I'll bet the one in your garden is Chinese yam, Dioscorea oppositifolia (widely known as D. batatas). It's a terrible pest in many areas and is widely considered to be "officially" invasive in some states. This species forms large tuberous roots and small aerial tubers. I've never seen a variegated form however. This species and D. japonica are similar, although web photos of D. japonica show very upright conspicuous spikes of flowers. The ones you saw with big aerial tubers might have been D. bulbifer. According to the wikipedia account, the tubers of this one range from edible if boiled to toxic. It is probably not hardy northward but is feral in Florida. Jim McKenney