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We propose a novel approach to reconstruct Hyperspectral images from very few number of noisy compressive measurements. Our reconstruction approach is based on a convex minimization which penalizes both the nuclear norm and the ℓ2,1 mixed-norm of the data matrix. Thus, the solution tends to have a simultaneous low-rank and joint-sparse structure. We explain how these two assumptions fit Hyperspectral data, and by severals simulations we show that our proposed reconstruction scheme significantly enhances the state-of-the-art tradeoffs between the reconstruction error and the required number of CS measurements.