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Statistical Inference via Convex Optimization

Anatoli Juditsky and Arkadi Nemirovski
Princeton University Press
Publication Date: 
Number of Pages: 
Princeton Series in Applied Mathematics
[Reviewed by
Brian Borchers
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This is not a book on convex optimization algorithms or theory, but rather a collection of results on statistical estimators that can be obtained by solving convex optimization problems.  Since these particular convex optimization problems can be efficiently solved, this is nearly as good as having a formula for the estimator.
The book deals with three major problems.  First, the authors consider the compressive sensing problem of recovering a sparse solution to a linear system of equations \( Ax=b \). The authors present results on considitons that ensure that a sparse solution will be recovered by minimizing \( \| x \|_{1} \) subject to \( Ax=b \).  This is a convex optimization problem that can be efficiently solved.  Next, they consider the problem of constructing risk optimal detectors for testing whether the parameters of a distribution lie within convex sets.  The optimal detector and a bound on the risk can be found by solving a convex optimization problem.  Finally, the authors consider the problem of constructing an optimal linear estimator for the recovery of a signal from noisy observations of a linear transformation of the signal.  Again, the optimal estimator can be found by solving a convex optimization problem.
The book has numerous examples and exercises.  Definitions and key results are presented separately from the proofs of the results.  The book should be reasonably accessible to readers with some graduate level background in mathematical statistics and convex optimization.  For example, For readers without background in convex optimization, Boyd and Vandenberghe's Convex Optimization would be an appropriate starting point.  Boyd and Vandenberghe cover the convex optimization theory and methods used in this book and also introduce the use of convex optimization in statistical estimation. 
Statistical Inference via Convex Optimization could be used as a textbook for an advanced graduate course.  It will also be useful as a reference for statisticians interested in using convex optimization based estimators.

Brian Borchers is a professor of mathematics at New Mexico Tech and the editor of MAA Reviews.
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