4,003 research outputs found
A weakly stable algorithm for general Toeplitz systems
We show that a fast algorithm for the QR factorization of a Toeplitz or
Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A.
Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx =
A^Tb, we obtain a weakly stable method for the solution of a nonsingular
Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the
solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further
details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm
Sparse Signal Recovery under Poisson Statistics
We are motivated by problems that arise in a number of applications such as
Online Marketing and explosives detection, where the observations are usually
modeled using Poisson statistics. We model each observation as a Poisson random
variable whose mean is a sparse linear superposition of known patterns. Unlike
many conventional problems observations here are not identically distributed
since they are associated with different sensing modalities. We analyze the
performance of a Maximum Likelihood (ML) decoder, which for our Poisson setting
involves a non-linear optimization but yet is computationally tractable. We
derive fundamental sample complexity bounds for sparse recovery when the
measurements are contaminated with Poisson noise. In contrast to the
least-squares linear regression setting with Gaussian noise, we observe that in
addition to sparsity, the scale of the parameters also fundamentally impacts
sample complexity. We introduce a novel notion of Restricted Likelihood
Perturbation (RLP), to jointly account for scale and sparsity. We derive sample
complexity bounds for regularized ML estimators in terms of RLP and
further specialize these results for deterministic and random sensing matrix
designs.Comment: 13 pages, 11 figures, 2 tables, submitted to IEEE Transactions on
Signal Processin
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