Designing structured tight frames via an alternating projection method
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 (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.
© Copyright 2005 IEEE. Reprinted with permission. Manuscript received December 8, 2003; revised September 10, 2004. [Posted online: 2005-01-10] The work of J.A. Tropp was supported by a National Science Foundation Graduate Fellowship. The work of I.S. Dhillon was supported by the National Science Foundation CAREER Award ACI-0093404. The work of R.W. Heath was supported by the Texas Advanced Technology Program under Grant 003658–0614-2001, the Samsung Advanced Institute of Technology, and the National Instruments Foundation. The work of T. Strohmer was supported in part by the National Science Foundation under Grant DMS–0208568. Communicated by G. Battail, Associate Editor At Large.