A patch was provided on the uptstreamed bug which resolves the issue. The issue seems to be that the 82801H (8086:284b) is really a rebranded mitac card but the driver doesn’t pick this up properly. This patch is included in Ubuntu now, but you have to append the following to your /etc/modprobe.d/alsa-base:
options snd-hda-intel model=mitac
For those on older Ubuntu versions like Hardy, you’ll need to hand-patch your alsa-driver to gain the support. See the upstream bug at https://bugtrack.alsa-project.org/alsa-bug/view.php?id=3886 for details.