A residual-iteration framework for alternating projections between affine subspaces

We reformulate the problem of alternating projections between two affine subspaces of a Hilbert space as the minimization of a least-squares functional associated with a bounded linear operator.

This viewpoint reveals that classical alternating projections coincide with the unit-step Landweber iteration and enables the introduction of a general residual-state iteration framework that encompasses Landweber, its steepest-descent variant, and the conjugate-gradient method.

Within this framework, we establish abstract convergence principles based on residual extinction and translation equivariance, allowing convergence analyses to be carried out once at the level of least-squares optimization and then transferred directly to alternating projection algorithms.

As applications, we obtain new variants of alternating projections accelerated by steepest descent and conjugate gradients, together with convergence guarantees in both the consistent and inconsistent settings.

We also establish linear convergence results under closed-range assumptions and express the convergence rates explicitly in terms of the Friedrichs angle and the largest principal angle between the underlying subspaces.