  With increasing concerns about security, the need for highly secure physical
biometrics-based authentication systems utilizing \emph{cancelable biometric}
technologies is on the rise. Because the problem of cancelable template
generation deals with the trade-off between template security and matching
performance, many state-of-the-art algorithms successful in generating high
quality cancelable biometrics all have random projection as one of their early
processing steps. This paper therefore presents a formal analysis of why random
projections is an essential step in cancelable biometrics. By formally defining
the notion of an \textit{Independent Subspace Structure} for datasets, it can
be shown that random projection preserves the subspace structure of data
vectors generated from a union of independent linear subspaces. The bound on
the minimum number of random vectors required for this to hold is also derived
and is shown to depend logarithmically on the number of data samples, not only
in independent subspaces but in disjoint subspace settings as well. The
theoretical analysis presented is supported in detail with empirical results on
real-world face recognition datasets.
