In this work, we present a new approach to analyze the gradient flow for a positive semi-definite matrix denoising problem in an extensive-rank and high-dimensional regime. We use recent linear pencil techniques of random matrix theory to derive fixed point equations which track the complete time evolution of the matrix-mean-square-error of the problem. The predictions of the resulting fixed point equations are validated by numerical experiments. In this short note we briefly illustrate a few predictions of our formalism by way of examples, and in particular we uncover continuous phase transitions in the extensive-rank and high-dimensional regime, which connect to the classical phase transitions of the low-rank problem in the appropriate limit. The formalism has much wider applicability than shown in this communication.

翻译：本文提出了一种新方法，用于分析广泛秩高维条件下正半定矩阵去噪问题中的梯度流。我们利用随机矩阵理论中最近发展的线性迹技术，推导了追踪问题矩阵均方误差完整时间演化的不动点方程。通过数值实验验证了所得不动点方程的预测结果。在这篇短文中，我们通过示例简要说明了该形式框架的若干预测，特别揭示了广泛秩高维条件下的连续相变现象，该现象在适当极限下与低秩问题的经典相变相关联。本文所述形式框架的适用范围远不止于本文展示的案例。