Exact rank of the general regression matrix

Determine the exact rank of the regression matrix A, and hence characterize the necessary and sufficient conditions for unique solvability of the joint least-squares problem, for arbitrary numbers of coherence intervals T and arbitrary system dimensions M, N, K, and J.

Background

The paper formulates joint estimation of the uplink data matrices and the slowly varying AP–AP interference channel as a least-squares problem whose vectorized regression matrix is A. Unique recovery requires A to have full column rank. The authors derive a general necessary condition on T and establish exact or sufficient rank results only in special cases, including T=1 and M=N=2K=2J.

For the general parameter regime, the rank is reduced to the rank of an auxiliary matrix, but the paper does not determine that rank exactly. Resolving this problem would provide complete identifiability conditions for the proposed AP–AP interference-mitigation method.