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I want to find the nearest (in the Frobenius sense) $n\times d$ (where $n > d$) matrix $X$ to a given matrix $A$ of the same size such that the $d$ columns of $X$ are orthonormal .

Since $X$ would lie on the Stiefel manifold, I can use polar or QR decomposition of $A$ to find $X$, but there is an additional constraint for this problem. I want the last row of $A$, which has a norm of one, to be preserved, i.e., I want the last row of $X$ to be the same as the last row of $A$.

Is there a way to do this?

Xander Henderson
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Arman
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1 Answers1

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Using the Frobenius norm,

$$ \begin{array}{ll} \underset {{\bf X} \in \Bbb R^{n \times d}} {\text{minimize}} & \| {\bf X} - {\bf A} \|_{\text F}^2 \\ \text{subject to} & {\bf X}^\top {\bf X} = {\bf I}_d \\ & {\bf e}_n^\top {\bf X} = {\bf e}_n^\top {\bf A} \end{array} $$

which can be rewritten as follows

$$ \begin{array}{ll} \underset {{\bf X} \in \Bbb R^{n \times d}} {\text{maximize}} & \langle {\bf A} , {\bf X} \rangle \\ \text{subject to} & {\bf X}^\top {\bf X} = {\bf I}_d \\ & {\bf e}_n^\top {\bf X} = {\bf e}_n^\top {\bf A} \end{array} $$

Relaxing$^\color{red}{\star}$ the Stiefel-related constraints,

$$ \begin{array}{ll} \underset {{\bf X} \in \Bbb R^{n \times d}} {\text{maximize}} & \langle {\bf A} , {\bf X} \rangle \\ \text{subject to} & {\bf X}^\top {\bf X} \preceq {\bf I}_d \\ & {\bf e}_n^\top {\bf X} = {\bf e}_n^\top {\bf A} \end{array} $$

which can be rewritten as the following semidefinite program (SDP)

$$ \color{blue}{\boxed{\begin{array}{ll} \underset {{\bf X} \in \Bbb R^{n \times d}} {\text{maximize}} & \langle {\bf A} , {\bf X} \rangle \\ \text{subject to} & \begin{bmatrix} {\bf I}_n & {\bf X} \\ {\bf X}^\top & {\bf I}_d\end{bmatrix} \succeq {\bf O}_{n+d} \\ & {\bf e}_n^\top {\bf X} = {\bf e}_n^\top {\bf A} \end{array}}} $$

Reference

$\color{red}{\star}\quad$ Kyle A. Gallivan, P.-A. Absil, Note on the convex hull of the Stiefel manifold, July 19, 2010.

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Rodrigo de Azevedo
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