Description
1. Consider the 4-by-3 matrix defined as V =
1 X X
X 2 4
−1 2 X
X −2 X
where X denotes missing
entries. Assume V is a rank-1 matrix.
a) Use what you know about the structure of rank-1 matrices to find the missing
entries.
b) What is the minimum number of missing entries for which you cannot complete
a 4-by-3 rank 1 matrix? Where are the missing entries in this case?
2. A data file is available that contains a rank-2, 16-by-16 matrix Xtrue with integer
entries and three versions of this matrix (Y1, Y2, and Y3) with differing numbers of
missing entries. The missing entries are indicated by NaN.
A script is provided to complete a matrix using iterative singular value thresholding.
The script contains a function that requires two inputs: i) the matrix with missing
entries, and ii) the rank.
a) Apply the iterative singular value thresholding function (provided in the script)
to the three incomplete matrices assuming the rank is 2. You will first need to
complete the line of code in the function. Compare your recovered completed matrices to Xtrue (Note: compare the output by subtracting the completed matrix
from the original matrix, and then displaying them). Does the number of missing
entries affect the accuracy of the completed matrix?
b) Now apply your routine to the three incomplete matrices assuming the rank is 3.
Compare your recovered completed matrices to Xtrue. Comment on the impact
of using the incorrect rank in the completion process.
