{"id":4370,"date":"2025-06-05T07:02:35","date_gmt":"2025-06-05T07:02:35","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/06\/05\/2506-03764\/"},"modified":"2025-06-05T07:02:35","modified_gmt":"2025-06-05T07:02:35","slug":"2506-03764","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/06\/05\/2506-03764\/","title":{"rendered":"Infinitesimal Higher-Order Spectral Variations in Rectangular Real Random Matrices"},"content":{"rendered":"

Infinitesimal Higher-Order Spectral Variations in Rectangular Real Random Matrices
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arXiv:2506.03764v1 Announce Type: new
\nAbstract: We present a theoretical framework for deriving the general $n$-th order Fr’echet derivatives of singular values in real rectangular matrices, by leveraging reduced resolvent operators from Kato’s analytic perturbation theory for self-adjoint operators. Deriving closed-form expressions for higher-order derivatives of singular values is notoriously challenging through standard matrix-analysis techniques. To overcome this, we treat a real rectangular matrix as a compact operator on a finite-dimensional Hilbert space, and embed the rectangular matrix into a block self-adjoint operator so that non-symmetric perturbations are captured. Applying Kato’s asymptotic eigenvalue expansion to this construction, we obtain a general, closed-form expression for the infinitesimal $n$-th order spectral variations. Specializing to $n=2$ and deploying on a Kronecker-product representation with matrix convention yield the Hessian of a singular value, not found in literature. By bridging abstract operator-theoretic perturbation theory with matrices, our framework equips researchers with a practical toolkit for higher-order spectral sensitivity studies in random matrix applications (e.g., adversarial perturbation in deep learning).<\/div>\n

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\n R’ois’in Luo
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Infinitesimal Higher-Order Spectral Variations in Rectangular Real Random Matrices arXiv:2506.03764v1 Announce Type: new Abstract: We present a theoretical framework for deriving the general $n$-th order Fr’echet derivatives of singular values in real rectangular matrices, by leveraging reduced resolvent operators from Kato’s analytic perturbation theory for self-adjoint operators. Deriving closed-form expressions for higher-order derivatives of singular […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,112],"tags":[419,1424,2865],"class_list":["post-4370","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-stat-ml","tag-matrix","tag-order","tag-rectangular"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/4370"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=4370"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/4370\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=4370"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=4370"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=4370"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}