Learning from Noise: Effective-Rank Collapse and Out-of-Distribution Rejection in Restricted Boltzmann Machines

The great success of neural networks primarily arises from the presence of the large number of weight parameters combined with nonlinearities in the input-output relationship of single neurons. In this work, we study the relationship between the statistical properties of the weights and the nonlinearity of the hidden unit in Restricted Boltzmann Machines (RBMs) on the one side, and the distribution induced on binary visible units. We do this for four commonly used activation functions: Linear, Step, ReLU, and Exponential, and make qualitative predictions about the ability of these models to learn distributions with strong higher order interactions over the visible nodes. We show that in general, in an ensemble of RBMs with Gaussian weights, these distributions are rare and hard to learn, except when the hidden unit activation function is an Exponential.

Observable Matrix Dynamics (OMD) is a diagnostic framework that probes the dynamics of high-dimensional internal representations of inputs by a neural network via a fixed-size $N \times N$ distance matrix $M(t)$ on a held set of $N$ inputs. OMD uses methods of random matrix theory and particle dynamics to explore spectral reorganisations that are missed by scalar loss functions, but are informative of the training process. We read $M(t)$ against a perturbative ambient-versus-latent decomposition extending the Bogomolny--Bohigas--Schmit (BBS) theory of random distance matrices, with per-snapshot diagnostics for the top-of-spectrum band structure and ambient noise, trajectory-level observables linking snapshots, and a 3D MDS embedding (bottom-three eigenvectors) rendering training as a moving particle cloud. Across seven experiments, diffusive regimes lack stable top-of-spectrum band structure, while sharp endogenous or externally driven reorganisations produce stable fingerprints: consistent with smooth or product latent geometries in BBS-adjacent cases, and with finite-cluster or Fourier-soliton structures otherwise. OMD thus reads the geometric regime of a representation rather than reporting a single intrinsic dimension.

Learning with noisy labels (LNL) is typically benchmarked by closed-set classification accuracy, yet deployment often requires classifiers to reject out-of-distribution (OOD) inputs. We present a learner-agnostic ACC-OOD benchmark that freezes LNL checkpoints and evaluates them with standardized near-/far-OOD routing and post-hoc scores across synthetic and real label noise. The benchmark reveals a recurring failure mode: high closed-set accuracy does not ensure OOD reliability, because low-confidence, misclassified in-distribution samples can overlap the score and feature regions occupied by OOD inputs under noisy training. We term this pathology uncertainty collapse. This structural overlap can make high-accuracy LNL methods lose separability at the ID-error/OOD interface under standard OOD scores. As an intervention, we study Virtual Margin Regularization (VMR), a lightweight repair probe demonstrated mainly with PSSCL that synthesizes boundary virtual outliers on trusted ID batches and widens the energy margin. VMR partially reduces the collapse-induced far-OOD failure without replacing the host objective or sacrificing closed-set accuracy in the tested settings. These results support LNL benchmarks that co-report closed-set generalization, open-world reliability, and structural overlap diagnostics.