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Evaluating the statistical dimension is a common tool to determine the asymptotic phase transition in compressed sensing problems with Gaussian ensemble. Unfortunately, the exact evaluation of the statistical dimension is very difficult and it has become standard to replace it with an upper-bound. To ensure that this technique is suitable, [1] has introduced an upper-bound on the gap between the statistical dimension and its approximation. In this work, we first show that the error bound in [1] in some low-dimensional models such as total variation and ell _{1} analysis minimization becomes poorly large. Next, we develop a new error bound which significantly improves the estimation gap... 

This work is about the total variation (TV) minimization which is used for recovering gradient-sparse signals from compressed measurements. Recent studies indicate that TV minimization exhibits a phase transition behavior from failure to success as the number of measurements increases. In fact, in large dimensions, TV minimization succeeds in recovering the gradient-sparse signal with high probability when the number of measurements exceeds a certain threshold; otherwise, it fails almost certainly. Obtaining a closed-form expression that approximates this threshold is a major challenge in this field and has not been appropriately addressed yet. In this work, we derive a tight lower-bound on...