In this paper, we contribute to solving a threefold problem: outlier rejection, true model reasoning and parameter estimation with a unified optimization modeling. To this end, we first pose this task as a sparse subspace recovering problem, to search a maximum of independent bases under an over-embedded data space. Then we convert the objective into a continuous optimization paradigm that estimates sparse solutions for both bases and errors. Wherein a fast and robust solver is proposed to accurately estimate the sparse subspace parameters and error entries, which is implemented by a proximal approximation method under the alternating optimization framework with the ``optimal'' sub-gradient descent. Extensive experiments regarding known and unknown model fitting on synthetic and challenging real datasets have demonstrated the superiority of our method against the state-of-the-art. We also apply our method to multi-class multi-model fitting and loop closure detection, and achieve promising results both in accuracy and efficiency. Code is released at: https://github.com/StaRainJ/DSP.