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We consider the problem of recovering sparse phenomena from projections of noisy data, a topic of interest in compressed sensing. We describe the problem in terms of sensing capacity, which we define as the supremum of the ratio of the number of signal dimensions that can be identified per projection. This notion quantifies minimum number of observations required to estimate a signal as a function of sensing channel, SNR, sensed environment(sparsity) as well as desired distortion up to which the sensed phenomena must be reconstructed. We first present bounds for two different sensing channels: (A) i.i.d. Gaussian observations (B) Correlated observations. We then extend the results derived for the correlated case to the problem of learning sparse graphical models. We then present convex programming methods for the different distortions for the correlated case. We then comment on the differences between the achievable bounds and the performance of convex programming methods.