Extrapolated Proximal Subgradient Algorithms for Nonconvex and Nonsmooth Fractional Programs

In this paper, we consider a broad class of nonsmooth and nonconvex fractional programs, which encompass many important modern optimization problems arising from diverse areas such as the recently proposed scale-invariant sparse signal reconstruction problem in signal processing. We propose a proximal subgradient algorithm with extrapolations for solving this optimization model and show that the iterated sequence generated by the algorithm is bounded and that any one of its limit points is a stationary point of the model problem. The choice of our extrapolation parameter is flexible and includes the popular extrapolation parameter adopted in the restarted fast iterative shrinking-threshold algorithm (FISTA). By providing a unified analysis framework of descent methods, we establish the convergence of the full sequence under the assumption that a suitable merit function satisfies the Kurdyka–Łojasiewicz property. Our algorithm exhibits linear convergence for the scale-invariant sparse signal reconstruction problem and the Rayleigh quotient problem over spherical constraint. When the denominator is the maximum of finitely many continuously differentiable weakly convex functions, we also propose another extrapolated proximal subgradient algorithm with guaranteed convergence to a stronger notion of stationary points of the model problem. Finally, we illustrate the proposed methods by both analytical and simulated numerical examples.

The inertial relaxed algorithm with Armijo-type line search for solving multiple-sets split feasibility problem

The multiple-sets split feasibility problem is the generalization of split feasibility problem, which has been widely used in fuzzy image reconstruction and sparse signal processing systems. In this paper, we present an inertial relaxed algorithm to solve the multiple-sets split feasibility problem by using an alternating inertial step. The advantage of this algorithm is that the choice of stepsize is determined by Armijo-type line search, which avoids calculating the norms of operators. The weak convergence of the sequence obtained by our algorithm is proved under mild conditions. In addition, the numerical experiments are given to verify the convergence and validity of the algorithm.

Contribution of linear and nonlinear mechanisms to predictive motion estimation

Successful behavior relies on the ability to use information obtained from past experience to predict what is likely to occur in the future. A salient example of predictive encoding comes from the vertebrate retina, where neural circuits encode information that can be used to estimate the trajectory of a moving object. Predictive computations should be a general property of sensory systems, but the features needed to identify these computations across neural systems are not well understood. Here, we identify several properties of predictive computations in the primate retina that likely generalize across sensory systems. These features include calculating the derivative of incoming signals, sparse signal integration, and delayed response suppression. These findings provide a deeper understanding of how the brain carries out predictive computations and identify features that can be used to recognize these computations throughout the brain.