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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.