arXiv:1604.06615v2 [math.OC] 25 Apr 2016 Construction of incoherent frames via preconditioning and convex optimization Pradip Sasmal, Prasad Theeda, C. S. Sastry, Phanindra Jampana Abstract—Unit norm tight frames (UNTFs) play a significant role in sparse representations, channel coding and communi- cations. One of the known techniques to produce UNTFs is through pre-conditioning. Motivated by existing pre-conditioning techniques and the application of incoherent tight frames in sparse representations, we propose a methodology to construct incoherent UNTFs via a convex optimization technique. In particular, starting with an initial frame, our method produces best possible incoherent unit norm frame that can be obtained through pre-conditioning. Our simulation results suggest that the proposed pre-conditioning improves sparse signal recovery by increasing the recovery region. Index Terms—Compressed Sensing, Convex Optimization, Pre- conditioning, Semi-definite Programming, Unit Norm Tight Frames. I. I NTRODUCTION F RAMES, overcomplete spanning systems, are introduced as a generalization of bases [1], [2], [4]. In signal processing, a frame provides a redundant and stable way of representing a signal [3], [22]. Frame theory has deep connections with Harmonic analysis, Operator theory, Coding theory [9] and Quantum theory [11]. Due to its widespread applicability, the theory of frames has become an active area of research for many researchers from several fields. A family of vectors {φ i } M i=1 in R m is called a frame for R m , if there exist constants 0 <A ≤ B< ∞ such that A ‖z ‖ 2 ≤ M  i=1 |〈z,φ i 〉| 2 ≤ B ‖z ‖ 2 , ∀z ∈ R m , (1) where A, B are called the lower and upper frame bounds respectively [1]. The matrix Φ m×M =[φ 1 ...φ M ] with φ i as columns is known as the frame synthesis operator. In this paper we refer to Φ as a frame and also as a matrix depending on the context. Coherence of a frame Φ is defined as the largest absolute normalized inner product between two distinct columns of Φ [12] µ(Φ) = max 1≤ i,j≤ M, i=j | φ T i φ j | ‖φ i ‖ 2 ‖φ j ‖ 2 . Frames with small coherence are known as incoherent. For a given M , frames with minimal coherence are called Grassmanian frames [5]. The lowest bound on the minimal Pradip Sasmal, Prasad Theeda, C. S. Sastry are with Department of Mathematics, Indian Institute of Technology Hyderabad, India, 502285. e- mail: {ma12p1005, ma13p1004, csastry}@iith.ac.in Phanindra Jampana is with Department of Chemical Engineering, Indian In- stitute of Technology Hyderabad, India, 502285. e-mail: pjampana@iith.ac.in achievable correlation for any frame, known as Welch bound [15], is given by µ Φ ≥  M − m m(M − 1) . (2) Incoherent Unit norm tight frames (UNTFs) have gained popularity for providing sparse representations[16], [7]. In [23], Christian Rusu has proposed a convex optimization technique directly on frame for designing incoherent frames. In [13], B. G. Bodmann et. al. have constructed UNTFs by solving a differential equation. P. G. Casazza et. al. [14] have used a gradient-descent-based algorithm for realizing UNTFs. In [24], E. Tsiligianni et. al. have proposed an iterative process on the Gram matrix of Φ to obtain an incoherent UNTF. In this paper, we propose a convex optimization technique that finds a pre-conditioner for frames yielding UNTFs with small coherence. The contributions of the present work are two-fold: • formulating a pre-conditioning technique for producing incoherent frames as a convex optimization problem. • studying numerically the relationship between the condi- tion number of pre-conditioner and coherence of resultant matrix. The paper is organized in several sections. In sections 2 and 3, we provide basics of sparse representations and UNTFs respectively. In sections 4 and 5, we present motivation and our proposed convex optimization method. In section 6, we study the relationship between condition number κ(G) and coherence µ GΦ of GΦ. While in section 7, we formulate the problems as semidefinite programming. In sections 8 and 9, we provide numerical results and the conclusion respectively. II. BASICS OF COMPRESSED SENSING The objective of Compressed Sensing (CS) is to recover x ∈ R M from a few of its linear measurements y ∈ R m through a stable and efficient reconstruction process via the concept of sparsity. From the measurement vector y and the sensing mechanism, one obtains a system y =Φx, where Φ is an m × M (m<M ) matrix. Sparsity is measured by ‖·‖ 0 norm and ‖x‖ 0 := |{j ∈{1, 2,...,M } : x j =0}|. Finding the sparsest solution can be formulated as the following minimization problem, generally denoted as a P 0 problem: P 0 : min x ‖x‖ 0 subject to Φx = y. This problem is a combinatorial minimization problem and is known to be NP-hard [17]. One may use greedy methods or 1