On the l 1 -Norm Invariant Convex k-sparse Decomposition of Signals

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1 On the l 1 -Norm Invariant Convex -Sparse Decomposition of Signals arxiv: v2 [cs.it] 11 Nov 2013 Guangwu Xu and Zhiqiang Xu Abstract Inspired by an interesting idea of Cai and Zhang, we formulate and prove the convex -sparse decomposition of vectors that is invariant with respect to the l 1 norm. This result fits well in discussing compressed sensing problems under RIP, but we believe it also has independent interest. As an application, a simple derivation of the RIP recovery condition δ +θ, < 1 is presented. Keywords: Convex -sparsedecomposition, l 1 minimization, restrictedisometry property, sparse recovery. 1 Introduction The Restricted Isometry Property (RIP) of Candès and Tao [7] is one of the most commonly used framewors for sparse recovery via l 1 minimization. For an n p matrix Φ R n p and an integer, 1 p, the -restricted isometry constant δ is the smallest constant such that 1 δ c 2 Φc 2 1+δ c 2 for every -sparse vector c (namely, c has at most nonzero components). If + p, the, -restricted orthogonality constant θ,, is the smallest number that satisfies Φc,Φc θ, c 2 c 2, for all -sparse vector c and -sparse vector c with disjoint supports. Department of EE & CS, University of Wisconsin-Milwauee, Milwauee, WI 53211, USA; e- mail: gxu4uwm@uwm.edu. Research supported in part by the National 973 Project of China (No. 2013CB834205). Inst. Comp. Math., Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China; xuzq@lsec.cc.ac.cn. Zhiqiang Xu was supported by NSFC grant , , and National Basic Research Program of China (973 Program 2010CB832702). 1

2 It has been shown that l 1 minimization can recover a sparse signal with a small or zeroerrorundervariousconditionsonδ andθ,, suchastheconditionδ +θ, +θ,2 < 1inCandèsandTao[7], andtheconditionδ < 0.307ofCai,WangandXu[6]. Recently, Cai and Zhang [2] established a sharp condition on δ for -sparse recovery: δ < 1 3. In the same paper, they also proved that δ 2 < 1 is sufficient for -sparse signal reconstruction. Cai and Zhang developed a marvelous technique in the proof of their 2 results. Inspired by the division lemma of Cai and Zhang [2], we formulate and prove the l 1 -norm invariant convex -sparse decomposition of vectors in this note. This result (Theorem 2.1) asserts that every vector is a convex combination of -sparse vectors with invariant l 1 norm. Such decomposition fits well in treating compressed sensing problems under RIP, as a tighter conversion between l 1 -norm and l 2 norm is desired. We shall demonstrate this by showing how to use the decomposition to derive the sparse recovery condition δ +θ, < 1 of Cai and Zhang [3] in a simple manner. However, we believe that this result is of independent interest for other applications. After the early appearance of this note (arxiv: , May 2013), we learned that Cai and Zhang [4] also established a similar decomposition and using it to derive some goodrip conditions (e.g., δ ). Using thel 1 -norminvariant convex -sparse decomposition, under the tight frame sparsification, Baer [1] obtained the condition δ for the Dictionary-Restricted Isometry Property. The paper is organized as follows. Section 2 presents the l 1 -norm invariant -sparse convex decomposition. As a consequence of this decomposition, we prove a useful result for comparing l p norms. In Section 3, the -sparse convex decomposition is used to give a simple derivation of the sparse recovery condition δ +θ, < 1 of Cai and Zhang. 2 Convex -Sparse Decomposition In this section, we prove that every vector is a convex combination of -sparse vectors with invariant l 1 norm. The formulation is inspired by the celebrated ideas from Cai and Zhang [2]. We also show that the l norm of the summand vectors is well behaved. More specifically, we have Theorem 2.1. For positive integers n, and positive constant C, let v R n be a vector with v 1 C and v C. Then there are -sparse vectors w 1,...,w M with w t 1 = v 1 and w t C for t = 1,,M, (1) 2

