Compression is essential for efficient storage and transmission of signals. One powerful method for compression is through the application of orthogonal transforms, which convert a group of N data samples into a group of N transform coefficients. In transform coding, the N samples are first transformed, and then the coefficients are individually quantized and entropy coded into binary bits. The transform serves two purposes: one is to compact the energy of the original N samples into coefficients with increasingly smaller variances so that removing smaller coefficients have negligible reconstruction errors, and another is to decorrelate the original samples so that the coefficients can be quantized and entropy coded individually without losing compression performance. 

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 [1] , [2] , typically considered in the context of classifiers.

The objectives of IEEE Signal Processing Magazine ( SPM ) are to propose, for any IEEE Signal Processing Society (SPS) member and beyond, a wide range of tutorial articles on both methods and applications in signal and image processing. The articles are divided into different categories: feature articles, column and forum articles, and articles in special issues, the specificities of which are detailed on the SPM webpage “Information for Authors - SPM”.

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.

In this article, we summarize the evolution of speech and language processing (SLP) in the past 25 years. We first provide a snapshot of popular research topics and the associated state of the art (SOTA) in various subfields of SLP 25 years ago, and then highlight the shift in research topics over the years. We describe the major breakthroughs in each of the subfields and the main driving forces that led us to the SOTA today. Societal impacts and potential future directions are also discussed.