Nonparametric Finite Mixture Models with Possible Shape Constraints: A Cubic Newton Approach

Understand

We explore computational aspects of maximum likelihood estimation of the mixture proportions of a nonparametric finite mixture model -- a convex optimization problem with old roots in statistics and a key member of the modern data analysis toolkit.

• Motivated by problems in shape constrained inference, we consider structured variants of this problem with additional convex polyhedral constraints.
• We propose a new cubic regularized Newton method for this problem and present novel worst-case and local computational guarantees for our algorithm.
• We extend earlier work by Nesterov and Polyak to the case of a self-concordant objective with polyhedral constraints, such as the ones considered herein.