Many tasks in signal processing and machine learning involve minimizing polynomials of degree four. These include phase retrieval, matrix factorization, sensor network localization and many more. In this talk, I will give an overview of the challenges of quartic minimization as well as some complexity results. In particular, we will focus on a particular class of convex quartic problems, and we analyze the notion of quartic condition number. We design an algorithm for reducing this condition number. To do so, we build a preconditioner using a generalized version of the Lewis weights (a.k.a leverage scores), and we show that it is optimal in some specific sense.

Based on joint work with Yurii Nesterov.