Introduction of additional critical points with respect to Euclidean setting

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Carlos Feres

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Mar 6, 2022, 7:34:25 PM3/6/22
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Hi all,

Let's say that for a particular function f: C^n->R over (complex) Euclidean space, the only known optimal solutions belong to a subset of the complex Stiefel manifold.

Given that the solutions are in a Riemannian manifold, we can optimize over the Sfiefel manifold using manifold optimization over the restriction of f on the Sfiefel manifold, denoted g=f |_{St} , and reduce the search space.

From the definition of Riemannian gradient, grad g = Proj (nabla f) , it is obvious that critical points of f are critical points of the Riemannian gradient, grad g. 

My question is: Is the converse true in general? Is it true on this particular case, where we know the solution space (and its dimension) beforehand?

Or is it possible that the restriction to the Stiefel manifold (or any manifold) introduces additional critical points? 

Thanks!

Nicolas Boumal

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Mar 7, 2022, 1:34:10 AM3/7/22
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Hello,

In general, restricting a function to a smaller set can introduce new critical points. Often times, this is desirable (but maybe not in your case).

Here is a simple example: let f(X) = real(Tr(A*X)) for some non-zero matrix A which has the same size as X, and A* is the conjugate-transpose.
Then, f does not have any critical points (grad f(X) = A \neq 0 for all X).
Yet, f does have a global minimizer and a global maximizer on St(n, p) (because the latter is compact and f is continuous), hence f must have critical points when restricted to St(n, p) (and likely that is what we would want in this scenario).

(I'm assuming your function f is defined on C^{n x p} and you restrict it to St(n, p) = {X : X*X = I}  as a subset of C^{n x p}.)

Best,
Nicolas

Carlos Feres

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Mar 7, 2022, 4:41:44 AM3/7/22
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Hi Nicolas,

Thanks for your quick answer. I figured that in general, the restriction introduces additional critical points. I am trying to prove that the critical set in Euclidean space is equal to the critical set in St(n,p) in my particular case, but it seems that I would need more information for a more formal argument. So far, my current take is that the critical set has dimension p, and the equation grad g = 0 should have 2np-p^2-p degrees of freedom, but I still need to formalize that to have a final answer.

Best,
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