Matrix-Free Least Squares Solvers: Values, Gradients, and What to Do With Them

Hrittik Roy; Søren Hauberg; Nicholas Krämer

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arXiv · 2510.19634

Matrix-Free Least Squares Solvers: Values, Gradients, and What to Do With Them

Hrittik Roy·Søren Hauberg·Nicholas Krämer

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Abstract

This paper argues that the method of least squares has significant unfulfilled potential in modern machine learning, far beyond merely being a tool for fitting linear models. To release its potential, we derive custom gradients that transform the solver into a differentiable operator, like a neural network layer, enabling many diverse applications. Empirically, we demonstrate: (i) scalability by enforcing weight sparsity on a 50 million parameter model; (ii) imposing conservativeness constraints in score-based generative models; and (iii) hyperparameter tuning of Gaussian processes based on predictive performance. By doing this, our work represents the next iteration in developing differentiable linear-algebra tools and making them widely accessible to machine learning practitioners.

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Hrittik Roy, Søren Hauberg, Nicholas Krämer. 2025-10-22. Matrix-Free Least Squares Solvers: Values, Gradients, and What to Do With Them. https://arxiv.org/abs/2510.19634

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