10 research outputs found
Clustering, Hamming Embedding, Generalized LSH and the Max Norm
We study the convex relaxation of clustering and hamming embedding, focusing
on the asymmetric case (co-clustering and asymmetric hamming embedding),
understanding their relationship to LSH as studied by (Charikar 2002) and to
the max-norm ball, and the differences between their symmetric and asymmetric
versions.Comment: 17 page
PRISMA: PRoximal Iterative SMoothing Algorithm
Motivated by learning problems including max-norm regularized matrix
completion and clustering, robust PCA and sparse inverse covariance selection,
we propose a novel optimization algorithm for minimizing a convex objective
which decomposes into three parts: a smooth part, a simple non-smooth Lipschitz
part, and a simple non-smooth non-Lipschitz part. We use a time variant
smoothing strategy that allows us to obtain a guarantee that does not depend on
knowing in advance the total number of iterations nor a bound on the domain
An inexact proximal path-following algorithm for constrained convex minimization
Many scientific and engineering applications feature large-scale non-smooth convex minimization problems over convex sets. In this paper, we address an important instance of this broad class where we assume that the non-smooth objective is equipped with a tractable proximity operator and that the convex constraints afford a self-concordant barrier. We provide a new joint treatment of proximal and self-concordant barrier concepts and illustrate that such problems can be efficiently solved without lifting problem dimensions. We propose an inexact path-following algorithmic framework and theoretically characterize the worst case convergence as well as computational complexity of this framework, and also analyze its behavior when the proximal subproblems are solved inexactly. To illustrate our framework, we apply its instances to both synthetic and real-world applications and illustrate their accuracy and scalability in large-scale settings. As an added bonus, we describe how our framework can obtain points on the Pareto frontier of regularized problems with self-concordant objectives in a tuning free fashion