  In this paper, we examine the problem of approximating a general linear
dimensionality reduction (LDR) operator, represented as a matrix $A \in
\mathbb{R}^{m \times n}$ with $m < n$, by a partial circulant matrix with rows
related by circular shifts. Partial circulant matrices admit fast
implementations via Fourier transform methods and subsampling operations; our
investigation here is motivated by a desire to leverage these potential
computational improvements in large-scale data processing tasks. We establish a
fundamental result, that most large LDR matrices (whose row spaces are
uniformly distributed) in fact cannot be well approximated by partial circulant
matrices. Then, we propose a natural generalization of the partial circulant
approximation framework that entails approximating the range space of a given
LDR operator $A$ over a restricted domain of inputs, using a matrix formed as a
product of a partial circulant matrix having $m '> m$ rows and a $m \times k$
'post processing' matrix. We introduce a novel algorithmic technique, based on
sparse matrix factorization, for identifying the factors comprising such
approximations, and provide preliminary evidence to demonstrate the potential
of this approach.
