A new form of the discrepancy principle for poisson inverse problems with compact operators is proposed and discussed in relation to various other proposals. It is shown that filter-induced spectral regularization wit...
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A new form of the discrepancy principle for poisson inverse problems with compact operators is proposed and discussed in relation to various other proposals. It is shown that filter-induced spectral regularization with a priori chosen smoothing parameter produces estimators that are rate-minimax under source conditions on the estimated function. With the discrepancy principle used for a posteriori choice of the smoothing parameter, the filter-induced solutions are consistent (in probability), but the convergence rates under source conditions are suboptimal, at least in the finitely smoothing case, which often happens when discrepancy principles are used in stochastic inverseproblems. Finite sample performance of the proposed procedure applied to a stereological problem of Spektor, Lord and Willis is illustrated with a simulation experiment.
The emerging theory of compressed sensing has been nothing short of a revolution in signal processing, challenging some of the longest-held ideas in signal processing and leading to the development of exciting new way...
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The emerging theory of compressed sensing has been nothing short of a revolution in signal processing, challenging some of the longest-held ideas in signal processing and leading to the development of exciting new ways to capture and reconstruct signals and images. Although the theoretical promises of compressed sensing are manifold, its implementation in many practical applications has lagged behind the associated theoretical development. Our goal is to elevate compressed sensing from an interesting theoretical discussion to a feasible alternative to conventional imaging, a significant challenge and an exciting topic for research in signal processing. When applied to imaging, compressed sensing can be thought of as a particular case of computational imaging, which unites the design of both the sensing and reconstruction of images under one design paradigm. Computational imaging tightly fuses modeling of scene content, imaging hardware design, and the subsequent reconstruction algorithms used to recover the images. This thesis makes important contributions to each of these three areas through two primary research directions. The first direction primarily attacks the challenges associated with designing practical imaging systems that implement incoherent measurements. Our proposed snapshot imaging architecture using compressive coded aperture imaging devices can be practically implemented, and comes equipped with theoretical recovery guarantees. It is also straightforward to extend these ideas to a video setting where careful modeling of the scene can allow for joint spatio-temporal compressive sensing. The second direction develops a host of new computational tools for photon-limited inverseproblems. These situations arise with increasing frequency in modern imaging applications as we seek to drive down image acquisition times, limit excitation powers, or deliver less radiation to a patient. By an accurate statistical characterization of the measurement process in
Many problems in machine learning write as the minimization of a sum of individual loss functions over the training examples. These functions are usually differentiable but, in some cases, their gradients are not Lips...
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Many problems in machine learning write as the minimization of a sum of individual loss functions over the training examples. These functions are usually differentiable but, in some cases, their gradients are not Lipschitz continuous, which compromises the use of (proximal) gradient algorithms. Fortunately, changing the geometry and using Bregman divergences can alleviate this issue in several applications, such as for poisson linear inverseproblems. However, the Bregman operation makes the aggregation of several points and gradients more involved, hindering the distribution of computations for such problems. In this paper, we propose an asynchronous variant of the Bregman proximal-gradient method, able to adapt to any centralized computing system. In particular, we prove that the algorithm copes with arbitrarily long delays and we illustrate its behaviour on distributed poisson inverse problems.
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