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For Stiefel manifold, it contains all the orthogonal column matrices

$$St(d,M) = \{X \in R^{M \times d} | X^TX = I\}$$

For Grassmann manifold, it is $$Gr(d,M) = \{col(X), X \in R^{M \times d}\}$$

For an objective problem $$ \min_{X \in St(d,M)} f(X)$$

can we solve above optimization using $$ \min_{X \in Gr(d,M)} f(X)$$

Is above transform legal? or $f(X)$ should satisfy some conditions?

jason
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    Yes. The Stiefel manifold is an $O(n)$ principle bundle over the Grassman manifold. https://en.wikipedia.org/wiki/Stiefel_manifold#As_a_principal_bundle ... for information on principle bundles these notes are good: https://www3.nd.edu/~mbehren1/18.906/prin.pdf – Elle Najt Oct 01 '18 at 20:00
  • I add some note above – jason Oct 01 '18 at 23:31

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Here $d\leq M$ and we work over $\mathbb{R}$. Your definition of $Gr$ is false; the $X$ must have full rank. On the other hand, it's not a good parametrization; you must add the quotient by the equivalence relation: $X\sim X'$ if and only if $X'=Xh$ where $h\in GL_d$.

Consider the canonical functions

$$f:O(M)\rightarrow St(d,M),g:O(M)\rightarrow Gr(d,M),h:St(d,M)\rightarrow Gr(d,m).$$

Since $f$, $g$, and $h$ are submersions, we deduce that

\begin{align} St(d,M) &\sim O(M)/O(M-d),Gr(d,M) \\ &\sim St(d,M)/O(d) \\ &\sim O(M)/(O(d)\times O(M-d)).\end{align}

Note also that $Gr(d,M)\sim Gr(M-d,M)$ and have dimension $d(M-d)$.

EDIT. Answer to OP. The problems $$\min_{U\in St(d,N)}\langle UU^T,L\rangle+\beta \|UU^T\|_1$$ and $$\min_{U\in Gr(d,N)}\langle UU^T,L\rangle+\beta \|UU^T\|_1$$ are equivalent under the condition that a representative of $U\in Gr$ is one of its orthonormal bases.

Xander Henderson
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