I'm not an expert on the genetic history of East Asian populations, but this might just simply mean that even though typical members of each population have 100% of their DNA from their native region, what geneticists and anthropologists probably mean when they say that Polynesians have an East Asian link is that when it comes to other populations outside of Polynesia, their closest immediate relatives are people from the East Asia region.

As an analogy, it's possible for someone who's Spanish to have 100% of their DNA coming from the Iberian Peninsula, but chances are when one goes back far enough, their closest relatives outside of the Iberian Peninsula are people from the Italy/Greece region. As a matter of fact, advancements in DNA testing and identification of specific genetic markers have allowed for further subdivision of "DNA Regions" if you will, such as 6 or so regions in what used to "West Africa," Italy/Greece and Iberian Peninsula for what used to be called "Southern Europe," Great Britain and Ireland for what used to be called "British Isles," etc.

I predict that within a year or so, East Asia will also be subdivided into such possible groups as Han Chinese, Japanese, etc. as more people from these countries get their DNA tested. Right now it seems like it's mostly people of European/Mediterranean heritage who have had their DNA analyzed, hence the ability to have identified a larger number of distinct regions within this general area.