Shape-Informed Clustering of Multi-Dimensional Functional Data via Deep Functional Autoencoders
Samuel Singh, Shirley Coyle, Mimi Zhang
We introduce FAEclust, a novel functional autoencoder framework for cluster analysis of multi-dimensional functional data, data that are random realizations of vector-valued random functions. Our framework features a functional encoder that captures complex nonlinear interdependencies among component functions and a universal approximator decoder capable of accurately reconstructing both Euclidean and manifold-valued functional data. Stability and robustness are enhanced through innovative regularization strategies applied to functional weights and biases.  Additionally, we incorporate a clustering loss into the network's training objective, promoting the learning of latent representations that are conducive to effective clustering. A key innovation is our shape-informed clustering objective, ensuring that the clustering results are resistant to phase variations in the functions.  We establish the universal approximation property of our non-linear decoder and validate the effectiveness of our model through extensive experiments.