Showing papers by "Joel A. Tropp published in 2005"

Designing structured tight frames via an alternating projection method

Abstract: 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.

Simultaneous sparse approximation via greedy pursuit

Abstract: A simple sparse approximation problem requests an approximation of a given input signal as a linear combination of T elementary signals drawn from a large, linearly dependent collection An important generalization is simultaneous sparse approximation Now one must approximate several input signals at once using different linear combinations of the same T elementary signals This formulation appears, for example, when analyzing multiple observations of a sparse signal that have been contaminated with noise A new approach to this problem is presented here: a greedy pursuit algorithm called simultaneous orthogonal matching pursuit The paper proves that the algorithm calculates simultaneous approximations whose error is within a constant factor of the optimal simultaneous approximation error This result requires that the collection of elementary signals be weakly correlated, a property that is also known as incoherence Numerical experiments demonstrate that the algorithm often succeeds, even when the inputs do not meet the hypotheses of the proof

Recovery of short, complex linear combinations via /spl lscr//sub 1/ minimization

Abstract: This note provides a condition under which lscr1 minimization (also known as basis pursuit) can recover short linear combinations of complex vectors chosen from fixed, overcomplete collection. This condition has already been established in the real setting by Fuchs, who used convex analysis. The proof given here is more direct

Generalized Finite Algorithms for Constructing Hermitian Matrices with Prescribed Diagonal and Spectrum

Abstract: In this paper, we present new algorithms that can replace the diagonal entries of a Hermitian matrix by any set of diagonal entries that majorize the original set without altering the eigenvalues of the matrix. They perform this feat by applying a sequence of (N-1) or fewer plane rotations, where N is the dimension of the matrix. Both the Bendel--Mickey and the Chan--Li algorithms are special cases of the proposed procedures. Using the fact that a positive semidefinite matrix can always be factored as $\mtx{X^\adj X}$, we also provide more efficient versions of the algorithms that can directly construct factors with specified singular values and column norms. We conclude with some open problems related to the construction of Hermitian matrices with joint diagonal and spectral properties.

Complex equiangular tight frames

Abstract: A complex equiangular tight frame (ETF) is a tight frame consisting of N unit vectors in C d whose absolute inner products are identical. One may view complex ETFs as a natural geometric generalization of an orthonormal basis. Numerical evidence suggests that these objects do not arise for most pairs ( d , N). The goal of this paper is to develop conditions on ( d , N) under which complex ETFs can exist. In particular, this work concentrates on the class of harmonic ETFs, in which the components of the frame vectors are roots of unity. In this case, it is possible to leverage field theory to obtain stringent restrictions on the possible values for ( d , N).

Applications of sparse approximation in communications

Abstract: Sparse approximation problems abound in many scientific, mathematical, and engineering applications. These problems are defined by two competing notions: we approximate a signal vector as a linear combination of elementary atoms and we require that the approximation be both as accurate and as concise as possible. We introduce two natural and direct applications of these problems and algorithmic solutions in communications. We do so by constructing enhanced codebooks from base codebooks. We show that we can decode these enhanced codebooks in the presence of Gaussian noise. For MIMO wireless communication channels, we construct simultaneous sparse approximation problems and demonstrate that our algorithms can both decode the transmitted signals and estimate the channel parameters

Average-case analysis of greedy pursuit

Abstract: Recent work on sparse approximation has focused on the theoretical performance of algorithms for random inputs. This average-case behavior is typically far better than the behavior for the worst inputs. Moreover, an average-case analysis fits naturally with the type of signals that arise in certain applications, such as wireless communications. This paper describes what is currently known about the performance of greedy prusuit algorithms with random inputs. In particular, it gives a new result for the performance of Orthogonal Matching Pursuit (OMP) for sparse signals contaminated with random noise, and it explains recent work on recovering sparse signals from random measurements via OMP. The paper also provides a list of open problems to stimulate further research.