The least trimmed squares (LTS) is a reasonable formulation of robust regression whereas it suffers from high computational cost due to the nonconvexity and nonsmoothness of its objective function. The most frequently used FAST-LTS algorithm is particularly slow when a sparsity-inducing penalty such as the $\ell_1$ norm is added. This paper proposes a computationally inexpensive algorithm for the sparse LTS, which is based on the proximal gradient method with a reformulation technique. Proposed method is equipped with theoretical convergence preferred over existing methods. Numerical experiments show that our method efficiently yields small objective value.

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