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  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
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\begin{center}
  4:15 p.m., Tuesday, November 14, 2006\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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\begin{center}
  \textsl{Emmanuel Candes} \\
Applied and Computational Mathematics\\
California Institute of Technology
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\begin{center}
  \textbf{The Dantzig Selector:  statistical estimation when p is larger than n}
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\noindent                      

In many important statistical applications, the number of variables or
parameters is much larger than the number of observations.  In
radiology and biomedical imaging for instance, one is typically able
to collect far fewer measurements about an image of interest than the
unknown number of pixels. Examples in functional MRI and tomography
immediately come to mind. Other examples of high-dimensional data in
genomics, signal processing and many other fields abound.  In the
context of mulitple linear regression for instance, a fundamental
question is whether it is possible to estimate a vector of parameters
of size p from a vector of observations of size n when n << p. This seems a
priori hopeless.

This talk introduces a new estimator, dubbed the ``Dantzig selector''
in honor of the late George Dantzig as it invokes linear programming,
and which enjoys remarkable statistical properties. Suppose that the
data or design matrix obeys a uniform uncertainty principle and that
the true parameter vector is sufficiently sparse or compressible which
roughly guarantees that the model is identifiable. Then the estimator
achieves an accuracy which nearly equals that one would achieve with
an oracle that would supply perfect information about which
coordinates of the unknown parameter vector are nonzero and which were
above the noise level. Our results connect with the important model
selection problem. In effect, the Dantzig Selector automatically
selects the subset of covariates with nearly the best predictive
power, by solving a convenient linear program.

The results are parts of a larger body of work perhaps best known as
"Compressive Sampling" or Compressed Sensing." If time allows, I will discuss
connections with other fields such as signal processing and coding theory.



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