Sparse clustering of functional data in presence of misalignment
Valeria Vitelli

Finding sparse solutions to clustering problems has emerged as a hot topic in statistics in recent years, due to the technological improvements in measurement systems leading to the spread of high-dimensional data in many real applications. This topic has started very recently to emerge in the literature on functional data, when it is often of much interest to select the curves’ most relevant features while jointly solving a classification problem. Functional sparse clustering can be analytically defined as a variational problem with a hard thresholding constraint ensuring the sparsity of the solution: this problem is shown to be well-posed, to have a unique optimal solution, and to provide good insights in real applications.
When dealing with curve clustering we cannot forget the presence of misalingment: this is a frequent situation in functional data analysis problems. Many methods to jointly cluster and align curves, which efficiently decouple amplitude and phase variability, have already been proposed in the literature on functional data. By focusing on one of these methods, I propose a possible approach to jointly deal with sparse functional clustering while also aligning the curves. I first frame a "brute force approach" to the problem of joint sparse clustering and alignment, and show some promising preliminary results on simulated data. I then try to formalize the necessary theoretical tools to solve this problem, and sketch the next steps.
I will conclude with a vision of the possible future research directions on the topic.