Designing Structured Tight Frames via an Alternating Projection Method

J.A. Tropp, I.S. Dhillon, Robert Heath, and Thomas Strohmer

Tight frames, also known as general Welch-Bound-Equality 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, which includes the frame design problem. To apply this method, one only needs to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is 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 article 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. Last, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for 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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