More great news, this time about Open Problem 10.1. of Ten Lectures and Forty-Two Open Problems in the Mathematics of Data Science: Nicolas Boumal (who has contributed to this blog!) has made significant progress on this problem on his new paper.
I will defer the details to both the notes and Nicolas’ paper. In the language of Open Problem 10.1, Nicolas shows that the Projected Power Method (with a slight technical twist) converges to the desired solution when . The bottleneck of the analysis seems to be related to Open Problem 10.2. in the same notes (in a similar way to the connection here). I highly recommend taking a look at Nicolas’ paper.