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

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.