Xiaomeng Ju

Hanchao Zhang, Xiaomeng Ju, Baoyi Shi et al.

This paper presents a new clustering algorithm for symmetric positive semi-definite (SPSD) matrices, called K-Tensors. The method identifies structured subsets of the SPSD cone characterized by common principal component (CPC) representations, where each subset corresponds to matrices sharing a common eigenstructure. Unlike conventional clustering approaches that rely on vectorization or transformations of SPSD matrices, thereby losing critical geometric and spectral information, K-Tensors introduces a divergence that respects the intrinsic geometry of SPSD matrices. This divergence preserves the shape and eigenstructure information and yields principal SPSD tensors, defined as a set of representative matrices that summarize the distribution of SPSD matrices. By exploring its theoretical properties, we show that the proposed clustering algorithm is self-consistent under mild distribution assumptions and converges to a local optimum. We demonstrate the use of the algorithm through an application to resting-state functional magnetic resonance imaging (rs-fMRI) data from the Human Connectome Project, where we cluster brain connectivity matrices to discover groups of subjects with shared connectivity structures.

Xiaomeng Ju, Matías Salibián-Barrera

Gradient boosting algorithms construct a regression predictor using a linear combination of ``base learners''. Boosting also offers an approach to obtaining robust non-parametric regression estimators that are scalable to applications with many explanatory variables. The robust boosting algorithm is based on a two-stage approach, similar to what is done for robust linear regression: it first minimizes a robust residual scale estimator, and then improves it by optimizing a bounded loss function. Unlike previous robust boosting proposals this approach does not require computing an ad-hoc residual scale estimator in each boosting iteration. Since the loss functions involved in this robust boosting algorithm are typically non-convex, a reliable initialization step is required, such as an L1 regression tree, which is also fast to compute. A robust variable importance measure can also be calculated via a permutation procedure. Thorough simulation studies and several data analyses show that, when no atypical observations are present, the robust boosting approach works as well as the standard gradient boosting with a squared loss. Furthermore, when the data contain outliers, the robust boosting estimator outperforms the alternatives in terms of prediction error and variable selection accuracy.