  The successive projection algorithm (SPA) has been known to work well for
separable nonnegative matrix factorization (NMF) problems arising in
applications, such as topic extraction from documents and endmember detection
in hyperspectral images. One of the reasons is in that the algorithm is robust
to noise. Gillis and Vavasis showed in [SIAM J. Optim., 25(1), pp. 677-698,
2015] that a preconditioner can further enhance its noise robustness. The proof
rested on the condition that the dimension $d$ and factorization rank $r$ in
the separable NMF problem coincide with each other. However, it may be
unrealistic to expect that the condition holds in separable NMF problems
appearing in actual applications; in such problems, $d$ is usually greater than
$r$. This paper shows, without the condition $d=r$, that the preconditioned SPA
is robust to noise.
