Constrained Variable Projection for Structured Problems

• URL: http://arxiv.org/abs/2606.23939v1
• Date: Mon, 22 Jun 2026 21:00:35 GMT
• Title: Constrained Variable Projection for Structured Problems
• Abstract summary: We develop a constrained variable-projection framework for data-science models.<n>We derive exact reduced-gradient formulas compatible with automatic differentiation.<n>Experiments on sparse autoencoding, dictionary learning, blind deconvolution, and few-shot learning suggest that the method can improve data efficiency.
• Score: 11.374309043794971
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Variable projection is a classical technique for separable nonlinear least-squares problems, in which variables that enter linearly are eliminated exactly, yielding a reduced nonlinear problem. By expressing this framework as a particular instance of a broader class of bilevel optimization problems, we develop a constrained variable-projection framework for data-science models, where the remaining variables are subject to convex constraints and the eliminated variables arise from a lower-level least-squares problem. In particular, by interpreting variable projection as a collapsed bilevel optimization problem, we derive exact reduced-gradient formulas compatible with automatic differentiation and propose a conditional-gradient algorithm for the resulting constrained reduced problem. We establish convergence guarantees under standard smoothness and compactness assumptions, and discuss extensions to structured lower-level variables. Numerical experiments on sparse autoencoding, dictionary learning, blind deconvolution, and few-shot learning suggest that the method can improve wall-clock efficiency and data efficiency relative to natural joint-optimization baselines.

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