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

The rapid growth of deep learning has brought both unprecedented capabilities and pressing challenges. On one hand, large neural networks deliver state-of-the-art performance across natural language processing, computer vision, and multimodal domains. On the other, their reliability is compromised by hallucinations and out-of-distribution errors, while their scale imposes severe efficiency and deployment barriers. This thesis develops a unifying spectral framework, grounded in Random Matrix Theory (RMT), to address both reliability and efficiency in modern AI systems. First, it introduces EigenTrack, a real-time detector of hallucination and distributional shift in large language and vision-language models. By extracting spectral statistics from sliding-window activation covariances, entropy, eigenvalue gaps, and divergence from the Marchenko–Pastur law, and modeling their temporal evolution with lightweight recurrent classifiers, EigenTrack detects anomalies before they manifest in model outputs. Results show state-of-the-art performance across multiple LLM and VLM families, offering interpretable signatures of failure dynamics. Second, the thesis presents RMT-KD, an iterative knowledge distillation method that applies random matrix principles to compress deep networks. By isolating outlier eigenvalues of hidden activations as carriers of causal structure, RMT-KD progressively projects models onto informative subspaces while preserving accuracy through self-distillation. Experiments on BERT, ResNet, and benchmark datasets demonstrate major parameter reduction with minimal accuracy loss, yielding faster inference and significant energy savings. These contributions establish spectral geometry as a principled lens for both diagnosing uncertainty and guiding compression. By linking eigenvalue dynamics to representation quality, the thesis advances interpretable, mathematically grounded methods that make AI systems simultaneously more trustworthy and more efficient.

Author dc:creator

• Davide Ettori (24398936)

• Deep Learning
• Large Language Models
• Mathematical Modeling

Statement dc:rights

• In Copyright

DOI dc:identifier

https://doi.org/10.25417/uic.32991827.v1

OAI identifier oai:identifier

oai:figshare.com:article/32991827