  We study algorithms for the Schatten-$p$ Low Rank Approximation (LRA)
problem. First, we show that by using fast rectangular matrix multiplication
algorithms and different block sizes, we can improve the running time of the
algorithms in the recent work of Bakshi, Clarkson and Woodruff (STOC 2022). We
then show that by carefully combining our new algorithm with the algorithm of
Li and Woodruff (ICML 2020), we can obtain even faster algorithms for
Schatten-$p$ LRA.
  While the block-based algorithms are fast in the real number model, we do not
have a stability analysis which shows that the algorithms work when implemented
on a machine with polylogarithmic bits of precision. We show that the LazySVD
algorithm of Allen-Zhu and Li (NeurIPS 2016) can be implemented on a floating
point machine with only logarithmic, in the input parameters, bits of
precision. As far as we are aware, this is the first stability analysis of any
algorithm using $O((k/\sqrt{\varepsilon})\text{poly}(\log n))$ matrix-vector
products with the matrix $A$ to output a $1+\varepsilon$ approximate solution
for the rank-$k$ Schatten-$p$ LRA problem.
