This study investigates the dynamics of alternating minimization applied to a bilinear regression task with normally distributed covariates, under the asymptotic system size limit where the number of parameters and observations diverge at the same rate. This is achieved by employing the replica method to a multi-temperature glassy system which unfolds the algorithm's time evolution. Our results show that the dynamics can be described effectively by a two-dimensional discrete stochastic process, where each step depends on all previous time steps, revealing the structure of the memory dependence in the evolution of alternating minimization. The theoretical framework developed in this work can be applied to the analysis of various iterative algorithms, extending beyond the scope of alternating minimization.

翻译：本研究探讨了在协变量服从正态分布条件下，应用于双线性回归任务的交替最小化算法的动力学行为，分析在渐近系统尺寸极限下（即参数数量与观测数量以相同速率发散时）的演化特性。通过将复本方法应用于多温度玻璃态系统，我们揭示了算法的时间演化过程。结果表明，该动力学可由一个二维离散随机过程有效描述，其中每一步都依赖于所有先前时间步，从而揭示了交替最小化演化过程中记忆依赖的结构特征。本研究建立的理论框架可推广应用于分析各类迭代算法，其适用范围超越了交替最小化算法本身。