Numerically stable variants of overrelaxation for operator Sinkhorn iteration
This work addresses a stability issue in accelerated optimization methods for completely positive maps, which is incremental but improves practical usability in numerical computations.
The paper tackled the numerical instability of accelerated operator Sinkhorn iteration (OSI) using nonlinear successive overrelaxation (SOR) for ill-conditioned scaling problems, and derived equivalent versions with on-the-fly scalings that enable stable SOR-acceleration, as confirmed by numerical experiments.
We consider accelerated versions of the operator Sinkhorn iteration (OSI) for solving scaling problems for completely positive maps. Based on the interpretation of OSI as alternating fixed point iteration, it has been recently proposed to achieve acceleration by means of nonlinear successive overrelaxation (SOR), e.g.~with respect to geodesics in Hilbert metric. The direct implementation of the proposed SOR algorithms, however, can be numerically unstable for ill-conditioned instances, limiting the achievable accuracy. Here we derive equivalent versions of OSI with SOR where, similar to the original OSI formulation, scalings are applied on the fly in order to take advantage of preconditioning effects. Numerical experiments confirm that this modification allows for numerically stable SOR-acceleration of OSI even in ill-conditioned cases.