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

• URL: http://arxiv.org/abs/2510.19634v1
• Date: Wed, 22 Oct 2025 14:31:51 GMT
• Title: Matrix-Free Least Squares Solvers: Values, Gradients, and What to Do With Them
• Authors: Hrittik Roy, Søren Hauberg, Nicholas Krämer,
• Abstract summary: This paper argues that the method of at least squares has significant unfulfilled potential in modern machine learning.<n>To release its potential, we derive custom gradients that transform the solver into a differentiable operator, like a neural network layer.
• Score: 17.808832664329426
• License: http://creativecommons.org/licenses/by/4.0/
• 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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