Restricted 𝑝-Isometry Properties of Partially Sparse Signal Recovery

Abstract

By generalizing the restricted 𝑝-isometry property to the partially sparse signal recovery problem, we give a sufficient condition for exactly recovering partially sparse signal via the partial 𝑙_𝑝 minimization (truncated 𝑙_𝑝 minimization) problem with 𝑝∈(0,1]. Based on this, we establish a simpler sufficient condition which can show how the 𝑝-RIP bounds vary corresponding to different 𝑝s.

1. Introduction

The partially sparse signal recovery (PSSR) is the problem of recovering a partially sparse signal from a certain number of linear measurements when the part of the signal is known to be sparse, which was coined by Bandeira et al. [1, 2]. This type of problems has many applications in signal and image processing, derivative-free optimizations, and so on; see, for example, [1–4]. Clearly, PSSR includes sparse signal recovery (SSR) as a special case. The latter is the well-known NP-hard problem in the compressed sensing (CS), which is also called cardinality minimization problem (CMP, or 𝑙₀-norm minimization problems); see, for example, [5–8]. In particular, Candés and Tao [8] introduced a restricted isometry property (RIP) of a sensing matrix which guarantees to recover a sparse solution of SSR by minimizing its convex relaxation (ℓ₁-norm minimization). However, there are some problems which cannot be reformulated as an SSR, but a PSSR. As we know, PSSR happens naturally in sparse Hessian recovery; see, for example, [2], where Bandeira et al. employed partially sparse recovery approach for building sparse quadratic interpolation models of functions with sparse Hessian. They have successfully applied the ℓ₁-norm minimization of PSSR in interpolation-based trust-region methods for derivative-free optimization. Vaswani and Lu [3] successfully applied modified CS (partially sparse recovery) in image reconstruction, where the sufficient RIP condition is weaker than the RIP for SSR. Moreover, Bandeira et al. [1] considered the RIP and null space properties (NSP) for PSSR and extended recovery results under noisy measurements to the partially sparse case, where partial NSP is a necessary and sufficient condition for PSSR. In [4], Jacques also established the partial RIP condition for PSSR with noise via its convex relaxation problem.

Note that in the CS context, the SSR problem can also be relaxed to a 𝑙_𝑝-norm minimization (truncated 𝑙_𝑝-minimization) problem with 0<𝑝<1; see, for example, [9–19]. It is well known that Chartrand [20] firstly show that fewer measurements are required for exact reconstruction if we replace 𝑙₁-norm with 𝑙_𝑝-norm (0<𝑝<1), and Chartrand and Staneva [10] established 𝑝-RIP conditions for exact SSR via 𝑙_𝑝-minimization. In particular, the numerical experiments in magnetic resonance imaging (MRI) showed that this approach works very efficiently; see [9] for details. Wang et al. [19] studied the performance of 𝑙_𝑝-minimization for strong recovery and weak recovery where we need to recover all the sparse vectors on one support with one sign pattern. Moreover, Saab et al. [16] provided a sufficient condition for SSR via 𝑙_𝑝-minimization and provided a lower bound of the support size up to which 𝑙_𝑝-minimization can recover all such sparse vectors, and Foucart and Lai [14] improved this bound by considering a generalized version of RIP condition. While SSR and 𝑙_𝑝-minimization have been the focus point of some recent research, there are fewer research related to PSSR and the partially 𝑙_𝑝-minimization. One may naturally wonder whether we can generalize the 𝑝-RIP conditions introduced by [10] from the SSR to the PSSR case. This paper will deal with this issue. We will give a different 𝑝-RIP recovery condition for PSSR via its nonconvex relaxation. Furthermore, based on the recent work by Oymak et al. [21], we also extend our result to the matrix setting.

In the next section, we give the PSSR model and review some preliminaries on 𝑝-RIP conditions. In Section 3, we establish the exact partially 𝑝-RIP recovery conditions for PSSR via its nonconvex 𝑙_𝑝-minimization. In Section 4, we give a sufficient condition for partially low-rank matrix recovery via the partially Schatten-𝑝 minimization problem.

