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Many scientific datasets are of high dimension, and the analysis usuallyrequires visual manipulation by retaining the most important structures ofdata. Principal curve is a widely used approach for this purpose. However, manyexisting methods work only for data with structures that are notself-intersected, which is quite restrictive for real applications. A fewmethods can overcome the above problem, but they either require complicatedhuman-made rules for a specific task with lack of convergence guarantee andadaption flexibility to different tasks, or cannot obtain explicit structuresof data. To address these issues, we develop a new regularized principal graphlearning framework that captures the local information of the underlying graphstructure based on reversed graph embedding. As showcases, models that canlearn a spanning tree or a weighted undirected $\ell_1$ graph are proposed, anda new learning algorithm is developed that learns a set of principal points anda graph structure from data, simultaneously. The new algorithm is simple withguaranteed convergence. We then extend the proposed framework to deal withlarge-scale data. Experimental results on various synthetic and six real worlddatasets show that the proposed method compares favorably with baselines andcan uncover the underlying structure correctly.

A Novel Regularized Principal Graph Learning Framework on Explicit Graph Representation