Min-max problems have broad applications in machine learning including learning with non-decomposable loss and learning with robustness to data's distribution. Although convex-concave min-max problems have been broadly studied with efficient algorithms and solid theories available, it still remains a challenge to design provably efficient algorithms for non-convex min-max problems. Motivated by the applications in machine learning, this paper studies a family of non-convex min-max problems, whose objective function is weakly convex in the variables of minimization and is concave in the variable of maximization. We propose a proximally guided stochastic subgradient method and a proximally guided stochastic variance-reduced method for this class of problems under different assumptions. We establish their time complexities for finding a nearly stationary point of the outer minimization problem corresponding to the min-max problem.
BackgroundIntegration of transcriptomic and metabolomic data improves functional interpretation of disease-related metabolomic phenotypes, and facilitates discovery of putative metabolite biomarkers and gene targets. For this reason, these data are increasingly collected in large (> 100 participants) cohorts, thereby driving a need for the development of user-friendly and open-source methods/tools for their integration. Of note, clinical/translational studies typically provide snapshot (e.g. one time point) gene and metabolite profiles and, oftentimes, most metabolites measured are not identified. Thus, in these types of studies, pathway/network approaches that take into account the complexity of transcript-metabolite relationships may neither be applicable nor readily uncover novel relationships. With this in mind, we propose a simple linear modeling approach to capture disease-(or other phenotype) specific gene-metabolite associations, with the assumption that co-regulation patterns reflect functionally related genes and metabolites.ResultsThe proposed linear model, metabolite ~ gene + phenotype + gene:phenotype, specifically evaluates whether gene-metabolite relationships differ by phenotype, by testing whether the relationship in one phenotype is significantly different from the relationship in another phenotype (via a statistical interaction gene:phenotype p-value). Statistical interaction p-values for all possible gene-metabolite pairs are computed and significant pairs are then clustered by the directionality of associations (e.g. strong positive association in one phenotype, strong negative association in another phenotype). We implemented our approach as an R package, IntLIM, which includes a user-friendly R Shiny web interface, thereby making the integrative analyses accessible to non-computational experts. We applied IntLIM to two previously published datasets, collected in the NCI-60 cancer cell lines and in human breast tumor and non-tumor tissue, for which transcriptomic and metabolomic data are available. We demonstrate that IntLIM captures relevant tumor-specific gene-metabolite associations involved in known cancer-related pathways, including glutamine metabolism. Using IntLIM, we also uncover biologically relevant novel relationships that could be further tested experimentally.ConclusionsIntLIM provides a user-friendly, reproducible framework to integrate transcriptomic and metabolomic data and help interpret metabolomic data and uncover novel gene-metabolite relationships. The IntLIM R package is publicly available in GitHub (https://github.com/mathelab/IntLIM) and includes a user-friendly web application, vignettes, sample data and data/code to reproduce results.Electronic supplementary materialThe online version of this article (10.1186/s12859-018-2085-6) contains supplementary material, which is available to authorized users.
Developing cheap and stable electrocatalysts is considered the key factor to achieve the large-scale application of fuel cells. In this paper, three-dimensional (3D) porous Co-doped vanadium nitride (VN) nanosheet-assembled microflowers are prepared with a facile solvothermal approach followed by nitridation at 500 °C in NH. It is found that the microflower morphology and the Co doping both significantly enhance the oxygen reduction reaction (ORR) performance of the materials. Because the unique 3D porous structure provides higher specific surface area and more active sites as well as enriching the d electrons of V via doping, Co also improves the intrinsic activity of VN. Our optimal VCoN microflowers achieve a half-wave potential for the ORR of up to 0.80 V in 0.1 M KOH solution, which is almost comparable to that of commercial 20% Pt/C. More importantly, the catalysts show superior durability with little current decline (less than 12%) during chronoamperometric evaluation for over 25 000 s. These features make the VCoN microflowers attractive for fuel cell applications.
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