Algorithm 1 Line search on Riemannian manifold

Input: A set of data = {s1, …, sn} on the manifold , coefficients C ∈ ℝn×m and initial dictionary atoms $a_1^0,\dots ,a_m^0$.

Output: The optimal dictionary atoms ( $a_1^{\ast },\dots ,a_m^{\ast }$) that minimize the cost function f(a1, …, am). Set scalars α > 0, β, σ ∈ (0, 1) and initialize k = 0. Compute grad $f(a_1^k,\dots ,a_m^k)=(\frac{\partial f(a_1^k)}{\partial a_1},\dots ,\frac{\partial f(a_m^k)}{\partial a_m})$ Pick, ${\eta }^k=({\eta }_1^k,\dots ,{\eta }_m^k)=-\text{grad}$ f, where ${\eta }_i^k\in T_{a_i^k}\mathcal{M}$. Find the smallest t such that $f({exp}_{a_1^k}(\alpha {\beta }^t{\eta }_1^k),\dots ,{exp}_{a_m^k}(\alpha {\beta }^t{\eta }_m^k))\le f(a_1^k,\dots ,a_m^k)-\sum _{i=1}^m\sigma \alpha {\beta }^t{\Vert {\eta }_i^k\Vert }_{a_i^k}$. Set, $a_i^{k+1}={exp}_{a_i^k}(\alpha {\beta }^t{\eta }_i^k),i=1,\dots ,m$. Stop if f does not change much, otherwise set k = k + 1 and go back to step 2.