Paper Corrections for Butterfly Factorization
This post lists corrections for the following paper.
 Y. Li, H. Yang, E. Martin, K. Ho and L. Ying, Butterfly Factorization, Multiscale Modeling and Simulation 13:2, 714732 (2015). [PDF, journal]
So far, I have found the following mistakes or typos.

Page 719, Section 2.2, last sentence of the first paragraph “… hence requires only $O(r(m+n))$ operations and memory.” should be “… hence requires only $O(r^2(m+n))$ operations and $O(r(m+n))$ memory …”

Page 718, Section 2.1, SVD via random matrixvector multiplication. The formula for the middle matrix $ M = Q^\star_{col} Z Q_{row} $ in the third step should not be used in the middle level factorization directly, since the matrix Z could not be applied to basis $ Q_{row} $ or $ Q_{col} $ again. The actual way to implement is that $ M = \left( R^\star_{row}Q_{col} \right)^\dagger R^\star_{row} Z R_{col} \left( Q^\star_{row} R_{col} \right)^\dagger $. Such the case $ Z R_{col} $ can be reused and all the computations are in the local block.
Yingzhou Li