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rather than random, for us to expect close convergence of the CUR decomposition to the
exact values.
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Figure 11.13
CUR-decomposition of the matrix of
Fig. 11.12
11.4.5
Eliminating Duplicate Rows and Columns
It is quite possible that a single row or column is selected more than once. There is no great
harm in using the same row twice, although the rank of the matrices of the decomposition
will be less than the number of row and column choices made. However, it is also possible
to combine
k
rows of
R
that are each the same row of the matrix
M
into a single row of
R
,
thus leaving
R
with fewer rows. Likewise,
k
columns of
C
that each come from the same
column of
M
can be combined into one column of
C
. However, for either rows or columns,
the remaining vector should have each of its elements multiplied by
When we merge some rows and/or columns, it is possible that
R
has fewer rows than
C
has columns, or vice versa. As a consequence,
W
will not be a square matrix. However,
we can still take its pseudoinverse by decomposing it into
W
=
X
Σ
Y
T
, where Σ is now a
diagonal matrix with some all-0 rows or columns, whichever it has more of. To take the
pseudoinverse of such a diagonal matrix, we treat each element on the diagonal as usual
(invert nonzero elements and leave 0 as it is), but then we must transpose the result.
EXAMPLE
11.17 Suppose
Then
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