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  \textbf{\Large{\textsc{STANFORD UNIVERSITY}}}\\[5pt]
  \textbf{\Large{\textsc{DEPARTMENT OF STATISTICS}}}\\[5pt]
  \Large{\textsc{DEPARTMENTAL SEMINAR}}
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  4:15 p.m., Tuesday, April 8, 2008\\
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  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{Art B. Owen} \\
  Department of Statistics\\
  Stanford University
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\begin{center}
  \subsection*{Transposably invariant sample reuse methods}
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Sample reuse methods like the bootstrap and cross-validation are
widely used in statistics and machine learning.  They
have some face value validity and they don't
depend on strong model assumptions.
These methods depend on repeating or omitting cases, while keeping
all the variables in those cases.  But for many data sets, it is 
not obvious whether the rows are cases and colunns are variables, 
or vice versa. For example, with movie ratings organized by movie and
customer, both movie and customer IDs can be thought of as variables.

This talk looks at bootstrap and cross-validation methods that
treat rows and columns of the matrix symmetrically.  We get the
same answer on X as on X'.  McCullagh has proved that no exact 
bootstrap exists in a certain framework of this type (crossed 
random effects).  
We show that a method based on resampling both rows and 
columns of the data matrix tracks the true error, for some simple 
statistics applied to large data matrices.

Similarly we look at a method of cross-validation that leaves
out blocks of the data matrix, generalizing a proposal due to
Gabriel that is used in the crop science literature.  We find 
empirically that this approach provides a good way to choose the 
number of terms in a truncated SVD model or a non-negative matrix 
factorization. We also apply some recent results in random matrix
theory to the truncated SVD case.

\par\noindent
This work is joint with Patrick Perry.

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