Marker code is an effective coding scheme to protect data from insertions and deletions. It has potential applications in future storage systems, such as DNA storage and racetrack memory. When decoding marker codes, perfect channel state information (CSI), i.e., insertion and deletion probabilities, are required to detect insertion and deletion errors. Sometimes, the perfect CSI is not easy to obtain or the accurate channel model is unknown. Therefore, it is deserved to develop detecting algorithms for marker code without the knowledge of perfect CSI. In this paper, we propose two CSI-agnostic detecting algorithms for marker code based on deep learning. The first one is a model-driven deep learning method, which deep unfolds the original iterative detecting algorithm of marker code. In this method, CSI become weights in neural networks and these weights can be learned from training data. The second one is a data-driven method which is an end-to-end system based on the deep bidirectional gated recurrent unit network. Simulation results show that error performances of the proposed methods are significantly better than that of the original detection algorithm with CSI uncertainty. Furthermore, the proposed data-driven method exhibits better error performances than other methods for unknown channel models.

Sparsity priors are commonly used in denoising and image reconstruction. For analysis-type priors, a dictionary defines a representation of signals that is likely to be sparse. In most situations, this dictionary is not known, and is to be recovered from pairs of ground-truth signals and measurements, by minimizing the reconstruction error. This defines a hierarchical optimization problem, which can be cast as a bi-level optimization. Yet, this problem is unsolvable, as reconstructions and their derivative wrt the dictionary have no closed-form expression. However, reconstructions can be iteratively computed using the Forward-Backward splitting (FB) algorithm. In this paper, we approximate reconstructions by the output of the aforementioned FB algorithm. Then, we leverage automatic differentiation to evaluate the gradient of this output wrt the dictionary, which we learn with projected gradient descent. Experiments show that our algorithm successfully learns the 1D Total Variation (TV) dictionary from piecewise constant signals. For the same case study, we propose to constrain our search to dictionaries of 0-centered columns, which removes undesired local minima and improves numerical stability.

We provide a comprehensive study of the convergence of forward-backward algorithm under suitable geometric conditions leading to fast rates. We present several new results and collect in a unified view a variety of results scattered in the literature, often providing simplified proofs. Novel contributions include the analysis of infinite dimensional convex minimization problems, allowing the case where minimizers might not exist. Further, we analyze the relation between different geometric conditions, and discuss novel connections with a priori conditions in linear inverse problems, including source conditions, restricted isometry properties and partial smoothness.