Abstract
Recursive feature machines (RFMs) learn representations of data by alternating between fitting a predictor to a dataset and updating features of that predictor using the average gradient outer product (AGOP). Connections between AGOPs and feature learning in neural networks motivate linear RFMs as a simple setting for analyzing how representations evolve during training. Here, we study the dynamics and statistics of linear RFM in noisy multi-output regression with isotropic sub-Gaussian input data and targets generated by a low-rank teacher matrix of dimension $d$. We extend the known connection between linear RFM and iteratively reweighted least squares from the interpolating setting to ridge-regularized multi-output regression with noise. We show that the learned feature matrix remains close to its infinite-data ideal counterpart at every iteration. Namely, for $n$ samples, we show the error in the feature matrix decays as $O(\sqrt{d/n})$ with high probability. Experiments on real-world text and single-cell gene-expression data illustrate the features learned by this simple linear model.
Keywords
Publication details
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Cite this article
APA 7
Cheng, A., Kiani, B. T., Lu, Y. M., & Radhakrishnan, A. (2026). Exact Dynamics and Finite-Sample Trajectory Recovery of Linear Recursive Feature Machines. https://omanscience.com/en/articles/exact-dynamics-and-finite-sample-trajectory-recovery-of-linear-recursive-feature-machines
MLA 9
Cheng, Andrew, et al. "Exact Dynamics and Finite-Sample Trajectory Recovery of Linear Recursive Feature Machines." https://omanscience.com/en/articles/exact-dynamics-and-finite-sample-trajectory-recovery-of-linear-recursive-feature-machines.
Chicago (author–date)
Cheng, Andrew, Bobak T. Kiani, Yue M. Lu, and Adityanarayanan Radhakrishnan. 2026. "Exact Dynamics and Finite-Sample Trajectory Recovery of Linear Recursive Feature Machines." https://omanscience.com/en/articles/exact-dynamics-and-finite-sample-trajectory-recovery-of-linear-recursive-feature-machines.
Harvard
Cheng, A., Kiani, B. T., Lu, Y. M. and Radhakrishnan, A. (2026) 'Exact Dynamics and Finite-Sample Trajectory Recovery of Linear Recursive Feature Machines', Available at: https://omanscience.com/en/articles/exact-dynamics-and-finite-sample-trajectory-recovery-of-linear-recursive-feature-machines.
Vancouver
Cheng A, Kiani BT, Lu YM, Radhakrishnan A. Exact Dynamics and Finite-Sample Trajectory Recovery of Linear Recursive Feature Machines. https://omanscience.com/en/articles/exact-dynamics-and-finite-sample-trajectory-recovery-of-linear-recursive-feature-machines
IEEE
A. Cheng, B. T. Kiani, Y. M. Lu, and A. Radhakrishnan, "Exact Dynamics and Finite-Sample Trajectory Recovery of Linear Recursive Feature Machines," https://omanscience.com/en/articles/exact-dynamics-and-finite-sample-trajectory-recovery-of-linear-recursive-feature-machines.