47 research outputs found
From seismography to compressed sensing and back: a brief history of optimization-based signal processing
In this talk we provide an overview of the history of l1-norm minimization applied to underdetermined inverse problems. In the 70s and 80s geophysicists proposed using l1-norm minimization for deconvolution from bandpass data in reflection seismography. In the 2000s, inspired by this approach and by magnetic resonance imaging, a method to provably recover sparse signals from random projections, known as compressed sensing, was developed. Theoretical insights used to analyze compressed sensing have recently been adapted to understand the potential and limitations of l1-norm minimization for deterministic problems. These include super-resolution from low-pass data and the deconvolution problem that originally motivated the geophysicists.Universidad de Málaga. Campus de Excelencia Internacional AndalucĂa Tech
Support detection in super-resolution
We study the problem of super-resolving a superposition of point sources from
noisy low-pass data with a cut-off frequency f. Solving a tractable convex
program is shown to locate the elements of the support with high precision as
long as they are separated by 2/f and the noise level is small with respect to
the amplitude of the signal
Towards a Mathematical Theory of Super-Resolution
This paper develops a mathematical theory of super-resolution. Broadly
speaking, super-resolution is the problem of recovering the fine details of an
object---the high end of its spectrum---from coarse scale information
only---from samples at the low end of the spectrum. Suppose we have many point
sources at unknown locations in and with unknown complex-valued
amplitudes. We only observe Fourier samples of this object up until a frequency
cut-off . We show that one can super-resolve these point sources with
infinite precision---i.e. recover the exact locations and amplitudes---by
solving a simple convex optimization problem, which can essentially be
reformulated as a semidefinite program. This holds provided that the distance
between sources is at least . This result extends to higher dimensions
and other models. In one dimension for instance, it is possible to recover a
piecewise smooth function by resolving the discontinuity points with infinite
precision as well. We also show that the theory and methods are robust to
noise. In particular, in the discrete setting we develop some theoretical
results explaining how the accuracy of the super-resolved signal is expected to
degrade when both the noise level and the {\em super-resolution factor} vary.Comment: 48 pages, 12 figure
Generative data-driven approaches for stochastic subgrid parameterizations in an idealized ocean model
Subgrid parameterizations of mesoscale eddies continue to be in demand for
climate simulations. These subgrid parameterizations can be powerfully designed
using physics and/or data-driven methods, with uncertainty quantification. For
example, Guillaumin and Zanna (2021) proposed a Machine Learning (ML) model
that predicts subgrid forcing and its local uncertainty. The major assumption
and potential drawback of this model is the statistical independence of
stochastic residuals between grid points. Here, we aim to improve the
simulation of stochastic forcing with generative models of ML, such as
Generative adversarial network (GAN) and Variational autoencoder (VAE).
Generative models learn the distribution of subgrid forcing conditioned on the
resolved flow directly from data and they can produce new samples from this
distribution. Generative models can potentially capture not only the spatial
correlation but any statistically significant property of subgrid forcing. We
test the proposed stochastic parameterizations offline and online in an
idealized ocean model. We show that generative models are able to predict
subgrid forcing and its uncertainty with spatially correlated stochastic
forcing. Online simulations for a range of resolutions demonstrated that
generative models are superior to the baseline ML model at the coarsest
resolution