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Frame Coherence and Sparse Signal Processing

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Frame Coherence and Sparse Signal Processing Frame Coherence and Sparse Signal Processing Dustin G. Mixon∗, Waheed U. Bajway, Robert Calderbanky ∗Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544 yDepartment of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708 Abstract—The sparse signal processing literature often uses condition is equivalent to having the spectral norm be as small random sensing matrices to obtain performance guarantees. as possible; for an M ×N unit norm frame F , this equivalently Unfortunately, in the real world, sensing matrices do not always means kF k2 = N . come from random processes. It is therefore desirable to evaluate 2 M whether an arbitrary matrix, or frame, is suitable for sensing Throughout the literature, applications require finite- sparse signals. To this end, the present paper investigates two dimensional frames that are nearly tight and have small worst- parameters that measure the coherence of a frame: worst-case case coherence [1]–[8]. Among these, a foremost application and average coherence. We first provide several examples of is sparse signal processing, where frames of small spectral frames that have small spectral norm, worst-case coherence, norm and/or small worst-case coherence are commonly used and average coherence. Next, we present a new lower bound on worst-case coherence and compare it to the Welch bound. Later, to analyze sparse signals [4]–[8]. Recently, [9] introduced we propose an algorithm that decreases the average coherence another notion of frame coherence called average coherence: of a frame without changing its spectral norm or worst-case N coherence. Finally, we use worst-case and average coherence, as 1 X νF := max hfi; fji : (2) opposed to the Restricted Isometry Property, to garner near- N−1 i2f1;:::;Ng j=1 optimal probabilistic guarantees on both sparse signal detection j6=i and reconstruction in the presence of noise. This contrasts with recent results that only guarantee noiseless signal recovery from Note that, in addition to having zero worst-case coherence, arbitrary frames, and which further assume independence across orthonormal bases also have zero average coherence. It was the nonzero entries of the signal—in a sense, requiring small established in [9] that when νF is sufficiently smaller than average coherence replaces the need for such an assumption. µF , a number of guarantees can be provided for sparse signal I. INTRODUCTION processing. It is therefore evident from [1]–[9] that there is a pressing need for nearly tight frames with small worst-case Many classical applications, such as radar and error- and average coherence, especially in the area of sparse signal correcting codes, make use of over-complete spanning systems processing. [1]. Oftentimes, we may view an over-complete spanning This paper offers four main contributions in this regard. system as a frame. Take F = ffigi2I to be a collection of First, we discuss three types of frames that exhibit small spec- vectors in some separable Hilbert space H. Then F is a frame tral norm, worst-case coherence, and average coherence: nor- if there exist frame bounds A and B with 0 < A ≤ B < 1 malized Gaussian, random harmonic, and code-based frames. 2 P 2 2 such that Akxk ≤ i2I jhx; fiij ≤ Bkxk for every With all three frame parameters provably small, these frames x 2 H. When A = B, F is called a tight frame. For finite- are guaranteed to perform well in relevant applications. Sec- dimensional unit norm frames, where I = f1;:::;Ng, the ond, performance in many applications is dictated by worst- worst-case coherence is a useful parameter: case coherence [1]–[8]. It is therefore particularly important to understand which worst-case coherence values are achievable. µF := max jhfi; fjij: (1) i;j2f1;:::;Ng To this end, the Welch bound [1] is commonly used in the i6=j arXiv:1105.4279v1 [math.FA] 21 May 2011 literature. However, the Welch bound is only tight when the Note that orthonormal bases are tight frames with A = B = 1 number of frame elements N is less than the square of the and have zero worst-case coherence. In both ways, frames spatial dimension M [1]. Another lower bound, given in [10] form a natural generalization of orthonormal bases. and [11], beats the Welch bound when there are more frame In this paper, we only consider finite-dimensional frames. elements, but it is known to be loose for real frames [12]. Those not familiar with frame theory can simply view a finite- Given this context, our next contribution is a new lower bound dimensional frame as an M × N matrix of rank