2. Preliminaries

In this section, we will review some basic concepts and results on the 𝑝-RIP recovery conditions for SSR and introduce the 𝑝-RIP definition for PSSR. We begin with defining the mathematical model of the PSSR problem as follows: min‖𝑥‖₀,s.t.𝐴₁𝑥+𝐴₂𝑦=𝑏, (1) where the 𝑙₀-norm ‖𝑥‖₀ is defined as ‖𝑥‖₀∶=|{𝑖∶𝑥_𝑖≠0}| (which is not really a norm since it is not positive homogeneous). For any positive number 𝑠, we say 𝑥 is 𝑠-sparse if ‖𝑥‖₀≤𝑠. 𝐴=[𝐴₁,𝐴₂]∈ℝ^𝑀×𝑁 is a sensing matrix with 𝐴₁∈ℝ^{𝑀×(𝑁−𝑟)},  𝐴₂∈ℝ^𝑀×𝑟, and 𝑏∈ℝ^𝑀. It means that the unknown vector consists of two parts (𝑥,𝑦), where 𝑥∈ℝ^𝑁−𝑟 is sparse and 𝑦∈ℝ^𝑟 is possibly dense. When 𝐴=𝐴₁, the previous problem reduces to the following 𝑙₀-norm minimization problem (sparse signal recovery, SSR): min‖𝑥‖₀,s.t.𝐴𝑥=𝑏. (2) The previous PSSR problem (1) is an NP-hard problem, since its special case SSR (2) is well-known NP-hard problem in the compressed sensing (CS). As we mentioned in Section 1, one popular and powerful approach is to solve it via ℓ₁-norm minimization (its convex relaxation), where the 𝑙₀-norm is replaced by the 𝑙₁-norm in SSR (2). Moreover, we can also use a nonconvex approach for exact reconstruction with fewer measurements than the convex relaxation; see, for example, [9, 10]. That is the 𝑙_𝑝-norm minimization problem with 0<𝑝<1, where we replace the 𝑙₀-norm with the 𝑙_𝑝-norm in (2) as follows: min‖𝑥‖^𝑝_𝑝,s.t.𝐴𝑥=𝑏. (3) Note that ‖⋅‖_𝑝 is not a norm when 𝑝∈(0,1), but it is much close to 𝑙₀-norm. Moreover, the numerical experiments in MRI showed that the approach via 𝑙_𝑝-minimization works very efficiently; see [9] for details. In particular, Chartrand and Staneva [10] introduced the concept of restricted isometry constant via 𝑙_𝑝-norm.

Definition 1 (𝑝-RIC [10]). Given a matrix 𝐴∈ℝ^𝑀×𝑁, where 𝑀<𝑁, 𝑠 is a positive number and 0<𝑝<1, then we say that 𝛿_𝑠 is the restricted 𝑝-isometry constant (or 𝑝-RIC) of order 𝑠 of the matrix 𝐴 if 𝛿_𝑠 is the smallest number, such that 1−𝛿_𝑠‖𝑥‖^𝑝₂≤‖𝐴𝑥‖^𝑝_𝑝≤1+𝛿_𝑠‖𝑥‖^𝑝₂, (4) for all 𝑠-sparse vectors 𝑥.

In the same paper, Chartrand and Staneva gave the following sufficient condition for exact SSR via 𝑙_𝑝-minimization.

Theorem 2 (see [10]). Let 𝐴∈ℝ^𝑀×𝑁, 𝑥∈ℝ^𝑁, and 𝑘=‖𝑥‖₀ be the size of the support of 𝑥, 0<𝑝<1, 𝑎₁>1, and 𝑎₂=𝑎₁^2/(2−𝑝), rounded up, so that 𝑎₂𝑘 is an integer (𝑎₂=⌈𝑎₁^2/(2−𝑝)𝑘⌉/𝑘). If 𝐴 satisfies 𝛿_𝑎2𝑘+𝑎₁𝛿_{(𝑎2+1)𝑘}<𝑎₁−1, (5) then 𝑥 is the unique minimizer of problem (2).

