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We investigate penalized maximum log-likelihood estimation for exponentialfamily distributions whose natural parameter resides in a reproducing kernelHilbert space. Key to our approach is a novel technique, doubly dual embedding,that avoids computation of the partition function. This technique also allowsthe development of a flexible sampling strategy that amortizes the cost ofMonte-Carlo sampling in the inference stage. The resulting estimator can beeasily generalized to kernel conditional exponential families. We establish aconnection between kernel exponential family estimation and MMD-GANs, revealinga new perspective for understanding GANs. Compared to the score matching basedestimators, the proposed method improves both memory and time efficiency whileenjoying stronger statistical properties, such as fully capturing smoothness inits statistical convergence rate while the score matching estimator appears tosaturate. Finally, we show that the proposed estimator empirically outperformsstate-of-the-art

Kernel Exponential Family Estimation via Doubly Dual Embedding