Bo Hu

Bo Hu, Yuheng Bu, José C. Príncipe

This paper proposes the Hierarchical Functional Maximal Correlation Algorithm (HFMCA), a hierarchical methodology that characterizes dependencies across two hierarchical levels in multiview systems. By framing view similarities as dependencies and ensuring contrastivity by imposing orthonormality, HFMCA achieves faster convergence and increased stability in self-supervised learning. HFMCA defines and measures dependencies within image hierarchies, from pixels and patches to full images. We find that the network topology for approximating orthonormal basis functions aligns with a vanilla CNN, enabling the decomposition of density ratios between neighboring layers of feature maps. This approach provides powerful interpretability, revealing the resemblance between supervision and self-supervision through the lens of internal representations.

Bo Hu, Shujian Yu, Jose C. Principe

Rényi's information provides a theoretical foundation for tractable and data-efficient non-parametric density estimation, based on pair-wise evaluations in a reproducing kernel Hilbert space (RKHS). This paper extends this framework to parametric probabilistic modeling, motivated by the fact that Rényi's information can be estimated in closed-form for Gaussian mixtures. Based on this special connection, a novel generative model framework called the structured generative model (SGM) is proposed that makes straightforward optimization possible, because costs are scale-invariant, avoiding high gradient variance while imposing less restrictions on absolute continuity, which is a huge advantage in parametric information theoretic optimization. The implementation employs a single neural network driven by an orthonormal input appended to a single white noise source adapted to learn an infinite Gaussian mixture model (IMoG), which provides an empirically tractable model distribution in low dimensions. To train SGM, we provide three novel variational cost functions, based on Rényi's second-order entropy and divergence, to implement minimization of cross-entropy, minimization of variational representations of $f$-divergence, and maximization of the evidence lower bound (conditional probability). We test the framework for estimation of mutual information and compare the results with the mutual information neural estimation (MINE), for density estimation, for conditional probability estimation in Markov models as well as for training adversarial networks. Our preliminary results show that SGM significantly improves MINE estimation in terms of data efficiency and variance, conventional and variational Gaussian mixture models, as well as the performance of generative adversarial networks.