Hyunmo Kang

Hyunmo Kang, Noam Itzhak Levi, Corinna Elena Wegner et al.

Sampling from learned high-dimensional distributions is a foundational computational problem. We introduce U-turn chains: Markov chains obtained by iterating short forward-backward steps of a diffusion model, in which each step proposes a move that remains on the learned data manifold and, paired with a Metropolis-Hastings correction, samples from energy-modified targets. For synthetic languages, we show that minimal U-turn dynamics undergoes an ergodicity-breaking phase transition driven by fragmentation of the data manifold; ergodicity is restored at larger U-turn magnitude. In the non-ergodic regime, low-level features relax faster than high-level ones, an ordering that inverts only at sufficiently large U-turn magnitude. We test these predictions on natural language and natural images. In both modalities, minimal U-turns relax slowly, especially for high-level features approximated by deep representations in CNNs or LLMs. The layer-ordering inversion appears only at large noise when mixing is efficient -- signatures consistent with strongly constrained, weakly mixing local dynamics. We discuss the implications of these results for sampling with diffusion models.

Hyunmo Kang, Abdulkadir Canatar, SueYeon Chung

Understanding representational similarity between neural recordings and computational models is essential for neuroscience, yet remains challenging to measure reliably due to the constraints on the number of neurons that can be recorded simultaneously. In this work, we apply tools from Random Matrix Theory to investigate how such limitations affect similarity measures, focusing on Centered Kernel Alignment (CKA) and Canonical Correlation Analysis (CCA). We propose an analytical framework for representational similarity analysis that relates measured similarities to the spectral properties of the underlying representations. We demonstrate that neural similarities are systematically underestimated under finite neuron sampling, mainly due to eigenvector delocalization. Moreover, for power-law population spectra, we show that the number of localized eigenvectors scales as the square root of the number of recorded neurons, providing a simple rule of thumb for practitioners. To overcome sampling bias, we introduce a denoising method to infer population-level similarity, enabling accurate analysis even with small neuron samples. Theoretical predictions are validated on synthetic and real datasets, offering practical strategies for interpreting neural data under finite sampling constraints.

Francesco Cagnetta, Hyunmo Kang, Matthieu Wyart

Recent theories suggest that Neural Scaling Laws arise whenever the task is linearly decomposed into power-law distributed units. Alternatively, scaling laws also emerge when data exhibit a hierarchically compositional structure, as is thought to occur in language and images. To unify these views, we consider classification and next-token prediction tasks based on probabilistic context-free grammars -- probabilistic models that generate data via a hierarchy of production rules. For classification, we show that having power-law distributed production rules results in a power-law learning curve with an exponent depending on the rules' distribution and a large multiplicative constant that depends on the hierarchical structure. By contrast, for next-token prediction, the distribution of production rules controls the local details of the learning curve, but not the exponent describing the large-scale behaviour.