In this paper, we consider the problem of identifying a linear map from measurements which are subject to intermittent and arbitrarily large errors. This is a fundamental problem in many estimation-related applications such as fault detection; state estimation in lossy networks, hybrid system identification, robust estimation, etc. The problem is hard because it exhibits some intrinsic combinatorial features. Therefore, obtaining an effective solution necessitates relaxations that are both solvable at a reasonable cost and effective in the sense that they can return the true parameter vector. The current paper discusses a nonsmooth convex optimization approach and provides a new analysis of its behavior. In particular, it is shown that under appropriate conditions on the data, an exact estimate can be recovered from data corrupted by a large (even infinite) number of gross errors. (C) 2016 Elsevier Ltd. All rights reserved.

The paper deals with the problem of finding sparse solutions to systems of polynomial equations possibly perturbed by noise. In particular, we show how these solutions can be recovered from group-sparse solutions of a derived system of linear equations. Then, two approaches are considered to find these group-sparse solutions. The first one is based on a convex relaxation resulting in a second-order cone programming formulation which can benefit from efficient reweighting techniques for sparsity enhancement. For this approach, sufficient conditions for the exact recovery of the sparsest solution to the polynomial system are derived in the noiseless setting, while stable recovery results are obtained for the noisy case. Though lacking a similar analysis, the second approach provides a more computationally efficient algorithm based on a greedy strategy adding the groups one-by-one. With respect to previous work, the proposed methods recover the sparsest solution in a very short computing time while remaining at least as accurate in terms of the probability of success. This probability is empirically analyzed to emphasize the relationship between the ability of the methods to solve the polynomial system and the sparsity of the solution.

The classical shift retrieval problem considers two signals in vector form that are related by a shift. This problem is of great importance in many applications and is typically solved by maximizing the cross-correlation between the two signals. Inspired by compressive sensing, in this paper, we seek to estimate the shift directly from compressed signals. We show that under certain conditions, the shift can be recovered using fewer samples and less computation compared to the classical setup. We also illustrate the concept of superresolution for shift retrieval. Of particular interest is shift estimation from Fourier coefficients. We show that under rather mild conditions only one Fourier coefficient suffices to recover the true shift.

Anomaly detection in large populations is a challenging but highly relevant problem. It is essentially a multi-hypothesis problem, with a hypothesis for every division of the systems into normal and anomalous systems. The number of hypothesis grows rapidly with the number of systems and approximate solutions become a necessity for any problem of practical interest. In this paper we take an optimization approach to this multi-hypothesis problem. It is first shown to be equivalent to a non-convex combinatorial optimization problem and then is relaxed to a convex optimization problem that can be solved distributively on the systems and that stays computationally tractable as the number of systems increase. An interesting property of the proposed method is that it can under certain conditions be shown to give exactly the same result as the combinatorial multi-hypothesis problem and the relaxation is hence tight.

This paper proposes a general convex framework for the identification of switched linear systems. The proposed framework uses over-parameterization to avoid solving the otherwise combinatorially forbidding identification problem, and takes the form of a least-squares problem with a sum-of-norms regularization, a generalization of the ℓ₁-regularization. The regularization constant regulates the complexity and is used to trade off the fit and the number of submodels.

Given a linear system in a real or complex domain, linear regression aims to recover the model parameters from a set of observations. Recent studies in compressive sensing have successfully shown that under certain conditions, a linear program, namely, l1-minimization, guarantees recovery of sparse parameter signals even when the system is underdetermined. In this paper, we consider a more challenging problem: when the phase of the output measurements from a linear system is omitted. Using a lifting technique, we show that even though the phase information is missing, the sparse signal can be recovered exactly by solving a semidefinite program when the sampling rate is sufficiently high. This is an interesting finding since the exact solutions to both sparse signal recovery and phase retrieval are combinatorial. The results extend the type of applications that compressive sensing can be applied to those where only output magnitudes can be observed. We demonstrate the accuracy of the algorithms through extensive simulation and a practical experiment.

While compressive sensing (CS) has been one of the most vibrant research fields in the past few years, most development only applies to linear models. This limits its application in many areas where CS could make a difference. This paper presents a novel extension of CS to the phase retrieval problem, where intensity measurements of a linear system are used to recover a complex sparse signal. We propose a novel solution using a lifting technique – CPRL, which relaxes the NP-hard problem to a nonsmooth semidefinite program. Our analysis shows that CPRL inherits many desirable properties from CS, such as guarantees for exact recovery. We further provide scalable numerical solvers to accelerate its implementation. 

Change detection has traditionally been seen as a centralized problem. Many change detection problems are however distributed in nature and the need for distributed change detection algorithms is therefore significant. In this paper a distributed change detection algorithm is proposed. The change detection problem is first formulated as a convex optimization problem and then solved distributively with the alternating direction method of multipliers (ADMM). To further reduce the computational burden on each sensor, a homotopy solution is also derived. The proposed method have interesting connections with Lasso and compressed sensing and the theory developed for these methods are therefore directly applicable.

Intrigued by some recent results on impulse response estimation by kernel and nonparametric techniques, we revisit the old problem of transfer function estimation from input-output measurements. We formulate a classical regularization approach, focused on finite impulse response (FIR) models, and find that regularization is necessary to cope with the high variance problem. This basic, regularized least squares approach is then a focal point for interpreting other techniques, like Bayesian inference and Gaussian process regression. The main issue is how to determine a suitable regularization matrix (Bayesian prior or kernel). Several regularization matrices are provided and numerically evaluated on a data bank of test systems and data sets. Our findings based on the data bank are as follows. The classical regularization approach with carefully chosen regularization matrices shows slightly better accuracy and clearly better robustness in estimating the impulse response than the standard approach - the prediction error method/maximum likelihood (PEM/ML) approach. If the goal is to estimate a model of given order as well as possible, a low order model is often better estimated by the PEM/ML approach, and a higher order model is often better estimated by model reduction on a high order regularized FIR model estimated with careful regularization. Moreover, an optimal regularization matrix that minimizes the mean square error matrix is derived and studied. The importance of this result lies in that it gives the theoretical upper bound on the accuracy that can be achieved for this classical regularization approach.