Inspired by the previous analysis, it is natural to give the partially 𝑙_𝑝-norm minimization problem for PSSR (1) as follows: min‖𝑥‖^𝑝_𝑝,s.t.𝐴₁𝑥+𝐴₂𝑦=𝑏. (6) In order to establish the link between the PSSR (1) and its partially 𝑙_𝑝-norm minimization problem, we need to give a partially 𝑝-RIC definition. Here we borrow the idea from Bandeira et al. [1]. Assume that 𝐴₂ is full column rank. For 𝐴=[𝐴₁,𝐴₂] as mentioned above, let 𝐵∶=𝐼−𝐴₂𝐴^𝑇₂𝐴₂⁻¹𝐴^𝑇₂, (7) which is the matrix of the orthogonal projection from ℝ^𝑁 to ℜ(𝐴₂)^⟂.

Definition 3 (Partially 𝑝-RIC). Let 𝐴=[𝐴₁,𝐴₂]∈ℝ^𝑀×𝑁, where 𝐴₁∈ℝ^{𝑀×(𝑁−𝑟)}, and 𝐴₂∈ℝ^𝑀×𝑟 is full column rank. We say that 𝛿^𝑟_𝑠−𝑟 is the Partially Restricted Isometry Constant (Partially 𝑝-RIC) of order 𝑠−𝑟 of the matrix 𝐴 if 𝛿^𝑟_𝑠−𝑟 is the 𝑝-RIC of order 𝑠−𝑟 of the matrix 𝐵𝐴₁; that is, 𝛿^𝑟_𝑠−𝑟 is the smallest number, such that 1−𝛿^𝑟_𝑠−𝑟‖𝑥‖^𝑝₂≤𝐵𝐴₁𝑥^𝑝_𝑝≤1+𝛿^𝑟_𝑠−𝑟‖𝑥‖^𝑝₂, (8) for all (𝑠−𝑟)-sparse vectors 𝑥, where 𝐵 is given by (7).

3. Main Results

We will give our main results which state sufficient 𝑝-RIP recovery conditions on the exact PSSR via the nonconvex 𝑙_𝑝-norm minimization. We begin with the following useful lemma.

Lemma 4. For 0<𝑝≤1, let 𝑐₁=𝑎₁^1−𝑝/2 and 𝑐₂=𝑎₂^1−𝑝/2 with 𝑎₁>0 and 𝑎₂>0. If (𝑐₁−1)/𝑐₂>|𝑎₁−𝑎₂|/𝑎₂, then 𝑎₁>1 and 𝑎₂>1.

Proof. In order to prove the lemma, we consider the following two cases.
Case  1 (𝑎₁≥𝑎₂). In this case, from the fact (𝑐₁−1)/𝑐₂>|𝑎₁−𝑎₂|/𝑎₂, we have 𝑎₁𝑎₂−𝑎₁𝑎₂^1−𝑝/21<1−𝑎₂^1−𝑝/2. (9) If 0<𝑎₂≤1, from the previous inequality we easily obtain 𝑎0<¹𝑎₂−𝑎₁𝑎₂^1−𝑝/21<1−𝑎₂^1−𝑝/2≤0, (10) which is a contradiction. Hence 𝑎₁≥𝑎₂>1.
Case  2 (𝑎₁<𝑎₂). Similarly, in this case, from (𝑐₁−1)/𝑐₂>|𝑎₁−𝑎₂|/𝑎₂, we obtain 𝑎0<1−¹𝑎₂<𝑐₁−1𝑐₂. (11) If 0<𝑎₁≤1, then 𝑐₁=𝑎₁^1−𝑝/2≤1. (12) Combining the previous inequalities we obtain 𝑎0<1−¹𝑎₂<𝑐₁−1𝑐₂≤0, (13) which is a contradiction. Hence 1<𝑎₁<𝑎₂.
Therefore, taking into account the previous two cases, we completed the proof.

