Clearly, (11) is a special case with L = 1.
with the definition of Ω in the previous section, the dimension of Ω x is d = O(n) which can be difficult to work with when n m. Proposition 4.1 allows us to reduce the dimension as long as the final matrix retains the form in (3) and has rank m. we can multiply Ω by any rank-m projection matrix Π ∈ R d×m on the right side: if Ω satisfies the properties in Proposition 4.1, then so does ΩΠ with m-dimensional rows Since rank(Ω) = m, a natural choice of Π is the projection onto the best-fit m-dimensional subspace of the row space of Ω.