Biology-inspired joint distribution neurons based on Hierarchical Correlation Reconstruction allowing for multidirectional neural networks
This work addresses the problem of making artificial neural networks more biologically plausible and flexible for researchers in machine learning and neuroscience, though it appears incremental as it builds on existing concepts like KANs with extensions to joint distributions.
The paper tackles the qualitative superiority of biological neural networks over artificial ones by proposing novel neurons based on Hierarchical Correlation Reconstruction (HCR) that model joint distributions and enable multidirectional signal propagation, resulting in a network capable of propagating probability distributions and supporting alternative training methods like information bottleneck optimization.
Biological neural networks seem qualitatively superior (e.g. in learning, flexibility, robustness) to current artificial like Multi-Layer Perceptron (MLP) or Kolmogorov-Arnold Network (KAN). Simultaneously, in contrast to them: biological have fundamentally multidirectional signal propagation \cite{axon}, also of probability distributions e.g. for uncertainty estimation, and are believed not being able to use standard backpropagation training \cite{backprop}. There are proposed novel artificial neurons based on HCR (Hierarchical Correlation Reconstruction) allowing to remove the above low level differences: with neurons containing local joint distribution model (of its connections), representing joint density on normalized variables as just linear combination of $(f_\mathbf{j})$ orthonormal polynomials: $ρ(\mathbf{x})=\sum_{\mathbf{j}\in B} a_\mathbf{j} f_\mathbf{j}(\mathbf{x})$ for $\mathbf{x} \in [0,1]^d$ and $B\subset \mathbb{N}^d$ some chosen basis. By various index summations of such $(a_\mathbf{j})_{\mathbf{j}\in B}$ tensor as neuron parameters, we get simple formulas for e.g. conditional expected values for propagation in any direction, like $E[x|y,z]$, $E[y|x]$, which degenerate to KAN-like parametrization if restricting to pairwise dependencies. Such HCR network can also propagate probability distributions (also joint) like $ρ(y,z|x)$. It also allows for additional training approaches, like direct $(a_\mathbf{j})$ estimation, through tensor decomposition, or more biologically plausible information bottleneck training: layers directly influencing only neighbors, optimizing content to maximize information about the next layer, and minimizing about the previous to remove noise, extract crucial information.