© 2020, Springer Science+Business Media, LLC, part of Springer Nature. The past several years have witnessed increasing research interest on covariance-based feature representation. Originally proposed as a region descriptor, it has now been used as a general representation in various recognition tasks, demonstrating promising performance. However, covariance matrix has some inherent shortcomings such as singularity in the case of small sample, limited capability in modeling complicated feature relationship, and a single, fixed form of representation. To achieve better recognition performance, this paper argues that more capable and flexible symmetric positive definite (SPD)-matrix-based representation shall be explored, and this is attempted in this work by exploiting prior knowledge of data and nonlinear representation. Specifically, to better deal with the issues of small number of feature vectors and high feature dimensionality, we propose to exploit the structure sparsity of visual features and exemplify sparse inverse covariance estimate as a new feature representation. Furthermore, to effectively model complicated feature relationship, we propose to directly compute kernel matrix over feature dimensions, leading to a robust, flexible and open framework of SPD-matrix-based representation. Through theoretical analysis and experimental study, the proposed two representations well demonstrate their advantages over the covariance counterpart in skeletal human action recognition, image set classification and object classification tasks.