Minimal set of Hyperrectangles covering an n-dimensional binary matrix with row permutations
If "n-partitioned hypergraphs" are defined as "hypergraphs together with a partition of their vertices into n subsets such that each hyperedge has only one vertex from each of those subsets" then your problem can be reformulated as  n-clique hyperdge-cover for n-uniform

n-partitioned hypergraphs, and that reformulation preserves approximations and the parameter

(when your problem is parametrized by number of hyperrectangles).

In particular, your problem is NP-hard to approximate, even for n=2.

Those problems are equivalent to:

Find [the assignment of one positive integer to each '1' in the matrix] such that with

C being the largest of those positive integers, [for all elements H of {1,2,3,...,C-1,C}

and all '0's in the matrix, there is a coordinate such that no '1' [with the

same value for that coordinate as the '0'] was assigned H], which minimizes C.

Thus, your problem can be solved in

((minimum_number_of_hyperrectangles)^(number_of_'1's)) * poly(input_size)

time with an embarassingly parallel workload.

Aside from the reformulation and the equivalent problem I mentioned,

I don't know of any problems that your problem is reducible to for any reason

other than your problem being in NPO and the other problem being NPO-hard.