Deep Unfolding: bridging the gap between model-based and data-driven approaches

This dissertation investigates recent machine learning architectures that aim to bridge traditional model-based methods with data-driven approaches, with a particular focus on interpretability, structure, and consistency. Central to this work is the paradigm of deep unfolding, which provides a unifying framework to reinterpret iterative algorithms as trainable neural networks. We show that deep unfolding naturally manifests through two complementary mechanisms: algorithm unrolling, which defines the network architecture by unfolding iterative procedures into layered structures, and backpropagation through time, which enables the optimization of temporally structured models. Within this framework, bilevel optimization emerges as a principled mathematical foundation that connects classical optimization algorithms with modern learning-based formulations, yielding neural architectures that remain grounded in well-understood mathematical principles. The first part of the dissertation focuses on audio processing applications, specifically source separation and detection. We revisit nonnegative matrix factorization (NMF) and its variants as a powerful model-based framework for time-frequency analysis. Algorithm unrolling is then employed to enhance NMF-based methods, leading to competitive deep architectures such as PAD-NMF and Deep-NMFD. These approaches overcome limitations of traditional NMF, including slow convergence and limited adaptability, while preserving interpretability. We further analyze how temporal dependencies are incorporated through different mechanisms, highlighting complementary strengths across practical scenarios. The second part of the dissertation addresses state estimation in dynamical systems, a domain traditionally dominated by Kalman filtering. After reviewing the Kalman filter, ensemble Kalman methods, and particle filters, we frame state estimation as a general data assimilation problem and explore learning-enhanced extensions of classical filtering techniques. Conditional diffusion models are presented as a data-driven alternative to particle filters, capable of sampling from complex conditional posteriors without requiring explicit likelihood models. In parallel, KalmanNet and the proposed Deep Kalman Filter are introduced as learned extensions of the Kalman filter that preserve its predictor-corrector structure while overcoming classical assumptions such as linearity and known noise statistics. Both methods exemplify deep unfolding, combining