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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 this threshold in case of any random measurement matrix whose null space is distributed uniformly with respect to the Haar measure. In contrast to the conventional TV phase transition results that depend on the simple gradient-sparsity level, our bound is highly affected by generalized notions of gradient-sparsity. Our proposed bound is very close to the true phase transition of TV minimization confirmed by simulation results.
Characterizing the phase transitions of convex optimizations in recovering structured signals or data is of central importance in compressed sensing, machine learning and statistics. The phase transitions of many convex optimization signal recovery m
This work considers the use of Total variation (TV) minimization in the recovery of a given gradient sparse vector from Gaussian linear measurements. It has been shown in recent studies that there exist a sharp phase transition behavior in TV minimiz
We consider the total variation (TV) minimization problem used for compressive sensing and solve it using the generalized alternating projection (GAP) algorithm. Extensive results demonstrate the high performance of proposed algorithm on compressive
We consider the classic joint source-channel coding problem of transmitting a memoryless source over a memoryless channel. The focus of this work is on the long-standing open problem of finding the rate of convergence of the smallest attainable expec
A closed-form expression for a lower bound on the per soliton capacity of the nonlinear optical fibre channel in the presence of (optical) amplifier spontaneous emission (ASE) noise is derived. This bound is based on a non-Gaussian conditional probab