{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/interpolating-the-trace-of-the-inverse-of","title":"Interpolating Log-Determinant and Trace of the Powers of Matrix $\\mathbf{A} + t \\mathbf{B}$","arxiv_id":"2009.07385","date":"2020-09-15","proceeding":null,"authors":["Siavash Ameli","Shawn C. Shadden"],"abstract":"We develop heuristic interpolation methods for the functions $t \\mapsto \\log \\det \\left( \\mathbf{A} + t \\mathbf{B} \\right)$ and $t \\mapsto \\operatorname{trace}\\left( (\\mathbf{A} + t \\mathbf{B})^{p} \\right)$ where the matrices $\\mathbf{A}$ and $\\mathbf{B}$ are Hermitian and positive (semi) definite and $p$ and $t$ are real variables. These functions are featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of sharp bounds for these functions. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method.","url_abs":"https://arxiv.org/abs/2009.07385v3","url_pdf":"https://arxiv.org/pdf/2009.07385v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"interpolating-the-trace-of-the-inverse-of","repo_url":"https://github.com/ameli/imate","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"interpolating-the-trace-of-the-inverse-of","repo_url":"https://github.com/ameli/ortho","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null},{"paper_slug":"interpolating-the-trace-of-the-inverse-of","repo_url":"https://github.com/ameli/detkit","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"regression-1","task_name":"regression"}],"methods":[{"method_slug":"gaussian-process","method_name":"Gaussian Process"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}