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Summary The fused lasso, also known as total-variation denoising, is a locally adaptive function estimator over a regular grid of design points. In this article, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to a method for nonparametric regression. This approach, which we call the $K$-nearest-neighbours fused lasso, involves computing the $K$-nearest-neighbours graph of the design points and then performing the fused lasso over this graph. We show that this procedure has a number of theoretical advantages over competing methods: specifically, it inherits local adaptivity from its connection to the fused lasso, and it inherits manifold adaptivity from its connection to the $K$-nearest-neighbours approach. In a simulation study and an application to flu data, we show that excellent results are obtained. For completeness, we also study an estimator that makes use of an $\epsilon$-graph rather than a $K$-nearest-neighbours graph and contrast it with the $K$-nearest-neighbours fused lasso. 

We consider the supervised classification setting, in which the data consist of p features measured on n observations, each of which belongs to one of K classes. Linear discriminant analysis (LDA) is a classical method for this problem. However, in the high dimensional setting where p≫n, LDA is not appropriate for two reasons. First, the standard estimate for the within-class covariance matrix is singular, and so the usual discriminant rule cannot be applied. Second, when p is large, it is difficult to interpret the classification rule that is obtained from LDA, since it involves all p features. We propose penalized LDA, which is a general approach for penalizing the discriminant vectors in Fisher’s discriminant problem in a way that leads to greater interpretability. The discriminant problem is not convex, so we use a minorization–maximization approach to optimize it efficiently when convex penalties are applied to the discriminant vectors. In particular, we consider the use of L1 and fused lasso penalties. Our proposal is equivalent to recasting Fisher’s discriminant problem as a biconvex problem. We evaluate the performances of the resulting methods on a simulation study, and on three gene expression data sets. We also survey past methods for extending LDA to the high dimensional setting and explore their relationships with our proposal.

 

We propose covariance-regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse covariance matrix of the features to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing the log-likelihood of the data, under a multivariate normal model, subject to a penalty; it is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyse gene expression data sets with multiple class and survival outcomes.

 

Brain networks are increasingly characterized at different scales, including summary statistics, community connectivity, and individual edges. While research relating brain networks to behavioral measurements has yielded many insights into brain‐phenotype relationships, common analytical approaches only consider network information at a single scale. Here, we designed, implemented, and deployed Multi‐Scale Network Regression (MSNR), a penalized multivariate approach for modeling brain networks that explicitly respects both edge‐ and community‐level information by assuming a low rank and sparse structure, both encouraging less complex and more interpretable modeling. Capitalizing on a large neuroimaging cohort(n= 1, 051), we demonstrate that MSNR recapitulates interpretable and statistically significant connectivity patterns associated with brain development, sex differences, and motion‐related artifacts. Compared to single‐scale methods, MSNR achieves a balance between prediction performance and model complexity, with improved interpretability. Together, by jointly exploiting both edge‐ and community‐level information, MSNR has the potential to yield novel insights into brain‐behavior relationships.