  Data dimensionality reduction in radio interferometry can provide savings of
computational resources for image reconstruction through reduced memory
footprints and lighter computations per iteration, which is important for the
scalability of imaging methods to the big data setting of the next-generation
telescopes. This article sheds new light on dimensionality reduction from the
perspective of compressed sensing theory and studies its interplay with imaging
algorithms designed in the context of convex optimization. We propose a
post-gridding linear data embedding to the space spanned by the left singular
vectors of the measurement operator, providing a dimensionality reduction below
image size. This embedding preserves the null space of the measurement operator
and hence also its sampling properties as per compressed sensing theory. We
show that this can be approximated by first computing the dirty image and then
applying a weighted subsampled discrete Fourier transform to obtain the final
reduced data vector. This Fourier dimensionality reduction model ensures a fast
implementation of the full measurement operator, essential for any iterative
image reconstruction method. The proposed reduction also preserves the i.i.d.
Gaussian properties of the original measurement noise. For convex
optimization-based imaging algorithms, this is key to justify the use of the
standard L2-norm as the data fidelity term. Our simulations confirm that this
dimensionality reduction approach can be leveraged by convex optimization
algorithms with no loss in imaging quality relative to reconstructing the image
from the complete visibility data set. Reconstruction results in simulation
settings with no direction dependent effects or calibration errors show
promising performance of the proposed dimensionality reduction. MATLAB code
implementing the proposed reduction method is available on GitHub.
