This paper addresses the problem of the clutter subspace projector estimation in the context of a disturbance composed of a low rank heterogeneous (Compound Gaussian) clutter and white Gaussian noise. In such a context, adaptive processing based on an estimated orthogonal projector onto the clutter subspace (instead of an estimated covariance matrix) requires less samples than classical methods. The clutter subspace estimate is usually derived from the eigenvalue decomposition of a covariance matrix estimate. However, it has been previously shown that a direct Maximum Likelihood Estimator of the clutter subspace projector can be obtained for the considered context. In this paper, we derive two algorithms based on the block majorization-minimization framework to reach this estimator. These algorithms are shown to be computationally faster than the state of the art, with guaranteed convergence. Finally, the performance of the related estimators is illustrated on realistic Space Time Adaptive Processing for airborne radar simulations.