Exponential families are widely used in machine learning; they include many distributions in continuous and discrete domains (e.g., Gaussian, Dirichlet, Poisson, and categorical distributions via the softmax transformation). Distributions in each of these families have fixed support. In contrast, for finite domains, there has been recent works on sparse alternatives to softmax (e.g. sparsemax, $alpha$-entmax, and fusedmax) and corresponding losses, which have varying support. This paper expands that line of work in several directions: first, it extends $Omega$-regularized prediction maps and Fenchel-Young losses to arbitrary domains (possibly countably infinite or continuous). For linearly parametrized families, we show that minimization of Fenchel-Young losses is equivalent to moment matching of the statistics, generalizing a fundamental property of exponential families. When $Omega$ is a Tsallis negentropy with parameter $alpha$, we obtain deformed exponential families, which include $alpha$-entmax and sparsemax ($alpha$ = 2) as particular cases. For quadratic energy functions in continuous domains, the resulting densities are $beta$-Gaussians, an instance of elliptical distributions that contain as particular cases the Gaussian, biweight, triweight and Epanechnikov densities, and for which we derive closed-form expressions for the variance, Tsallis entropy, and Fenchel-Young loss. When $Omega$ is a total variation or Sobolev regularizer, we obtain a continuous version of the fusedmax. Finally, we introduce continuous-domain attention mechanisms, deriving efficient gradient backpropagation algorithms for $alpha in {1, 4/3, 3/2, 2}$. Using them, we demonstrate our sparse continuous distributions for attention-based audio classification and visual question answering, showing that they allow attending to time intervals and compact regions.
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