Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It is ll known that compressed sensing problems reduce to solving large under-determined systems of equations. If  choose the compressed measurement matrix according to some appropriate distribution and the signal is sparse enough the l 1  optimization can exactly recover the ideally sparse signal with overwhelming probability by Candes, E. and Tao, T., [2], [1]. In the current paper,  will consider the case of the so-called approximately sparse signals. These signals are a generalized version of the ideally sparse signals. Letting the zero valued components of the ideally sparse signals to take the values of certain small magnitude one can construct the approximately sparse signals. Using a different but simple proof technique  show that the claims similar to those of [2] and [1] related to the proportionality of the number of large components of the signals to the number of measurements, hold for approximately sparse signals as ll. Furthermore, using the same technique  compute the explicit values of what this proportionality can be if the compressed measurement matrix A has a rotationally invariant distribution of the null-space.  also give the quantitative tradeoff beten the signal sparsity and the recovery robustness of the l 1  minimization. As it will turn out in an asymptotic case of the number of measurements the threshold result of [1] corresponds to a special case of our result.
['Mihailo Stojnic', 'Weiyu Xu', 'Babak Hassibi']
I am writing a research paper on a idea Compressed sensing of approximately sparse signals
[{"from": "human", "value": "I am writing a research paper on a idea Compressed sensing of approximately sparse signals"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It is ll known that compressed sensing problems reduce to solving large under-determined systems of equations. If  choose the compressed measurement matrix according to some appropriate distribution and the signal is sparse enough the l 1  optimization can exactly recover the ideally sparse signal with overwhelming probability by Candes, E. and Tao, T., [2], [1]. In the current paper,  will consider the case of the so-called approximately sparse signals. These signals are a generalized version of the ideally sparse signals. Letting the zero valued components of the ideally sparse signals to take the values of certain small magnitude one can construct the approximately sparse signals. Using a different but simple proof technique  show that the claims similar to those of [2] and [1] related to the proportionality of the number of large components of the signals to the number of measurements, hold for approximately sparse signals as ll. Furthermore, using the same technique  compute the explicit values of what this proportionality can be if the compressed measurement matrix A has a rotationally invariant distribution of the null-space.  also give the quantitative tradeoff beten the signal sparsity and the recovery robustness of the l 1  minimization. As it will turn out in an asymptotic case of the number of measurements the threshold result of [1] corresponds to a special case of our result."}]