Matrix Completion with Noisy Entries and Outliers

This paper considers the problem of matrix completion when the observed entries are noisy and contain outliers. It begins with introducing a new optimization criterion for which the recovered matrix is defined as its solution. This criterion uses the celebrated Huber function from the robust statistics literature to downweigh the effects of outliers. A practical algorithm is developed to solve the optimization involved. This algorithm is fast, straightforward to implement, and monotonic convergent. Furthermore, the proposed methodology is theoretically shown to be stable in a well defined sense. Its promising empirical performance is demonstrated via a sequence of simulation experiments, including image inpainting.

XMM-Newton and Chandra Observations of Abell 2626: Interacting Radio Jets and Cooling Core with Jet Precession?

We present a detailed analysis of the XMM-Newton and Chandra observations of Abell 2626 focused on the X-ray and radio interactions. Within the region of the radio mini-halo (~70 kpc), there are substructures which are probably produced by the central radio source and the cooling core. We find that there is no obvious correlation between the radio bars and the X-ray image. The morphology of Abell 2626 is more complex than that of the standard X-ray radio bubbles seen in other cool core clusters. Thus, Abell 2626 provides a challenge to models for the cooling flow -- radio source interaction. We identified two soft X-ray (0.3--2 keV) peaks with the two central cD nuclei; one of them has an associated hard X-ray (2--10 keV) point source. We suggest that the two symmetric radio bars can be explained by two precessing jets ejected from an AGN. Beyond the central regions, we find two extended X-ray sources to the southwest and northeast of the cluster center which are apparently associated with merging subclusters. The main Abell 2626 cluster and these two subclusters are extended along the direction of the Perseus-Pegasus supercluster, and we suggest that Abell 2626 is preferentially accreting subclusters and groups from this large-scale structure filament. We also find an extended X-ray source associated with the cluster S0 galaxy IC 5337; the morphology of this source suggests that it is infalling from the west, and is not associated with the southwest subcluster, as had been previously suggested.