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In many applications, extremely large problem sizes are envisioned, with at least tens of thousands of equations and hundreds of thousands of unknowns. For such Deanna needell thesis sizes, low computational complexity is paramount.
Iterative thresholding algorithms have been proposed to address this problem. In this paper we want to analyze two of these algorithms theoretically, and give sufficient conditions under which they recover the sparsest solution.
The abstract This paper studies the question of how well a signal can be reprsented by a sparse linear combination of reference signals from an overcomplete dictionary. When the dictionary size is exponential in the dimension of signal, then the exact characterization of the optimal distortion is given as a function of the dictionary size exponent and the number of reference signals for the linear representation.
Roughly speaking, every signal is sparse if the dictionary size is exponentially large, no matter how small the exponent is.
Furthermore, an iterative method similar to matching pursuit that successively finds the best reference signal at each stage gives asymptotically optimal representations. This method is essentially equivalent to successive refinement for multiple descriptions and provides a simple alternative proof of the successive refinability of white Gaussian sources.
Although the residual method, or constrained regularization, is frequently used in applications, a detailed study of its properties is still missing. In particular, the questions of stability and convergence rates have hardly been treated in the literature.
This sharply contrasts the progress of the theory of Tikhonov regularization, where for instance the notion of the Bregman distance has recently led to a series of new results for regularization in Banach spaces.
The present paper intends to bridge the gap between the existing theories as far as possible. We develop a stability and convergence theory for the residual method in general topological spaces.
In addition, we prove convergence rates in terms of generalized Bregman distances. Exemplarily, we apply our theory to compressed sensing. Here, we show the well-posedness of the method and derive convergence rates both for convex and non-convex regularization under rather weak conditions.
It is for instance shown that in the non-convex setting the linear convergence of the regularized solutions already follows from the sparsity of the true solution and the injectivity of the linear operator in the equation to be solved.
This paper is concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to data and a total variation constraint.
To our knowledge, this is the first successful attempt of addressing such strategy for the nonlinear, nonadditive, and nonsmooth problem of total variation minimization.
We propose a new method of image fusion that utilizes the recently developed theory of compressive sensing.
Compressive sensing indicates that a signal that is sparse in an appropriate set of basis vectors may be recovered almost exactly from a few samples via l1-minimization if the system matrix satisfies some conditions.Mar 05, · Deanna Needell gave background on the Kaczmarz Algorithm which gives alternative way to motivate importance sampling results.
In particular, first few slides illustrate why the order matters. She also gives analytic expression to find the best next point to sample for the quadratic case.
Deanna Needell is an American applied mathematician at University of California, Los Angeles. Education [ edit ] Deanna Needell received her PhD in mathematics from the . Nov 25, · Deanna needell thesis Test and two full-length practice tests that reflect the actual ap psychology exam in length, and an analysis of the test’s essay question with a sample essay.
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