Here's a graphic of Antarctic ice MELT ending in 2009 by Tedesco and Monaghan. Note that because it is melt, the decline means that more ice is NOT melting.
Here is the extent of sea ice in the southern hemisphere, as measured by satellite. The increase is highly significant (at the .0001) level. These are from Cryosphere Today and are updated every few days.
Then there are the data from the GRACE satellite, which measures the gravity anomaly (thicker ice means more gravity). This is only from 2002 to 2008 and shows a net loss of ice from Antarctica if 87+/- 30 billion tons per year. That's a very small amount, actually, and raises sea level by the paltry amount of .25 millimeters per year. This works out to a grand total of ONE INCH PER CENTURY of sea level rise.
Given the fact that the melt--which is measured at the surface--is decreasing, but that there is also a (small) net loss of ice measured by GRACE would mean that the loss is likely to be originating from underneath the ice shelves that abutt the land. This results in a slight increase in the flow of the great ice sheets which is where the net loss is measured by GRACE. which is what is implied in the paper related to the article you sent. Keep in mind that it is VERY small--in fact, too small to be able to measure reliably.
Further evidence for the melt-from-underneath hypothesis can be found in the satellite-sensed temperature data for Antarctica and surrounding areas. You can see that there is some (very slight) warming in the ocean near Antarctica--even as much of the land itself is cooling.
The synthesis is that it is likely that there is some slight melting from undeneath, but that the sea ice--which does not affect sea level--is also increasing. That may be because snowfall appears to be increasing over the land areas that are adjacent to this warming, which is what would be expected. After all, it's still way below freezing, and warming the water a teense means there is more water vapor which is going to fall as snow when condensed. All of the numbers, except for the snowfall increase over the Antarctic peninsula (the point which juts out towards South America) are very, very small.