用深度自编码器分析多维函数数据，自动捕捉形状特征实现稳定聚类。
Shape-Informed Clustering of Multi-Dimensional Functional Data via Deep Functional Autoencoders
- 通过非线性编码器学习多维函数间的复杂依赖关系。
- 引入形状感知聚类目标，有效抵抗函数相位变化影响。
- 适用于具有相位差异的多维函数数据聚类任务。
我们提出FAEclust，一种用于多维函数数据分析的新颖函数自编码器框架，该数据为向量值随机函数的随机实现。框架包含通用逼近编码器，可捕捉各分量函数间的复杂非线性依赖；以及通用逼近解码器，能准确重构欧氏与流形值函数数据。通过针对函数权重和偏置的创新正则化策略提升模型稳定性和鲁棒性。同时，在训练目标中加入聚类损失，促进学习有利于聚类的潜在表示。关键创新在于形状感知聚类目标，确保聚类结果对函数相位变化不敏感。我们建立了非线性解码器的通用逼近性质，并通过大量实验验证了模型的有效性。
原文摘要 · Abstract (English)
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.
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