We below propose a general recovery condition for PSSR via its 𝑝-norm minimization.

Theorem 5. Let 𝐴=[𝐴₁,𝐴₂]∈ℝ^𝑀×𝑁 with 𝐴₁∈ℝ^{𝑀×(𝑁−𝑟)} and 𝐴₂∈ℝ^𝑀×𝑟. Suppose that 𝐴₂ is full column rank, and let 𝐴₁𝑥+𝐴₂𝑦=𝑏 with ‖𝑥‖₀=𝑘. For 0<𝑝≤1, 𝑎₁>0, and 𝑎₂>0, let 𝑐₁=𝑎₁^1−𝑝/2, 𝑐₂=𝑎₂^1−𝑝/2 with (𝑐₁−1)/𝑐₂>|𝑎₁−𝑎₂|/𝑎₂. If 𝑎₂𝑐₁𝛿_{(𝑎2+1)𝑘}+𝑎₁−𝑎₂𝑐₂+𝑎₂𝛿_𝑎1𝑘<𝑎₂𝑐₁−𝑎₁−𝑎₂𝑐₂−𝑎₂, (14) then (𝑥,𝑦) is the unique minimizer of problem(6)

Proof. Note that (𝑥,𝑦) is a feasible solution to optimization problem (6). We remain to show that the solution set is a singleton {(𝑥,𝑦)}. This proof generally modifies that of [10], but under different assumptions. (Specifically, we use a different way to arrange the elements of 𝑇^𝐶₀ in the following.) Let (𝑢,𝑣) be an arbitrary solution to problem (6). we will show that 𝑢=𝑥 and 𝑦=𝑣. We will prove 𝑢=𝑥 firstly. Taking ℎ=𝑢−𝑥, we will show that ℎ=0. Let Φ=𝐵𝐴₁. For 𝑇∈{1,…,𝑁−𝑟}, Φ_𝑇 denotes the matrix equaling Φ in those columns whose indices belong to 𝑇 and otherwise zero. Similarly, we define the vector ℎ_𝑇. Let 𝑇₀ be the support of 𝑥. Then, the supports of 𝑥 and ℎ_𝑇^𝐶₀ are disjoint since 𝑇₀⋂𝑇^𝐶₀=∅. From direct calculation, we obtain ‖𝑥‖^𝑝_𝑝≥‖𝑢‖^𝑝_𝑝=‖𝑥+ℎ‖^𝑝_𝑝=𝑥+ℎ_𝑇0+ℎ_𝑇^𝐶₀^𝑝_𝑝=𝑥+ℎ_𝑇0‖^𝑝_𝑝+‖ℎ_𝑇^𝐶₀^𝑝_𝑝≥‖𝑥‖^𝑝_𝑝−ℎ_𝑇0‖^𝑝_𝑝+‖ℎ_𝑇^𝐶₀^𝑝_𝑝, (15) where the first inequality holds because (𝑢,𝑣) solves (6), and the last one holds by the triangle inequality for ‖⋅‖^𝑝_𝑝. Then we have ℎ_𝑇^𝐶₀^𝑝_𝑝≤ℎ_𝑇0^𝑝_𝑝. (16)
