Accelerating the alternating projection algorithm for the case of affine subspaces using supporting hyperplanes

The von Neumann-Halperin method of alternating projections converges strongly to the projection of a given point onto the intersection of finitely many closed affine subspaces. We propose acceleration schemes making use of two ideas: Firstly, each projection onto an affine subspace identifies a hyperplane of codimension 1 containing the intersection, and secondly, it is easy to project onto a finite intersection of such hyperplanes. We give conditions for which our accelerations converge strongly. Finally, we perform numerical experiments to show that these accelerations perform well for a matrix model updating problem.Comment: 16 pages, 3 figures (Corrected minor typos in Remark 2.2, Algorithm 2.5, proof of Theorem 3.12, as well as elaborated on certain proof

Card-Shuffling via Convolutions of Projections on Combinatorial Hopf Algebras

Recently, Diaconis, Ram and I created Markov chains out of the coproduct-then-product operator on combinatorial Hopf algebras. These chains model the breaking and recombining of combinatorial objects. Our motivating example was the riffle-shuffling of a deck of cards, for which this Hopf algebra connection allowed explicit computation of all the eigenfunctions. The present note replaces in this construction the coproduct-then-product map with convolutions of projections to the graded subspaces, effectively allowing us to dictate the distribution of sizes of the pieces in the breaking step of the previous chains. An important example is removing one "vertex" and reattaching it, in analogy with top-to-random shuffling. This larger family of Markov chains all admit analysis by Hopf-algebraic techniques. There are simple combinatorial expressions for their stationary distributions and for their eigenvalues and multiplicities and, in some cases, the eigenfunctions are also calculable.Comment: 12 pages. This is an extended abstract, to appear in Proceedings of the 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC). Comments are very welcom