2024/12/04 by Kamyar Moshksar, Moshksar, Kamyar · 2 citations
Physics and Astronomy · #Chaos control and synchronization
paper · pdf · doi:10.48550/arxiv.2412.03774
The paper studies concentration of measure for Gaussian quadratic chaos in the non-asymptotic regime where existing bounds are improved and new bounds are proposed. We begin by slightly tightening Hanson-Wright inequality (HWI) by increasing its absolute constant from the largest known value of 0.125 to at least 0.145 in the symmetric case. A sharper version of an inequality of Laurent and Massart (LMI) is presented. It results in an increase in the absolute constant in HWI from the largest available value of 1-(√(3))/(2) due to LMI to (9-√(17))/(32) in the positive-semidefinite case. Moving beyond HWI, we develop a sequence of inequalities indexed by m≥1 that involves Schatten norms of the underlying symmetric matrix. The case m=1 recovers HWI and the case m=∞ leads to a novel bound called the m_∞-bound. Avoiding Markov's inequality, we introduce the strong χ2-inequality and its loosened version, the weak χ2-inequality. To investigate the m_∞-bound, we explore all concentration bounds that only involve the operator norm of the underlying positive-definite matrix. Five candidates are examined, namely, the m_∞-bound, relaxed versions of HWI and LMI, the weak χ2-bound and the large deviations bound. The sharpest among these bounds is either the m_∞-bound or the weak χ2-bound. If the matrix dimension is n=2,4,6, the weak χ2-bound is tighter than the m_∞-bound. For even n≥8, the m_∞-bound is sharper than the weak χ2-bound if and only if the ratio of the tail parameter over the operator norm lies inside an open interval which expands indefinitely as n grows. Modified versions of HW, m_∞ and strong χ2 inequalities of various orders are proposed. Their effectiveness is demonstrated by two applications in signal detection for sonar and wireless communications.