Abstract: The pertinent structure of real-world data, in particular in high dimensional ambient space, is often significantly simpler than the ambient dimension would suggest. Low complexity models capturing this simpler structure can be used in tasks like detecting the boundaries of a region of interest in an image. For high-dimensional inverse problems, low complexity models are often used to make problems well-posed and result in regularized solutions. Such low complexity models can take the form of union-of-subspaces models (e.g., sparsity or low-rank assumptions) or have non-linear structure (e.g., low dimensional manifolds) and may involve statistical elements. Further, enforcing low complexity models is crucial in machine learning tasks to prevent overfitting. This talk will cover the basic theory and applications of low complexity models.