  We present a new method for exploring cancer gene expression data based on
tools from algebraic topology. Our method selects a small relevant subset from
tens of thousands of genes while simultaneously identifying nontrivial higher
order topological features, i.e., holes, in the data. We first circumvent the
problem of high dimensionality by dualizing the data, i.e., by studying genes
as points in the sample space. Then we select a small subset of the genes as
landmarks to construct topological structures that capture persistent, i.e.,
topologically significant, features of the data set in its first homology
group. Furthermore, we demonstrate that many members of these loops have been
implicated for cancer biogenesis in scientific literature. We illustrate our
method on five different data sets belonging to brain, breast, leukemia, and
ovarian cancers.
