This study investigates the asymptotic dynamics of alternating minimization applied to optimize a bilinear non-convex function with normally distributed covariates. We employ the replica method from statistical physics in a multi-step approach to precisely trace the algorithm's evolution. Our findings indicate that the dynamics can be described effectively by a two--dimensional discrete stochastic process, where each step depends on all previous time steps, revealing a memory dependency in the procedure. The theoretical framework developed in this work is broadly applicable for the analysis of various iterative algorithms, extending beyond the scope of alternating minimization.

翻译：本研究探讨了应用于优化具有正态分布协变量的双线性非凸函数的交替最小化的渐近动力学。我们采用统计物理学中的副本方法，通过多步途径精确追踪算法的演化过程。研究结果表明，该动力学可由一个二维离散随机过程有效描述，其中每一步依赖于所有先前的时间步，揭示了过程中的记忆依赖性。本文开发的理论框架广泛适用于多种迭代算法的分析，超越了交替最小化的范畴。