This research paper delves into the annealed complexity of Gaussian random homogeneous polynomials, specifically examining their behavior in the presence of deterministic perturbations. The study derives formulas for the exponential asymptotics of critical points and local maxima by combining the Kac-Rice formula with determinant asymptotics for finite-rank perturbations of Gaussian Wigner matrices. A key finding is the identification of a topological phase transition where new zero-complexity regions emerge above a certain threshold in external parameters, potentially indicating critical points with large correlations to perturbation vectors. AI
IMPACT This research contributes to the theoretical understanding of random polynomials and their critical points, which can have implications for fields like machine learning where complex functions and their optimization landscapes are studied.
RANK_REASON The item is an academic paper published on arXiv. [lever_c_demoted from research: ic=1 ai=0.4]
AI-generated summary · Google Gemini · from 1 sources. How we write summaries →
