We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real @math , let us call a stochastic game @math -ergodic, if its values from any two initial positions differ by at most @math . The proposed new algorithm outputs for every @math in finite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an @math -range, or identifies two initial positions @math and @math and corresponding stationary strategies for the players proving that the game values starting from @math and @math are at least @math apart. In particular, the above result shows that if a stochastic game is @math -ergodic, then there are stationary strategies for the players proving @math -ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (1980) claiming that if a stochastic game is @math -ergodic, then there are @math -optimal stationary strategies for every @math . The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game.
Interestingly, potentials appear in @cite_16 implicitly, as the differences of local values of positions, as well as in @cite_17 , as the dual variables to linear programs corresponding to the controlled Markov processes, which appear when a player optimizes his strategy against a given strategy of the opponent. Yet, the potential transformation is not considered explicitly in these papers.
@cite_16 @cite_17
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Markov Random Field is now ubiquitous in many formulations of various vision problems. Recently, optimization of higher-order potentials became practical using higher-order graph cuts: the combination of i) the fusion move algorithm, ii) the reduction of higher-order binary energy minimization to first-order, and iii) the QPBO algorithm. In the fusion move, it is crucial for the success and efficiency of the optimization to provide proposals that fits the energies being optimized. For higher-order energies, it is even more so because they have richer class of null potentials. In this paper, we focus on the efficiency of the higher-order graph cuts and present a simple technique for generating proposal labelings that makes the algorithm much more efficient, which we empirically show using examples in stereo and image denoising. Abstract We study the multi-channel Gelʼfand–Calderon inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation − Δ ψ + v ( x ) ψ = 0 , x ∈ D , where v is a smooth matrix-valued potential defined on a bounded planar domain D . We give an exact global reconstruction method for finding v from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide. In cryo-electron microscopy, the three-dimensional (3D) electric potentials of an ensemble of molecules are projected along arbitrary viewing directions to yield noisy two-dimensional images. The volume maps representing these potentials typically exhibit a great deal of structural variability, which is described by their 3D covariance matrix. Typically, this covariance matrix is approximately low rank and can be used to cluster the volumes or estimate the intrinsic geometry of the conformation space. We formulate the estimation of this covariance matrix as a linear inverse problem, yielding a consistent least-squares estimator. For @math images of size @math -by- @math pixels, we propose an algorithm for calculating this covariance estimator with computational complexity @math , where the condition number @math is empirically in the range 10--200. Its efficiency relies on the observation that the normal equations are equivalent to a deconvolution problem in six dimensions.... This paper develops the application of the alternating direction method of multipliers (ADMM) to optimize a dynamic objective function in a decentralized multi-agent system. At each time slot, agents in the network observe local functions and cooperate to track the optimal time-varying argument of the sum objective. This cooperation is based on maintaining local primal variables that estimate the value of the optimal argument and auxiliary dual variables that encourage proximity with neighboring estimates. Primal and dual variables are updated by an ADMM iteration that can be implemented in a distributed manner whereby local updates require access to local variables and the most recent primal variables from adjacent agents. For objective functions that are strongly convex and have Lipschitz continuous gradients, the distances between the primal and dual iterates to their corresponding time-varying optimal values are shown to converge to a steady state gap. This gap is explicitly characterized in terms of the condition number of the objective function, the condition number of the network that is defined as the ratio between the largest and smallest nonzero Laplacian eigenvalues, and a bound on the drifts of the optimal primal variables and the optimal gradients. Numerical experiments corroborate theoretical findings and show that the results also hold for non-differentiable and non-strongly convex primal objectives.