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The first observation we made was the obvious -- the mean squared error for matrices that had unobserved indices which were completely non-uniform was very high, and had a large standard deviation, whereas the perfectly uniform matrices had a low mean squared error with a small standard deviation.
These results confirmed that our implementation of the algorithm and the code for the other parts of our project worked correctly.
The more surprising result of our tests, showed that matrix completion can tolerate a high level of non-uniformity. Our graph showed that at, according to our model, at about 15% uniformity, matrix completion returns results that are comparable to 100% uniformity.
Even the visual depictions of this kind of uniformity are interesting to ponder. At 15%, there is still a clear accumulation of unobserved indices at the center of the matrix, yet the algorithm works just as well.
The implications for this are tremendous, and the results show that uniformity isn’t a huge concern when applying matrix completion to real world problems. Unless the matrix is completely non-uniform, matrix completion will provide valid results.
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