Abstract

In a series of papers from 2004-7, Candes and Tao, in conjunction with others, obtained some remarkable results for signal recovery and, more generally, for large p small n problems. The work culminated with the paper that introduced the Dantzig selector.

I will discuss just a few of the results from these papers. In the noise-free case, they show that exact signal reconstruction is possible from highly incomplete data [1]. I will illustrate this with an example and outline when one can achieve exact reconstruction. It turns out the results are robust to noise and the Dantzig selector is the approach suggested in such a scenario [2]. I will run through the proofs for the accuracy of the resulting estimators in the sparse case.

[1] Candes, E. J., Romberg, J. and Tao, T. Robust uncertainty principles: exact sig- nal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory (2006) 52 489-509

http://www.acm.caltech.edu/~emmanuel/papers/ExactRecovery.pdf

[2] Candes, E. J. and Tao, T. The Dantzig selector: statistical estimation when p is much larger than n. Annals of Statistics (2007) vol. 35 (6) pp. 2313-2351

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aos/1201012958