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Designing Structured Tight Frames Via an Alternating Projection Method Joel A

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Designing Structured Tight Frames Via an Alternating Projection Method Joel A 1 Designing Structured Tight Frames via an Alternating Projection Method Joel A. Tropp, Student Member, IEEE, Inderjit S. Dhillon, Member, IEEE, Robert W. Heath Jr., Member, IEEE and Thomas Strohmer Abstract— Tight frames, also known as general Welch-Bound- tight frame to a given ensemble of structured vectors; then Equality sequences, generalize orthonormal systems. Numer- it finds the ensemble of structured vectors nearest to the ous applications—including communications, coding and sparse tight frame; and it repeats the process ad infinitum. This approximation—require finite-dimensional tight frames that pos- sess additional structural properties. This paper proposes a technique is analogous to the method of projection on convex versatile alternating projection method that is flexible enough to sets (POCS) [4], [5], except that the class of tight frames is solve a huge class of inverse eigenvalue problems, which includes non-convex, which complicates the analysis significantly. Nev- the frame design problem. To apply this method, one only needs ertheless, our alternating projection algorithm affords simple to solve a matrix nearness problem that arises naturally from implementations, and it provides a quick route to solve difficult the design specifications. Therefore, it is fast and easy to develop versions of the algorithm that target new design problems. frame design problems. This article offers extensive numerical Alternating projection is likely to succeed even when algebraic evidence that our method succeeds for several important cases. constructions are unavailable. We argue that similar techniques apply to a huge class of To demonstrate that alternating projection is an effective tool inverse eigenvalue problems. for frame design, the article studies some important structural There is a major conceptual difference between the use properties in detail. First, it addresses the most basic de- sign problem—constructing tight frames with prescribed vector of finite models in the numerical calculation of infinite- norms. Then, it discusses equiangular tight frames, which are dimensional frames and the design of finite-dimensional natural dictionaries for sparse approximation. Last, it examines frames. In the former case, the finite model plays only an tight frames whose individual vectors have low peak-to-average- auxiliary role in the approximate computation of an infinite- power ratio (PAR), which is a valuable property for CDMA appli- dimensional tight frame [1]. In the latter case, the problem cations. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices thoroughly under consideration is already finite-dimensional, thus it does investigate the convergence properties of the algorithm. not involve discretization issues. In this paper, we consider only finite-dimensional tight frames. Index Terms— Tight frames, general Welch-Bound-Equality sequences, alternating projection, inverse eigenvalue problems, Frame design is essentially an algebraic problem. It boils DS-CDMA, signature sequences, equiangular lines, Grassman- down to producing a structured matrix with certain spectral nian packing, peak-to-average-power ratio, point-to-set maps properties, which may require elaborate discrete and combina- torial mathematics. Alternating projection is powerful because I. INTRODUCTION it reduces this fiendish algebra to a simple analytic question: How does one find an ensemble of structured vectors nearest IGHT FRAMES provide a natural generalization of or- to a given tight frame? This minimization problem can usually T thonormal systems, and they arise in numerous practical be dispatched with standard tools, such as differential calculus and theoretical contexts [1]. There is no shortage of tight or Karush–Kuhn–Tucker theory. frames, and applications will generally require that the vectors In the past, most design methods have employed algebraic comprising the frame have some kind of additional structure. techniques. To appreciate the breadth of this literature, one For example, it might be necessary for the vectors to have might peruse Sarwate’s recent survey paper about tight frames specific Euclidean norms, or perhaps the vectors should have with unit-norm vectors [6]. The last few years have also seen small mutual inner products. Thus arises a design problem: some essentially algebraic algorithms that can construct tight How do you build a structured tight frame? frames with non-constant vector norms [7]–[9]. To address the question, this article proposes a numerical When algebraic methods work, they work brilliantly. A method based on alternating projection that builds on our numerical approach like alternating projection can hardly hope work in [2], [3]. The algorithm alternately finds the nearest to compete with the most profound insights of engineers and J. A. Tropp is with the Inst. for Computational Engineering and Sci- mathematicians. On