3 such that v = x t w t (2) for some nonnegative real numbers x 1,...,x M with M t=1 x t = 1. t=1 Proof. If = n, or v is already -sparse, then there is nothing to do. Assume now n >. Without loss of generality, we may consider the case that all components of v are positive 1 (the general case can be argued easily, as (2) still holds by multiplying 1 to the ith components of both sides ) v(1) v(2) v(n) > 0. For each j = 1,...,, let η j := C v(j). Since η j = C v(1) v() v( +1)+ +v(n), so we have η j 0 and η j > 0. Let λ i := η i η, i = 1,2,...,, j Then i=1 λ i = 1. We shall construct +1 vectors g 0,...,g ; each has n 1 nonzero components and satisfies g t 1 = v 1 and g t C for t = 0,1,...,. Furthermore, v is a convex combination of g 0,...,g. In the following construction, we will use v {j} to denote the vector whose j th component is v(j) and other components arezero, and use {e 1,...,e n } to denote the canonical basis. The +1 vectors g 0,...,g are g 0 = g t = (v(j)+λ j v(n))e j +v {+1} + +v {n 1}, (v(j)+λ j v(n))e j +(v(t)+λ t v(n))e n +v {+1} + +v {n 1},1 t. j t Let y 1 := λ 1 v(n) v(1)+λ 1 v(n), y 2 := λ 2 v(n) v(2)+λ 2 v(n),...,y := λ v(n) v()+λ v(n). Then we see that y i < λ i. So, by setting y 0 = 1 y 1 y, we have y 0 > 0 and y 0 +y 1 + +y = 1. 1 In this note, we will treat a vector of R n as a function from {1,,n} to R. 3

4 It is also straightforward to verify v = y 0 g 0 +y 1 g 1 + +y g. For example, the first component of y 0 g 0 +y 1 g 1 + +y g is y 0 (v(1)+λ 1 v(n))+y 2 (v(1)+λ 1 v(n))+ +y (v(1)+λ 1 v(n)) The other requirements for g t = (v(1)+λ 1 v(n))(1 y 1 ) = v(1). (t = 0,1,...,) are 1. g t 1 = v 1. This is certainly true. 2. g t (i) C. To see this, we note that 0 if i = t or if t = 0,i = n; v(i)+λ i v(n) if 1 i and i t; g t (i) = v(i) if < i < n; v(t)+λ t v(n) if t > 0,i = n. For 1 i n, C g t(i) C (v(i)+λ iv(n)) = η i λ i v(n) = η i η iv(n) η j η i η i v(n) v( +1)+ +v(n) 0. If n 1 >, we repeat this process for each g t, and so on, until a -sparse convex combination is reached. Remar 2.2. The proof of Theorem 2.1 in fact presents a method to construct the vectors w t,t = 1,...,M with M = ( n ). Using this method, the time complexity to construct the M vectors is O( n ). It will be interesting to design efficient algorithms to construct the vectors. 3 RIP Conditions in Compressed Sensing In this section, we shall use the -sparse convex decomposition to describe a short proof of the following results of Cai and Zhang [3]. Our proof follows an approach similar to that in [5, 6]. We first consider the recovery of -sparse signal: Theorem 3.1. Let β be a -sparse signal and y = Φβ where Φ satisfies Let Then β = ˆβ. δ +θ, < 1. (3) ˆβ = argmin γ R p { γ 1 subject to y = Φγ} 4

5 Proof. Let h := ˆβ β. We need to show that h = 0. Otherwise, we assume h(1) h(2) h(p) > 0. Denote T := {1,2,,}, S := { +1,+2,,p}, then as in [6], the minimality of ˆβ yields h S 1 h T 1, where h Q = hi Q and I Q denotes the indicator function of the set Q (namely, I Q (j) = 1 if j Q and 0 if j / Q). From the assumption, we also have h S (j) h T 1 for all j S, i.e. h S h T 1. Therefore, by theorem 2.1, h S can be written as h S = x j w j where x j 0 and q x j = 1, with each w j is -sparse and supported on S, and w j 1 = h S 1, w j h T 1. As h T and w j have disjoint supports and w j 2 ( h T 1 ) 2 h T 2, we get (1 δ ) h T 2 2 Φh T 2 2 = Φh T,Φh S x j θ, h T 2 w j 2 = θ, h T 2 2. We have reached a contradiction. Hence h = 0. x j Φh T,Φw j x j θ, h T 2 h T 2 Remar 3.2. We state the proof for the -sparse signal. In fact, one also can extend the proof to the noise case easily. In [1], Baer stated such a proof for the case where the signals are sparse in a redundant dictionary. References [1] C. A. Baer, A Note on Sparsification by Frames, August, 2013, [2] T. Cai and A. Zhang, Sharp RIP bound for sparse signal and low-ran matrix recovery, Applied and Computational Harmonic Analysis, 35(2013),

6 [3] T. Cai and A. Zhang, Compressed sensing and affine ran minimization under restricted isometry, IEEE Transactions on Signal Processing, to appear. [4] T. Cai and A. Zhang, Sparse representation of a polytope and recovery of sparse signals and low-ran matrices, June, 2013, [5] T. Cai, L. Wang, and G. Xu, Stable recovery of sparse signals and an oracle inequality, IEEE Transactions on Information Theory, 56(2010), [6] T. Cai, L. Wang, and G. Xu, New Bounds for restricted isometry constants, IEEE Transactions on Information Theory, 56(2010), [7] E. J. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51(2005)