We consider statistical inference in a semi-low-dimensional approach to the analysis of high-dimensional data. The relationship between this semi-low-dimensional approach and regularized estimation of high-dimensional objects is parallel to the more familiar one between semiparametric analysis and nonparametric estimation. Low-dimensional projection methods are used to correct the bias of regularized high-dimensional estimators, leading to efficient point and interval estimation. Bootstrap can be used to carry out simultaneous inference. Only a small fraction of labelled data are needed in a semisupervised setting. Examples include regression and graphical models for continuous and binary data.