The reason to focus on DAE priors is explained in line 54-56 (also in line 14 and 35 of this rebuttal).

There are two major routes to address linear inverse problems. Whereas regularization-based approaches build estimators as solutions of penalized regression optimization problems, Bayesian estimators rely on the posterior distribution of the unknown, given some assumed family of priors. While these may seem radically different approaches, recent results have shown that, in the context of additive white Gaussian denoising, the Bayesian conditional mean estimator is always the solution of a penalized regression problem. The contribution of this paper is twofold. First, we extend the additive white Gaussian denoising results to general linear inverse problems with colored Gaussian noise. Second, we characterize conditions under which the penalty function associated to the conditional mean estimator can satisfy certain popular properties such as convexity, separability, and smoothness. This sheds light on some tradeoff between computational efficiency and estimation accuracy in sparse regularization, and draws some connections between Bayesian estimation and proximal optimization.

We study the problem of single-channel source separation (SCSS), and focus on cyclostationary signals, which are particularly suitable in a variety of application domains. Unlike classical SCSS approaches, we consider a setting where only examples of the sources are available rather than their models, inspiring a data-driven approach. For source models with underlying cyclostationary Gaussian constituents, we establish a lower bound on the attainable mean squared error (MSE) for any separation method, model-based or data-driven. Our analysis further reveals the operation for optimal separation and the associated implementation challenges. As a computationally attractive alternative, we propose a deep learning approach using a U-Net architecture, which is competitive with the minimum MSE estimator. We demonstrate in simulation that, with suitable domain-informed architectural choices, our U-Net method can approach the optimal performance with substantially reduced computational burden.

Despite the simplicity and intuitive interpretation of Minimum Mean Squared Error (MMSE) estimators, their effectiveness in certain scenarios is questionable. Indeed, minimizing squared errors on average does not provide any form of stability, as the volatility of the estimation error is left unconstrained. When this volatility is statistically significant, the difference between the average and realized performance of the MMSE estimator can be drastically different. To address this issue, we introduce a new risk-aware MMSE formulation which trades between mean performance and risk by explicitly constraining the expected predictive variance of the involved squared error. We show that, under mild moment boundedness conditions, the corresponding risk-aware optimal solution can be evaluated explicitly, and has the form of an appropriately biased nonlinear MMSE estimator. We further illustrate the effectiveness of our approach via several numerical examples, which also showcase the advantages of risk-aware MMSE estimation against risk-neutral MMSE estimation, especially in models involving skewed, heavy-tailed distributions.

We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator---that is, a measurable function of the observation---and a fictitious adversary choosing a prior---that is, a pair of signal and noise distributions ranging over independent Wasserstein balls---with the goal to minimize and maximize the expected squared estimation error, respectively. We show that if the Wasserstein balls are centered at normal distributions, then the zero-sum game admits a Nash equilibrium, where the players' optimal strategies are given by an {\em affine} estimator and a {\em normal} prior, respectively. We further prove that this Nash equilibrium can be computed by solving a tractable convex program. Finally, we develop a Frank-Wolfe algorithm that can solve this convex program orders of magnitude faster than state-of-the-art general purpose solvers. We show that this algorithm enjoys a linear convergence rate and that its direction-finding subproblems can be solved in quasi-closed form.

We study a distributionally robust mean square error estimation problem over a nonconvex Wasserstein ambiguity set containing only normal distributions. We show that the optimal estimator and the least favorable distribution form a Nash equilibrium. Despite the non-convex nature of the ambiguity set, we prove that the estimation problem is equivalent to a tractable convex program. We further devise a Frank-Wolfe algorithm for this convex program whose direction-searching subproblem can be solved in a quasi-closed form. Using these ingredients, we introduce a distributionally robust Kalman filter that hedges against model risk.

We study a distributionally robust mean square error estimation problem over a nonconvex Wasserstein ambiguity set containing only normal distributions. We show that the optimal estimator and the least favorable distribution form a Nash equilibrium. Despite the non-convex nature of the ambiguity set, we prove that the estimation problem is equivalent to a tractable convex program. We further devise a Frank-Wolfe algorithm for this convex program whose direction-searching subproblem can be solved in a quasi-closed form. Using these ingredients, we introduce a distributionally robust Kalman filter that hedges against model risk.

Efficient and reliable estimation in many signal processing problems encountered in applications requires adopting sparsity prior in a suitable basis on the signals and using techniques from compressed sensing (CS). In this paper, we study a CS problem known as Multiple Measurement Vectors (MMV) problem, which arises in joint estimation of multiple signal realizations when the signal samples have a common (joint) support over a fixed known dictionary. Although there is a vast literature on the analysis of MMV, it is not yet fully known how the number of signal samples and their statistical correlations affects the performance of the joint estimation in MMV. Moreover, in many instances of MMV the underlying sparsifying dictionary may not be precisely known, and it is still an open problem to quantify how the dictionary mismatch may affect the estimation performance. In this paper, we focus on $\ell_{2,1}$-norm regularized least squares ($\ell_{2,1}$-LS) as a well-known and widely-used MMV algorithm in the literature. We prove an interesting decoupling property for $\ell_{2,1}$-LS, where we show that it can be decomposed into two phases: i) use all the signal samples to estimate the signal covariance matrix (coupled phase), ii) plug in the resulting covariance estimate as the true covariance matrix into the Minimum Mean Squared Error (MMSE) estimator to reconstruct each signal sample individually (decoupled phase). As a consequence of this decomposition, we are able to provide further insights on the performance of $\ell_{2,1}$-LS for MMV. In particular, we address how the signal correlations and dictionary mismatch affects its estimation performance. We also provide numerical simulations to validate our theoretical results.

Network growth processes can be understood as generative models of the structure and history of complex networks. This point of view naturally leads to the problem of network archaeology: Reconstructing all the past states of a network from its structure---a difficult permutation inference problem. In this paper, we introduce a Bayesian formulation of network archaeology, with a generalization of preferential attachment as our generative mechanism. We develop a sequential importance sampling algorithm to evaluate the posterior averages of this model, as well as an efficient heuristic that uncovers the history of a network in linear time. We use these methods to identify and characterize a phase transition in the quality of the reconstructed history, when they are applied to artificial networks generated by the model itself. Despite the existence of a no-recovery phase, we find that non-trivial inference is possible in a large portion of the parameter space as well as on empirical data.