This is done as follows (Deerwester et al., 1990;Bellegarda, 2000), again with the objective of minimizing the reconstruction error.
first, note that since U is column-orthonormal, (1) implies that if we notionally index a new document by p, we proceed by forming a new column (document) vector W p using the standard term-weighting, and then find its LSA-space representation V p using (2).