The problem of robust sparse coding is considered. It is defined as finding linear reconstruction coefficients that minimize the sum of absolute values of the errors, instead of the more typically used sum of squares of the errors. This change lowers the influence of large errors and enhances the robustness of the solution to noise in the data. Sparsity is enforced by limiting the sum of absolute values of the coefficients. We present an algorithm that finds the path traced by the coefficients when the sparsity-inducing constraint is varied. The optimality conditions are derived and included in the algorithm to speed its execution. The proposed method is validated on the problem of robust face recognition.