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 universal-approximator 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.

翻译：我们提出FAEclust，一种用于多维函数数据聚类分析的新型函数自编码器框架，该数据是向量值随机函数的随机实现。我们的框架采用具有通用逼近能力的编码器来捕捉分量函数间复杂的非线性相互依赖关系，以及能够精确重构欧几里得和流形值函数数据的通用逼近解码器。通过对函数权重和偏置应用创新的正则化策略，增强了模型的稳定性和鲁棒性。此外，我们在网络训练目标中引入了聚类损失，以促进学习有利于有效聚类的潜在表示。一个关键创新是我们的形状感知聚类目标，该目标确保聚类结果对函数的相位变化具有鲁棒性。我们建立了非线性解码器的通用逼近特性，并通过大量实验验证了模型的有效性。