Human activity analysis in videos has increasingly attracted attention in computer vision research with the massive number of videos now accessible online. Although many recognition algorithms have been reported recently, activity representation is challenging. Recently, manifold regularized sparse coding has obtained promising performance in action recognition, because it simultaneously learns the sparse representation and preserves the manifold structure. In this paper, we propose a generalized version of Laplacian regularized sparse coding for human activity recognition called   $p$  -Laplacian regularized sparse coding (pLSC). The proposed method exploits   $p$   -Laplacian regularization to preserve the local geometry. The   $p$   -Laplacian is a nonlinear generalization of standard graph Laplacian and has tighter isoperimetric inequality. As a result, pLSC provides superior theoretical evidence than standard Laplacian regularized sparse coding with a proper   $p$  . We also provide a fast iterative shrinkage-thresholding algorithm for the optimization of pLSC. Finally, we input the sparse codes learned by the pLSC algorithm into support vector machines and conduct extensive experiments on the unstructured social activity attribute dataset and human motion database (HMDB51) for human activity recognition. The experimental results demonstrate that the proposed pLSC algorithm outperforms the manifold regularized sparse coding algorithms including the standard Laplacian regularized sparse coding algorithm with a proper    $p$  .
['Weifeng Liu', 'Zheng-Jun Zha', 'Yanjiang Wang', 'Ke Lu', 'Dacheng Tao']
$p$ -Laplacian Regularized Sparse Coding for Human Activity Recognition