I need to find a way to parametrise a matrix that is both sparse (to some degree) and orthogonal, i.e., I am looking for a parametrisation that describes $A \in \mathbb{R}^{n\times m}$ such that $AA^𝑇=𝐼$ and $||A||_0 << nm$.

It is not hard to generate orthogonal matrices through an SVD, but the orthogonal components of a sparse matrix are not guaranteed to be sparse themselves. Similarly, priors like spike and slab can lead to sparse matrices, but those are usually non-orthogonal.

Thus, is anyone aware of any work that presents a way of generating matrices that are both sparse and orthogonal?