joint work with Andy Sun
This work is motivated by the desire to solve network constrained nonconvex programs with convergence guarantees. We propose a two-level algorithm with an inner level of ADMM in an augmented Lagrangian framework that can guarantee global convergence for a wide range of smooth and nonsmooth nonconvex constrained programs. We demonstrate the effectiveness of the algorithm with comparison to existing algorithms on some important classes of problems and discuss its convergence rate properties.
The non-negative matrix factorization (NMF) model with an additional orthogonality constraint on one of the factor matrices, called the orthogonal NMF (ONMF), has been found to provide improved clustering performance over the K-means. We propose an equivalent problem reformulation that transforms the orthogonality constraint into a set of norm-based non-convex equality constraints. We then apply one smooth and one non-smooth penalty approach to handle these non-convex constraints, which result in desirable separable structures for efficient parallel computations.
joint work with Stephen Vavasis, Chen Wen (Sabrina) Zhai
Sum-of-norms clustering is a novel clustering method using convex optimization. Recently, Panahi et al. proved that sum-of norms clustering is guaranteed to recover a mixture of Gaussians under the restriction that the sample size is not too large. This paper lifts the restriction and shows that sum-of-norms clustering with equal weights recovers a mixture of Gaussians even as the number of samples tends to infinity. Our proof relies on an interesting characterization of clusters computed by sum-of-norms clustering developed inside a proof of the agglomeration conjecture by Chiquet et al.