Non-convex optimization plays important role in wide range of areas of science and engineering — from learning feature representations for visual classification, to reconstructing images in biology, medicine and astronomy, to disentangling spikes from multiple neurons. The worst case theory for non-convex optimization is dismal: in general, even guaranteeing a local minimum is NP hard. However, in these and other applications, very simple iterative methods such as gradient descent often perform surprisingly well.
In this talk, I will discuss examples of nonconvex optimization problems that can be solved to global optimality using simple iterative methods, which succeed independent of initialization. These include variants of the sparse blind deconvolution problem, sparse dictionary learning problem, image recovery from certain types of phaseless measurements. These problems possess a characteristic structure, in which (i) all local minima are global, and (ii) the energy landscape does not have any “flat” saddle points. For each of the aforementioned problems, this geometric structure allows us to obtain new types of performance guarantees. I will motivate these problems from applications in imaging and computer vision, and describe how this viewpoint leads to new approaches to analyzing electron microscopy data.
Joint work with Ju Sun (Stanford), Qing Qu (Columbia), Yuqian Zhang (Columbia), Yenson Lau (Columbia) Sky Cheung, (Columbia), Abhay Pasupathy (Columbia).
John Wright, Columbia University