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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.26318 |
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| _version_ | 1866916046757167104 |
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| author | Machado, Ananias Fampa, Marcia Lee, Jon |
| author_facet | Machado, Ananias Fampa, Marcia Lee, Jon |
| contents | Generalized inverses play a fundamental role in numerical linear algebra, particularly when matrices are rectangular, singular, or rank deficient. Even when the input matrix is sparse, generalized inverses such as the M-P pseudoinverse are typically dense, leading to high storage requirements, expensive matrix-vector multiplications, and reduced efficiency. We investigate computing sparse symmetric generalized inverses for sparse symmetric matrices, extending previous work on sparse generalized inverses for rectangular rank-deficient matrices. We consider minimizing the vector 1-norm over the set of symmetric generalized inverses, using vector 1-norm minimization as a surrogate for sparsity promotion. We give a new characterization of symmetric generalized inverses of symmetric matrices, which yields a compact affine reformulation of the problem. From this, we develop a Douglas-Rachford splitting (DRS) algorithm equipped with a closed-form projection onto the feasible affine space. Computational experiments compare the proposed DRS approach with exact linear-optimization formulations solved by a commercial optimizer, as well as with local-search heuristics. The results demonstrate that the proposed formulation and algorithm produce generalized inverses that are substantially sparser than the Moore-Penrose pseudoinverse while maintaining significantly smaller 1-norms than competing approaches. Moreover, the DRS algorithm exhibits superior scalability relative to exact linear-optimization formulations, successfully solving instances far beyond the reach of commercial solvers. As an application, we investigate the computation of least-squares solutions for problems involving many right-hand-side vectors and sparse rank-deficient design matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26318 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sparse symmetric generalized inverses for sparse symmetric matrices Machado, Ananias Fampa, Marcia Lee, Jon Optimization and Control Generalized inverses play a fundamental role in numerical linear algebra, particularly when matrices are rectangular, singular, or rank deficient. Even when the input matrix is sparse, generalized inverses such as the M-P pseudoinverse are typically dense, leading to high storage requirements, expensive matrix-vector multiplications, and reduced efficiency. We investigate computing sparse symmetric generalized inverses for sparse symmetric matrices, extending previous work on sparse generalized inverses for rectangular rank-deficient matrices. We consider minimizing the vector 1-norm over the set of symmetric generalized inverses, using vector 1-norm minimization as a surrogate for sparsity promotion. We give a new characterization of symmetric generalized inverses of symmetric matrices, which yields a compact affine reformulation of the problem. From this, we develop a Douglas-Rachford splitting (DRS) algorithm equipped with a closed-form projection onto the feasible affine space. Computational experiments compare the proposed DRS approach with exact linear-optimization formulations solved by a commercial optimizer, as well as with local-search heuristics. The results demonstrate that the proposed formulation and algorithm produce generalized inverses that are substantially sparser than the Moore-Penrose pseudoinverse while maintaining significantly smaller 1-norms than competing approaches. Moreover, the DRS algorithm exhibits superior scalability relative to exact linear-optimization formulations, successfully solving instances far beyond the reach of commercial solvers. As an application, we investigate the computation of least-squares solutions for problems involving many right-hand-side vectors and sparse rank-deficient design matrices. |
| title | Sparse symmetric generalized inverses for sparse symmetric matrices |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2605.26318 |