A new algorithmic framework for sparse channel identification is proposed. Although the focus of this paper is on sparse underwater acoustic channels, this framework can be applied in any field where sequential noisy signal samples are obtained from a linear time-varying system. A suit of new algorithms is derived by minimizing a differentiable cost function that utilizes the underlying Riemannian structure of the channel as well as the L0-norm of the complex-valued channel taps. The sparseness effect of the proposed algorithms is successfully demonstrated by estimating a mobile shallow-water acoustic channel. The clear superiority of the new algorithms over state-of-the-art sparse adaptive algorithms is shown. Moreover, the proposed algorithms are employed by a channel-estimate-based decision-feedback equalizer (CEB DFE). These CEB DFE structures are compared with a direct-adaptation DFE (DA DFE), which is based on sparse and nonsparse adaptation. Our results confirm the improved error-rate performance of the new CEB DFEs when the channel is sparse.