Mathematical optimization is an important branch of applied mathematics. Different classes
of optimization problems are categorized based on their problem structures. While there are …

We review probabilistic principal components, principal fitted components, sufficient
dimension reduction, and envelopes, arguing that at their core they are all based on …

Riemannian optimization is the task of finding an optimum of a real-valued function defined
on a Riemannian manifold. Riemannian optimization has been a topic of much interest over …

Abstract Recently, the approximate Karush–Kuhn–Tucker (AKKT) conditions, also called the
sequential optimality conditions, have been proposed for nonlinear optimization in …

Sufficient dimension reduction methods are flexible tools for data visualization and
exploratory analysis, typically in a regression of a univariate response on a multivariate …

Motivated by brain connectivity analysis and many other network data applications, we study
the problem of estimating covariance and precision matrices and their differences across …

In this paper, we propose a new line-search method that improves the ordinary Armijo line-
search in Riemannian optimization. For optimization problems on Riemannian manifolds …

Sufficient dimension reduction methods aim to reduce the dimensionality of predictors while
preserving regression information relevant to the response. In this article, we develop …

Envelope model is a nascent dimension reduction technique. We focus on extending the
envelope methodology to broader applications. In the first part of this thesis we propose a …