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Positive Solutions for Large Random Linear Systems

Abstract : Consider a large linear system with random underlying matrix: xn = 1n + 1/(αn √βn) Mn xn, where xn is the unknown, 1n is a vector of ones, Mn is a random matrix and αn, βn are scaling parameters to be specified. We investigate the componentwise positivity of the solution x n depending on the scaling factors, as the dimensions of the system grow to infinity. We consider 2 models of interest: The case where matrix Mn has independent and identically distributed standard Gaussian random variables, and a sparse case with a growing number of vanishing entries.In each case, there exists a phase transition for the scaling parameters below which there is no positive solution to the system with growing probability and above which there is a positive solution with growing probability.These questions arise from feasibility and stability issues for large biological communities with interactions.
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Submitted on : Monday, January 4, 2021 - 4:25:16 PM
Last modification on : Friday, December 3, 2021 - 11:43:51 AM
Long-term archiving on: : Monday, April 5, 2021 - 9:07:51 PM

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Pierre Bizeul, Maxime Clenet, Jamal Najim. Positive Solutions for Large Random Linear Systems. ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 2020, Barcelona, Spain. pp.8777-8781, ⟨10.1109/ICASSP40776.2020.9053593⟩. ⟨hal-03093082⟩

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