The physical world surrounding us exhibits an enormous degree of complexity and richness of phenomena. It has long been noted that this complexity we encounter - both in the classical and the quantum world - can arise from simple rules, say particles interacting locally. Still, the simplicity of the rules does not mean that an actual description in the huge configuration spaces arising is feasible. This is specifically true in quantum mechanics, where the vast dimension of the involved spaces is an enormous challenge to our description of the physical world.
In this proposal, we set out for the bold undertaking of searching for underlying laws, models and descriptions directly. We advocate a radically novel paradigm that looks at nature not in the tradition of first trying to formulate basic models and then to test predictions arising of these models. Rather conversely, we aim at identifying and simplifying the models from the outset, either from data or basic principles. We do so by combining a powerful machinery from signal processing, tools such as non-commutative compressed sensing and bi-linear form estimation, with tensor network descriptions. We apply this mindset to complex models in quantum chemistry, topologically ordered and other interacting quantum systems.