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Linear Recursive Feature Machines provably recover low-rank matrices.

Adityanarayanan Radhakrishnan^1,2, Mikhail Belkin³, Dmitriy Drusvyatskiy⁴

¹Applied Math, Harvard University, MA 02138.

Proceedings of the National Academy of Sciences of the United States of America

|March 28, 2025

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Recursive Feature Machines (RFMs) offer a new approach to understanding neural network feature learning. This method explicitly performs dimensionality reduction, outperforming deep linear networks in sparse recovery tasks.

Area of Science:

Machine Learning
Statistical Inference
Computational Theory

Background:

Neural networks excel at predictions, seemingly defying the curse of dimensionality.
Feature learning, a form of dimensionality reduction, is a hypothesized reason for this success.
The average gradient outer product (AGOP) is a statistical estimator linked to feature learning.

Purpose of the Study:

To provide theoretical guarantees for Recursive Feature Machines (RFMs) in dimensionality reduction.
To connect feature learning in neural networks with classical sparse recovery algorithms.
To develop a scalable and efficient implementation of RFMs for practical applications.

Main Methods:

Analyzing Recursive Feature Machines (RFMs) for overparameterized problems in sparse linear regression and low-rank matrix recovery.

Keywords:

feature learning matrix sensing neural networks sparse recovery

Related Experiment Videos

Demonstrating that linear RFMs (lin-RFMs) are equivalent to a variant of Iteratively Reweighted Least Squares (IRLS).

Implementing lin-RFM to handle large-scale matrices with numerous missing entries.

Main Results:

Theoretical guarantees are established for RFM's dimensionality reduction capabilities.
Lin-RFMs are shown to be a variant of the IRLS algorithm.
The developed lin-RFM implementation is faster than standard IRLS and outperforms deep linear networks.

Conclusions:

RFMs provide explicit feature learning, offering insights into neural network behavior.
The study bridges the gap between neural network feature learning and classical sparse recovery.
The efficient lin-RFM implementation enables practical application in large-scale sparse recovery and matrix completion.