80,958 research outputs found
Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation
We apply multivariate Lagrange interpolation to synthesize polynomial
quantitative loop invariants for probabilistic programs. We reduce the
computation of an quantitative loop invariant to solving constraints over
program variables and unknown coefficients. Lagrange interpolation allows us to
find constraints with less unknown coefficients. Counterexample-guided
refinement furthermore generates linear constraints that pinpoint the desired
quantitative invariants. We evaluate our technique by several case studies with
polynomial quantitative loop invariants in the experiments
Iterative greedy algorithm for solving the FIR paraunitary approximation problem
In this paper, a method for approximating a multi-input multi-output (MIMO) transfer function by a causal finite-impulse response (FIR) paraunitary (PU) system in a weighted least-squares sense is presented. Using a complete parameterization of FIR PU systems in terms of Householder-like building blocks, an iterative algorithm is proposed that is greedy in the sense that the observed mean-squared error at each iteration is guaranteed to not increase. For certain design problems in which there is a phase-type ambiguity in the desired response, which is formally defined in the paper, a phase feedback modification is proposed in which the phase of the FIR approximant is fed back to the desired response. With this modification in effect, it is shown that the resulting iterative algorithm not only still remains greedy, but also offers a better magnitude-type fit to the desired response. Simulation results show the usefulness and versatility of the proposed algorithm with respect to the design of principal component filter bank (PCFB)-like filter banks and the FIR PU interpolation problem. Concerning the PCFB design problem, it is shown that as the McMillan degree of the FIR PU approximant increases, the resulting filter bank behaves more and more like the infinite-order PCFB, consistent with intuition. In particular, this PCFB-like behavior is shown in terms of filter response shape, multiresolution, coding gain, noise reduction with zeroth-order Wiener filtering in the subbands, and power minimization for discrete multitone (DMT)-type transmultiplexers
Relative entropy and the multi-variable multi-dimensional moment problem
Entropy-like functionals on operator algebras have been studied since the
pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The most
well-known are the von Neumann entropy and a
generalization of the Kullback-Leibler distance , refered to as quantum relative entropy and used to quantify
distance between states of a quantum system. The purpose of this paper is to
explore these as regularizing functionals in seeking solutions to
multi-variable and multi-dimensional moment problems. It will be shown that
extrema can be effectively constructed via a suitable homotopy. The homotopy
approach leads naturally to a further generalization and a description of all
the solutions to such moment problems. This is accomplished by a
renormalization of a Riemannian metric induced by entropy functionals. As an
application we discuss the inverse problem of describing power spectra which
are consistent with second-order statistics, which has been the main motivation
behind the present work.Comment: 24 pages, 3 figure
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