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 $AA^=$ 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?



Parametrising a sparse orthogonal matrix
The smallest number of nonzero entries in an $n\times n$ fully indecomposable$^\*$ orthogonal matrix is $4n−4$. A method to construct such a matrix is described in [Sparse orthogonal matrices](https://core.ac.uk/download/pdf/82782666.pdf) (2003).


$^\*$ A fully indecomposable matrix does not have a $p\times q$ zero submatrix with $p+q=n$.


