An inequality for the matrix pressure function and applications

We prove an a priori lower bound for the pressure, or $p$-norm joint spectral radius, of a measure on the set of $d \times d$ real matrices which parallels a result of J. Bochi for the joint spectral radius. We apply this lower bound to give new proofs of the continuity of the affinity dimension of a self-affine set and of the continuity of the singular-value pressure for invertible matrices, both of which had been previously established by D.-J. Feng and P. Shmerkin using multiplicative ergodic theory and the subadditive variational principle. Unlike the previous proof, our lower bound yields algorithms to rigorously compute the pressure, singular value pressure and affinity dimension of a finite set of matrices to within an a priori prescribed accuracy in finitely many computational steps. We additionally deduce a related inequality for the singular value pressure for measures on the set of $2 \times 2$ real matrices, give a precise characterisation of the discontinuities of the singular value pressure function for two-dimensional matrices, and prove a general theorem relating the zero-temperature limit of the matrix pressure to the joint spectral radius.Comment: To appear in Advances in Mathematic

Continuity of the Effective Path Delay Operator for Networks Based on the Link Delay Model

This paper is concerned with a dynamic traffic network performance model, known as dynamic network loading (DNL), that is frequently employed in the modeling and computation of analytical dynamic user equilibrium (DUE). As a key component of continuous-time DUE models, DNL aims at describing and predicting the spatial-temporal evolution of traffic flows on a network that is consistent with established route and departure time choices of travelers, by introducing appropriate dynamics to flow propagation, flow conservation, and travel delays. The DNL procedure gives rise to the path delay operator, which associates a vector of path flows (path departure rates) with the corresponding path travel costs. In this paper, we establish strong continuity of the path delay operator for networks whose arc flows are described by the link delay model (Friesz et al., 1993). Unlike result established in Zhu and Marcotte (2000), our continuity proof is constructed without assuming a priori uniform boundedness of the path flows. Such a more general continuity result has a few important implications to the existence of simultaneous route-and-departure choice DUE without a priori boundedness of path flows, and to any numerical algorithm that allows convergence to be rigorously analyzed.Comment: 12 pages, 1 figur

Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation

In this paper, we aim at recovering an unknown signal x0 from noisy L1measurements y=Phi*x0+w, where Phi is an ill-conditioned or singular linear operator and w accounts for some noise. To regularize such an ill-posed inverse problem, we impose an analysis sparsity prior. More precisely, the recovery is cast as a convex optimization program where the objective is the sum of a quadratic data fidelity term and a regularization term formed of the L1-norm of the correlations between the sought after signal and atoms in a given (generally overcomplete) dictionary. The L1-sparsity analysis prior is weighted by a regularization parameter lambda>0. In this paper, we prove that any minimizers of this problem is a piecewise-affine function of the observations y and the regularization parameter lambda. As a byproduct, we exploit these properties to get an objectively guided choice of lambda. In particular, we develop an extension of the Generalized Stein Unbiased Risk Estimator (GSURE) and show that it is an unbiased and reliable estimator of an appropriately defined risk. The latter encompasses special cases such as the prediction risk, the projection risk and the estimation risk. We apply these risk estimators to the special case of L1-sparsity analysis regularization. We also discuss implementation issues and propose fast algorithms to solve the L1 analysis minimization problem and to compute the associated GSURE. We finally illustrate the applicability of our framework to parameter(s) selection on several imaging problems

Discrete-time twisting controller without numerical chattering: analysis and experimental results with an implicit method

International audienceIn this note, we present an implementation of the twisting controller on an electropneumatic plant for a tracking control problem. Implicit and explicit discrete-time twisting controllers are considered, and some implementation details are provided. Experimental results are provided and analyzed. They sustain the theoretical superiority of the implicitly discretized version, as shown in previous work. The main advantages of the implicit method are better tracking and drastic reduction in the input and output chattering. This is achieved without modifying the controller structure compared to its continuous-time version