A Mapper Algorithm with implicit intervals and its optimization

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.

A distribution-guided Mapper algorithm

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.