If the router's actually rebooting, then it's either receiving a remote firmware update, or it's programmed to reboot for some reason, or there's a fault (power supply dropping out?).
If the line is dropping sync, that could be initiated by the other end. BT wholesale lines have something called Dynamic Line Management where they automagically re-sync with different parameters in order to optimise the stability/performance of the line. If it's re-syncing with different throughput and margins, that's probably what's doing it. It should settle down eventually, unless the properties of the line are changing (eg. we used[1] to get a spate of DLM-induced re-syncs at the start and end of university term, when all the students and their electronics moved out/in.)
Alternatively, the line could be losing sync for some other reason. Eg. a dodgy thermostat might be providing a burst of RF noise every day at about the same time and causing the line to drop.
If PPP is dropping but the line is staying in sync, that's likely to be jibble at your ISPs end. LNSes rebooting or similar.
[1] We're now on a TalkTalk wholesale line, where I can nail the parameters at a known-good value.