ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: select dimensions corresponding to non-zero diagonal values of . A similar selection of dimensions on gives of size below the dimension of the oversampled discrete Fourier grid, thus leading to the dimensionality reduction operator We see that the full measurement operator preserves the null space of , following from , thus retaining any original NSP. The diagonal dominated nature of has already been shown, which leads to an appropriate modelling of the noise. Also, applying is fast owing to the sparsity of . Note that in the context of this dimensionality reduction with , the weighting matrix , in fact designed for optimal weighting of the embedded visibilities, also operates as uniform weighting. Indeed, the diagonal values of are a measure of the density of continuous visibilities at each discrete grid point. As with , we also approximate the initial noise covariance matrix in this case by assuming , leading to the dimensionality reduction . This variant is included in comparisons of reconstruction quality. 3.4 Feature comparison A comparison of the different dimensionality reduction methods , and is shown in Table 1 . We note in the listing that provides a good combination of the desired final dimension, the guarantees for compressed sensing-based imaging to reconstruct images and a largely diagonal noise covariance matrix that enables us to embed this technique in the convex optimization algorithm we employ for imaging. embeds the data to image size while maintaining any original NSP of , and has a fast implementation; however, it fails to appropriately model the noise and is therefore less suitable for the minimization problem which requires i.i.d. Gaussian embedded noise to enable a simple 2 -norm data fidelity term. continues to maintain the NSP of and the i.i.d. Gaussian properties of the noise, which are essential for image reconstruction using our convex optimization algorithms. However, the data reduction is limited to a size . As mentioned in Sections 3.2 and 3.3 , for each of , and , an attempt is also made to further approximate the initial noise covariance matrix by the identity matrix in order to render the application of the respective dimensionality reduction methods even faster. Equations ( 19 ), ( 22 ) and ( 25 ) are then simplified to , and , respectively. However, this approximation is seen to be inappropriate, leading to poorer modelling of the noise and consequently lowering image reconstruction quality. , , and their respective variants are used to reduce dimensionality before performing image reconstruction in different settings and with varying data sizes. 3.5 Further reduction by thresholding As a conservative dimensionality reduction method, would embed to a final data size under the assumption that there are very few zero-valued singular values of , within limits of numerical precision, and all corresponding singular vectors are thus necessary to retain the information content of . Similarly, embeds to a size