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In this paper, we examine the problem of approximating a general lineardimensionality reduction (LDR) operator, represented as a matrix $A \in\mathbb{R}^{m \times n}$ with $m < n$, by a partial circulant matrix with rowsrelated by circular shifts. Partial circulant matrices admit fastimplementations via Fourier transform methods and subsampling operations; ourinvestigation here is motivated by a desire to leverage these potentialcomputational improvements in large-scale data processing tasks. We establish afundamental result, that most large LDR matrices (whose row spaces areuniformly distributed) in fact cannot be well approximated by partial circulantmatrices. Then, we propose a natural generalization of the partial circulantapproximation framework that entails approximating the range space of a givenLDR operator $A$ over a restricted domain of inputs, using a matrix formed as aproduct of a partial circulant matrix having $m '> m$ rows and a $m \times k$'post processing' matrix. We introduce a novel algorithmic technique, based onsparse matrix factorization, for identifying the factors comprising suchapproximations, and provide preliminary evidence to demonstrate the potentialof this approach.

On Convolutional Approximations to Linear Dimensionality Reduction Operators for Large Scale Data Processing