I tend to agree with the previous poster. Even if unattended upgrades is ON I doubt that the PC will turn itself off (this isn't Windows XP after all
)
Have you overclocked? If so first thing to do is back off on the speed.
If the PC were rebooting I would suspect memory. In this case however I would first check hardware in this order:
1. Examine the mainboard capacitors. If any are domed or swollen out of a perfectly cylindrical shape then it's time for a new one. (which usually implies a new CPU and RAM if the MB is elderly)
2. Monitor CPU temperatures, either via lmsensors or else reboot and immediately select setup and the screen showing hardware info. If you're within 10 deg of the CPU limit (remember the CPU will be cooling as you reboot) then the problem may be here.
Is the CPU cooler dust free? You may need to remove it, remove the fan and clean out the radiator. Remember to use new thermal grease when you replace it.
3. At the hardware setup screen check the voltages. Tolerances on the lower voltages are very tight.
4. Is the PSU dust free? Check the holes inside the case for fluff. Look through the fan blades. Use a vacuum cleaner to clean it out. Don't open the PSU unless you know what you're doing. Linux users are valuable and we like to keep them
.
5.Check all the air inlets to the case - this includes the one in front, hidden by the front cover. Clean if necessary.
One of these is very likely to be the cause of the problem.
Good luck.