In this paper we propose a sparse solution to the inverse problem of nonlinear dimensionality reduction (NLDR), which holds potential high-performance applications in data representation, compression, generation, and visualization. Firstly, the sparse solution model of the inverse problem of NLDR is established, which consists of four components: classical NLDR, sparse dictionary learning, NLDR embedding, and sparse NLDR reconstruction. Secondly, the special sparse solution to the inverse problem of isometric feature mapping (ISOMAP), a classical NDLR algorithm, is presented. ISOMAP embedding and sparse ISOMAP reconstruction algorithms are raised, and the alternating directions method of multipliers (ADMM) is adopted to resolve the minimization problem of the special sparse solution. Finally, it is revealed by the experimental results that, in the situation of very low dimensional representation, the proposed method is superior to the state of the art methods, such as discrete cosine transformation (DCT) and sparse representation (SR), in the reconstruction performance of image and video data.