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I have the following optimization problem in $X \in \mathbb R^{r \times n}$

$$\begin{array}{ll} \text{minimize} & \| A - B X \|_F + \beta \, \mbox{Tr}(X^TCX)\\ \text{subject to} & X^T X = I_n\end{array}$$

where matrices $A \in \mathbb R^{m \times n}$, $B \in \mathbb R^{m \times r}$, $C \in \mathbb R^{r \times r}$ are given, and $r \ll \min(m,n)$. $A$ is a binary matrix, $C$ and $B$ are positive. $I$ is identity matrix. $\mbox{Tr}$ is the trace operator and $\beta > 0$. How to solve above optimization problem with equality constraint (e.g. orthogonal constraint). Can we use gradient descent or project gradient descent?

Rodrigo de Azevedo
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jason
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  • At least you need to specify the space $A,B,C,X$ lives in. Your $\beta$ should also be greater than $0$ and is a data to this optimization problem. You may search the term \textbf{Stiefel manifold optimization} for more information. – Brian Ding Mar 07 '18 at 02:40
  • Thanks for the suggestion. – jason Mar 07 '18 at 02:47
  • Take a look at this. – Rodrigo de Azevedo Mar 07 '18 at 17:15
  • @RodrigodeAzevedo, Thanks, I will take a look – jason Mar 07 '18 at 17:18
  • @Brian: \textit and \textbf don't work here, use *one asterisk* for italics and **two asterisks** for bold. –  Mar 07 '18 at 17:49
  • Fatness and orthogonality do not quite mix. Suppose $r=3$ and $n=10$. The constraint $X^\top X = I_{10}$ requires that the $10$ columns of $X$ be orthonormal. However, how can one find $10$ orthogonal vectors in $3$-dimensional space? – Rodrigo de Azevedo Mar 08 '18 at 08:51

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