Matrix Completion

 Here, we are trying to utilize patters to attempt to fill out the missing entries of a matrix. 

We are trying to use patterns to predict the missing entries of a matrix.

Ω = {i, j: Xij given} are the indices of the known values. 

The next operation that I can perform is rank minimization. This is denoted by the following equation:

X = argmin (rank M) such that Mij = Xij. Where Ai, j is in Ω. Find the minimum number of pattern giving values (rank M = #{l : σl > 0}).

The other thing is nuclear norm minimization. 

My prof explained this pretty well in his lecture in how to complete matrices. 

This method actually guarantees minimal rank solutions, and is an effective singular value threshold algorithm for completion. 

The following pseudocode is the method for Iterative Singular Value Thresholding, given that certain matrix values are equivalent to 0.

1. Set all unknown entries to 0.

1. Take the singular value decomposition matrix, and truncate it, to find the best rank approximation.

2. Reset the known entries to the original values.

3. Repeat steps 1 and 2 and keep multiplying the Matrix's transpose by the matrix itself (A^T * A).