We present a novel learning-based method for generating optimal motion plans for high-dimensional motion planning problems. In order to cope with the curse of dimensional- ity, our method proceeds in a fashion similar to block co- ordinate descent in finite-dimensional optimization: at each iteration, the motion is optimized over a lower dimensional subspace while leaving the path fixed along the other dimen- sions. Naive implementations of such an idea can produce vastly suboptimal results. In this work, we show how a prof- itable set of directions in which to perform this dimensional descent procedure can be learned efficiently. We provide suf- ficient conditions for global optimality of dimensional de- scent in this learned basis, based upon the low-dimensional structure of the planning cost function. We also show how this dimensional descent procedure can easily be used for problems that do not exhibit such structure with monotonic convergence. We illustrate the application of our method to high dimensional shape planning and arm trajectory planning problems.