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The multiple measurement vector (MMV) problem is applicable in a wide range of applications such as photoplethysmography (PPG), remote PPG measurement, heart rate estimation, and directional arrival estimation of multiple sources. Measurements in the aforementioned applications exhibit a dependency structure, which is not considered in the general MMV algorithms. Modeling the dependency or the correlation structure of the solution matrix to MMV problems can increase the recovery performance. The solution matrix <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> can be decomposed into a mixing matrix <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> and a sparse matrix with independent columns <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>. The key idea of this model is that the matrix S can be sparser than the mixing matrix <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula>. Previous MMV algorithms did not consider such a structure for <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula>. This paper proposes two algorithms, which are based on orthogonal matching pursuit and basis pursuit, and derives the exact recovery guarantee conditions for both approaches. We compare the simulation results of the proposed algorithms with the conventional algorithms and show that the proposed algorithms outperform previous algorithms especially in the case of the low number of measurements.