MOTIVATION: Marker genes, defined as genes that are expressed primarily in a single-cell type, can be identified from the single-cell transcriptome; however, such data are not always available for the many uses of marker genes, such as deconvolution of bulk tissue. Marker genes for a cell type, however, are highly correlated in bulk data, because their expression levels depend primarily on the proportion of that cell type in the samples. Therefore, when many tissue samples are analyzed, it is possible to identify these marker genes from the correlation pattern. RESULTS: To capitalize on this pattern, we develop a new algorithm to detect marker genes by combining published information about likely marker genes with bulk transcriptome data in the form of a semi-supervised algorithm. The algorithm then exploits the correlation structure of the bulk data to refine the published marker genes by adding or removing genes from the list. AVAILABILITY AND IMPLEMENTATION: We implement this method as an R package markerpen, hosted on CRAN (https://CRAN.R-project.org/package=markerpen). SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

While machine learning has proven to be a powerful data-driven solution to many real-life problems, its use in sensitive domains has been limited due to privacy concerns. A popular approach known as differential privacy offers provable privacy guarantees, but it is often observed in practice that it could substantially hamper learning accuracy. In this paper we study the learnability (whether a problem can be learned by any algorithm) under Vapnik's general learning setting with differential privacy constraint, and reveal some intricate relationships between privacy, stability and learnability. In particular, we show that a problem is privately learnable if an only if there is a private algorithm that asymptotically minimizes the empirical risk (AERM). In contrast, for non- private learning AERM alone is not sufficient for learnability. This result suggests that when searching for private learning algorithms, we can restrict the search to algorithms that are AERM. In light of this, we propose a conceptual procedure that always finds a universally consistent algorithm whenever the problem is learnable under privacy constraint. We also propose a generic and practical algorithm and show that under very general conditions it privately learns a wide class of learning problems. Lastly, we extend some of the results to the more practical $(\epsilon,\delta)$-differential privacy and establish the existence of a phase-transition on the class of problems that are approximately privately learnable with respect to how small $\delta$ needs to be.

We propose a novel convex relaxation of sparse principal subspace estimation based on the convex hull of rank-$d$ projection matrices (the Fantope). The convex problem can be solved efficiently using alternating direction method of multipliers (ADMM). We establish a near-optimal convergence rate, in terms of the sparsity, ambient dimension, and sample size, for estimation of the principal subspace of a general covariance matrix without assuming the spiked covariance model. In the special case of $d=1$, our result implies the near- optimality of DSPCA even when the solution is not rank 1. We also provide a general theoretical framework for analyzing the statistical properties of the method for arbitrary input matrices that extends the applicability and provable guarantees to a wide array of settings. We demonstrate this with an application to Kendall's tau correlation matrices and transelliptical component analysis.