Multiple Penalized Principal Curves: Analysis and Computation

Abstract

We study the problem of determining the one-dimensional structure that best represents a given data set. More precisely, we take a variational approach to approximating a given measure (data) by curves. We consider an objective functional whose minimizers are a regularization of principal curves and introduce a new functional which allows for multiple curves. We prove existence of minimizers and investigate their properties. While both of the functionals used are non-convex, we show that enlarging the configuration space to allow for multiple curves leads to a simpler energy landscape with fewer undesirable (high-energy) local minima. We provide an efficient algorithm for approximating minimizers of the functional and demonstrate its performance on real and synthetic data. The numerical examples illustrate the effectiveness of the proposed approach in the presence of substantial noise, and the viability of the algorithm for high-dimensional data.

Principal Manifold Estimation Via Model Complexity Selection

“…Future work may investigate the generalization of our proposed framework, for which the HS principal curve algorithm and PCA are special cases, to counterparts on Riemannian manifolds and in Wasserstein spaces. Kirov and Slepčev (2017) introduced a multiple penalized principal curve framework permitting a fitted 1‐dimensional structure to consist of several disconnected curves. Our proposed framework may be generalized to fit a high‐dimensional structure consisting of several disconnected components.…”

Analyzing Multimodal Probability Measures with Autoencoders

“…We present here such a strategy. Related strategies were suggested for principal curves (see Section II.B of ref and Section 3.5 of ref for instance), and actually tested in ref .…”

Average-distance problem with curvature penalization for data parameterization: regularity of minimizers

“…In terms of the regularity of curvature these results are similar (while the techniques are different) to results in [28,29] where the penalized problem was studied. Kirov and one of authors [24] relaxed the Problem 1.2 to allow for multiple curves, and developed an efficient algorithm for both Problem 1.2 and the relaxed problem. Delicado [14] introduced a new notion of principal curves based on the so-called "principal oriented points".…”