Learning and Prediction Theory of Distributed Least Squares

With the fast development of the sensor and network technology, distributed estimation has attracted more and more attention, due to its capability in securing communication, in sustaining scalability, and in enhancing safety and privacy. In this paper, we consider a least-squares (LS)-based distributed algorithm build on a sensor network to estimate an unknown parameter vector of a dynamical system, where each sensor in the network has partial information only but is allowed to communicate with its neighbors. Our main task is to generalize the well-known theoretical results on the traditional LS to the current distributed case by establishing both the upper bound of the accumulated regrets of the adaptive predictor and the convergence of the distributed LS estimator, with the following key features compared with the existing literature on distributed estimation: Firstly, our theory does not need the previously imposed independence, stationarity or Gaussian property on the system signals, and hence is applicable to stochastic systems with feedback control. Secondly, the cooperative excitation condition introduced and used in this paper for the convergence of the distributed LS estimate is the weakest possible one, which shows that even if any individual sensor cannot estimate the unknown parameter by the traditional LS, the whole network can still fulfill the estimation task by the distributed LS. Moreover, our theoretical analysis is also different from the existing ones for distributed LS, because it is an integration of several powerful techniques including stochastic Lyapunov functions, martingale convergence theorems, and some inequalities on convex combination of nonnegative definite matrices.Comment: 14 pages, submitted to IEEE Transactions on Automatic Contro

Renormalization Group in Quantum Mechanics

The running coupling constants are introduced in Quantum Mechanics and their evolution is described by the help of the renormalization group equation. The harmonic oscillator and the propagation on curved spaces are presented as examples. The hamiltonian and the lagrangian scaling relations are obtained. These evolution equations are used to construct low energy effective models.Comment: Updated and extended version to appear in Annals of Physics, 24pg, TEX fil