Now we arrange the elements of 𝑇^𝐶₀ in order of decreasing magnitude of |ℎ| and partition into 𝑇^𝐶₀=𝑇₁⋃𝑇₂⋃⋯⋃𝑇_𝐽, where 𝑇₁ has 𝑎₂𝑘 elements and 𝑇_𝑗  (𝑗≥2) each has 𝑎₁𝑘 elements (except possibly 𝑇_𝐽). Set 𝑇₀₁=𝑇₀⋃𝑇₁. Note that 𝐵𝐴₂=𝐼−𝐴₂𝐴^𝑇₂𝐴₂⁻¹𝐴^𝑇₂𝐴₂=0. (17) Direct calculations yield 0=𝐵𝐴₁𝑥+𝐴₂𝑦−𝐴₁𝑢−𝐴₂𝑣^𝑝_𝑝=𝐵𝐴₁𝑥−𝐵𝐴₁𝑢^𝑝_𝑝=‖Φ𝑥−Φ𝑢‖^𝑝_𝑝=‖Φℎ‖^𝑝_𝑝=Φℎ_𝑇01+_𝑗≥2Φℎ_𝑇𝑗^𝑝_𝑝≥Φℎ_𝑇01^𝑝_𝑝−_𝑗≥2Φℎ_𝑇𝑗^𝑝_𝑝≥Φℎ_𝑇01^𝑝_𝑝−_𝑗≥2Φℎ_𝑇𝑗^𝑝_𝑝≥1−𝛿_{(𝑎2+1)𝑘}ℎ_𝑇01^𝑝₂−1+𝛿_𝑎1𝑘×_𝑗≥2ℎ_𝑇𝑗^𝑝₂. (18) Now we discuss the relation between 𝑙₂-norm and 𝑙_𝑝-norm. For each 𝑡∈𝑇_𝑗 and 𝑠∈𝑇_𝑗−1, it holds |ℎ_𝑡|≤|ℎ_𝑠|. So, we have for 𝑗=2,ℎ_𝑡^𝑝≤ℎ_𝑇1^𝑝_𝑝𝑎₂𝑘⟹ℎ_𝑡²≤ℎ_𝑇1²_𝑝𝑎₂𝑘^2/𝑝⟹ℎ_𝑇2²₂𝑎₁𝑘≤ℎ_𝑇1²_𝑝𝑎₂𝑘^2/𝑝⟹ℎ_𝑇2^𝑝₂≤𝑎^𝑝/2𝑎₂𝑘^1−𝑝/2ℎ_𝑇1^𝑝_𝑝. (19) Similarly, we obtain that for 𝑗≥3, ℎ_𝑇𝑗^𝑝₂≤1𝑎₁𝑘^1−𝑝/2ℎ_𝑇𝑗−1^𝑝_𝑝. (20) Applying the Holder's inequality, we obtain ℎ_𝑇0^𝑝_𝑝=_𝑡∈𝑇0ℎ_𝑡^𝑝≤⋅1_𝑡∈𝑇0ℎ_𝑡^2𝑝/2_𝑡∈𝑇01^1−𝑝/2=ℎ_𝑇0^𝑝₂⋅𝑘^1−𝑝/2. (21) Similarly, we have ℎ_𝑇1^𝑝_𝑝≤ℎ_𝑇1^𝑝₂⋅𝑎₂𝑘^1−𝑝/2. (22) Therefore, _𝑗≥2ℎ_𝑇𝑗^𝑝₂≤𝑎₁^𝑝/2𝑎₂𝑘^1−𝑝/2ℎ_𝑇1^𝑝_𝑝+1𝑎₁𝑘^1−𝑝/2_𝑗≥2ℎ_𝑇𝑗^𝑝_𝑝By(19)and()=120𝑎₁𝑘^1−𝑝/2_𝑗≥1ℎ_𝑇𝑗^𝑝_𝑝+𝑎₁−𝑎₂𝑎₁^𝑝/2𝑎₁𝑎₂𝑘^1−𝑝/2ℎ_𝑇1^𝑝_𝑝=1𝑎₁𝑘^1−𝑝/2ℎ_𝑇^𝐶₀^𝑝_𝑝+𝑎₁−𝑎₂𝑎₁^𝑝/2𝑎₁𝑎₂𝑘^1−𝑝/2ℎ_𝑇1^𝑝_𝑝≤1𝑎₁𝑘^1−𝑝/2ℎ_𝑇0^𝑝_𝑝+𝑎₁−𝑎₂𝑎₁^𝑝/2𝑎₁𝑎₂𝑘^1−𝑝/2ℎ_𝑇1^𝑝_𝑝By()≤𝑘16_1−𝑝/2𝑎₁𝑘^1−𝑝/2ℎ_𝑇0^𝑝₂+𝑎₁−𝑎₂𝑎₂𝑘^1−𝑝/2𝑎₂𝑎₁𝑘^1−𝑝/2ℎ_𝑇1^𝑝₂By(21)and()=122𝑐₁ℎ_𝑇0^𝑝₂+𝑎₁−𝑎₂𝑐₂𝑎₂𝑐₁ℎ_𝑇1^𝑝₂≤1𝑐₁ℎ_𝑇0^𝑝₂+𝑎₁−𝑎₂𝑐₂𝑎₂𝑐₁ℎ_𝑇1^𝑝₂≤1𝑐₁ℎ_𝑇01^𝑝₂+𝑎₁−𝑎₂𝑐₂𝑎₂𝑐₁ℎ_𝑇01^𝑝₂. (23) Thus by (18) and (23), we have 0≥1−𝛿_{(𝑎2+1)𝑘}ℎ_𝑇01^𝑝₂−1+𝛿_{𝑎1𝑘𝑗≥2}ℎ_𝑇𝑗^𝑝₂By()≥181−𝛿_{(𝑎2+1)𝑘}ℎ_𝑇01^𝑝₂−1+𝛿_𝑎1𝑘𝑐₁ℎ_𝑇01^𝑝₂−𝑎₁−𝑎₂𝑐₂1+𝛿_𝑎1𝑘𝑎₂𝑐₁ℎ_𝑇01^𝑝₂By(=23)1−𝛿_{(𝑎2+1)𝑘}−1+𝛿_𝑎1𝑘𝑐₁−𝑎₁−𝑎₂𝑐₂1+𝛿_𝑎1𝑘𝑎₂𝑐₁×ℎ_𝑇01^𝑝₂. (24) Clearly, the assumption ensures that the scalar factor is positive, and hence we obtain ℎ_𝑇01=0. That means ℎ_𝑇0=0. Using ‖ℎ_𝑇^𝐶₀‖^𝑝_𝑝≤‖ℎ_𝑇0‖^𝑝_𝑝, we obtain ℎ_𝑇^𝐶₀=0. Therefore, ℎ=0, which means 𝑥=𝑢.
Now we remain to show that 𝑦=𝑣. It is obvious that 𝐴₁𝑥+𝐴₂𝑦=𝐴₁𝑢+𝐴₂𝑣. Since 𝑥=𝑢, we have 𝐴₂(𝑦−𝑣)=0. Then 𝑦=𝑣 because 𝐴₂ is full column rank.

