Definition 2.16 in the paper "Universality of Approximate Message Passing Algorithms and Tensor Networks" (PDF link via arXiv.org) by Wang, et al., describes a rectangular generalized invariant matrix:
I was wondering if this type of matrix captures either of the following random binary matrices:
- A doubly regular binary matrix with fixed column and row weights, where the matrix is sampled uniformly at random from all such doubly matrices.
- A near-constant column weight matrix, which is defined as a matrix where a fixed number of entries of each column of are selected uniformly at random with replacement and set to 1, with independence between columns. The remaining entries of the matrix are set to 0.
Due to my non-existent knowledge of random matrix theory, I wasn't able to identify if the following matrices above belong to the class of rectangular generalized invariant matrix, and appreciate it if someone could point it out to me. Thanks.
Remark: The paper mentions that such a matrix applies to deterministic Hadamard matrices, so I was wondering if it could also apply to random binary matrices? I thought maybe it might work if we replace zeros and ones in the binary matrices with -1 and 1 respectively.