On Random Construction of a Bipolar Sensing Matrix with Compact Representation

Research in compressed sensing [2] [3] is expanding rapidly. The sufficient condition for ℓ 1 -recovery based on the Restricted Isometry Property (RIP) [3] [4] is one of the celebrated results in this field. The design of sensing matrices with small RIP constants is a theoretically interesting and challenging problem. Currently, random constructions provide the strongest results, and the analysis of random constructions is based on large deviations of maximum and minimum singular values of random matrices [5] [3].

In the present paper, a random construction of bipolar sensing matrices based on binary linear codes is introduced and its RIP is analyzed. The column vectors of the proposed sensing matrix are nonzero codewords of a randomly chosen binary linear code. Using a generator matrix, a p × m sensing matrix can be represented by O(p log 2 m)-bits. The existence of sensing matrices with the RIP is shown based on an argument on the ensemble average of the weight distribution of binary linear codes.

The symbols R and F 2 represent the field of real numbers and the finite field with two elements {0, 1}, respectively. The set of all p × m real matrices is denoted by R p×m . In the present paper, the notation x ∈ R p indicates that x is a column vector of length p. The notation

The ℓ 0 -norm is defined by

where supp(x) denotes the index set of nonzero components of x. The functions w h (•) and d h (•, •) are the Hamming weight and Hamming distance functions, respectively.

Let

represents the set of consecutive integers from a to b.

The restricted isometry property of Φ introduced by Candes and Tao [3] plays a key role in a sufficient condition of ℓ 1recovery.

Definition 1:

for any S-sparse vector x ∈ R m , then we say that Φ has the RIP of order S. If Φ has the RIP of order S, then the smallest constant satisfying (3) is called the RIP constant of Φ, which is denoted by δ S .

Assume that Φ has the RIP with small δ S . In such a case, any sub-matrix composed from Q-columns (1 ≤ Q ≤ S) of Φ is nearly orthonormal. Recently, Candes [4] reported the relation between the RIP and the ℓ 1 -recovery property. A portion of the main results of [4] is summarized as follows. Let S ∈ [1, m], and assume that Φ has the RIP with

For any S-sparse vector e ∈ R m (i.e., ||e|| 0 ≤ S), the solution of the following ℓ 1 -minimization problem

coincides exactly with e, where s = Φe. Note that [4] considers stronger reconstruction results (i.e., robust reconstruction). The matrix Φ in ( 5) is called a sensing matrix.

The incoherence of Φ defined below and the RIP constant are closely related.

Definition 2: The incoherence of Φ is defined by

The following lemma shows the relation between the incoherence and the RIP constant. Similar bounds are well known (e.g., [9]).

Lemma 1:

An elementary proof (different from that in [9]) is presented in Appendix.

In this section, we present a construction method for sensing matrices based on binary linear codes. A sensing matrix obtained from this construction has a concise description. A sensor can store a generator matrix of a binary linear code, instead of the entire sensing matrix.

The function β p : F p 2 → R p is called a binary to bipolar conversion function defined by

where e is an all-one column vector of length p. Namely, using the binary to bipolar conversion function, a binary sequence is converted to a {+1/ √ p, -1/ √ p}-sequence. The following lemma demonstrates that the inner product of two bipolar sequences β p (a) and β p (b) is determined from the Hamming distance between the binary sequences a and b.

Lemma 2: For any a, b ∈ F p 2 , the inner product of β p (a) and β p (b) is given by 9) is derived as follows:

It is easy to confirm that β p (a) is normalized, i.e., ||β p (a)|| 2 = 1, for any a ∈ F p 2 .

Let H ∈ F r×p 2 (p > r) be a binary r×p parity check matrix where 2 p-r ≥ p holds. The binary linear coded C(H) defined by H is given by

where 0 r is a zero-column vector of length r. The following definition gives the construction of sensing matrices. Definition 3: Assume that all of the nonzero codewords of C(H) are denoted by c 1 , c 2 , . . . , c M (based on any predefined order), where M = 2 p-rank(H) -1 ≥ 2 p-r -1. The sensing matrix Φ(H) ∈ R p×m is defined by

where m = 2 p-r -1. If Φ(H) has the RIP of order S, the RIP constant corresponding to Φ(H) is denoted by δ S (H).

Since the order of the columns is unimportant, we do not distinguish between sensing matrices of different column order (or choice of codewords from C(H)).

If the weights of all nonzero codewords of C(H) are very close to p/2, then the incoherence of Φ(H) becomes small, as described in detail in the following lemma.

Lemma 3: Assume that ǫ(0 < ǫ < 1) is given and that

due to the linearity of C(H). This means that

where

The definition of incoherence and the above inequalities lead to an upper bound on the incoherence:

We here consider binary linear codes whose weight distribution

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