Subspace decomposition in regularized least-squares: solution properties, restricted coercivity and beyond

Feng Xue; Hui Zhang

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arXiv · 2507.20686

Subspace decomposition in regularized least-squares: solution properties, restricted coercivity and beyond

Feng Xue·Hui Zhang

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Abstract

We investigate the solution properties of the regularized least-squares problem. Using a subspace decomposition technique, we derive expressions for the solution set in terms of the conjugate function, from which various properties, including existence, compactness and uniqueness, can then be easily analyzed. A key distinction of our approach from existing works is the separate treatment of existence and compactness. We unify many existing results based on recession cones and sublevel sets, and link them to our findings by connecting the recession function with the recession cone of the subdifferential of the conjugate function. In particular, the concept of restricted coercivity is developed and discussed in various aspects. The associated linearly constrained counterpart is discussed in a similar manner. Its connections to regularized least-squares are further established via the exactness of infimal postcomposition. Our results are supported by numerous examples, among which the geometric interpretation of the lasso solution deserves further investigations in near future.

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Feng Xue, Hui Zhang. 2025-07-28. Subspace decomposition in regularized least-squares: solution properties, restricted coercivity and beyond. https://arxiv.org/abs/2507.20686

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