Eigenvalue Calibration for Semantic Embeddings of Large Language Models

Official code for our UAI 2026 paper "Eigenvalue Calibration for Semantic Embeddings of Large Language Models".

Sebastian G. Gruber^†, Nassim Walha^†, Francis Bach, Florian Buettner (^† equal contribution)

Uncertainty quantification for LLMs increasingly relies on the eigenvalues of density matrices built from semantic embeddings of generated answers — but conventional calibration theory for classification probabilities does not transfer to eigenvalues. This repository provides the code for our framework that:

• interprets an LLM + semantic embedding model as a density matrix predictor, and defines matrix and eigenvalue calibration for it,
• proves an entropy–risk equivalence under calibration and a central eigenvalue calibration inequality,
• shows that matrix temperature scaling of eigenvalues provably optimizes calibration when minimizing proper matrix score risk, and
• introduces a clustering-based reliability diagram for visualizing eigenvalue calibration.

Across TriviaQA and Natural Questions with Phi-4, Phi-4 Mini, and Llama4 Maverick, we find that current LLMs are systematically overconfident in their maximum eigenvalue, and that our temperature scaling reduces this overconfidence while also improving downstream AUROC in most settings.

Key idea

We treat the eigenvalues of the density matrix $d(x) = \mathbb{E}_{a \sim f(x)}[e(a)e(a)^\intercal]$ (built from an LLM $f$'s sampled answers and their semantic embeddings $e$) as probabilities over latent semantic outcomes, and calibrate them via temperature scaling on the spectrum.

Normalised semantic embeddings reside on a hypersphere (left); we interpret the eigenvalues of their density matrix as probabilities of latent semantic outcomes (right).

LLMs are consistently overconfident in their predicted maximum eigenvalue — our matrix temperature scaling corrects this, lowering the expected calibration error (ECE) of the reliability diagram:

Reliability diagrams for Phi-4 Mini on TriviaQA, before (left, ECE 0.36) and after (right, ECE 0.21) matrix temperature scaling.

Citation

If you find this work useful, please cite:

@inproceedings{gruber2026eigenvalue,
  title     = {Eigenvalue Calibration for Semantic Embeddings of Large Language Models},
  author    = {Gruber, Sebastian G. and Walha, Nassim and Bach, Francis and Buettner, Florian},
  booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence (UAI)},
  year      = {2026}
}

To Reproduce our Results

All evaluation outputs are presented in the Jupyter Notebooks main.ipynb and rebuttal_experiments.ipynb for an easier and more interactive exploration of our experiments. Set up the environment first, either via conda (conda env create -f environment.yml) or pip (pip install -r requirements.txt); both contain the minimal set of packages needed to run the notebooks and scripts. The simulations are light-weight and can be easily run on a low-budget laptop. If you want to run the LLM answer-generation scripts on GPU, install a CUDA-enabled build of torch matching your driver from pytorch.org.

Real-world Experiments

Before you run the respective cells with real-world experiments in the notebook, you can download the respective answers and embeddings as a dataset to avoid computational costs. Either manually download from the link in the following, or install conda install -c conda-forge gdown. Then, to download the zipped LLM answers, run gdown https://drive.google.com/uc?id=1PZmFnbQuLqr_WGxpASPugeM4IzpAoHjY.

To extract them into the folder data, run unzip data.zip -d ..