Abstract
This paper presents a sharp geometric analysis of the recovery performance of
sparse regularization. More specifically, we analyze the BLASSO method which
estimates a sparse measure (sum of Dirac masses) from randomized sub-sampled
measurements. This is a "continuous", often called off-the-grid, extension of
the compressed sensing problem, where the $\ell^1$ norm is replaced by the
total variation of measures. This extension is appealing from a numerical
perspective because it avoids to discretize the the space by some grid. But
more importantly, it makes explicit the geometry of the problem since the
positions of the Diracs can now freely move over the parameter space. On a
methodological level, our contribution is to propose the Fisher geodesic
distance on this parameter space as the canonical metric to analyze
super-resolution in a way which is invariant to reparameterization of this
space. Switching to the Fisher metric allows us to take into account
measurement operators which are not translation invariant, which is crucial for
applications such as Laplace inversion in imaging, Gaussian mixtures estimation
and training of multilayer perceptrons with one hidden layer. On a theoretical
level, our main contribution shows that if the Fisher distance between spikes
is larger than a Rayleigh separation constant, then the BLASSO recovers in a
stable way a stream of Diracs, provided that the number of measurements is
proportional (up to log factors) to the number of Diracs. We measure the
stability using an optimal transport distance constructed on top of the Fisher
geodesic distance. Our result is (up to log factor) sharp and does not require
any randomness assumption on the amplitudes of the underlying measure. Our
proof technique relies on an infinite-dimensional extension of the so-called
"golfing scheme" which operates over the space of measures and is of general
interest.
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