I did not have time to dig out the purple potatoes I've planted during the spring, now it's early winter, not much frost is expected in this area. Will they survive if I leave them in the soil and will they grow in the spring again? Shall I dig them out now? Shall I harvest them during next summer?
Yes, they will growth again. Potatoes are the winter house of potato plant (Solanum tuberosum). But you will have too many plants.
I think in spring, you should harvest them and replant some of them (as the original distance). If you will do this early in spring, you can eventually eat the rest of potatoes, you should check visually that they are still full and no green parts. This was a "traditional" way to store potatoes (and other vegetables) in winter: putting them under dirt/sable. But usually it is done inside, humid but not wet. So you should check. In any case, you can use all of them as seed potato, for next harvest.
The stem and leaves die at -2°C (28°F), so it's unlikely they will make it until spring. Also, soil humidity might help the tubers to rot, so in my opinion you won't have much to harvest after the winter passes. If the soil isn't frozen and if the stems are alive, you still have a chance to harvest them.
In the late 60's early 70's in fruitvale b.c. canada my family left potatoes in the ground and dug them all winter , even through 3 feet of snow . deep snow kept the ground from freezing, and the potatoes tasted great, unt5il they were all eaten . Just watch out for voles,those little grubbers. get the mouse traps.