  Applying compressive sensing (CS) allows for sub-Nyquist sampling in several
application areas in 5G and beyond. This reduces the associated training,
feedback, and computation overheads in many applications. However, the
applicability of CS relies on the validity of a signal sparsity assumption and
knowing the exact sparsity level. It is customary to assume a foreknown
sparsity level. Still, this assumption is not valid in practice, especially
when applying learned dictionaries as sparsifying transforms. The problem is
more strongly pronounced with multidimensional sparsity. In this paper, we
propose an algorithm for estimating the composite sparsity lying in multiple
domains defined by learned dictionaries. The proposed algorithm estimates the
sparsity level over a dictionary by inferring it from its counterpart with
respect to a compact discrete Fourier basis. This inference is achieved by a
machine learning prediction. This setting learns the intrinsic relationship
between the columns of a dictionary and those of such a fixed basis. The
proposed algorithm is applied to estimating sparsity levels in wireless
channels, and in cognitive radio spectra. Extensive simulations validate a high
quality of sparsity estimation leading to performances very close to the
impractical case of assuming known sparsity.
