2,691 research outputs found
Projected gradient descent for non-convex sparse spike estimation
We propose a new algorithm for sparse spike estimation from Fourier
measurements. Based on theoretical results on non-convex optimization
techniques for off-the-grid sparse spike estimation, we present a projected
gradient descent algorithm coupled with a spectral initialization procedure.
Our algorithm permits to estimate the positions of large numbers of Diracs in
2d from random Fourier measurements. We present, along with the algorithm,
theoretical qualitative insights explaining the success of our algorithm. This
opens a new direction for practical off-the-grid spike estimation with
theoretical guarantees in imaging applications
Blind Two-Dimensional Super-Resolution and Its Performance Guarantee
In this work, we study the problem of identifying the parameters of a linear
system from its response to multiple unknown input waveforms. We assume that
the system response, which is the only given information, is a scaled
superposition of time-delayed and frequency-shifted versions of the unknown
waveforms. Such kind of problem is severely ill-posed and does not yield a
unique solution without introducing further constraints. To fully characterize
the linear system, we assume that the unknown waveforms lie in a common known
low-dimensional subspace that satisfies certain randomness and concentration
properties. Then, we develop a blind two-dimensional (2D) super-resolution
framework that applies to a large number of applications such as radar imaging,
image restoration, and indoor source localization. In this framework, we show
that under a minimum separation condition between the time-frequency shifts,
all the unknowns that characterize the linear system can be recovered precisely
and with very high probability provided that a lower bound on the total number
of the observed samples is satisfied. The proposed framework is based on 2D
atomic norm minimization problem which is shown to be reformulated and solved
efficiently via semidefinite programming. Simulation results that confirm the
theoretical findings of the paper are provided
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