Trying to understand a paper on ksvd algorithm (dictionary learning) by Elad, et al
By examples that use the atom $d_k$,i.e. those where $x_T^{k} (i)$ is nonzero, the authors are referring to the nonzero elements of the coefficient matrix $X$ because only nonzero elements play a role in multiplying two matrices. Since we seek to have a sparse representation, we only pay attention to the nonzero elements of $X$ and keep the zero elements intact.