Rigorous characterization of AM dynamics as a two-dimensional Gaussian process with memory

Establish a rigorous proof that, for alternating minimization applied to the bilinear regression model with square loss and i.i.d. Gaussian covariates in the full-batch setting, the algorithm’s asymptotic dynamics admits a statistical characterization as an explicit two-dimensional discrete Gaussian process with memory dependence, i.e., that the joint law of per-coordinate iterates across time is equivalently described by the specified Gaussian recursion and associated fixed-point order parameters derived via the replica method.

Background

The paper analyzes alternating minimization (AM) for a bilinear non-convex regression model under Gaussian design, using a nonrigorous replica-method framework to derive fixed-point equations and an effective stochastic description of the iterates. The authors present a closed-form characterization of the algorithm’s asymptotic dynamics and introduce an explicit two-dimensional discrete Gaussian process whose recursion incorporates memory terms dependent on all previous steps.

Because the derivation relies on the replica method, the result is presented as a conjectural statistical characterization rather than a rigorous theorem. A formal proof would validate the claimed equivalence between the AM iterates and the proposed Gaussian process, including the memory structure encoded by the order parameters.