  Matrix factorization methods are important tools in data mining and analysis.
They can be used for many tasks, ranging from dimensionality reduction to
visualization. In this paper we concentrate on the use of matrix factorizations
for finding patterns from the data. Rather than using the standard algebra --
and the summation of the rank-1 components to build the approximation of the
original matrix -- we use the subtropical algebra, which is an algebra over the
nonnegative real values with the summation replaced by the maximum operator.
Subtropical matrix factorizations allow "winner-takes-it-all" interpretations
of the rank-1 components, revealing different structure than the normal
(nonnegative) factorizations. We study the complexity and sparsity of the
factorizations, and present a framework for finding low-rank subtropical
factorizations. We present two specific algorithms, called Capricorn and
Cancer, that are part of our framework. They can be used with data that has
been corrupted with different types of noise, and with different error metrics,
including the sum-of-absolute differences, Frobenius norm, and Jensen--Shannon
divergence. Our experiments show that the algorithms perform well on data that
has subtropical structure, and that they can find factorizations that are both
sparse and easy to interpret.
