Nonparametric Finite Mixture Models with Possible Shape Constraints: A Cubic Newton Approach
Name
21m1430972.pdf
Description
Published version
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770.5 KB
Format
Adobe PDF
Checksum (MD5)
182432f66b0bb2dc553df9375847e6e9
Author(s)
Wang, Haoyue
Ibrahim, Shibal
Mazumder, Rahul
Date Issued
March 2025
Journal
SIAM Journal on Mathematics of Data Science
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Version
Final published version
Abstract
We explore computational aspects of maximum likelihood estimation of the mixture proportions in a nonparametric finite mixture model—a convex optimization problem with old roots in statistics. Motivated by problems in shape constrained inference, we also 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. We propose a Frank–Wolfe method to solve the cubic regularized Newton subproblem and derive efficient solutions for the linear optimization oracles that may be of independent interest. In the particular case of Gaussian mixtures without shape constraints, we derive bounds on how well the finite mixture problem approximates the infinite-dimensional Kiefer–Wolfowitz maximum likelihood estimator. Experiments on synthetic and real datasets suggest that our proposed algorithms exhibit improved runtimes and scalability features over prior benchmarks.
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DOI of Published Version
https://doi.org/10.1137/21M1430972
