2021/08/05 by António Farinhas, Wilker Aziz, Farinhas, António +5
Computer Science · Mathematics · #Adversarial Robustness in Machine Learning #Applied mathematics #Artificial intelligence #Artificial neural network #Computer science #Differentiable function #Domain Adaptation and Few-Shot Learning #FOS: Computer and information sciences #Generative Adversarial Networks and Image Synthesis #Latent variable #MNIST database #Machine Learning (cs.LG) #Mathematics #Probability distribution #Pure mathematics #Random variable #Statistics #Theoretical computer science #cs.LG
paper · pdf · doi:10.48550/arxiv.2108.02658
Accepted for oral presentation at ICLR 2022
openalex publication_date 2021/08/05 · arxiv created 2022/02/11 · arxiv updated 2022/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Neural networks and other machine learning models compute continuous representations, while humans communicate mostly through discrete symbols. Reconciling these two forms of communication is desirable for generating human-readable interpretations or learning discrete latent variable models, while maintaining end-to-end differentiability. Some existing approaches (such as the Gumbel-Softmax transformation) build continuous relaxations that are discrete approximations in the zero-temperature limit, while others (such as sparsemax transformations and the Hard Concrete distribution) produce discrete/continuous hybrids. In this paper, we build rigorous theoretical foundations for these hybrids, which we call "mixed random variables." Our starting point is a new "direct sum" base measure defined on the face lattice of the probability simplex. From this measure, we introduce new entropy and Kullback-Leibler divergence functions that subsume the discrete and differential cases and have interpretations in terms of code optimality. Our framework suggests two strategies for representing and sampling mixed random variables, an extrinsic ("sample-and-project") and an intrinsic one (based on face stratification). We experiment with both approaches on an emergent communication benchmark and on modeling MNIST and Fashion-MNIST data with variational auto-encoders with mixed latent variables.