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Frequency-Selective Vandermonde Decomposition of Toeplitz Matrices with Applications
The classical result of Vandermonde decomposition of positive semidefinite
Toeplitz matrices, which dates back to the early twentieth century, forms the
basis of modern subspace and recent atomic norm methods for frequency
estimation. In this paper, we study the Vandermonde decomposition in which the
frequencies are restricted to lie in a given interval, referred to as
frequency-selective Vandermonde decomposition. The existence and uniqueness of
the decomposition are studied under explicit conditions on the Toeplitz matrix.
The new result is connected by duality to the positive real lemma for
trigonometric polynomials nonnegative on the same frequency interval. Its
applications in the theory of moments and line spectral estimation are
illustrated. In particular, it provides a solution to the truncated
trigonometric -moment problem. It is used to derive a primal semidefinite
program formulation of the frequency-selective atomic norm in which the
frequencies are known {\em a priori} to lie in certain frequency bands.
Numerical examples are also provided.Comment: 23 pages, accepted by Signal Processin
Poisson Matrix Completion
We extend the theory of matrix completion to the case where we make Poisson
observations for a subset of entries of a low-rank matrix. We consider the
(now) usual matrix recovery formulation through maximum likelihood with proper
constraints on the matrix , and establish theoretical upper and lower bounds
on the recovery error. Our bounds are nearly optimal up to a factor on the
order of . These bounds are obtained by adapting
the arguments used for one-bit matrix completion \cite{davenport20121}
(although these two problems are different in nature) and the adaptation
requires new techniques exploiting properties of the Poisson likelihood
function and tackling the difficulties posed by the locally sub-Gaussian
characteristic of the Poisson distribution. Our results highlight a few
important distinctions of Poisson matrix completion compared to the prior work
in matrix completion including having to impose a minimum signal-to-noise
requirement on each observed entry. We also develop an efficient iterative
algorithm and demonstrate its good performance in recovering solar flare
images.Comment: Submitted to IEEE for publicatio
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