We propose a novel graphical model selection scheme for high-dimensional stationary time series or discrete time processes. The method is based on a natural generalization of the graphical LASSO algorithm, introduced originally for the case of i.i.d. samples, and estimates the conditional independence graph of a time series from a finite length observation. The graphical LASSO for time series is defined as the solution of an ℓ 1 -regularized maximum (approximate) likelihood problem. We solve this optimization problem using the alternating direction method of multipliers. Our approach is nonparametric as we do not assume a finite dimensional parametric model, but only require the process to be sufficiently smooth in the spectral domain. For Gaussian processes, we characterize the performance of our method theoretically by deriving an upper bound on the probability that our algorithm fails. Numerical experiments demonstrate the ability of our method to recover the correct conditional independence graph from a limited amount of samples.

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The "least absolute shrinkage and selection operator" (Lasso) method has been adapted recently for network-structured datasets. In particular, this network Lasso method allows to learn graph signals from a small number of noisy signal samples by using the total variation of a graph signal for regularization. While efficient and scalable implementations of the network Lasso are available, only little is known about the conditions on the underlying network structure which ensure network Lasso to be accurate. By leveraging concepts of compressed sensing, we address this gap and derive precise conditions on the underlying network topology and sampling set which guarantee the network Lasso for a particular loss function to deliver an accurate estimate of the entire underlying graph signal. We also quantify the error incurred by network Lasso in terms of two constants which reflect the connectivity of the sampled nodes.

We consider unbiased estimation of a sparse nonrandom vector corrupted by additive white Gaussian noise. We show that while there are infinitely many unbiased estimators for this problem, none of them has uniformly minimum variance. Therefore, we focus on locally minimum variance unbiased (LMVU) estimators. We derive simple closed-form lower and upper bounds on the variance of LMVU estimators or, equivalently, on the Barankin bound (BB). Our bounds allow an estimation of the threshold region separating the low-SNR and high-SNR regimes, and they indicate the asymptotic behavior of the BB at high SNR. We also develop numerical lower and upper bounds which are tighter than the closed-form bounds and thus characterize the BB more accurately. Numerical studies compare our characterization of the BB with established biased estimation schemes, and demonstrate that while unbiased estimators perform poorly at low SNR, they may perform better than biased estimators at high SNR. An interesting conclusion of our analysis is that the high-SNR behavior of the BB depends solely on the value of the smallest nonzero component of the sparse vector, and that this type of dependence is also exhibited by the performance of certain practical estimators. Index TermsSparsity, unbiased estimation, denoising, Cramér-Rao bound, Barankin bound, Hammersley-Chapman-Robbins bound, locally minimum variance unbiased estimator.where A ∈ R M ×N (M ≥ N ) is a known matrix with orthonormal columns, i.e., A T A = I N , and n ∼ N (0, σ 2 I M ) denotes zero-mean white Gaussian noise with known variance σ 2 (here, I N denotes the identity matrix of size N × N ). Note that without loss of generality we can then assume that A = I N and M = N , i.e., y = x + n, since premultiplication of the model (1) by A T will reduce the estimation problem to an equivalent problem y ′ = A ′ x + n ′ in which A ′ = A T A = I N and the noise n ′ = A T n is again zero-mean white Gaussian with variance σ 2 . Such a sparse signal model can be used, e.g., for channel estimation [16] when the channel consists only of few significant taps and an orthogonal training signal is used [17]. Another application that fits our scope is image denoising using an orthonormal wavelet basis [3]. We note that parts of this work were previously presented in [18].The estimation problem (1) with A = I N was studied by Donoho and Johnstone [19,20]. Their work was aimed at demonstrating asymptotic minimax optimality, i.e., they considered estimators having optimal worst-case behavior when the problem dimensions N, S tend to infinity. By contrast, we consider the finite-dimensional setting, and attempt to characterize the performance at each value of x, rather than analyzing worst-case behavior. Such a "pointwise" approach was also advocated by the authors of [21,22], who studied the CRB for the sparse linear model (1) with arbitrary A. However, the CRB is a local bound, in the sense that the performance characterization it provides is only based on the statistical properties in the neighborhood of...