12,322 research outputs found
Renormalization and Computation II: Time Cut-off and the Halting Problem
This is the second installment to the project initiated in [Ma3]. In the
first Part, I argued that both philosophy and technique of the perturbative
renormalization in quantum field theory could be meaningfully transplanted to
the theory of computation, and sketched several contexts supporting this view.
In this second part, I address some of the issues raised in [Ma3] and provide
their development in three contexts: a categorification of the algorithmic
computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra
renormalization of the Halting Problem.Comment: 28 page
Approximating orthogonal matrices by permutation matrices
Motivated in part by a problem of combinatorial optimization and in part by
analogies with quantum computations, we consider approximations of orthogonal
matrices U by ``non-commutative convex combinations'' A of permutation matrices
of the type A=sum A_sigma sigma, where sigma are permutation matrices and
A_sigma are positive semidefinite nxn matrices summing up to the identity
matrix. We prove that for every nxn orthogonal matrix U there is a
non-commutative convex combination A of permutation matrices which approximates
U entry-wise within an error of c n^{-1/2}ln n and in the Frobenius norm within
an error of c ln n. The proof uses a certain procedure of randomized rounding
of an orthogonal matrix to a permutation matrix.Comment: 18 page
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