Seminar Details
2017-02-28 (14:00) : On Non-Convex Approaches to Blind Calibration for Compressed Sensing
At Euler building (room A.002)
Organized by Mathematical Engineering
Speaker :
Cambareri, Valerio
Abstract :
Blind Calibration is a bilinear inverse problem that arises in the
recovery of signals and images from some observations, as gathered by
computational sensing models (e.g., those of Compressed Sensing) subject to
parametric uncertainty. As an optimisation problem, blind calibration is
intrinsically non-convex, although it can be convexified through the so-called
"lifting" approach at the expense of high computational complexity.
In this talk we will present some fast algorithms for blind calibration that
attempt to recover an unknown signal from its projections on sub-Gaussian random
vectors, each of which is subject to an unknown multiplicative factor (or gain).
To jointly solve this optimisation problem with respect to the signal and gains
we will resort to a simple projected gradient descent algorithm, which starts
from a carefully chosen initialisation point and runs the descent on a
non-convex objective. Most importantly, an analysis of this scheme under
independent draws of the sensing model allows us to prove that the algorithm
actually convergences to the exact solution provided a sample complexity
requirement is met, i.e., depending on the amount of information collected by
the sensing model. Interestingly, we will show that this requirement grows
linearly (up to log factors) in the number of unknowns of the problem, and how
it depends on the priors assumed on the recovered signal and gains. Moreover, a
straightforward modification of our algorithm will allow us to enforce sparsity
in the signal domain, thus enabling blind calibration for Compressed Sensing.
Our theory is completed by some numerical evidence on the phase transition of
these algorithms, highlighting a regime in which exact recovery occurs almost
always; we shall conclude our talk with a practical application of blind
calibration in the recovery of the gains in an imaging sensor. This is a joint
work with Prof. Laurent Jacques.