Exact Dynamics and Finite-Sample Trajectory Recovery of Linear Recursive Feature Machines

2026-10-06

AI Summary

This study addresses the unclear dynamical and statistical properties of linear Recursive Feature Machines (RFMs) in noisy multi-output regression. To overcome existing interpolation limitations, this work extends RFMs to noisy ridge regression, leveraging average gradient outer products and sub-Gaussian analysis to establish finite-sample probabilistic bounds while revealing an intrinsic connection to iteratively reweighted least squares. Theoretically, it proves that feature matrix errors converge at a rate of O(√d/n) under finite samples, rigorously characterizing the dynamics of feature evolution. Empirically, the effectiveness of the proposed framework is validated on real-world text and single-cell gene expression datasets.

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.

Problem

• Linear Recursive Feature Machines
• Multi-output Regression
• Feature Learning
• Finite-Sample Analysis
• Training Dynamics

Innovation

• Recursive Feature Machines
• Average Gradient Outer Product
• Iteratively Reweighted Least Squares
• Finite-Sample Trajectory Recovery
• Ridge-Regularized Regression