Abstract
Multidimensional datapoint clouds representing large datasets are frequently characterized by non-trivial low-dimensional geometry and topology which can be recovered by unsupervised machine learning approaches, in particular, by principal graphs. Principal graphs approximate the multivariate data by a graph injected into the data space with some constraints imposed on the node mapping. Here we present ElPiGraph, a scalable and robust method for constructing principal graphs. ElPiGraph exploits and further develops the concept of elastic energy, the topological graph grammar approach, and a gradient descent-like optimization of the graph topology. The method is able to withstand high levels of noise and is capable of approximating data point clouds via principal graph ensembles. This strategy can be used to estimate the statistical significance of complex data features and to summarize them into a single consensus principal graph. ElPiGraph deals efficiently with large datasets in various fields such as biology, where it can be used for example with single-cell transcriptomic or epigenomic datasets to infer gene expression dynamics and recover differentiation landscapes.
Highlights
One of the major trends in modern machine learning is the increasing use of ideas borrowed from multi-dimensional and information geometry
Topological data analysis (TDA) focuses on extracting persistent homologies of simplicial complexes derived from a datapoint cloud and exploits data space geometry [3]
ElPiGraph is able to construct an appropriate data approximator when no information is available on the underlying topology of the data by using the consensus principal graph approach
Summary
One of the major trends in modern machine learning is the increasing use of ideas borrowed from multi-dimensional and information geometry. Geometric data analysis (GDA) approaches treats datasets of various nature (multi-modal measurements, images, graph embeddings) as clouds of points in multi-dimensional space equipped with appropriate distance or similarity measures [1]. One of the useful concepts in modern data analysis is the complementarity principle formulated in [7]: the data space can be split into a low volume (low dimensional) subset, which requires nonlinear methods for constructing complex data approximators, and a high-dimensional subset, characterized by measure concentration and simplicity allowing the effective application of linear methods. Manifold learning methods and their generalizations using different mathematical objects (such as graphs) aim at revealing and characterizing the geometry and topology of the low-dimensional part of the datapoint cloud
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
