  We consider classification problems in which the label space has structure. A
common example is hierarchical label spaces, corresponding to the case where
one label subsumes another (e.g., animal subsumes dog). But labels can also be
mutually exclusive (e.g., dog vs cat) or unrelated (e.g., furry, carnivore). To
jointly model hierarchy and exclusion relations, the notion of a HEX (hierarchy
and exclusion) graph was introduced in [7]. This combined a conditional random
field (CRF) with a deep neural network (DNN), resulting in state of the art
results when applied to visual object classification problems where the
training labels were drawn from different levels of the ImageNet hierarchy
(e.g., an image might be labeled with the basic level category "dog", rather
than the more specific label "husky"). In this paper, we extend the HEX model
to allow for soft or probabilistic relations between labels, which is useful
when there is uncertainty about the relationship between two labels (e.g., an
antelope is "sort of" furry, but not to the same degree as a grizzly bear). We
call our new model pHEX, for probabilistic HEX. We show that the pHEX graph can
be converted to an Ising model, which allows us to use existing off-the-shelf
inference methods (in contrast to the HEX method, which needed specialized
inference algorithms). Experimental results show significant improvements in a
number of large-scale visual object classification tasks, outperforming the
previous HEX model.
