Let's say I have a 4-dimensional subspace of a 300 dimensional ambient space, represented by an orthonormal basis matrix $A$.
What I essentially want is to rotate this basis matrix, so that the individual basis vectors are still orthogonal and unit length, but more closely aligned to the axes of the ambient space. In other words, I want to find a basis matrix that spans the same space, but is more "sparse" than $A$ (sparse meaning many small values, not necessarily exactly 0 values).
I was thinking this could be solved by multiplying in a rotation matrix (that is constrained to never rotate outside of the subspace) and then optimizing the parameters of that rotation matrix such that the L1-norm of the resulting rotated basis matrix is minimized. Is this a reasonable approach? Is there something more obvious that I might be missing?
For background: I want to adjust a large set of basis matrices so that when I multiply together any pair ($A$, $B$) of them, the matrix $A \times B^T$ is as sparse as possible, i.e. certain the dot-products of the most similar vectors are as large as possible and the others are as small as possible.
Edit: as a trivial example, consider this 2-d subspace basis of R3: \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \end{bmatrix}
Trivially, this could be "aligned" to produce e.g. \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \end{bmatrix} I'm interested in a method to generalize this, so that a basis matrix is made as similar as possible to the identity matrix. I'm thinking "similar" in this case may mean maximally close along the diagonal, but I'm not sure about this intuition (especially since basis vectors can be permuted without changing the spanned space).