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We review connections between phase transitions in high-dimensional combinatorial geometry and phase transitions occurring in modern high-dimensional data analysis and signal processing.
[] Donoho, D. L. 2005 a a High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discrete Comput. Geom
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[] Donoho, D. L. 2005 b b Neighborly polytopes and sparse solution of underdetermined linear equations. Technical Report, Statistics Department, Stanford University
2005
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[] Donoho, D. L. & Tanner, J. 2005 a a Sparse nonnegative solutions of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. USA
2005
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[] Donoho, D. L. & Tanner, J. 2005 b b Neighborliness of randomly-projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA
2005
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[] Donoho, D. L. & Stodden, V. 2006 Breakdown point of model selection when the number of variables exceeds the number of observations. Proceedings of the International Joint Conference on Neural Networks
2006
Earlier work this paper cites.
[] Stodden, V. 2006 Breakdown point of model selection when there are more variables than observations
2006
Cited alongside, same era.
[] Berinde, R., Gilbert, A. C., Indyk, P., Karloff, H. & Strauss, M. J. 2008 Combining geometry and combinatorics: a unified approach to sparse signal recovery
2008
Cited alongside, same era.
[] Donoho, D. L. & Tanner, J. 2008 a a Counting the faces of randomly-projected hypercubes and orthants, with applications
2008
Cited alongside, same era.
[] Donoho, D. L. & Tanner, J. 2008 b b Exponential bounds implying construction of compressed sensing matrices, error-correcting codes and neighborly polytopes by random sampling
2008
Cited alongside, same era.
[] CVX optimization software. http://www.stanford.edu/ ~ \;\tilde{}\; boyd/cvx/
Cited in the paper.
[] Adamczak, R., Litvak, A. E., Pajor, A. & Tomczak-Jaegermann, N. 2009 Compressed sensing matrices with independent columns and neighborly polytopes by random sampling
2009
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[] Donoho, D. L. & Jin, J. 2009 Feature selection by higher criticism thresholding achieves the optimal phase diagram. Phil. Trans. A
2009
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[] Donoho, D. L. & Tanner, J. 2009 a a Counting faces of randomly projected polytopes when the projection radically lowers dimension. J. AMS
2009
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[] Maleki, A. & Donoho, D. L. 2009 Optimally tuned iterative reconstruction algorithms for compressed sensing
2009
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[] Mosek optimization software. http://www.mosek.com
Cited in the paper.