M whose on the worst-case coherence of real frames. Our bound beats columns are the frame elements. With this view, the tightness both the Welch bound and the bound in [10] and [11] when the number of frame elements far exceeds the spatial dimension. This work was supported by the Office of Naval Research under Grant N00014-08-1-1110, by the Air Force Office of Scientific Research under Third, since average coherence is new to the frame theory Grants FA9550-09-1-0551 and FA 9550-09-1-0643 and by NSF under Grant literature, we investigate how it relates to worst-case coherence DMS-0914892. Mixon was supported by the A.B. Krongard Fellowship. The and spectral norm. In particular, we want average coherence to views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of satisfy the following property, which is used in [9] to provide Defense, or the U.S. Government. various guarantees for sparse signal processing: Definition 1. We say an M × N unit norm frame F satisfies However, to the best of our knowledge, there is no result in the the Strong Coherence Property if literature that shows that random harmonic frames have small worst-case coherence. To fill this gap, the following theorem (SCP-1) µ ≤ 1 and (SCP-2) ν ≤ pµF ; F 164 log N F M characterizes the spectral norm and the worst-case and average where µF and νF are given by (1) and (2), respectively. coherence of random harmonic frames. Since average coherence is so new, there is currently no in- Theorem 3 (Geometry of random harmonic frames). Let U be tuition as to when (SCP-2) is satisfied. As a third contribution, an N × N non-normalized discrete Fourier transform matrix, 2πik`=N this paper shows how to transform a frame that satisfies (SCP- explicitly, Uk` := e for each k; ` = 0;:::;N − 1. Next, N 1) into another frame with the same spectral norm and worst- let fBigi=1 be a collection of independent Bernoulli random M case coherence that additionally satisfies (SCP-2). Finally, this variables with mean N , and take M := fi : Bi = 1g. paper uses the Strong Coherence Property to provide new Finally, construct an jMj × N harmonic frame F by col- guarantees on both sparse signal detection and reconstruction lecting rows of U which correspond to indices in M and in the presence of noise. These guarantees are related to those normalize the columns. Then F is a unit norm tight frame: 2 N N in [4], [5], [7], and we elaborate on this relationship in Section kF k2 = jMj . Furthermore, provided 16 log N ≤ M ≤ 3 , the V. In the interest of space, the proofs have been omitted following inequalities simultaneously hold with probability throughout, but they can be found in [13]. exceeding 1 − 4N −1 − N −2: 1 3 (i) 2 M ≤ jMj ≤ 2 M, II. FRAME CONSTRUCTIONS µF (ii) νF ≤ p , Many applications require nearly tight frames with small jMj q 118(N−M) log N worst-case and average coherence. In this section, we give (iii) µF ≤ MN . three types of frames that satisfy these conditions. C. Code-based frames A. Normalized Gaussian frames Many structures in coding theory are also useful for con- Construct a matrix with independent, Gaussian distributed structing frames. Here, we build frames from a code that entries that have zero mean and unit variance. By normalizing originally emerged with Berlekamp in [20], and found recent the columns, we get a matrix called a normalized Gaussian reincarnation with [21]. We build a 2m × 2(t+1)m frame, frame. This is perhaps the most widely studied type of frame indexing rows by elements of F2m and indexing columns in the signal processing and statistics literature. by (t + 1)-tuples of elements from F2m . For x 2 F2m and To be clear, the term “normalized” is intended to distinguish t+1 α 2 F2m , the corresponding entry of the matrix F is the results presented here from results reported in earlier i Trα x+Pt α x2 +1 works, such as [9], [14]–[16], which only ensure that the frame p1 0 i=1 i Fxα = m (−1) ; (3) elements of Gaussian frames have unit norm in expectation. 2 In other words, normalized Gaussian frames are frames with where Tr : F2m ! F2 denotes the trace map, defined by Pm−1 2i individual frame elements independently and uniformly dis- Tr(z) = i=0 z . The following theorem gives the spectral tributed on the unit hypersphere in RM . norm and the worst-case and average coherence of this frame. That said, the following theorem characterizes the spectral Theorem 4 (Geometry of code-based frames). The 2m × norm and the worst-case and average coherence of normalized 2(t+1)m frame defined by (3) is unit norm and tight, i.e., Gaussian frames. kF k2 = 2tm, with worst-case coherence µ ≤ p 1 and 2 F 2m−2t−1 Theorem 2 (Geometry of normalized Gaussian frames).
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