The Euclidean gradient on Stiefel manifold

[grandowife]

Hello,

My objective function is min_{U \in S(d, p)} f(U) = || U * U' * X - X ||_{F}^{2}, where S(d, p) means the Stiefel manifold, || · ||_{F} refers to the Frobenius norm of matrix, and X is the d x n data matrix.

My problem is: can I use the orthogonal property of U when calculating the Euclidean norm? That is, using the property for the red block in the following figure.

Best 

Qiuying Shi

[Nicolas Boumal]

Hello,

Yes, you can. If $U$ is a point on your manifold and you compute the value or any derivative of $f$ at $U$, then you are allowed to use the fact that $U$ is ont that manifold in order to simplify expressions.

Best,

Nicolas