A Scalable Operator-Valued Kernel Learning Framework for Continuous Multi-Way Regression
Continuous multi-way data arise in applications such as climate modeling, neuroimaging, and multimedia analytics, where both covariates and responses are indexed over structured spatial or temporal continuous domains. Existing methods discretize such data as fixed-grid arrays and use tensor regression methods. Functional regression instead models observations directly over continuous domains. Yet existing methods are primarily developed for one-way predictors and responses and rely heavily on finite-dimensional basis representations. To fill this gap, we develop FRCOMA, a scalable operator-valued kernel learning framework for nonlinear regression with heterogeneous continuous multi-way covariates and responses. The proposed method uses separable operator-valued kernels to decouple input similarity from output functional dependence, leading to a Kronecker-structured kernel system that can be solved through spectral computation. We further introduce sparse kernel weights to identify informative covariates and develop a block coordinate descent algorithm for joint estimation. We establish a representer theorem, support recovery guarantees, and algorithmic convergence results under regularity conditions. Numerical studies and applications to movie-trailer engagement prediction and climate-field modeling show that FRCOMA improves prediction and variable selection relative to functional and tensor regression baselines, especially when continuity and heterogeneous multi-way structure are important.