In the decoder of live video streaming, compressed video often suffers from block loss during recovery. The spatial error concealment utilizes the correlation between blocks in the current frame for the recovery of error image, without requiring information from other frames. Among many spatial error concealment algorithms, sparse representation mechanism further utilizes the sparsity of an image and achieve better recovery quality than pixel-wise interpolation mechanism. The current sparse representation algorithms still face challenges such as inaccurate selection of candidate subregions and parameter sensitivity of correlation model. Therefore, this paper studies the dual space regularization framework according to sparse representation, and focuses on optimizing such two stages as local region matching and local linear correlation modeling in this framework. We proposes a double-sparse spatial error concealment algorithm with adaptive threshold and α-ML kernel function. During the stage of local region matching, the proposed algorithm designs a local region matching method with adaptive threshold, which can flexibly adapt to the missing subregions with different characteristics, and provide more accurate observation space and potential space for dictionary construction and local linear correlation modeling. During the stage of local linear correlation modeling, the proposed algorithm utilizes a kernel ridge regression method with α-ML kernel function as the local linear correlation model. Compared with the Gaussian kernel function, the α-ML kernel function has low parameter sensitivity and good flexibility. Experimental results show that in typical block loss modes, the proposed algorithm outperforms other existing spatial error concealment algorithms in terms of recovery quality.