🤖 AI Summary
This paper addresses functional linear regression under discrete, sparse, and incomplete observations in distributed settings. To tackle the practical challenges of irregular sampling and limited data per node, we propose a distributed spectral learning algorithm based on Sobolev kernels, enabling accurate modeling from finite-point observations. Our key contributions are threefold: (i) We establish the first tight convergence bounds—matching upper and lower rates—for distributed spectral learning in Sobolev spaces, explicitly characterizing the fundamental interplay between function regularity and convergence rate; (ii) We develop a regularization framework tailored to discrete observations, achieving minimax-optimal convergence rates in the Sobolev norm; (iii) Theoretical analysis validates the appropriateness of our regularity assumptions and demonstrates superior tightness compared to existing distributed functional learning methods, significantly enhancing both theoretical precision and practical applicability.
📝 Abstract
By selecting different filter functions, spectral algorithms can generate various regularization methods to solve statistical inverse problems within the learning-from-samples framework. This paper combines distributed spectral algorithms with Sobolev kernels to tackle the functional linear regression problem. The design and mathematical analysis of the algorithms require only that the functional covariates are observed at discrete sample points. Furthermore, the hypothesis function spaces of the algorithms are the Sobolev spaces generated by the Sobolev kernels, optimizing both approximation capability and flexibility. Through the establishment of regularity conditions for the target function and functional covariate, we derive matching upper and lower bounds for the convergence of the distributed spectral algorithms in the Sobolev norm. This demonstrates that the proposed regularity conditions are reasonable and that the convergence analysis under these conditions is tight, capturing the essential characteristics of functional linear regression. The analytical techniques and estimates developed in this paper also enhance existing results in the previous literature.