Suppose we want to solve the following optimization problem in $S \in \mathbb{R}^{N \times T}$, where $T \gg S$,
$$\min_{S} f(S) \mbox{ subject to } SS^T \mbox{is diagonal}$$
which means rows of the matrix $S$ are mutually orthogonal.
I am suggested to solve this alternative problem by the following method:
$$\min_{S} f(S) + \|\mathcal{P}(SS^T)\|_1$$
in which $\mathcal{P}$ is a projection onto the off-diagonal indexes.
But I don't this $\ell_1$ penalty will promote the orthogonality between each rows by simply promote the sparsity of the off-diagonal elements, since we are not doing any actions like block coordinate descent to promote the orthogonality.
Any suggestions?