Tirza Routtenberg

Yakov Gusakov, Osvaldo Simeone, Tirza Routtenberg et al.

Deep neural network (DNN)-based receivers offer a powerful alternative to classical model-based designs for wireless communication, especially in complex and nonlinear propagation environments. However, their adoption is challenged by the rapid variability of wireless channels, which makes pre-trained static DNN-based receivers ineffective, and by the latency and computational burden of online stochastic gradient descent (SGD)-based learning. In this work, we propose an online learning framework that enables rapid low-complexity adaptation of DNN-based receivers. Our approach is based on two main tenets. First, we cast online learning as Bayesian tracking in parameter space, enabling a single-step adaptation, which deviates from multi-epoch SGD . Second, we focus on modular DNN architectures that enable parallel, online, and localized variational Bayesian updates. Simulations with practical communication channels demonstrate that our proposed online learning framework can maintain a low error rate with markedly reduced update latency and increased robustness to channel dynamics as compared to traditional gradient descent based method.

Nir Shlezinger, Tirza Routtenberg

Inference tasks in signal processing are often characterized by the availability of reliable statistical modeling with some missing instance-specific parameters. One conventional approach uses data to estimate these missing parameters and then infers based on the estimated model. Alternatively, data can also be leveraged to directly learn the inference mapping end-to-end. These approaches for combining partially-known statistical models and data in inference are related to the notions of generative and discriminative models used in the machine learning literature, typically considered in the context of classifiers. The goal of this lecture note is to introduce the concepts of generative and discriminative learning for inference with a partially-known statistical model. While machine learning systems often lack the interpretability of traditional signal processing methods, we focus on a simple setting where one can interpret and compare the approaches in a tractable manner that is accessible and relevant to signal processing readers. In particular, we exemplify the approaches for the task of Bayesian signal estimation in a jointly Gaussian setting with the mean-squared error (MSE) objective, i.e., a linear estimation setting.

Itay Buchnik, Guy Sagi, Nimrod Leinwand et al.

Dynamic systems of graph signals are encountered in various applications, including social networks, power grids, and transportation. While such systems can often be described as state space (SS) models, tracking graph signals via conventional tools based on the Kalman filter (KF) and its variants is typically challenging. This is due to the nonlinearity, high dimensionality, irregularity of the domain, and complex modeling associated with real-world dynamic systems of graph signals. In this work, we study the tracking of graph signals using a hybrid model-based/data-driven approach. We develop the GSP-KalmanNet, which tracks the hidden graphical states from the graphical measurements by jointly leveraging graph signal processing (GSP) tools and deep learning (DL) techniques. The derivations of the GSP-KalmanNet are based on extending the KF to exploit the inherent graph structure via graph frequency domain filtering, which considerably simplifies the computational complexity entailed in processing high-dimensional signals and increases the robustness to small topology changes. Then, we use data to learn the Kalman gain following the recently proposed KalmanNet framework, which copes with partial and approximated modeling, without forcing a specific model over the noise statistics. Our empirical results demonstrate that the proposed GSP-KalmanNet achieves enhanced accuracy and run time performance as well as improved robustness to model misspecifications compared with both model-based and data-driven benchmarks.

Nir Shlezinger, Santiago Segarra, Yi Zhang et al.

Optimization methods play a central role in signal processing, serving as the mathematical foundation for inference, estimation, and control. While classical iterative optimization algorithms provide interpretability and theoretical guarantees, they often rely on surrogate objectives, require careful hyperparameter tuning, and exhibit substantial computational latency. Conversely, machine learning (ML ) offers powerful data-driven modeling capabilities but lacks the structure, transparency, and efficiency needed for optimization-driven inference. Deep unfolding has recently emerged as a compelling framework that bridges these two paradigms by systematically transforming iterative optimization algorithms into structured, trainable ML architectures. This article provides a tutorial-style overview of deep unfolding, presenting a unified perspective of methodologies for converting optimization solvers into ML models and highlighting their conceptual, theoretical, and practical implications. We review the foundations of optimization for inference and for learning, introduce four representative design paradigms for deep unfolding, and discuss the distinctive training schemes that arise from their iterative nature. Furthermore, we survey recent theoretical advances that establish convergence and generalization guarantees for unfolded optimizers, and provide comparative qualitative and empirical studies illustrating their relative trade-offs in complexity, interpretability, and robustness.

Yakov Medvedovsky, Eran Treister, Tirza Routtenberg

The Laplacian-constrained Gaussian Markov Random Field (LGMRF) is a common multivariate statistical model for learning a weighted sparse dependency graph from given data. This graph learning problem can be formulated as a maximum likelihood estimation (MLE) of the precision matrix, subject to Laplacian structural constraints, with a sparsity-inducing penalty term. This paper aims to solve this learning problem accurately and efficiently. First, since the commonly used $\ell_1$-norm penalty is inappropriate in this setting and may lead to a complete graph, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to existing first-order methods for this problem, we develop a second-order proximal Newton approach to obtain an efficient solver, utilizing several algorithmic features, such as using Conjugate Gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of both computational complexity and graph learning accuracy compared to existing methods.

Morad Halihal, Tirza Routtenberg, H. Vincent Poor

In this paper, we analyze the performance of the estimation of Laplacian matrices under general observation models. Laplacian matrix estimation involves structural constraints, including symmetry and null-space properties, along with matrix sparsity. By exploiting a linear reparametrization that enforces the structural constraints, we derive closed-form matrix expressions for the Cramer-Rao Bound (CRB) specifically tailored to Laplacian matrix estimation. We further extend the derivation to the sparsity-constrained case, introducing two oracle CRBs that incorporate prior information of the support set, i.e. the locations of the nonzero entries in the Laplacian matrix. We examine the properties and order relations between the bounds, and provide the associated Slepian-Bangs formula for the Gaussian case. We demonstrate the use of the new CRBs in three representative applications: (i) topology identification in power systems, (ii) graph filter identification in diffused models, and (iii) precision matrix estimation in Gaussian Markov random fields under Laplacian constraints. The CRBs are evaluated and compared with the mean-squared-errors (MSEs) of the constrained maximum likelihood estimator (CMLE), which integrates both equality and inequality constraints along with sparsity constraints, and of the oracle CMLE, which knows the locations of the nonzero entries of the Laplacian matrix. We perform this analysis for the applications of power system topology identification and graphical LASSO, and demonstrate that the MSEs of the estimators converge to the CRB and oracle CRB, given a sufficient number of measurements.

Julian P. Merkofer, Guy Revach, Nir Shlezinger et al.

Direction of arrival (DoA) estimation of multiple signals is pivotal in sensor array signal processing. A popular multi-signal DoA estimation method is the multiple signal classification (MUSIC) algorithm, which enables high-performance super-resolution DoA recovery while being highly applicable in practice. MUSIC is a model-based algorithm, relying on an accurate mathematical description of the relationship between the signals and the measurements and assumptions on the signals themselves (non-coherent, narrowband sources). As such, it is sensitive to model imperfections. In this work we propose to overcome these limitations of MUSIC by augmenting the algorithm with specifically designed neural architectures. Our proposed deep augmented MUSIC (DA-MUSIC) algorithm is thus a hybrid model-based/data-driven DoA estimator, which leverages data to improve performance and robustness while preserving the interpretable flow of the classic method. DA-MUSIC is shown to learn to overcome limitations of the purely model-based method, such as its inability to successfully localize coherent sources as well as estimate the number of coherent signal sources present. We further demonstrate the superior resolution of the DA-MUSIC algorithm in synthetic narrowband and broadband scenarios as well as with real-world data of DoA estimation from seismic signals.