Truncated Sparse Approximation Property and Truncated q-Norm Minimization
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This paper considers approximately sparse signal and low-rank matrix's recovery via truncated norm minimization _xx_T_q and _XX_T_S_q from noisy measurements. We first introduce truncated sparse approximation property, a more general robust null space property, and establish the stable recovery of signals and matrices under the truncated sparse approximation property. We also explore the relationship between the restricted isometry property and truncated sparse approximation property. And we also prove that if a measurement matrix A or linear map A satisfies truncated sparse approximation property of order k, then the first inequality in restricted isometry property of order k and of order 2k can hold for certain different constants δ_k and δ_2k, respectively. Last, we show that if δ_t(k+|T^c|)<√((t-1)/t) for some t≥ 4/3, then measurement matrix A and linear map A satisfy truncated sparse approximation property of order k. Which should point out is that when T^c=∅, our conclusion implies that sparse approximation property of order k is weaker than restricted isometry property of order tk.
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