What we would really like to say is that the best rank one approximation B (1) (under some norm) of B is ∝ P (w i ) P (w i−1 ).
while this statement is not true under the 2 norm, it is true under generalized KL divergence (Lee and Seung, 2001): generalized KL divergence preserves row and column sums: if M (κ) is the best rank κ approximation of M under gKL then the row sums and column sums of M (κ) and M are equal (Ho and Van Dooren, 2008).