摘要
This paper considers approximately sparse signal and low-rank matrix’s recovery via truncated norm minimization minx∥xT∥q and minX∥XT∥Sq 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 constantsδk andδ2k,respectively.Last,we show that ifδs(k+|T^c|)<√(s-1)/s for some s≥4/3,then measurement matrix A and linear map A satisfy truncated sparse approximation property of order k.It should be pointed out that when Tc=Ф,our conclusion implies that sparse approximation property of order k is weaker than restricted isometry property of order sk.
This paper considers approximately sparse signal and low-rank matrix’s recovery via truncated norm minimization minx ||xT||q and minX||XT||Sq 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 2 k can hold for certain different constants δk and δ2k, respectively. Last, we show that if δs（k+|Tc|） <（（s-1）/s）1/2 for some s≥ 4/3, then measurement matrix A and linear map A satisfy truncated sparse approximation property of order k. It should be pointed out that when Tc= Ф, our conclusion implies that sparse approximation property of order k is weaker than restricted isometry property of order sk.
基金
supported by the National Natural Science Foundation of China(11871109)
NSAF(U1830107)
the Science Challenge Project(TZ2018001)
