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The principal component prsuit with reduced linear measurements (PCP_RLM) has gained great attention in applications, such as machine learning, video, and aligning multiple images. The recent research shows that strongly convex optimization for compressive principal component pursuit can guarantee the exact low-rank matrix recovery and sparse matrix recovery as well. In this paper, we prove that the operator of PCP_RLM satisfies restricted isometry property (RIP) with high probability. In addition, we derive the bound of parameters depending only on observed quantities based on RIP property, which will guide us how to choose suitable parameters in strongly convex programming.

Matrix completion is a signal processing technique, which recovers an unknown low-rank or approximately low-rank matrix from a sampling of its entries. Recently, this technique has been applied in many fields, such as medical [

These fundamental results have a great impact in engineering and applied mathematics. However, these results are limited into convex optimization; that is, nuclear norm based convex optimization leads to the exact low-rank matrix recovery under suitable conditions. It is well known that strongly convex optimization has many intrinsic advantages, such as the uniqueness of optimal solution. Especially, the strongly convex optimization was widely used in designing efficient and effective algorithm in the literature of compressive sensing and low-rank matrix recovery. Zhang et al. extend them; they proved that strongly convex optimizations can guarantee the exact low-rank matrix recovery as well in the paper [

Pertaining to the strongly convex optimization for compressive principal component pursuit, we had studied it in detail in our former work in [

For convenience, we will give here a brief summary of the notations that will be used throughout the paper. We denoted the operator norm of matrix by

In this paper, we only address the strongly convex programming of principal component pursuit with reduced linear measurements, which is considered in paper [

When

Fix any

The value of

When

Assuming

Pertaining to the projection operator

Suppose that

According to Lemma

Supposing that

It is obvious that the band of

In this paper, we first prove that although the operator

The main center of the paper provides the proof of Theorem

In this section, we will provide the proof of Theorem

We will provide some important lemmas, which are used in the main theorem.

Let

If

Let

In other words,

Theorem 8(a).

According to triangle inequality, we have

Theorem 8(b).

For

Suppose that

With high probability, one has

According to the definition of

Assume that

According to Lemma

Under the assumption of Theorem

Note that

Next, we will bound the behavior of

Theorem 13(a).

As pointed out by one reviewer, Lemma 15 in [

According to the definition of

According to Lemma

Theorem 13(b).

The main idea of the proof follows the arguments of Theorem 8(b).

According to Theorems 8 and 13 and the assumption of Lemma

In this paper, we address the problem of principal component pursuit with reduced linear measurements (PCP_RLM). In order to obtain an easy handed band of

The authors would like to thank the anonymous reviewers for their comments that helped to improve the quality of the paper. This research was supported by the National Natural Science Foundation of China (NSFC) under Grant 61172140, and “985” Key Projects for the excellent teaching team support (postgraduate) under Grant A1098522-02. Haiwen Xu is supported by Joint Fund of the National Natural Science Foundation of China and the Civil Aviation Administration of China (Grant no. U1233105).