Sparse Clustering of Functional Data

TL;DR

This paper introduces a novel method for clustering functional data that incorporates feature selection through a sparsity constraint, with proven existence and uniqueness of solutions in finite and infinite dimensions, demonstrating improved performance over existing methods.

Contribution

It is the first to define and solve a sparse clustering problem specifically for functional data using a variational approach with hard thresholding.

Findings

Improved clustering accuracy on simulated data.

Proven existence and uniqueness of solutions.

Successful application to real functional data.

Abstract

We consider the problem of clustering functional data while jointly selecting the most relevant features for classification. This problem has never been tackled before in the functional data context, and it requires a proper definition of the concept of sparsity for functional data. Functional sparse clustering is here analytically defined as a variational problem with a hard thresholding constraint ensuring the sparsity of the solution. First, a unique solution to sparse clustering with hard thresholding in finite dimensions is proved to exist. Then, the infinite dimensional generalization is given and proved to have a unique solution. Both the multivariate and the functional version of sparse clustering with hard thresholding exhibits improvements on other standard and sparse clustering strategies on simulated data. A real functional data application is also shown.