  With the growing interest and applications in machine learning and data
science, finding an efficient method to sparse analysis the high-dimensional
data and optimizing a dimension reduction model to extract lower dimensional
features has becoming more and more important. Orthogonal constraints (Stiefel
manifold) is a commonly met constraint in these applications, and the sparsity
is usually enforced through the element-wise L1 norm. Many applications can be
found on optimization over Stiefel manifold within the area of physics and
machine learning. In this paper, we propose a novel idea by tackling the
Stiefel manifold through an nonlinear eigen-approach by first using ADMM to
split the problem into smooth optimization over manifold and convex non-smooth
optimization, and then transforming the former into the form of nonlinear
eigenvalue problem with eigenvector dependency (NEPv) which is solved by
self-consistent field (SCF) iteration, and the latter can be found to have an
closed-form solution through proximal gradient. Compared with existing methods,
our proposed algorithm takes the advantage of specific structure of the
objective function, and has efficient convergence results under mild
assumptions.
