  Identifying important components or factors in large amounts of noisy data is
a key problem in machine learning and data mining. Motivated by a pattern
decomposition problem in materials discovery, aimed at discovering new
materials for renewable energy, e.g. for fuel and solar cells, we introduce
CombiFD, a framework for factor based pattern decomposition that allows the
incorporation of a-priori knowledge as constraints, including complex
combinatorial constraints. In addition, we propose a new pattern decomposition
algorithm, called AMIQO, based on solving a sequence of (mixed-integer)
quadratic programs. Our approach considerably outperforms the state of the art
on the materials discovery problem, scaling to larger datasets and recovering
more precise and physically meaningful decompositions. We also show the
effectiveness of our approach for enforcing background knowledge on other
application domains.
