arXiv subjects

Ziyad Oulhaj

Publications and source records attributed to Ziyad Oulhaj.

3 recordsLinked to original sources

arXiv2026

Gromov-Wasserstein bound between Reeb and Mapper graphs

Since its introduction as a computable approximation of the Reeb graph, the Mapper graph has become one of the most popular tools from topological data analysis for performing data visualization and inference. However, finding an appropriate metric (that is, a tractable metric with theoretical guarantees) for comparing Reeb and Mapper graphs, in order to, e.g., quantify the rate of convergence of the Mapper graph to the Reeb graph, is a difficult problem. While several metrics have been proposed in the literature, none is able to incorporate measure information, when data points are sampled according to an underlying probability measure. The resulting Reeb and Mapper graphs are therefore purely deterministic and combinatorial, and substantial effort is thus required to ensure their statistical validity. In this article, we handle this issue by treating Reeb and Mapper graphs as metric measure spaces. This allows us to use Gromov-Wasserstein metrics to compare these graphs directly in order to better incorporate the probability measures that data points are sampled from. Then, we describe the geometry that arises from this perspective, and we derive rates of convergence of the Mapper graph to the Reeb graph in this context. Finally, we showcase the usefulness of such metrics for Reeb and Mapper graphs in a few numerical experiments.

math.ST↗

arXiv2024

Deep Mapper: Efficient Visualization of Plausible Conformational Pathways

Acquiring plausible pathways on high-dimensional structural distributions is beneficial in several domains. For example, in the drug discovery field, a protein conformational pathway, i.e. a highly probable sequence of protein structural changes, is useful to analyze interactions between the protein and the ligands, helping to create new drugs. Recently, a state-of-the-art method in drug discovery was presented, which efficiently computes protein pathways using latent variables obtained from an isometric auto-encoding of the space of 3D density maps associated to protein conformations. However, our preliminary experiments show that there is room to significantly reduce the computing time. In this study, we use the Mapper algorithm, which is a Topological Data Analysis method, and present a novel variant to extract plausible conformational pathways from the isometric latent space with comparatively short running time. The extracted pathways are visualized as paths on the resulting Mapper graph. The methodological novelties are described as follows: firstly, the filter function of the Mapper algorithm is optimized so as to extract the pathways via minimization of an energy loss defined on the Mapper graph itself, while filter functions taken in the classical Mapper algorithm are fixed beforehand. The optimization is with respect to parameters of a deep neural network in the filter. Secondly, the clustering method, which defines the vertices and edges of the Mapper graph, of our algorithm, is designed by incorporating domain prior knowledge to assist the extraction. In our numerical experiments, based on an isometric latent space built on the common 50S-ribosomal dataset, the resulting Mapper graph successfully includes all the well-recognized plausible pathways. Moreover, our running time is much shorter than the above state-of-the-art counterpart.

q-bio.QM↗

arXiv2024

Differentiable Mapper For Topological Optimization Of Data Representation

Unsupervised data representation and visualization using tools from topology is an active and growing field of Topological Data Analysis (TDA) and data science. Its most prominent line of work is based on the so-called Mapper graph, which is a combinatorial graph whose topological structures (connected components, branches, loops) are in correspondence with those of the data itself. While highly generic and applicable, its use has been hampered so far by the manual tuning of its many parameters-among these, a crucial one is the so-called filter: it is a continuous function whose variations on the data set are the main ingredient for both building the Mapper representation and assessing the presence and sizes of its topological structures. However, while a few parameter tuning methods have already been investigated for the other Mapper parameters (i.e., resolution, gain, clustering), there is currently no method for tuning the filter itself. In this work, we build on a recently proposed optimization framework incorporating topology to provide the first filter optimization scheme for Mapper graphs. In order to achieve this, we propose a relaxed and more general version of the Mapper graph, whose convergence properties are investigated. Finally, we demonstrate the usefulness of our approach by optimizing Mapper graph representations on several datasets, and showcasing the superiority of the optimized representation over arbitrary ones.

cs.LG↗