Timeline for Fastest algorithm for finding the closest semi-definite matrix?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| May 5, 2023 at 18:49 | comment | added | Steve Huntsman | nhigham.com/2021/01/26/… | |
| May 5, 2023 at 15:51 | comment | added | Rodrigo de Azevedo | @AlecJacobson This is mentioned in section 8.1.1 (page 399) of Boyd & Vandenberghe's Convex Optimization. | |
| May 5, 2023 at 14:53 | comment | added | Alec Jacobson | This also seems to appear in [Sheung Hun Cheng and Nicholas Higham, A Modified Cholesky Algorithm Based on a Symmetric Indefinite Factorization, SIAM J. Matrix Anal. Appl. 19(4), 1097–1110, 1998] perhaps with a more solid proof. | |
| May 5, 2023 at 14:52 | comment | added | Alec Jacobson | Here's my proof ‖ X - A ‖_F^2 = ‖ Vᵀ X - Vᵀ A ‖_F^2 = ‖ Vᵀ X V - Vᵀ A V ‖_F^2 = ‖ Vᵀ X V - Vᵀ VΛVᵀ V ‖_F^2 = ‖ Vᵀ X V - Λ ‖_F^2. This implies that VᵀXV should be diagonal: X→VΩVᵀ with Ω diagonal and in turn X ≽ 0 implies Ω ≥ 0. So we have ‖ Vᵀ VΩVᵀ V - Λ ‖_F^2 = ‖ Ω - Λ ‖_F^2 which now has a simple minimizer: Ωᵢᵢ = max( Λᵢᵢ, 0) apologies for no newlines in this comment. | |
| May 5, 2023 at 14:44 | history | edited | Rodrigo de Azevedo |
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| May 5, 2023 at 14:40 | comment | added | Neal | Sorry if I'm being dull - why is the "clamped matrix" $V\operatorname{max}(\Lambda,0)V^T$ the closest positive semidefinite matrix to $A$? | |
| May 5, 2023 at 14:16 | history | asked | Alec Jacobson | CC BY-SA 4.0 |