The main reason for Ground Speed Mini is to guard against low airspeed in windshear / microburst / CB activity.
Take the microburst case, you fly the approach at say a constant 130kts IAS and into an increasing headwind component, so the Gspd decreases at the same rate. As you come out of the positive shear and into the negative shear, the headwind rapidly changes to a tailwind and suddenly your IAS falls rapidly and you have BIG problems (a la Delta (?) Tristar 15 years ago or so into DFW (?)).
With Groundspeed Mini, your initial 130kts IAS would increase by an amount equal to the increase in headwind component, giving a constant groundspeed, maybe to say 170kts IAS. On coming out of the positive shear your IAS would reduce rapidly but from a higher value, back towards the datum of 130kts. Result aircraft safe(r).
Some problem areas have been found to exist with the A319 current software and it can cause some flap limiting speed issues and be fun when asked for 160 to 4d when you resume 'managed' speed at 4 miles for the speed to increase to 175 or so!
Hope this helps