Deep learning (DL) has been wildly successful in practice, and most of the state-of-the-art machine learning methods are based on neural networks (NNs). Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep NNs (DNNs). In this article, we present a relatively new mathematical framework that provides the beginning of a deeper understanding of DL. This framework precisely characterizes the functional properties of NNs that are trained to fit to data. The key mathematical tools that support this framework include transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory, which are all techniques deeply rooted in signal processing.

It is our great pleasure to introduce the second part of this special issue to you! The IEEE Signal Processing Society (SPS) has completed 75 years of remarkable service to the signal processing community. The eight selected articles included in this second part are clear portraits of that. As the review process for these articles took longer, however, they could not be included in the first part of the special issue, and we are glad to bring them to you now.

Twenty-five years ago, the field of computational imaging arguably did not exist, at least not as a standalone arena of research activity and technical development. Of course, the idea of using computation to form images had been around for several decades, largely thanks to the development of medical imaging—such as magnetic resonance imaging (MRI) and X-ray tomography - in the 1970s and synthetic-aperture radar (SAR) even earlier. 

Audio signal processing has passed many landmarks in its development as a research topic. Many are well known, such as the development of the phonograph in the second half of the 19th century and technology associated with digital telephony that burgeoned in the late 20th century and is still a hot topic in multiple guises. Interestingly, the development of audio technology has been fueled not only by advancements in the capabilities of technology but also by high consumer expectations and customer engagement.

The ICASSP 2023 conference in Rhodes, Greece, was remarkable from multiple perspectives. Notably, this was the first fully in-person ICASSP after three consecutive virtual conferences, which were necessitated by the COVID-19 pandemic. Attendees fully embraced the opportunity to engage in live interactions and reestablish their networks.