Modified Adaptive Basis Pursuits for Recovery of Correlated Sparse Signals Sathiya Narayanan, Sujit Kumar Sahoo and Anamitra Makur School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore Abstract—In Distributed Compressive Sensing (DCS), corre- lated sparse signals refer to as an ensemble of signals char- acterized by presenting a sparse correlation. If one signal is known apriori, the remaining signals in the ensemble can be reconstructed using l1-minimization with far fewer measurements compared to separate CS reconstruction. Reconstruction of such correlated signals is possible via Modified-CS and Regularized- Modified-BP but these methods are greatly influenced by the support set of the known signal which also includes locations irrelavant to the target signal. While recovering each signal, prior to Modified-CS or Regularized-Modified-BP, we propose an adaptation step to retain only the sparse locations significant to that signal. We call our proposed methods as Modified- Adaptive-BP and Regularized-Modified-Adaptive-BP. Theoretical guarantees and experimental results show that our proposed method provides efficient recovery compared to that of the Modified-CS and its regularized version. Index Terms—Distributed Compressive Sensing, Correlated sparse signals, Adaptation, Modified-Adaptive-BP, Regularized- Modified-Adaptive-BP. I. I NTRODUCTION Compressive sensing (CS) ensures the recovery of a sparse signal x ∈ R n using a small number of linear observations of the form y = Ax ∈ R m , where A ∈ R m×n is a known matrix with m ≪ n. If the signal x is S-sparse, in the sense that there are S non-zero entries in x, then exact recovery is possible through l 1 -minimization given below provided the number of measurements m = O(S log(n/S)) [1] [2]: min β ∥β∥ ℓ1 s.t. Aβ = y. (1) Robustness of CS can be studied using the restricted isometry property (RIP) of the sensing matrix. For all S-sparse x,a sensing matrix A is said to follow RIP if (1 − δ S )∥x∥ 2 ℓ2 ≤∥Ax∥ 2 ℓ2 ≤ (1 + δ S )∥x∥ 2 ℓ2 (2) Matrix A is said to be obeying RIP if the restricted isometry constant δ S is not close to one [3]. RIP implies that all subsets of S columns of A will be nearly orthogonal to each other. A. Motivation and Relation to Prior Work Distributed Compressive Sensing (DCS) exploits both intra- signal and inter-signal correlation structures. DCS encodes each signal individually by projecting it onto another, inco- herent, random basis and then transmits just a few of the resulting coefficients to the decoder. Therefore, a decoder can reconstruct all the correlated signals precisely. As the This work is technically supported by the project RG 27/10. signals are sparse, one could encode and decode each of them separately using the CS framework. If one signal is known apriori, motivated by the idea of using side information in DCS, the remaining signals in the ensemble can be recon- structed using l 1 -minimization with far fewer measurements compared to seperate CS reconstruction. Reconstruction of such correlated signals is possible via Modified-CS (MOD- CS) [4] and Regularized-Modified-BP (Reg-MOD-BP) [5]. However, these methods are greatly influenced by the support set of the known signal which also includes locations irrelavant to the target signal. Therefore, we propose an adaptation step, prior to MOD-CS or Reg-MOD-BP, which tries to retain only those locations that are relevant to the target signal. We call our proposed methods as Modified-Adaptive-BP (MABP) and Regularized-Modified-Adaptive-BP (RMABP). B. Paper Outline The rest of this paper is organized as follows. In section 2, we discuss the correlated sparse signal model and two existing BP based recovery techniques. In section 3, we propose our modified adaptive basis pursuit methods for correlated sparse signals and the theoretical guarantees are given in section 4. In section 5, we present the simulation results of our proposed methods and compare its performance to that of the existing methods. Section 6 concludes the paper. II. RECOVERY OF CORRELATED SPARSE SIGNALS In this section, we introduce the signal model of correlated sparse signals and discuss two Basis Pursuit based CS recon- struction algorithms. In the correlated sparse signal ensemble, each signal consists of two components: a common sparse component that is present in all of the signals, and a sparse innovation component that is unique to each signal. This is similar to the Joint Sparse Model (JSM) analyzed in [6]. Let us denote the J signals in the ensemble by x (t) , t ∈ 1, 2, ..., J . Assume that x (t) ∈ R n and it has a sparse representation in basis Ψ. The correlated sparse signal model is, x (t) = z + z (t) t ∈ 1, 2, ..., J (3) with z = ΨΘ z , ∥Θ z ∥ 0 = K and z (t) = ΨΘ (t) , ∥Θ (t) ∥ 0 = K (t) . Let A be the i.i.d. Gaussian measurement matrix for signal x (t) such that y (t) = Ax (t) (4) gives m ≪ n incoherent measurements of x (t) . In the reconstruction part, we need to estimate every n-length sparse