN layers are linked by a realization function, Φ (an affine transformation) and a component-wise activation function, ρ. Consider the fully connected feedforward neural network depicted in Figure 2. The network architecture can be described by defining the number of layers N, L, the number of neurons, and the activation function.

We introduce giotto-tda, a Python library that integrates high-performance topological data analysis with machine learning via a scikit-learn-compatible API and state-of-the-art C++ implementations. The library's ability to handle various types of data is rooted in a wide range of preprocessing techniques, and its strong focus on data exploration and interpretability is aided by an intuitive plotting API. Source code, binaries, examples, and documentation can be found at https://github.com/giotto-ai/giotto-tda.

--T opological data analysis aims to extract topological quantities from data, which tend to focus on the broader global structure of the data rather than local information. The Mapper method, specifically, generalizes clustering methods to identify significant global mathematical structures, which are out of reach of many other approaches. We propose a classifier based on applying the Mapper algorithm to data projected onto a latent space. We obtain the latent space by using PCA or autoencoders. Notably, a classifier based on the Mapper method is immune to any gradient based attack, and improves robustness over traditional CNNs (convolutional neural networks). We report theoretical justification and some numerical experiments that confirm our claims. I NTRODUCTION Deep neural networks [1], [2] are well known to be not robust with respect to input image perturbations, which are designed by adding to images perturbations that are typically non-perceptible by humans [3]-[5]. In this paper we explore opportunities for combining deep learning techniques with a well known topological data analysis (TDA) algorithm - the Mapper algorithm [6], which we use to create classifiers with improved robustness. First, the training data is projected onto a latent space. The latent space in the simplest variant is constructed using PCA components, and we also use nonlinear projections by utilizing various autoencoders [1], [7]-[9]. Then, a discrete graph representation (Mapper) is assigned to the training data projected onto the latent space .