In this paper, we consider a class of orthogonal constrained optimization problems, the feasible region of which is called the Stiefel manifold. Our new proposed framework combines a function value reduction step with a symmetrization step. Different from the existing approaches, the function value reduction is conducted in the Euclidean space instead of the Stiefel manifold or its tangent spac...

In this paper we present an unsupervised learning algorithm of neural networks with p inputs and m outputs whose weight vectors have orthonormal constraints. In this setting the learning algorithm can be regarded as optimization posed on the Stiefel manifold, and we generalize the natural gradient method to this case based on geodesics. By exploiting its geometric property as a quotient space: ...

We derive a family of high-order, structure-preserving approximations of the Riemannian exponential map on several matrix manifolds, including the group of unitary matrices, the Grassmannian manifold, and the Stiefel manifold. Our derivation is inspired by the observation that if Ω is a skew-Hermitian matrix and t is a sufficiently small scalar, then there exists a polynomial of degree n in tΩ ...

In this paper we give methods to construct zero divisors in the Cayley–Dickson algebras An=R 2 for n larger than 4. Also we relate the set of zero divisors with suitable Stiefel manifolds.

A Stiefel manifold of the compact type is often encountered in many fields of Engineering including, signal and image processing, machine learning, numerical optimization and others. The Stiefel manifold is a Riemannian homogeneous space but not a symmetric space. In previous work, researchers have defined probability distributions on symmetric spaces and performed statistical analysis of data ...

Using search directions of a recent class of three--term conjugate gradient methods, modified versions of the Hestenes-Stiefel and Polak-Ribiere-Polyak methods are proposed which satisfy the sufficient descent condition. The methods are shown to be globally convergent when the line search fulfills the (strong) Wolfe conditions. Numerical experiments are done on a set of CUTEr unconstrained opti...