Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 367, Issue 1906, 2009, Pages 4273-4293

  Donoho, David ^a   Tanner, Jared ^b  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Combinatorial geometry; Compressed sensing; High dimension low sample size datasets; High throughput measurements; Robust linear models

Indexed keywords

ASYMPTOTIC ANALYSIS; COMPUTATIONAL GEOMETRY; CURVE FITTING; DATA HANDLING; MATHEMATICAL MODELS; MATRIX ALGEBRA; SIGNAL PROCESSING; SIGNAL RECONSTRUCTION; TOPOLOGY;

COMBINATORIAL GEOMETRY; COMPRESSED SENSING; DATA SETS; HIGH DIMENSIONS; HIGH-THROUGHPUT MEASUREMENTS; LINEAR MODEL; SAMPLE SIZES;

PHASE TRANSITIONS;

EID: 73149095252     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2009.0152     Document Type: Article

References (12)

1. Adamczak, R., Litvak, A. E., Pajor, A. & Tomczak-Jaegermann, N. 2009 Compressed sensing matrices with independent columns and neighborly polytopes by random sampling. (http:// arxiv.org/abs/0904.4723).
2. Berinde, R., Gilbert, A. C., Indyk, P., Karloff, H. & Strauss, M. J. 2008 Combining geometry and combinatorics: a unified approach to sparse signal recovery. See http://front.math.ucdavis.edu/0804.4666.
3. Donoho, D. L. 2005a High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discrete Comput. Geom. 35, 617-652. (doi:10.1007/s00454-005- 1220-0.
4. Donoho, D. L. 20056 Neighborly polytopes and sparse solution of underdetermined linear equations. See http://www-stat.stanford.edu/~donoho/ Reports/2005/NPaSSULE-01-28-05.pdf.
5. Donoho, D. & Jin, J. 2009 Feature selection by higher criticism thresholding achieves the optimal phase diagram. Phil. Trans. R. Soc. A 367, 4449-4470. (doi:10.1098/rsta.2009.0129).
6. Donoho, D. L. & Tanner, J. 2005a Sparse nonnegative solutions of underdetermined linear equations by linear programming. Proc. Natl Acad. Sci. USA 102, 9446-9451. (doi:10.1073/pnas. 0502269102).
7. Donoho, D. L. & Tanner, J. 20056 Neighborliness of randomly-projected simplices in high dimensions. Proc. Natl Acad. Sci. USA 102, 9452-9457. (doi:10.1073/pnas.0502258102). (Pubitemid 40981702)
8. Donoho, D. L. & Tanner, J. 2008a Counting the faces of randomly-projected hypercubes and orthants, with applications. See http://front.math.ucdavis.edu/0807.3590.
9. Donoho, D. L. & Tanner, J. 20086 Exponential bounds implying construction of compressed sensing matrices, error-correcting codes and neighborly polytopes by random sampling. See http://www.maths.ed.ac.uk/~tanner/ DoTa-Finite.pdf.
10. Donoho, D. L. & Tanner, J. 2009 Counting faces of randomly projected polytopes when the projection radically lowers dimension. J. Am. Math. Soc. 22, 1-53. (doi:10.1090/s0894-0347-08-00600-0).
11. Maleki, A. & Donoho, D. L. 2009 Optimally tuned iterative reconstruction algorithms for compressed sensing. See http://www.stanford.edu/ ∼arianm/SP-OptimalTuning.pdf.
12. Stodden, V. 2006 Breakdown point of model selection when there are more variables than observations. PhD thesis, Stanford University.