As is well known, traditional spectral clustering (SC) methods are developed based on the manifold assumption, namely, that two nearby data points in the high-density region of a low-dimensional data manifold have the same cluster label. But, for some high-dimensional and sparse data, such an assumption might be invalid. Consequently, the clustering performance of SC will be degraded sharply in this case. To solve this problem, in this paper, we propose a general spectral embedded framework, which embeds the true cluster assignment matrix for high-dimensional data into a nonlinear space by a predefined embedding function. Based on this framework, several algorithms are presented by using different embedding functions, which aim at learning the final cluster assignment matrix and a transformation into a low dimensionality space simultaneously. More importantly, the proposed method can naturally handle the out-of-sample extension problem. The experimental results on benchmark datasets demonstrate that the proposed method significantly outperforms existing clustering methods.