the other hand, algebraic and combina- ences (ICES), The University of Texas at Austin, Austin, TX 78712 USA, toric tools are not always effective. For example, we might <[email protected]>. require a structured tight frame for a vector space whose I. S. Dhillon is with the Dept. of Computer Sciences, The University of Texas at Austin, Austin, TX 78712 USA, <[email protected]>. dimension is not a prime-power. Even in these situations, R. W. Heath is with the Dept. of Electrical and Computer Engi- alternating projection will usually succeed. Moreover, it can neering, The University of Texas at Austin, Austin, TX 78712 USA, help researchers develop the insight necessary for completing <[email protected]>. T. Strohmer is with the Dept. of Mathematics, The University of California an algebraic construction. at Davis, Davis, CA 95616 USA, <[email protected]>. The literature does not offer many numerical approaches 2 to frame design. It appears that most of the current al- A. Frames gorithms can be traced to the discovery by Rupf–Massey Let α and β be positive constants. A finite frame for [10] and Viswanath–Anantharam [7] that tight frames with 1 d the complex Hilbert space C is a sequence of N vectors prescribed column norms are the optimal sequences for direct- N d {xn}n=1 drawn from C that satisfies a generalized Parseval spread, synchronous code-division multiple access systems condition: (DS-CDMA). The DS-CDMA application prompted a long N series of papers, including [11]–[15], that describe iterative 2 X 2 2 d α kvk2 ≤ |hv, xni| ≤ β kvk2 for all v ∈ C . (1) methods for constructing tight frames with prescribed column n=1 norms. These techniques are founded on an oblique char- h·, ·i acterization of tight frames as the minimizers of a quantity We denote the Euclidean inner product with , and we write k·k α β called total squared correlation (TSC). It is not clear how 2 for the associated norm. The numbers and are called one could generalize these methods to solve different types of the lower and upper frame bounds. The number of vectors in frame design problems. Moreover, the alternating projection the frame may be no smaller than the dimension of the space N ≥ d approach that we propose significantly outperforms at least (i.e. ). α = β one of the TSC-minimization algorithms. Two of the algebraic If it is possible to take , then we have a tight frame or α methods that we mentioned above, [7] and [9], were also an -tight frame. When the frame vectors all have unit norm, kx k ≡ 1 designed with the DS-CDMA application in mind, while the i.e. n 2 , the system is called a unit-norm frame. Unit- third algebraic method [8] comes from the soi-disant frame norm tight frames are also known as Welch-Bound-Equality community. We are not aware of any other numerical methods sequences [17], [25]. Tight frames with non-constant vector for frame design. norms have also been called general Welch-Bound-Equality sequences [7]. Finite- and infinite-dimensional tight frames are phenome- nally useful. Finite tight frames have applications in coding theory [16], in communications [17] and in sparse approxi- B. Associated Matrices mation [18]–[20]. These applications motivated many of the Form a d×N matrix with the frame vectors as its columns: design criteria considered in this paper. Tight Gabor frames, which are infinite-dimensional, can be used to analyze pseudo- X = x1 x2 x3 ... xN . differential operators [21] and to design filter banks and trans- This matrix is referred to as the frame synthesis operator, but mission pulses for wireless communications [22], [23]. Shift- we shall usually identify the synthesis operator with the frame invariant tight frames, which are also infinite-dimensional, itself. arise in sampling theory and signal reconstruction [24]. Two other matrices arise naturally in connection with the frame. We first define the Gram matrix as G def= X ∗X . (The ∗ A. Outline symbol indicates conjugate transposition of matrices and vectors.) The diagonal entries of the Gram matrix equal the Section II continues with a short introduction to tight squared norms of the frame vectors, and the off-diagonal frames. Then, in Section III, we state two formal frame entries of the Gram matrix equal the inner products between design problems. Connections among frame design problems, distinct frame vectors. The Gram matrix is always Hermitian inverse eigenvalue problems and matrix nearness problems are and positive semi-definite. established. This provides a natural segue to the alternating The positive-definite matrix XX ∗ is usually called the frame projection algorithm. Afterward, we apply the basic algorithm operator. Since to design three different types of structured frames, in order N of increasing implementation difficulty. Section IV discusses X 2 v∗(XX ∗) v = |hv, x i| tight frames with prescribed column norms; Section V covers n n=1 equiangular tight frames; and Section VI constructs tight frames whose individual vectors have low peak-to-average- we can rewrite (1) as power ratio. Each of these sections contains numerical exper- v∗(XX ∗) v α ≤ ≤ β.
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