We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the analysis of spectral and optimization algorithms, which require understanding the spectrum of a random matrix depending on data obtained as independent samples. Using ideas of decoupling and linearization from analysis, we show a simple way of expressing norm bounds for such matrices, in terms of matrices of lower-degree polynomials corresponding to derivatives. Iterating this method gives a simple bound with an elementary proof, which can recover many bounds previously required more involved techniques.
@InProceedings{tulsiani_et_al:LIPIcs.ITCS.2025.91, author = {Tulsiani, Madhur and Wu, June}, title = {{Simple Norm Bounds for Polynomial Random Matrices via Decoupling}}, booktitle = {16th Innovations in Theoretical Computer Science Conference (ITCS 2025)}, pages = {91:1--91:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-361-4}, ISSN = {1868-8969}, year = {2025}, volume = {325}, editor = {Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.91}, URN = {urn:nbn:de:0030-drops-227194}, doi = {10.4230/LIPIcs.ITCS.2025.91}, annote = {Keywords: Matrix Concentration, Decoupling, Graph Matrices} }
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