This paper gives new bounds for restricted isometry constant(RIC)in compressed sensing.LetΦbe an m×n real matrix and k be a positive integer with k≤n.The main results of this paper show that if the restricted isometry constant ofΦsat-isfiesδ8ak<1 andδk+ak<3/2−1+√(4a+3)^(2)−8/8aforα>3/8,then k-sparse solution can be recovered exactly via l1 minimization in the noiseless case.In particular,whenα=1,1.5,2 and3,we haveδ2k<0.5746 andδ8k<1,orδ2.5k<0.7046 andδ12k<1,orδ3k<0.7731 andδ16k<1 orδ4k<0.8445 andδ24k<1.
This paper aims at achieving a simultaneously sparse and low-rank estimator from the semidefinite population covariance matrices.We first benefit from a convex optimization which develops l1-norm penalty to encourage the sparsity and nuclear norm to favor the low-rank property.For the proposed estimator,we then prove that with high probability,the Frobenius norm of the estimation rate can be of order O(√((slgg p)/n))under a mild case,where s and p denote the number of nonzero entries and the dimension of the population covariance,respectively and n notes the sample capacity.Finally,an efficient alternating direction method of multipliers with global convergence is proposed to tackle this problem,and merits of the approach are also illustrated by practicing numerical simulations.
Sheng-Long ZhouNai-Hua XiuZi-Yan LuoLing-Chen Kong
