The density peaks (DP) algorithm for cluster analysis, introduced by Rodriguez and Laio in 2014, has proven empirically competitive or superior in multiple aspects to other contemporary clustering algorithms. Yet, it suffers from certain drawbacks and limitations when used for clustering high-dimensional data. We introduce SD-DP, the sparse dual version of DP. While following the DP principle and maintaining its appealing properties, we establish a sparse descriptor of local density as a robust representation. By analyzing and exploiting the consequential properties, we are able to use sparse graph-matrix expressions and operations throughout the clustering process. As a result, SD-DP has provably linear-scaling computation complexity under practical conditions. We show, with experimental results on several real-world high-dimensional datasets, that SD-DP outperforms DP in robustness, accuracy, self-governance, and efficiency.