Isometric feature mapping (ISOMAP) is one of the classical methods of nonlinear dimensionality reduction (NLDR) and seeks for low dimensional (LD) structure of high dimensional (HD) data. However, the inverse problem of ISOMAP has never been investigated, which recovers the HD sample from the related LD sample, and its application prospect in data representation, generation, compression and visualization will be very brilliant. Because the inverse problem of ISOMAP is ill-posed and undetermined, the sparsity of HD data is employed to reconstruct the HD data from the corresponding LD data. The theoretical architecture of sparse reconstruction of ISOMAP representation comprises the original ISOMAP algorithm, learning algorithm of sparse dictionary, general ISOAMAP embedding algorithm and sparse ISOMAP reconstruction algorithm. The sparse ISOMAP reconstruction algorithm is an optimization problem with sparse priors of the HD data, which is resolved by the alternating directions method of multipliers (ADMM). It is uncovered from the experimental results that, in the case of very LD ISOMAP representation, the proposed method outperforms the state-of-the-art methods, such as discrete cosine transformation (DCT) and sparse representation (SR), in the reconstruction performance of signal, image and video data.