Nonconvex variants of support vector machines (SVMs) have been developed for various purposes. For example, robust SVMs attain robustness to outliers by using a nonconvex loss function, while extended [Formula: see text]-SVM (E[Formula: see text]-SVM) extends the range of the hyperparameter by introducing a nonconvex constraint. Here, we consider an extended robust support vector machine (ER-SVM), a robust variant of E[Formula: see text]-SVM. ER-SVM combines two types of nonconvexity from robust SVMs and E[Formula: see text]-SVM. Because of the two nonconvexities, the existing algorithm we proposed needs to be divided into two parts depending on whether the hyperparameter value is in the extended range or not. The algorithm also heuristically solves the nonconvex problem in the extended range. In this letter, we propose a new, efficient algorithm for ER-SVM. The algorithm deals with two types of nonconvexity while never entailing more computations than either E[Formula: see text]-SVM or robust SVM, and it finds a critical point of ER-SVM. Furthermore, we show that ER-SVM includes the existing robust SVMs as special cases. Numerical experiments confirm the effectiveness of integrating the two nonconvexities.

BACKGROUND: In high-throughput experimental biology, it is widely acknowledged that while expression levels measured at the levels of transcriptome and the corresponding proteome do not, in general, correlate well, messenger RNA levels are used as convenient proxies for protein levels. Our interest is in developing data-driven computational models that can bridge the gap between these two levels of measurement at which different mechanisms of regulation may act on different molecular species causing any observed lack of correlations. To this end, we build data-driven predictors of protein levels using mRNA levels and known proxies of translation efficiencies as covariates. Previous work showed that in such a setting, outliers with respect to the model are reliable candidates for post-translational regulation. RESULTS: Here, we present and compare two novel formulations of deriving a protein concentration predictor from which outliers may be extracted in a systematic manner. The first approach, outlier rejecting regression, allows explicit specification of a certain fraction of the data as outliers. In a regression setting, this is a non-convex optimization problem which we solve by deriving a difference of convex functions algorithm (DCA). With post-translationally regulated proteins, one expects their concentrations to be affected primarily by disruption of protein stability. Our second algorithm exploits this observation by minimizing an asymmetric loss using quantile regression and extracts outlier proteins whose measured concentrations are lower than what a genome-wide regression would predict. We validate the two approaches on a dataset of yeast transcriptome and proteome. Functional annotation check on detected outliers demonstrate that the methods are able to identify post-translationally regulated genes with high statistical confidence.

Financial risk measures have been used recently in machine learning. For example, ν-support vector machine ν-SVM) minimizes the conditional value at risk (CVaR) of margin distribution. The measure is popular in finance because of the subadditivity property, but it is very sensitive to a few outliers in the tail of the distribution. We propose a new classification method, extended robust SVM (ER-SVM), which minimizes an intermediate risk measure between the CVaR and value at risk (VaR) by expecting that the resulting model becomes less sensitive than ν-SVM to outliers. We can regard ER-SVM as an extension of robust SVM, which uses a truncated hinge loss. Numerical experiments imply the ER-SVM's possibility of achieving a better prediction performance with proper parameter setting.