A powerful concept behind much of the recent progress in machine learning is the extraction of common features across data from heterogeneous sources or tasks. Intuitively, using all of one's data to learn a common representation function benefits both computational effort and statistical generalization by leaving a smaller number of parameters to fine-tune on a given task. Toward theoretically grounding these merits, we propose a general setting of recovering linear operators $M$ from noisy vector measurements $y = Mx + w$, where the covariates $x$ may be both non-i.i.d. and non-isotropic. We demonstrate that existing isotropy-agnostic representation learning approaches incur biases on the representation update, which causes the scaling of the noise terms to lose favorable dependence on the number of source tasks. This in turn can cause the sample complexity of representation learning to be bottlenecked by the single-task data size. We introduce an adaptation, $\texttt{De-bias & Feature-Whiten}$ ($\texttt{DFW}$), of the popular alternating minimization-descent scheme proposed independently in Collins et al., (2021) and Nayer and Vaswani (2022), and establish linear convergence to the optimal representation with noise level scaling down with the $\textit{total}$ source data size. This leads to generalization bounds on the same order as an oracle empirical risk minimizer. We verify the vital importance of $\texttt{DFW}$ on various numerical simulations. In particular, we show that vanilla alternating-minimization descent fails catastrophically even for iid, but mildly non-isotropic data. Our analysis unifies and generalizes prior work, and provides a flexible framework for a wider range of applications, such as in controls and dynamical systems.
College of Computer Science and Technology, Zhejiang University（浙江大学计算机科学与技术学院）；School of Public Affairs, Zhejiang University（浙江大学公共管理学院）；Tongdun Technology（同盾科技）
Kernel Regression in Structured Non-IID Settings: Theory and Implications for Denoising Score Learning
Institute of Data Science, The University of Hong Kong（数据科学研究所，香港大学）；Computer Sciences Department, University of Wisconsin-Madison（计算机科学系，威斯康星大学麦迪逊分校）；Institute of Data Science and School of Computing & Data Science, The University of Hong Kong（数据科学研究所和计算与数据科学学院，香港大学）；School of Computing & Data Science and Institute of Data Science, The University of Hong Kong（计算与数据科学学院和数据科学研究所，香港大学）
