Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Consider the  n -dimensional vector  y = X β+e where β ∈ \BBR p  has only  k  nonzero entries and e ∈ \BBR n  is a Gaussian noise. This can be vied as a linear system with sparsity constraints corrupted by noise, where the objective is to estimate the sparsity pattern of β given the observation vector  y  and the measurement matrix  X . First,  derive a nonasymptotic upper bound on the probability that a specific wrong sparsity pattern is identified by the maximum-likelihood estimator.  find that this probability depends (inversely) exponentially on the difference of || X β|| 2  and the  l  2  -norm of  X β projected onto the range of columns of  X  indexed by the wrong sparsity pattern. Second, when  X  is randomly drawn from a Gaussian ensemble,  calculate a nonasymptotic upper bound on the probability of the maximum-likelihood decoder not declaring (partially) the true sparsity pattern. Consequently,  obtain sufficient conditions on the sample size  n  that guarantee almost surely the recovery of the true sparsity pattern.  find that the required growth rate of sample size  n  matches the growth rate of previously established necessary conditions.
['Kamiar Rahnama Rad']
I am writing a research paper on a idea Nearly Sharp Sufficient Conditions on Exact Sparsity Pattern Recovery
[{"from": "human", "value": "I am writing a research paper on a idea Nearly Sharp Sufficient Conditions on Exact Sparsity Pattern Recovery"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Consider the  n -dimensional vector  y = X \u03b2+e where \u03b2 \u2208 \\BBR p  has only  k  nonzero entries and e \u2208 \\BBR n  is a Gaussian noise. This can be vied as a linear system with sparsity constraints corrupted by noise, where the objective is to estimate the sparsity pattern of \u03b2 given the observation vector  y  and the measurement matrix  X . First,  derive a nonasymptotic upper bound on the probability that a specific wrong sparsity pattern is identified by the maximum-likelihood estimator.  find that this probability depends (inversely) exponentially on the difference of || X \u03b2|| 2  and the  l  2  -norm of  X \u03b2 projected onto the range of columns of  X  indexed by the wrong sparsity pattern. Second, when  X  is randomly drawn from a Gaussian ensemble,  calculate a nonasymptotic upper bound on the probability of the maximum-likelihood decoder not declaring (partially) the true sparsity pattern. Consequently,  obtain sufficient conditions on the sample size  n  that guarantee almost surely the recovery of the true sparsity pattern.  find that the required growth rate of sample size  n  matches the growth rate of previously established necessary conditions."}]