Theorem 5 states a different sufficient condition for the exactly PSSR via its nonconvex relaxation from the existing conditions for SSR.

Theorem 6. Let 𝐴=[𝐴₁,𝐴₂]∈ℝ^𝑀×𝑁 with 𝐴₁∈ℝ^{𝑀×(𝑁−𝑟)} and 𝐴₂∈ℝ^𝑀×𝑟. Suppose that 𝐴₂ is full column rank, and let 𝐴₁𝑥+𝐴₂𝑦=𝑏 with ‖𝑥‖₀=𝑘. For 0<𝑝≤1 and 𝑎>1, if 𝛿_(𝑎+1)𝑘<𝑎^1−𝑝/2−1𝑎^1−𝑝/2,+1 (25) then (𝑥,𝑦) is the unique minimizer of problem (6). Specifically, for all 0<𝑝≤1, if 𝛿_(𝑎+1)𝑘<√𝑎−1√,𝑎+1 (26) then (𝑥,𝑦) is the unique minimizer of problem (6).

Proof. Applying Theorem 5, here we only need to show that if (25) holds, we can find 𝑎₁ and 𝑎₂, such that (14) holds. We consider the three cases in the following.
Case  i (𝑎₁≥𝑎₂+1). In this case, we easily obtain 𝑎₁−𝑎₂≥1 and 𝛿_𝑎1𝑘≥𝛿_{(𝑎2+1)𝑘}. Therefore the following condition can guarantee the inequality (14): 𝑎₂𝑐₁𝛿_𝑎1𝑘+𝑎₁−𝑎₂𝑐₂+𝑎₂𝛿_𝑎1𝑘<𝑎₂𝑐₁+𝑐₂−1−𝑎₁𝑐₂. (27) Simplifying the previous inequality, we obtain 𝛿_𝑎1𝑘<𝑎₂𝑐₁−𝑎₁−𝑎₂𝑐₂−𝑎₂𝑎₂𝑐₁+𝑎₁−𝑎₂𝑐₂+𝑎₂=𝑎₁^1−𝑝/2−𝑎₁−𝑎₂𝑎₂^−𝑝/2−1𝑎₁^1−𝑝/2+𝑎₁−𝑎₂𝑎₂^−𝑝/2.+1 (28) In this case, employing 𝑎₂^−𝑝/2>0, we easily get that 𝑎₁=𝑎₂+1 gives the maximum value of the right of the inequality (the strongest result) which satisfies the condition (14). That is, 𝛿_{(𝑎2+1)𝑘}<𝑎₂+1^1−𝑝/2−𝑎₂^−𝑝/2−1𝑎₂+1^1−𝑝/2+𝑎₂^−𝑝/2.+1 (29)
Case  ii (𝑎₂≤𝑎₁<𝑎₂+1). In this case, we can get that 0≤𝑎₁−𝑎₂<1 and 𝛿_𝑎1𝑘<𝛿_{(𝑎2+1)𝑘}. Similarly, the following condition can guarantee the inequality (14): 𝑎₂𝑐₁𝛿_{(𝑎2+1)𝑘}+𝑎₁−𝑎₂𝑐₂+𝑎₂𝛿_{(𝑎2+1)𝑘}<𝑎₂𝑐₁+𝑐₂−1−𝑎₁𝑐₂. (30) Simplifying the previous inequality, we obtain 𝛿_{(𝑎2+1)𝑘}<𝑎₁^1−𝑝/2−𝑎₁−𝑎₂𝑎₂^−𝑝/2−1𝑎₁^1−𝑝/2+𝑎₁−𝑎₂𝑎₂^−𝑝/2.+1 (31) In this case, employing 𝑎₂^−𝑝/2>0, we get that 𝑎₁=𝑎₂ give the maximum value of the right of the inequality; that is, 𝛿_{(𝑎2+1)𝑘}<𝑎₂^1−𝑝/2−1𝑎₂^1−𝑝/2.+1 (32)
