Restricted Isometry Property Of Order Research Articles

Improved sufficient condition of ℓ 1–2 ‐minimisation for robust signal recovery

By means of the restricted isometry property of order k and the restricted orthogonality property of order (k,k), this Letter mainly establishes an improved sufficient condition for a recently emerged l 1 − 2 -minimisation to guarantee the robust signal recovery. The obtained condition is proved to be much better than the state-of-the-art ones for almost all parameters k.

Truncated sparse approximation property and truncated q-norm minimization

This paper considers approximately sparse signal and low-rank matrix's recovery via truncated norm minimization $\min_{x}\|x_T\|_q$ and $\min_{X}\|X_T\|_{S_q}$ 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 $\mathcal{A}$ 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 $\delta_{k}$ and $\delta_{2k}$, respectively. Last, we show that if $\delta_{t(k+|T^c|)}<\sqrt{(t-1)/t}$ for some $t\geq 4/3$, then measurement matrix $A$ and linear map $\mathcal{A}$ satisfy truncated sparse approximation property of order $k$. Which should point out is that when $T^c=\emptyset$, our conclusion implies that sparse approximation property of order $k$ is weaker than restricted isometry property of order $tk$.

A Proof of Conjecture on Restricted Isometry Property Constants $\delta _{tk}\ \left(0&lt;t&lt;\frac {4}{3}\right)$

In this paper, we give a complete answer to the conjecture on restricted isometry property (RIP) constants $\delta _{tk} (0 , which was proposed by T. Cai and A. Zhang. We have shown that when $0 , the condition $\delta _{tk} is sufficient to guarantee the exact recovery for all $k$ -sparse signals in the noiseless case via the constrained $\ell _{1}$ -norm minimization. These bounds are sharp in the sense that for any $\epsilon >0,\,\,\delta _{tk} cannot guarantee the exact recovery of some $k$ -sparse signals. Furthermore, it will be shown that similar characterizations also hold for low-rank matrix recovery. Thus, combined with T. Cai and A. Zhang’s work, a complete characterization for sharp RIP constants $\delta _{tk}$ for all $t > 0$ is obtained to guarantee the exact recovery of all $k$ -sparse signals and matrices with rank at most $k$ by $\ell _{1}$ -norm minimization and nuclear norm minimization, respectively. Noisy cases and approximately sparse cases are also considered. To solve the conjecture, we construct a few identities so that RIP of order $tk$ , which is the target of our main results, can be perfectly applied to them.

Improved bounds on restricted isometry constant for orthogonal multi matching pursuit

Orthogonal Multi Matching Pursuit (OMMP) is a super greedy-type algorithm for sparse approximation. We an- alyze the convergence property of OMMP based on Restricted Isometry Property (RIP). Our main conclusion is that if the sampling matrix Φ satisfies the Restricted Isometry Property of order (sK) with isometry constant δ< 1+1 /2 K/s − 1/2 K/s +4 K/s, then OMMP (s) can exactly recover an arbitrary K-sparse signal x from y =Φ x in at most K steps. where y ∈ R M is an observation vector and Φ ∈ R M ×N is a given M × N matrix with M< N ,w e usually call it measurement matrix or sampling matrix. Our goal is to recover x based on Φ and y. In this paper, we study the efficiency of compressed sensing via Orthogonal Matching Multi Pursuit (OMMP). OMMP (s) is a natural generation of Orthogonal Matching Pursuit (OMP) in the sense that it selects s candidates at each step while OMP selects one. Let us begin with the demonstration of the use of super greedy algorithms in the compressed sensing problem. Now let us recall some necessary concepts of CS. Assume that 1 K M N and 0 <δ< 1 .L et||x||0 denotes the number of nonzero coordinates of a signal x =( xj) N=1 ∈ R N , we say that the signal x is K-sparse if ||x||0 K.

New and Improved Johnson–Lindenstrauss Embeddings via the Restricted Isometry Property

Consider an $m \times N$ matrix $\Phi$ with the restricted isometry property of order k and level $\delta$; that is, the norm of any k-sparse vector in $\mathbb{R}^N$ is preserved to within a multiplicative factor of $1 \pm \delta$ under application of $\Phi$. We show that by randomizing the column signs of such a matrix $\Phi$, the resulting map with high probability embeds any fixed set of $p = O(e^k)$ points in $\mathbb{R}^N$ into $\mathbb{R}^m$ without distorting the norm of any point in the set by more than a factor of $1 \pm 4 \delta$. Consequently, matrices with the restricted isometry property and with randomized column signs provide optimal Johnson–Lindenstrauss embeddings up to logarithmic factors in N. In particular, our results improve the best known bounds on the necessary embedding dimension m for a wide class of structured random matrices; for partial Fourier and partial Hadamard matrices, we improve the recent bound $m \gtrsim \delta^{-4} \log(p) \log^4(N)$ given by Ailon and Liberty to $m \gtrsim \delta^{-2} \log(p) \log^4(N)$, which is optimal up to the logarithmic factors in N. Our results also have a direct application in the area of compressed sensing for redundant dictionaries.

Analysis of Orthogonal Matching Pursuit Using the Restricted Isometry Property

Orthogonal matching pursuit (OMP) is the canonical greedy algorithm for sparse approximation. In this paper we demonstrate that the restricted isometry property (RIP) can be used for a very straightforward analysis of OMP. Our main conclusion is that the RIP of order K+1 (with isometry constant δ <; [ 1/( 3√K)]) is sufficient for OMP to exactly recover any K-sparse signal. The analysis relies on simple and intuitive observations about OMP and matrices which satisfy the RIP. For restricted classes of K-sparse signals (those that are highly compressible), a relaxed bound on the isometry constant is also established. A deeper understanding of OMP may benefit the analysis of greedy algorithms in general. To demonstrate this, we also briefly revisit the analysis of the regularized OMP (ROMP) algorithm.