Relaxing Bijectivity Constraints with Continuously Indexed Normalising Flows

Jul 12, 2020

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We show that the bijectivity of normalising flows means they are misspecified for modelling target densities whose support has a different topology from the prior. In this case, we prove that the flow must become arbitrarily close to noninvertible in order even to approximate the target closely. This result has implications for all flow-based models, and particularly residual flows (ResFlows), which explicitly control the Lipschitz constant of the bijection used. To address this, we propose continuously indexed flows (CIFs), which replace the single bijection used by normalising flows with a continuously indexed family of bijections, and which intuitively allow rerouting mass that would be misplaced by a single bijection. We prove that CIFs can exactly match the support of the target even when its topology differs from the prior, and obtain empirically better performance for a variety of models on a variety of benchmarks.

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The International Conference on Machine Learning (ICML) is the premier gathering of professionals dedicated to the advancement of the branch of artificial intelligence known as machine learning. ICML is globally renowned for presenting and publishing cutting-edge research on all aspects of machine learning used in closely related areas like artificial intelligence, statistics and data science, as well as important application areas such as machine vision, computational biology, speech recognition, and robotics. ICML is one of the fastest growing artificial intelligence conferences in the world. Participants at ICML span a wide range of backgrounds, from academic and industrial researchers, to entrepreneurs and engineers, to graduate students and postdocs.

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