Obtaining accurate channel state estimates at reasonable training overheads remains a big challenge for the applicability of multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM). Recently, the exploitation of channel sparsity has led to sub-Nyquist channel sampling thereby reducing the channel training overhead. Still, there is a growing belief in channel sparsity appearance in many dimensions; time, frequency, angle, and space. Accordingly, this paper proposes an algorithm for channel estimation where sparsity in multidimensions is simultaneously exploited. Also, the applicability of sparse coding relies on the validity of a signal sparsity assumption and knowing the exact sparsity level. However, 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 also propose an algorithm for estimating the composite sparsity lying in multiple domains defined by learned dictionaries. Simulations validate a substantial channel estimation quality attained by the proposed algorithm as compared to the existing algorithms. The simulations also validate a high quality of sparsity estimation leading to performances close to the impractical case of assuming known sparsity.