Avalanches at rough surfaces

We describe the surface properties of a simple lattice model of a sandpile that includes evolving structural disorder. We present a dynamical scaling hypothesis for generic sandpile automata, and additionally explore the kinetic roughening of the sandpile surface, indicating its relationship with the sandpile evolution. Finally, we comment on the surprisingly good agreement found between this model, and a previous continuum model of sandpile dynamics, from the viewpoint of critical phenomena.Comment: 8 Pages, 7 Figures (in 15 parts); accepted for publication in Physical Review

An Optimal Control Derivation of Nonlinear Smoothing Equations

The purpose of this paper is to review and highlight some connections between the problem of nonlinear smoothing and optimal control of the Liouville equation. The latter has been an active area of recent research interest owing to work in mean-field games and optimal transportation theory. The nonlinear smoothing problem is considered here for continuous-time Markov processes. The observation process is modeled as a nonlinear function of a hidden state with an additive Gaussian measurement noise. A variational formulation is described based upon the relative entropy formula introduced by Newton and Mitter. The resulting optimal control problem is formulated on the space of probability distributions. The Hamilton's equation of the optimal control are related to the Zakai equation of nonlinear smoothing via the log transformation. The overall procedure is shown to generalize the classical Mortensen's minimum energy estimator for the linear Gaussian problem.Comment: 7 pages, 0 figures, under peer reviewin

On the Sample Complexity of Predictive Sparse Coding

The goal of predictive sparse coding is to learn a representation of examples as sparse linear combinations of elements from a dictionary, such that a learned hypothesis linear in the new representation performs well on a predictive task. Predictive sparse coding algorithms recently have demonstrated impressive performance on a variety of supervised tasks, but their generalization properties have not been studied. We establish the first generalization error bounds for predictive sparse coding, covering two settings: 1) the overcomplete setting, where the number of features k exceeds the original dimensionality d; and 2) the high or infinite-dimensional setting, where only dimension-free bounds are useful. Both learning bounds intimately depend on stability properties of the learned sparse encoder, as measured on the training sample. Consequently, we first present a fundamental stability result for the LASSO, a result characterizing the stability of the sparse codes with respect to perturbations to the dictionary. In the overcomplete setting, we present an estimation error bound that decays as \tilde{O}(sqrt(d k/m)) with respect to d and k. In the high or infinite-dimensional setting, we show a dimension-free bound that is \tilde{O}(sqrt(k^2 s / m)) with respect to k and s, where s is an upper bound on the number of non-zeros in the sparse code for any training data point.Comment: Sparse Coding Stability Theorem from version 1 has been relaxed considerably using a new notion of coding margin. Old Sparse Coding Stability Theorem still in new version, now as Theorem 2. Presentation of all proofs simplified/improved considerably. Paper reorganized. Empirical analysis showing new coding margin is non-trivial on real dataset

Stability Margin Scaling Laws for Distributed Formation Control as a Function of Network Structure

We consider the problem of distributed formation control of a large number of vehicles. An individual vehicle in the formation is assumed to be a fully actuated point mass. A distributed control law is examined: the control action on an individual vehicle depends on (i) its own velocity and (ii) the relative position measurements with a small subset of vehicles (neighbors) in the formation. The neighbors are defined according to an information graph. In this paper we describe a methodology for modeling, analysis, and distributed control design of such vehicular formations whose information graph is a D-dimensional lattice. The modeling relies on an approximation based on a partial differential equation (PDE) that describes the spatio-temporal evolution of position errors in the formation. The analysis and control design is based on the PDE model. We deduce asymptotic formulae for the closed-loop stability margin (absolute value of the real part of the least stable eigenvalue) of the controlled formation. The stability margin is shown to approach 0 as the number of vehicles N goes to infinity. The exponent on the scaling law for the stability margin is influenced by the dimension and the structure of the information graph. We show that the scaling law can be improved by employing a higher dimensional information graph. Apart from analysis, the PDE model is used for a mistuning-based design of control gains to maximize the stability margin. Mistuning here refers to small perturbation of control gains from their nominal symmetric values. We show that the mistuned design can have a significantly better stability margin even with a small amount of perturbation. The results of the analysis with the PDE model are corroborated with numerical computation of eigenvalues with the state-space model of the formation.Comment: This paper is the expanded version of the paper with the same name which is accepted by the IEEE Transactions on Automatic Control. The final version is updated on Oct. 12, 201