Sparse clustering of functional data

We consider the problem of clustering functional data while jointly selecting the most relevant features for classification. 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 under reasonable assumptions. Both the multivariate and the functional versions of sparse clustering with hard thresholding exhibit improvements on other standard and sparse clustering strategies on simulated data. A real functional data application is also shown.