Inés García‐Redondo

Aideen Fay, Inés García-Redondo, Qiquan Wang et al.

Existing interpretability methods for Large Language Models (LLMs) predominantly capture linear directions or isolated features. This overlooks the high-dimensional, relational, and nonlinear geometry of model representations. We apply persistent homology (PH) to characterize how adversarial inputs reshape the geometry and topology of internal representation spaces of LLMs. This phenomenon, especially when considered across operationally different attack modes, remains poorly understood. We analyze six models (3.8B to 70B parameters) under two distinct attacks, indirect prompt injection and backdoor fine--tuning, and show that a consistent topological signature persists throughout. Adversarial inputs induce topological compression, where the latent space becomes structurally simpler, collapsing the latent space from varied, compact, small-scale features into fewer, dominant, large-scale ones. This signature is architecture-agnostic, emerges early in the network, and is highly discriminative across layers. By quantifying the shape of activation point clouds and neuron-level information flow, our framework reveals geometric invariants of representational change that complement existing linear interpretability methods.

Charlie B. Tan, Inés García-Redondo, Qiquan Wang et al.

Bounding and predicting the generalization gap of overparameterized neural networks remains a central open problem in theoretical machine learning. There is a recent and growing body of literature that proposes the framework of fractals to model optimization trajectories of neural networks, motivating generalization bounds and measures based on the fractal dimension of the trajectory. Notably, the persistent homology dimension has been proposed to correlate with the generalization gap. This paper performs an empirical evaluation of these persistent homology-based generalization measures, with an in-depth statistical analysis. Our study reveals confounding effects in the observed correlation between generalization and topological measures due to the variation of hyperparameters. We also observe that fractal dimension fails to predict generalization of models trained from poor initializations. We lastly reveal the intriguing manifestation of model-wise double descent in these topological generalization measures. Our work forms a basis for a deeper investigation of the causal relationships between fractal geometry, topological data analysis, and neural network optimization.