Case  iii (𝑎₁<𝑎₂). In this case, it is clear that 0<𝑎₂−𝑎₁, 𝑎₁<𝑎₂+1, and 𝛿_𝑎1𝑘<𝛿_{(𝑎2+1)𝑘}. So the following condition can guarantee the inequality (14): 𝑎₂𝑐₁𝛿_{(𝑎2+1)𝑘}+𝑎₂−𝑎₁𝑐₂+𝑎₂𝛿_{(𝑎2+1)𝑘}<𝑎₂𝑐₁−𝑐₂−1+𝑎₁𝑐₂. (33) Simplifying the previous inequality, we obtain 𝛿_{(𝑎2+1)𝑘}<𝑎₁^1−𝑝/2−𝑎₂−𝑎₁𝑎₂^−𝑝/2−1𝑎₁^1−𝑝/2+𝑎₂−𝑎₁𝑎₂^−𝑝/2.+1 (34) In this case, employing 𝑎₂^−𝑝/2>0, we chose 𝑎₂−𝑎₁→0 to give the maximum value of the right of the inequality. That is, 𝛿_{(𝑎2+1)𝑘}<𝑎₂^1−𝑝/2−1𝑎₂^1−𝑝/2.+1 (35)
It is easy to see that ((𝑎₂+1)^1−𝑝/2−𝑎₂^−𝑝/2−1)/((𝑎₂+1)^1−𝑝/2+𝑎₂^−𝑝/2+1)<(𝑎₂^1−𝑝/2−1)/(𝑎₁^1−𝑝/2+1). In fact, (𝑎₂+1)^1−𝑝/2+𝑎₂^−𝑝/2+1>𝑎₂^1−𝑝/2+1. On the other hand, (𝑎₂+1)^1−𝑝/2−𝑎₂^−𝑝/2−1−(𝑎₂^1−𝑝/2−1)=(𝑎₂+1)[(𝑎₂+1)^−𝑝/2−𝑎₂^−𝑝/2]<0, which means (𝑎₂+1)^1−𝑝/2−𝑎₂^−𝑝/2−1<𝑎₂^1−𝑝/2−1.
Therefore, combining the previous three cases, we obtain that one can choose 𝑎₁=𝑎₂ to get the weakest sufficient condition. It is easy to see that 𝑎₁=𝑎₂ satisfying the assumptions of Theorem 5.
After the previous discussion, using condition (25) and choosing 𝑎₁=𝑎₂=𝑎, we can derive condition (14).
Specifically, we consider the following function: 𝑓𝑝=𝑎^1−𝑝/2−1𝑎^1−𝑝/22+1=1−𝑎^1−𝑝/2.+1 (36) Clearly, 𝑓^′𝑝𝑎=−^1−𝑝/2ln𝑎𝑎^1−𝑝/2+1²<0, (37) and hence 𝑓(𝑝) is a decreasing function of 𝑝. Thus, for all 0<𝑝≤1, condition 𝛿_(𝑎+1)𝑘<√𝑎−1√𝑎+1 (38) can guarantee condition (14).
The proof is completed.

