Yes you can! In fact that's my preferred method of doing it . Any sweet potato leaves are edible but not all are worth eating so this way I know what I'm getting when I plant them :)

The sweet potato is very easy to plant through cuttings that is sold in Asian markets. 1newport is right, some sweet potato are cultivated just for the leaves. My method is the easiest because I just plant the stem, the leaves has been stripped, directly to the soil. I didn't even have to prepare the soil with additives; it was planted in clay soil and it will do fine. Keep the soil moist and the cuttings will root easily and form new leaves. I did realize later that adding mulch is a huge benefit--it keeps the soil moist and prevents the dirt off the leaves when watering.