  Matrix completion is a ubiquitous tool in machine learning and data analysis.
Most work in this area has focused on the number of observations necessary to
obtain an accurate low-rank approximation. In practice, however, the cost of
observations is an important limiting factor, and experimentalists may have on
hand multiple modes of observation with differing noise-vs-cost trade-offs.
This paper considers matrix completion subject to such constraints: a budget is
imposed and the experimentalist's goal is to allocate this budget between two
sampling modalities in order to recover an accurate low-rank approximation.
Specifically, we consider that it is possible to obtain low noise, high cost
observations of individual entries or high noise, low cost observations of
entire columns. We introduce a regression-based completion algorithm for this
setting and experimentally verify the performance of our approach on both
synthetic and real data sets. When the budget is low, our algorithm outperforms
standard completion algorithms. When the budget is high, our algorithm has
comparable error to standard nuclear norm completion algorithms and requires
much less computational effort.
