682 research outputs found
Sparse Spikes Deconvolution on Thin Grids
This article analyzes the recovery performance of two popular finite
dimensional approximations of the sparse spikes deconvolution problem over
Radon measures. We examine in a unified framework both the L1 regularization
(often referred to as Lasso or Basis-Pursuit) and the Continuous Basis-Pursuit
(C-BP) methods. The Lasso is the de-facto standard for the sparse
regularization of inverse problems in imaging. It performs a nearest neighbor
interpolation of the spikes locations on the sampling grid. The C-BP method,
introduced by Ekanadham, Tranchina and Simoncelli, uses a linear interpolation
of the locations to perform a better approximation of the infinite-dimensional
optimization problem, for positive measures. We show that, in the small noise
regime, both methods estimate twice the number of spikes as the number of
original spikes. Indeed, we show that they both detect two neighboring spikes
around the locations of an original spikes. These results for deconvolution
problems are based on an abstract analysis of the so-called extended support of
the solutions of L1-type problems (including as special cases the Lasso and
C-BP for deconvolution), which are of an independent interest. They precisely
characterize the support of the solutions when the noise is small and the
regularization parameter is selected accordingly. We illustrate these findings
to analyze for the first time the support instability of compressed sensing
recovery when the number of measurements is below the critical limit (well
documented in the literature) where the support is provably stable
Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem
We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re
equation with second boundary value condition, provided the target is a convex
set. This yields a fast adaptive method to numerically solve the Optimal
Transport problem between two absolutely continuous measures, the second of
which has convex support. The proposed numerical method actually captures a
specific Brenier solution which is minimal in some sense. We prove the
convergence of the method as the grid stepsize vanishes and we show with
numerical experiments that it is able to reproduce subtle properties of the
Optimal Transport problem
Convergence of Entropic Schemes for Optimal Transport and Gradient Flows
Replacing positivity constraints by an entropy barrier is popular to
approximate solutions of linear programs. In the special case of the optimal
transport problem, this technique dates back to the early work of
Schr\"odinger. This approach has recently been used successfully to solve
optimal transport related problems in several applied fields such as imaging
sciences, machine learning and social sciences. The main reason for this
success is that, in contrast to linear programming solvers, the resulting
algorithms are highly parallelizable and take advantage of the geometry of the
computational grid (e.g. an image or a triangulated mesh). The first
contribution of this article is the proof of the -convergence of the
entropic regularized optimal transport problem towards the Monge-Kantorovich
problem for the squared Euclidean norm cost function. This implies in
particular the convergence of the optimal entropic regularized transport plan
towards an optimal transport plan as the entropy vanishes. Optimal transport
distances are also useful to define gradient flows as a limit of implicit Euler
steps according to the transportation distance. Our second contribution is a
proof that implicit steps according to the entropic regularized distance
converge towards the original gradient flow when both the step size and the
entropic penalty vanish (in some controlled way)
Functionalized polyhydroquinolines from amino acids using a key one-pot cyclization cascade and application to the synthesis of (±)-Δ7-mesembrenone
Substituted polyhydroquinolines are ubiquitous skeletal cores found in drugs and bioactive natural products. As a new route to access this motif, we successfully developed a one-pot cyclization cascade with high chemocontrol and diastereoselectivi-ty. The sequence generates two cycles, three carbon-carbon bonds, and an all-carbon quaternary center in a highly conver-gent process. Functionalized polyhydroquinolines and congeners are accessible from commercially available amino acids. This versatile and robust strategy was applied to the synthesis of (±)-D7-mesembrenon
Une approche épigraphique pour le théorème des représentants
Describing the solutions of inverse problems arising in signal or image processing is an important issue both for theoretical and numerical purposes. We propose a principle which describes the solutions to convex variational problems involving a finite number of measurements. We discuss its optimality on various problems concerning the recovery of Radon measures
A characterization of the Non-Degenerate Source Condition in Super-Resolution
In a recent article, Schiebinger et al. provided sufficient conditions for the noiseless recovery of a signal made of M Dirac masses given 2M + 1 observations of, e.g. , its convolution with a Gaussian filter, using the Basis Pursuit for measures. In the present work, we show that a variant of their criterion provides a necessary and sufficient condition for the Non-Degenerate Source Condition (NDSC) which was introduced by Duval and Peyré to ensure support stability in super-resolution. We provide sufficient conditions which, for instance, hold unconditionally for the Laplace kernel provided one has at least 2M measurements. For the Gaussian filter, we show that those conditions are fulfilled in two very different configurations: samples which approximate the uniform Lebesgue measure or, more surprisingly, samples which are all confined in a sufficiently small interval
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