Expressing a matrix as the sum of a low-rank matrix plus a sparse matrix is a flexible model capturing global and local features in data. This model is the foundation of robust principle component analysis [1,2], and popularized by dynamic-foreground/static-background separation [3] amongst other applications. Compressed sensing, matrix completion, and their variants [4,5] have established that data satisfying low complexity models can be efficiently measured and recovered from a number of measurements proportional to the model complexity rather than the ambient dimension. This manuscript develops similar guarantees showing that m × n matrices that can be expressed as the sum of a rank-r matrix and a s-sparse matrix can be recovered by computationally tractable methods from O(r(m + n − r) + s) log(mn/s) linear measurements. More specifically, we establish that the restricted isometry constants for the aforementioned matrices remain bounded independent of problem size provided p/mn, s/p, and r(m + n − r)/p reman fixed. Additionally, we show that semidefinite programming and two hard threshold gradient descent algorithms, NIHT and NAHT, converge to the measured matrix provided the measurement operator's RIC's are sufficiently small. Numerical experiments illustrating these results are shown for synthetic problems, dynamic-foreground/staticbackground separation, and multispectral imaging.

Robust Principal Component Analysis (PCA) (Candès et al., 2011) and low-rank matrix completion (Recht et al., 2010) are extensions of PCA that allow for outliers and missing entries respectively. It is well-known that solving these problems requires a low coherence between the low-rank matrix and the canonical basis, since in the extreme cases -when the low-rank matrix we wish to recover is also sparse -there is an inherent ambiguity. However, the well-posedness issue in both problems is an even more fundamental one: in some cases, both Robust PCA and matrix completion can fail to have any solutions due to the set of low-rank plus sparse matrices not being closed, which in turn is equivalent to the notion of the matrix rigidity function not being lower semicontinuous (Kumar et al., 2014). By constructing infinite families of matrices, we derive bounds on the rank and sparsity such that the set of low-rank plus sparse matrices is not closed. We also demonstrate numerically that a wide range of non-convex algorithms for both Robust PCA and matrix completion have diverging components when applied to our constructed matrices. An analogy can be drawn to the case of sets of higher order tensors not being closed under canonical polyadic (CP) tensor rank, rendering the best low-rank tensor approximation unsolvable (De Silva and Lim, 2008) and hence encourage the use of multilinear tensor rank (De Lathauwer, 2000).

Snapshot mosaic multispectral imagery acquires an undersampled data cube by acquiring a single spectral measurement per spatial pixel. Sensors which acquire p frequencies, therefore, suffer from severe 1/p undersampling of the full data cube. We show that the missing entries can be accurately imputed using non-convex techniques from sparse approximation and matrix completion initialised with traditional demosaicing algorithms. In particular, we observe the peak signalto-noise ratio can typically be improved by 2 dB to 5 dB over current state-of-the-art methods when simulating a p = 16 mosaic sensor measuring both high and low altitude urban and rural scenes as well as ground-based scenes.

In many cancers, significantly down- or upregulated genes are found within chromosomal regions with DNA copy number alteration opposite to the expression changes. Generally, this paradox has been overlooked as noise, but can potentially be a consequence of interference of epigenetic regulatory mechanisms, including microRNA-mediated control of mRNA levels.To explore potential associations between microRNAs and paradoxes in non-small-cell lung cancer (NSCLC) we curated and analyzed lung adenocarcinoma (LUAD) data, comprising gene expressions, copy number aberrations (CNAs) and microRNA expressions. We integrated data from 1,062 tumor samples and 241 normal lung samples, including newly-generated array comparative genomic hybridization (aCGH) data from 63 LUAD samples.We identified 85 “paradoxical” genes whose differential expression consistently contrasted with aberrations of their copy numbers. Paradoxical status of 70 out of 85 genes was validated on sample-wise basis using The Cancer Genome Atlas (TCGA) LUAD data. Of these, 41 genes are prognostic and form a clinically relevant signature, which we validated on three independent datasets. By meta-analysis of results from 9 LUAD microRNA expression studies we identified 24 consistently-deregulated microRNAs. Using TCGA-LUAD data we showed that deregulation of 19 of these microRNAs explains differential expression of the paradoxical genes.Our results show that deregulation of paradoxical genes is crucial in LUAD and their expression pattern is maintained epigenetically, defying gene copy number status.