Designing Structured Tight Frames Via an Alternating Projection Method

Joel Tropp, Inderjit Dhillon, Robert Heath Jr., Thomas Strohmer

Abstract:   Tight frames, also known as general Welch-boundequality sequences, generalize orthonormal systems. Numerous applications—including communications, coding, and sparse approximation—require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm.

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  • Designing Structured Tight Frames Via an Alternating Projection Method (pdf, software)
    J. Tropp, I. Dhillon, R. Jr., T. Strohmer.
    IEEE Transactions on Information Theory 51(1), pp. 188-209, January 2005.