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Friday, February 28, 2014

Sparse Estimation From Noisy Observations of an Overdetermined Linear System / On the Randomized Kaczmarz Algorithm

I wonder why I  never see the Donoho-Tanner phase transition in these overdetermined linear systems papers. Anyway today, we have some solvers in that area: 

Sparse Estimation From Noisy Observations of an Overdetermined Linear System by Liang Dai, Kristiaan Pelckmans
This note studies a method for the efficient estimation of a finite number of unknown parameters from linear equations, which are perturbed by Gaussian noise.
In case the unknown parameters have only few nonzero entries, the proposed estimator performs more efficiently than a traditional approach. The method consists of three steps:
(1) a classical Least Squares Estimate (LSE),
(2) the support is recovered through a Linear Programming (LP) optimization problem which can be computed using a soft-thresholding step,
(3) a de-biasing step using a LSE on the estimated support set.
The main contribution of this note is a formal derivation of an associated ORACLE property of the final estimate.
That is, when the number of samples is large enough, the estimate is shown to equal the LSE based on the support of the {\em true} parameters.

On the Randomized Kaczmarz Algorithm by Liang Dai, Mojtaba Soltanalian, Kristiaan Pelckmans
The Randomized Kaczmarz Algorithm is a randomized method which aims at solving a consistent system of over determined linear equations. This note discusses how to find an optimized randomization scheme for this algorithm, which is related to the question raised by \cite{c2}. Illustrative experiments are conducted to support the findings.

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Anonymous said...

Seems to me that you aren't seeing these types of phase transition diagrams, because the Wu-Verdu result shows that you can get stable recovery for far lower measurement rates. The regime discussed here is interesting only because the noise is large.

Igor said...

I am sure I have seen it, this one come to my mind: