This article is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit ; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the ℓ 1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces.

We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal x ∈ C n about which we have phaseless samples of the form y r = ⟨a r , x⟩ 2 , r = 1, . . . , m (knowledge of the phase of these samples would yield a linear system). This paper develops a non-convex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval.

The three members of the T1R class of taste-specific G proteincoupled receptors have been hypothesized to function in combination as heterodimeric sweet taste receptors. Here we show that human T1R2͞T1R3 recognizes diverse natural and synthetic sweeteners. In contrast, human T1R1͞T1R3 responds to the umami taste stimulus L-glutamate, and this response is enhanced by 5-ribonucleotides, a hallmark of umami taste. The ligand specificities of rat T1R2͞T1R3 and T1R1͞T1R3 correspond to those of their human counterparts. These findings implicate the T1Rs in umami taste and suggest that sweet and umami taste receptors share a common subunit. Large-scale sequencing of a subtracted cDNA library derived from rat taste tissue identified a new C-family G proteincoupled receptor, T1R1, that is expressed selectively in taste receptor cells; T1R1-based degenerate PCR led to the identification of a related taste-specific receptor, T1R2 (1). Recently, a third and possibly final member of the T1R family, T1R3, was identified in the human DNA databank (2-7). Tellingly, mouse T1R3 maps to a genomic interval containing Sac, a locus that influences sweet taste in the mouse (8, 9). Recent highresolution genetic mapping and complementation studies have strengthened the connection between mouse T1R3 and Sac (2-7). Although T1R1 and T1R2 appear to be expressed in predominantly nonoverlapping regions of the tongue, they each are coexpressed with T1R3 (1, 3, 4, 6). These overlapping expression patterns and precedent from the structurally related heterodimeric ␥-aminobutyric acid type B receptor (10-13) suggested that T1R1 and T1R2 may combine with T1R3 to form heterodimeric sweet taste receptors. Indeed, rat T1R2 has been shown recently to function in combination with T1R3 to recognize a subset of sweet taste stimuli, a finding that has been proposed to reflect the involvement of additional combinations of T1Rs in sweet taste (6). In this study we cloned and functionally expressed human and rat T1Rs. Human and rat T1R2͞ T1R3 recognized all sweet taste stimuli tested, and human and rat T1R1͞T1R3 recognized umami taste stimuli. These findings suggest that different combinations of T1Rs function as heterodimeric sweet and umami taste receptors.Material and Methods T1R Cloning. Intronless human T1R expression constructs were generated in a pEAK10-derived vector (Edge Biosystems, Gaithersburg, MD) by a combination of cDNA-based and genomic DNA-based methods. To generate the full-length T1R1 expression construct, two 5Ј coding exons identified in a cloned T1R1 interval (GenBank accession no. AL159177) were combined by PCR overlap and then joined to a 5Ј-truncated testis cDNA clone. The T1R2 expression construct was generated from a partially sequenced T1R2 genomic interval. Two missing T1R2 5Ј introns were identified by screening shotgun libraries of the cloned genomic interval using probes derived from the corresponding rat coding sequence. Coding exons then were combined by PCR overlap to produce the full-length expression construct. The T1R3 e...

ABSTRACT:Introduction: Sclerosteosis is a rare high bone mass genetic disorder in humans caused by inactivating mutations in SOST, the gene encoding sclerostin. Based on these data, sclerostin has emerged as a key negative regulator of bone mass. We generated SOST knockout (KO) mice to gain a more detailed understanding of the effects of sclerostin deficiency on bone. Materials and Methods: Gene targeting was used to inactivate SOST and generate a line of SOST KO mice. Radiography, densitometry, CT, histomorphometry, and mechanical testing were used to characterize the impact of sclerostin deficiency on bone in male and female mice. Comparisons were made between same sex KO and wildtype (WT) mice. Results:The results for male and female SOST KO mice were similar, with differences only in the magnitude of some effects. SOST KO mice had increased radiodensity throughout the skeleton, with general skeletal morphology being normal in appearance. DXA analysis of lumbar vertebrae and whole leg showed that there was a significant increase in BMD (>50%) at both sites. CT analysis of femur showed that bone volume was significantly increased in both the trabecular and cortical compartments. Histomorphometry of trabecular bone revealed a significant increase in osteoblast surface and no significant change in osteoclast surface in SOST KO mice. The bone formation rate in SOST KO mice was significantly increased for trabecular bone (>9-fold) at the distal femur, as well as for the endocortical and periosteal surfaces of the femur midshaft. Mechanical testing of lumbar vertebrae and femur showed that bone strength was significantly increased at both sites in SOST KO mice. Conclusions: SOST KO mice have a high bone mass phenotype characterized by marked increases in BMD, bone volume, bone formation, and bone strength. These results show that sclerostin is a key negative regulator of a powerful, evolutionarily conserved bone formation pathway that acts on both trabecular and cortical bone.

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