Low rank approximation of binary valued matrix How does one get the low rank approximation of binary matrix? Is the low rank approximation also a binary matrix? 
Note - Here binary matrix just means that any entry of the matrix can either be 0 or 1. 
 A: A naive approach might be using a plain ol' SVD with thresholds. Choose the thresholds that maximize classification accuracy, or some other sensitivity/specificity tradeoff. 
Thresholding might up the rank, but it begs the question of why one would need to a low rank approximation of a binary matrix. Usually, low rank approximations are convenient for scale development and identification of subscales which is strongly based on assumptions about additive variance (this holds for normally distributed random variables but not Bernoulli random variables).
To me, I would consider creating separate models for the marginal cell counts as well as the odds ratios in a log linear model describing the assocations. That would create a large number of parameters, obviously. If I wanted something that were computationally more efficient and "easier to digest" I might consider using sparse estimation with a LASSO or other tool.
A: One idea is to compute the Hamming distance between data vectors and then use this as a pairwise distance metric. If the number of dimensions is much larger than the number of data points, this may make sense. I am not sure if this reduces rank for certain, but it's a more compact representation of the structure.
