Feature Learning in Image Hierarchies using Functional Maximal Correlation

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

The Cross Density Kernel Function: A Novel Framework to Quantify Statistical Dependence for Random Processes

This paper proposes a novel multivariate definition of statistical dependence using a functional methodology inspired by Alfred R\'enyi. We define a new symmetric and self-adjoint cross density kernel through a recursive bidirectional statistical mapping between conditional densities of continuous random processes, which estimates their statistical dependence. Therefore, the kernel eigenspectrum is proposed as a new multivariate statistical dependence measure, and the formulation requires fewer assumptions about the data generation model than current methods. The measure can also be estimated from realizations. The proposed functional maximum correlation algorithm (FMCA) is applied to a learning architecture with two multivariate neural networks. The FMCA optimal solution is an equilibrium point that estimates the eigenspectrum of the cross density kernel. Preliminary results with synthetic data and medium size image datasets corroborate the theory. Four different strategies of applying the cross density kernel are thoroughly discussed and implemented to show the versatility and stability of the methodology, and it transcends supervised learning. When two random processes are high-dimensional real-world images and white uniform noise, respectively, the algorithm learns a factorial code i.e., the occurrence of a code guarantees that a certain input in the training set was present, which is quite important for feature learning