Activation Functions, Statistics and Learning of Higher-Order Interactions in Restricted Boltzmann Machines

Restricted Boltzmann machines (RBMs) represent data by shaping an energy landscape over visible and hidden configurations, but their discriminative use is fragile under out-of-distribution (OOD) inputs: samples outside the training distribution can be absorbed into one of the learned class basins rather than rejected. Here, we analyze this failure mode through the spectrum of the induced visible--visible interaction $J=WW^{T}$, where $W$ is the visible--hidden weight matrix. Relative to a Marchenko--Pastur random-matrix reference, conventional training spreads spectral weight into many weak, bulk-compatible directions, increasing the effective rank of $J$. When auxiliary random binary images are assigned to a rejection label during training, the learned interaction undergoes effective-rank collapse: weak bulk-like modes are depleted, spectral weight concentrates into fewer dominant eigendirections, and the effective rank of $J$ approaches that of the empirical data covariance matrix. The resulting RBM rejects structured OOD image datasets while preserving MNIST classification accuracy, showing that random auxiliary exposure can reshape both the interaction spectrum and the free-energy landscape of an energy-based classifier.

Learning in neural systems arises from synaptic changes that reshape the representations underlying behavior. While low-rank recurrent neural networks (RNNs) have emerged as a powerful framework for linking connectivity to function, a theoretical understanding of their learning process remains elusive. Here, we extend the low-rank framework from activity to learning by deriving gradient-descent dynamics directly in a reduced overlap space. We formulate a closed-form, low-dimensional system of ODEs that governs learning in this space, exact for linear RNNs and asymptotically exact for nonlinear RNNs in the large-$N$ Gaussian limit. Central to our analysis is a distinction between two classes of overlaps: loss-visible overlaps, which fully determine network activity, output, and loss, and loss-invisible overlaps, which do not affect function but are required to describe learning. We illustrate the consequences of this decomposition through two phenomena. First, we show that learning can serve as a perturbation that exposes differences in connectivity between functionally equivalent networks. Second, we show that loss-invisible overlaps can act as memory variables that encode training history, and characterize the conditions under which this occurs. Finally, we present several testable predictions for biological learning experiments derived from our theory.

Deep neural networks exhibit regular macroscopic behavior despite highly nonlinear dynamics in vast parameter spaces. We develop a statistical-mechanical description of learning directly in function space, treating parameter configurations as microscopic realizations and functions with their dynamical operators as macroscopic variables. For mean-squared loss, the exact error dynamics are governed by the learning operator $M=JJ^\ast$. Combining the dynamical Boltzmann weight of the conditional stochastic dynamics with the parameter-space density of states, whose local curvature defines a statistical operator $B$, and integrating over local fluctuations yields $\Phi_{\mathrm{fluc}}(M;B)=\frac{\sigma_\xi^2}{2}\log\det(M^{-1}+B)+\mathrm{const}.$ At fixed spectrum, this term is rotationally stationary when $[M,B]=0$, is minimized by pairing large eigenvalues of $M$ with small eigenvalues of $B$, and generates a local restoring contribution against rotational mismatch. For ReLU-type function spaces under mild stable statistical conditions, $B=\sigma_\xi^2L^\ast\mathcal K L$, where $L$ measures coarse-grained second-order structure. Thus the low-$B$ sector corresponds, up to bounded anisotropy of $\mathcal K$, to low structural curvature, implying a preference for faster relaxation along smooth, data-adaptive directions. These results identify function space as a natural macroscopic level for studying stable collective organization in learning.