The Convex Geometry of Linear Inverse Problems

In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered are those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases such as sparse vectors and low-rank matrices, as well as several others including sums of a few permutations matrices, low-rank tensors, orthogonal matrices, and atomic measures. The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery. Thus this work extends the catalog of simple models that can be recovered from limited linear information via tractable convex programming

Common variants near MC4R are associated with fat mass, weight and risk of obesity.

To identify common variants influencing body mass index (BMI), we analyzed genome-wide association data from 16,876 individuals of European descent. After previously reported variants in FTO, the strongest association signal (rs17782313, P = 2.9 x 10(-6)) mapped 188 kb downstream of MC4R (melanocortin-4 receptor), mutations of which are the leading cause of monogenic severe childhood-onset obesity. We confirmed the BMI association in 60,352 adults (per-allele effect = 0.05 Z-score units; P = 2.8 x 10(-15)) and 5,988 children aged 7-11 (0.13 Z-score units; P = 1.5 x 10(-8)). In case-control analyses (n = 10,583), the odds for severe childhood obesity reached 1.30 (P = 8.0 x 10(-11)). Furthermore, we observed overtransmission of the risk allele to obese offspring in 660 families (P (pedigree disequilibrium test average; PDT-avg) = 2.4 x 10(-4)). The SNP location and patterns of phenotypic associations are consistent with effects mediated through altered MC4R function. Our findings establish that common variants near MC4R influence fat mass, weight and obesity risk at the population level and reinforce the need for large-scale data integration to identify variants influencing continuous biomedical traits

No evidence that protein truncating variants in BRIP1 are associated with breast cancer risk: implications for gene panel testing.

BACKGROUND: BRCA1 interacting protein C-terminal helicase 1 (BRIP1) is one of the Fanconi Anaemia Complementation (FANC) group family of DNA repair proteins. Biallelic mutations in BRIP1 are responsible for FANC group J, and previous studies have also suggested that rare protein truncating variants in BRIP1 are associated with an increased risk of breast cancer. These studies have led to inclusion of BRIP1 on targeted sequencing panels for breast cancer risk prediction. METHODS: We evaluated a truncating variant, p.Arg798Ter (rs137852986), and 10 missense variants of BRIP1, in 48 144 cases and 43 607 controls of European origin, drawn from 41 studies participating in the Breast Cancer Association Consortium (BCAC). Additionally, we sequenced the coding regions of BRIP1 in 13 213 cases and 5242 controls from the UK, 1313 cases and 1123 controls from three population-based studies as part of the Breast Cancer Family Registry, and 1853 familial cases and 2001 controls from Australia. RESULTS: The rare truncating allele of rs137852986 was observed in 23 cases and 18 controls in Europeans in BCAC (OR 1.09, 95% CI 0.58 to 2.03, p=0.79). Truncating variants were found in the sequencing studies in 34 cases (0.21%) and 19 controls (0.23%) (combined OR 0.90, 95% CI 0.48 to 1.70, p=0.75). CONCLUSIONS: These results suggest that truncating variants in BRIP1, and in particular p.Arg798Ter, are not associated with a substantial increase in breast cancer risk. Such observations have important implications for the reporting of results from breast cancer screening panels.The COGS project is funded through a European Commission's Seventh Framework Programme grant (agreement number 223175 - HEALTH-F2-2009-223175). BCAC is funded by Cancer Research UK [C1287/A10118, C1287/A12014] and by the European Community´s Seventh Framework Programme under grant agreement number 223175 (grant number HEALTH-F2-2009-223175) (COGS). Funding for the iCOGS infrastructure came from: the European Community's Seventh Framework Programme under grant agreement n° 223175 (HEALTH-F2-2009-223175) (COGS), Cancer Research UK (C1287/A10118, C1287/A 10710, C12292/A11174, C1281/A12014, C5047/A8384, C5047/A15007, C5047/A10692, C8197/A16565), the National Institutes of Health (CA128978) and Post-Cancer GWAS initiative (1U19 CA148537, 1U19 16 CA148065 and 1U19 CA148112 - the GAME-ON initiative), the Department of Defense (W81XWH-10-1- 0341), the Canadian Institutes of Health Research (CIHR) for the CIHR Team in Familial Risks of Breast Cancer, Komen Foundation for the Cure, the Breast Cancer Research Foundation, and the Ovarian Cancer Research Fund. This study made use of data generated by the Wellcome Trust Case Control consortium. Funding for the project was provided by the Wellcome Trust under award 076113. The results published here are in part based upon data generated by The Cancer Genome Atlas Project established by the National Cancer Institute and National Human Genome Research Institute.This is the author accepted manuscript. The final version is available from BMJ Group at http://dx.doi.org/10.1136/jmedgenet-2015-103529

Functional mechanisms underlying pleiotropic risk alleles at the 19p13.1 breast-ovarian cancer susceptibility locus

A locus at 19p13 is associated with breast cancer (BC) and ovarian cancer (OC) risk. Here we analyse 438 SNPs in this region in 46,451 BC and 15,438 OC cases, 15,252 BRCA1 mutation carriers and 73,444 controls and identify 13 candidate causal SNPs associated with serous OC (P=9.2 × 10-20), ER-negative BC (P=1.1 × 10-13), BRCA1-associated BC (P=7.7 × 10-16) and triple negative BC (P-diff=2 × 10-5). Genotype-gene expression associations are identified for candidate target genes ANKLE1 (P=2 × 10-3) and ABHD8 (P<2 × 10-3). Chromosome conformation capture identifies interactions between four candidate SNPs and ABHD8, and luciferase assays indicate six risk alleles increased transactivation of the ADHD8 promoter. Targeted deletion of a region containing risk SNP rs56069439 in a putative enhancer induces ANKLE1 downregulation; and mRNA stability assays indicate functional effects for an ANKLE1 3′-UTR SNP. Altogether, these data suggest that multiple SNPs at 19p13 regulate ABHD8 and perhaps ANKLE1 expression, and indicate common mechanisms underlying breast and ovarian cancer risk

Genetic loci influencing kidney function and chronic kidney disease

Using genome-wide association, we identify common variants at 2p12-p13, 6q26, 17q23 and 19q13 associated with serum creatinine, a marker of kidney function (P = 10 10 to 10 15). Of these, rs10206899 (near NAT8, 2p12-p13) and rs4805834 (near SLC7A9, 19q13) were also associated with chronic kidney disease (P = 5.0 × 10 5 and P = 3.6 × 10 4, respectively). Our findings provide insight into metabolic, solute and drug-transport pathways underlying susceptibility to chronic kidney disease