  Mapping between discrete and continuous distributions is a difficult task and
many have had to resort to heuristical approaches. We propose a
tessellation-based approach that directly learns quantization boundaries in a
continuous space, complete with exact likelihood evaluations. This is done
through constructing normalizing flows on convex polytopes parameterized using
a simple homeomorphism with an efficient log determinant Jacobian. We explore
this approach in two application settings, mapping from discrete to continuous
and vice versa. Firstly, a Voronoi dequantization allows automatically learning
quantization boundaries in a multidimensional space. The location of boundaries
and distances between regions can encode useful structural relations between
the quantized discrete values. Secondly, a Voronoi mixture model has
near-constant computation cost for likelihood evaluation regardless of the
number of mixture components. Empirically, we show improvements over existing
methods across a range of structured data modalities.
