SMALL will develop a new foundational framework for processing signals, using adaptive sparse structured representations.A key discriminating feature of sparse representations, which opened up the horizons to new ways of thinking in signal processing including compressed sensing, has been the focus on developing reliable algorithms with provable performance and bounded complexity. Yet, such approaches are simply inapplicable in many scenarios for which no suitable sparse model is known. Moreover, the success of sparse models heavily depends on the choice of a "dictionary" to reflect the natural structures of a class of data, but choosing a dictionary is currently something of an "art", using expert knowledge rather than automatically applicable principles. Inferring a dictionary from training data is key to the extension of sparse models for new exotic types of data.SMALL will explore new generations of provably good methods to obtain inherently data-driven sparse models, able to cope with large-scale and complicated data much beyond state-of-the-art sparse signal modelling. The project will develop a foundational theoretical framework for the dictionary-learning problem, and scalable algorithms for the training of structured dictionaries. SMALL algorithms will be evaluated against state-of-the art alternatives and we will demonstrate our approach on a range of showcase applications. We will organise two open workshops to disseminate our results and get feedback from the research community.The proposed framework will deeply impact the research landscape since the new models, approaches and algorithms will be generically applicable to a wide variety of signal processing problems, including acquisition, enhancement, manipulation, interpretation and coding. This new line of attack will lead to many new theoretical and practical challenges, with a potential to reshape both the signal processing research community and the burgeoning compressed sensing industry.