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13.4 Network Capacity of Compressive Data Gathering 279
sparsity and favorable structure of the overcomplete system. Moreover, they also
proved that stable recovery of the sparse signal in an overcomplete dictionary also
works for noisy data, and the optimally sparse approximation to the noisy data, to
within the noise level, differs from the optimally sparse decomposition of the ideal
noiseless signal by at most a constant multiple of the noise level.
Suppose
˜
x is a vector of length 2N, and is the solution to the l
1
-minimization
problem defined in Eq. (13.8) when an overcomplete dictionary is used. Similarly,
the original sensor readings can be reconstructed by
˜
d = Ψ
0
˜
x. Denote
˜
x
s
as an N-
dimensional vector composed of the last N elements of ...