I've been experiencing this problem as well, in my case with a 32 GB 4th generation Apple TV.
This may sound strange, but I think I've isolated the trigger for these random power-ons to the new "Siri Remote." The last two times my Apple TV started up on its own, I noticed a small, greyed out overlay on the upper right corner of my TV's screen indicating that the connection to the remote had been lost (this message disappears after a few seconds, so you may not have noticed it). I should note that on all of these occasions, the Siri Remote has been within a few feet of my Apple TV, so it's not a matter of the Remote being out of Bluetooth range.
Whether this is indeed the cause of my random power-ons or not, the trouble is that I don't think there's much I can do about it outside of waiting for Apple to release a software update. I've got my Remote charging now to see if its battery state has anything to do with this behavior and will report back if it appears to help at all (FWIW, my Apple TV turned on by itself when I plugged my Siri Remote into a live lightening cable), but I'm not holding my breath.
But if anyone else here has any thoughts on how to keep the Apple TV from losing its connection to the Siri Remote, I'd love to hear them!