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Semi-Discrete Normalizing Flows through Differentiable Tessellation
Ricky T. Q. Chen · Brandon Amos · Maximilian Nickel

Thu Dec 01 02:00 PM -- 04:00 PM (PST) @ Hall J #429

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.

Author Information

Ricky T. Q. Chen (FAIR Labs, Meta AI)
Brandon Amos (Facebook AI Research)
Maximilian Nickel (Facebook)

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