The title really says it all.
I am looking to solve the following problem:
$$ \underset{U}{\text{minimize}} \; || \mathbf{A} - \mathbf{U}||_{F} $$ $$ \text{subject to} \; \; [\mathbf{U}]_{i,j} \in \{-1, 0, +1 \}$$ $$ \; \mathbf{U}^{T} \mathbf{U} = \mathbf{I} $$
So I think I can use an alternating projection algorithm here, but I am not sure.
Algorithm:
- Solve only with constraint $\mathbf{U}^{T} \mathbf{U} = \mathbf{I}$: I.e. $\hat{\mathbf{U}} = \mathbf{W} \mathbf{V}^{T}$, where $\mathbf{W}, \mathbf{V}$ are the left and right eigenvector matrices in the SVD of $\mathbf{A}$.
- Project onto the first constraint set: I.e. $[\hat{\mathbf{U}}]_{i,j} = 1 \; \text{if} \;[\hat{\mathbf{U}}]_{i,j} > \epsilon \; \text{else if} \; [\hat{\mathbf{U}}]_{i,j} = 0 \; \text{if} \;|[\hat{\mathbf{U}}]_{i,j}| \leq \epsilon \; \text{else} -1 $ for $\epsilon$ a small positive number.
- Repeat until convergence, using the last found $\hat{\mathbf{U}}$ instead of $\mathbf{A}$ in 1.
Will this work? Are my projection operators correct?
Its sort of a von-Neumann type iteration. Not sure if I am going to get the closest matrix to $\mathbf{A}$ in this intersection though...