Applying Theorem 6, we understand how the 𝑝-RIP bounds related to 𝑝 as in Figure 1. From Figure 1, it is easy to give a stronger result (i.e., weaker sufficient condition) for smaller 𝑝. Moreover, by taking different values of 𝑝 with 𝑝=1/4,1/2,3/4, we obtain some interesting 𝑝-RIP bounds as in Table 1.

Table 1 

Bounds comparison on different values of 𝑝.

Figure 1 

The upper bound of 𝑝-RIC of order (𝑎+1)𝑘 with particular values of 𝑝.

4. Final Remark

In this paper, we studied the restricted 𝑝-isometry property to the partially sparse signal recovery problem and proposed a sufficient 𝑝-RIP condition for exactly recovering partially sparse signal via the partially 𝑙_𝑝-minimization problem with 𝑝∈(0,1]. It is worth generalizing the 𝑝-RIP condition from the vector case to the matrix case. Note that the well-known low-rank matrix recovery (LMR) problem has many applications and appeared in the literature of a diverse set of fields including matrix completion, quantum state tomography, face recognition, magnetic resonance imaging (MRI), computer vision, and system identification and control; see, for example, [21, 22] for more details and the reference therein. In particular, Oymak et al. [21] showed that several sufficient RIP recovery conditions for 𝑘 sparse vector are also sufficient for recovery of matrices of rank up to 2𝑘 via Schatten 𝑝-norm minimization. According to our approach to extend the 𝑝-RIP bound from SSP to partially SSR, we can obtain some different restricted 𝑝-isometry properties for LMR problem by using the idea in [21].

Acknowledgments

The work was supported in part by the National Basic Research Program of China (2010CB732501), the National Natural Science Foundation of China (11171018), and the Fundamental Research Funds for the Central Universities (2011JBM128, 2013JBM095).