Next-generation radio-interferometers face a computing challenge with respect to the imaging techniques that can be applied in the big data setting in which they are designed. Dimensionality reduction can thus provide essential savings of computing resources, allowing imaging methods to scale with data. The work presented here approaches dimensionality reduction from a compressed sensing theory perspective, and links to its role in convex optimization-based imaging algorithms. We describe a novel linear dimensionality reduction technique consisting of a linear embedding to the space spanned by the left singular vectors of the measurement operator. A subsequent approximation of this embedding is shown to be practically implemented through a weighted subsampled Fourier transform of the dirty image. Preliminary results on simulated data with realistic coverages suggest that this approach provides significant reduction of data dimension to well below image size, while achieving comparable image quality to that obtained from the complete data set.