Structure and Redundancy in Large Language Models: A Spectral Study via Random Matrix Theory

This thesis addresses two persistent and closely related challenges in modern deep learning, reliability and efficiency, through a unified framework grounded in Spectral Geometry and Random Matrix Theory (RMT). As deep networks and large language models continue to scale, their internal behavior becomes increasingly opaque, leading to hallucinations, fragile generalization under distribution shift, and growing computational and energy demands. By analyzing the eigenvalue dynamics of hidden activations across layers and inputs, this work shows that spectral statistics provide a compact, stable, and interpretable lens on model behavior, capable of separating structured, causal representations from noise-dominated variability. Within this framework, the first contribution, EigenTrack, introduces a real-time method for detecting hallucinations and out-of-distribution behavior in large language and vision-language models. EigenTrack transforms streaming activations into spectral descriptors such as entropy, variance, and deviations from the Marchenko-Pastur baseline, and models their temporal evolution using lightweight recurrent classifiers, enabling early detection of reliability failures before they appear in model outputs while offering interpretable insight into representation dynamics. The second contribution, RMT-KD, presents a principled approach to compressing deep networks via random matrix theoretic knowledge distillation. By interpreting outlier eigenvalues in activation spectra as carriers of task-relevant information, RMT-KD progressively projects networks onto lower-dimensional subspaces through iterative self-distillation, yielding significantly more compact and energy-efficient models while preserving accuracy and dense, hardware-friendly structure.

TRACER: Trajectory Risk Aggregation for Critical Episodes in Agentic Reasoning

Estimating uncertainty for AI agents in real-world multi-turn tool-using interaction with humans is difficult because failures are often triggered by sparse critical episodes (e.g., looping, incoherent tool use, or user-agent miscoordination) even when local generation appears confident. Existing uncertainty proxies focus on single-shot text generation and therefore miss these trajectory-level breakdown signals. We introduce TRACER, a trajectory-level uncertainty metric for dual-control Tool-Agent-User interaction. TRACER combines content-aware surprisal with situational-awareness signals, semantic and lexical repetition, and tool-grounded coherence gaps, and aggregates them using a tail-focused risk functional with a MAX-composite step risk to surface decisive anomalies. We evaluate TRACER on $τ^2$-bench by predicting task failure and selective task execution. To this end, TRACER improves AUROC by up to 37.1% and AUARC by up to 55% over baselines, enabling earlier and more accurate detection of uncertainty in complex conversational tool-use settings. Our code and benchmark are available at https://github.com/sinatayebati/agent-tracer.

Spectral Geometry for Deep Learning: Compression and Hallucination Detection via Random Matrix Theory

Large language models and deep neural networks achieve strong performance but suffer from reliability issues and high computational cost. This thesis proposes a unified framework based on spectral geometry and random matrix theory to address both problems by analyzing the eigenvalue structure of hidden activations. The first contribution, EigenTrack, is a real-time method for detecting hallucinations and out-of-distribution behavior in language and vision-language models using spectral features and their temporal dynamics. The second contribution, RMT-KD, is a principled compression method that identifies informative spectral components and applies iterative knowledge distillation to produce compact and efficient models while preserving accuracy. Together, these results show that spectral statistics provide interpretable and robust signals for monitoring uncertainty and guiding compression in large-scale neural networks.

EigenTrack: Spectral Activation Feature Tracking for Hallucination and Out-of-Distribution Detection in LLMs and VLMs

Large language models (LLMs) offer broad utility but remain prone to hallucination and out-of-distribution (OOD) errors. We propose EigenTrack, an interpretable real-time detector that uses the spectral geometry of hidden activations, a compact global signature of model dynamics. By streaming covariance-spectrum statistics such as entropy, eigenvalue gaps, and KL divergence from random baselines into a lightweight recurrent classifier, EigenTrack tracks temporal shifts in representation structure that signal hallucination and OOD drift before surface errors appear. Unlike black- and grey-box methods, it needs only a single forward pass without resampling. Unlike existing white-box detectors, it preserves temporal context, aggregates global signals, and offers interpretable accuracy-latency trade-offs.

RMT-KD: Random Matrix Theoretic Causal Knowledge Distillation

Large deep learning models such as BERT and ResNet achieve state-of-the-art performance but are costly to deploy at the edge due to their size and compute demands. We present RMT-KD, a compression method that leverages Random Matrix Theory (RMT) for knowledge distillation to iteratively reduce network size. Instead of pruning or heuristic rank selection, RMT-KD preserves only informative directions identified via the spectral properties of hidden representations. RMT-based causal reduction is applied layer by layer with self-distillation to maintain stability and accuracy. On GLUE, AG News, and CIFAR-10, RMT-KD achieves up to 80% parameter reduction with only 2% accuracy loss, delivering 2.8x faster inference and nearly halved power consumption. These results establish RMT-KD as a mathematically grounded approach to network distillation.

SPARC: Subspace-Aware Prompt Adaptation for Robust Continual Learning in LLMs

We propose SPARC, a lightweight continual learning framework for large language models (LLMs) that enables efficient task adaptation through prompt tuning in a lower-dimensional space. By leveraging principal component analysis (PCA), we identify a compact subspace of the training data. Optimizing prompts in this lower-dimensional space enhances training efficiency, as it focuses updates on the most relevant features while reducing computational overhead. Furthermore, since the model's internal structure remains unaltered, the extensive knowledge gained from pretraining is fully preserved, ensuring that previously learned information is not compromised during adaptation. Our method achieves high knowledge retention in both task-incremental and domain-incremental continual learning setups while fine-tuning only 0.04% of the model's parameters. Additionally, by integrating LoRA, we enhance adaptability to computational constraints, allowing for a tradeoff between accuracy and training cost. Experiments on the SuperGLUE benchmark demonstrate that our PCA-based prompt tuning combined with LoRA maintains full knowledge retention while improving accuracy, utilizing only 1% of the model's parameters. These results establish our approach as a scalable and resource-efficient solution for continual learning in LLMs.