Properties of the Concrete distribution
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We examine properties of the Concrete (or Gumbel-softmax) distribution on the simplex. Using the natural vector space structure of the simplex, the Concrete distribution can be regarded as a transformation of the uniform distribution through a reflection and a location-scale transformation. The Fisher information is computed and the corresponding information metric is hyperbolic space. We explicitly give an explicit transformation of the parameters of the distribution to Poincaré half-space coordinates, which correspond to an orthogonal parameterization, and the Fisher-Rao geodesic distance is computed.
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