Unitary and non-unitary

Question

I have a problem where the optimum is achieved when non-unitary is equal to the unitary? Given the unitary matrix $\mathbf{U} \in \mathcal{C}^{N \times d}, N>d$, $\mathbf{G} \in \mathcal{C}^{N \times M}$ and $\mathbf{V} \in \mathcal{C}^{M \times d}$ is another unitary matrix. The optimal solution is obtained when $(\mathbf{I}_N-\mathbf{U}\mathbf{U}^H)\mathbf{G}\mathbf{V}\approx 0$, where $\mathbf{I}_N$ is $N \times N$ identity matrix and $.^H$ represents the conjugate transpose. This is achieved when $\mathbf{G}\mathbf{V}\approx \mathbf{U}$ but I know $\mathbf{G}\mathbf{V}$ is not unitary because $\mathbf{G}$ is a random matrix, considered to be known. So I asked myself is there anyway to obtain $\mathbf{G}\mathbf{V}\approx \mathbf{U}$. Thank you for answering.