  This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an $n
\times q$ rank-$r$ matrix $X^*$ from $y_k = |A_k^\top x^*_k|$, $k=1, 2,..., q$,
when each $y_k$ is an m-length vector containing independent phaseless linear
projections of $x^*_k$. The different matrices $A_k$ are i.i.d. and each
contains i.i.d. standard Gaussian entries. We obtain an improved guarantee for
AltMinLowRaP, which is an Alternating Minimization solution to LRPR that was
introduced and studied in our recent work. As long as the right singular
vectors of $X^*$ satisfy the incoherence assumption, we can show that the
AltMinLowRaP estimate converges geometrically to $X^*$ if the total number of
measurements $mq \gtrsim nr^2 (r + \log(1/\epsilon))$. In addition, we also
need $m \gtrsim max(r, \log q, \log n)$ because of the specific asymmetric
nature of our problem. Compared to our recent work, we improve the sample
complexity of the AltMin iterations by a factor of $r^2$, and that of the
initialization by a factor of $r$. We also extend our result to the noisy case;
we prove stability to corruption by small additive noise.
