What do Geometric Hallucination Detection Metrics Actually Measure?

Hallucination remains a barrier to deploying generative models in high-consequence applications. This is especially true in cases where external ground truth is not readily available to validate model outputs. This situation has motivated the study of geometric signals in the internal state of an LLM that are predictive of hallucination and require limited external knowledge. Given that there are a range of factors that can lead model output to be called a hallucination (e.g., irrelevance vs incoherence), in this paper we ask what specific properties of a hallucination these geometric statistics actually capture. To assess this, we generate a synthetic dataset which varies distinct properties of output associated with hallucination. This includes output correctness, confidence, relevance, coherence, and completeness. We find that different geometric statistics capture different types of hallucinations. Along the way we show that many existing geometric detection methods have substantial sensitivity to shifts in task domain (e.g., math questions vs. history questions). Motivated by this, we introduce a simple normalization method to mitigate the effect of domain shift on geometric statistics, leading to AUROC gains of +34 points in multi-domain settings.

Can Neural Networks Learn Small Algebraic Worlds? An Investigation Into the Group-theoretic Structures Learned By Narrow Models Trained To Predict Group Operations

While a real-world research program in mathematics may be guided by a motivating question, the process of mathematical discovery is typically open-ended. Ideally, exploration needed to answer the original question will reveal new structures, patterns, and insights that are valuable in their own right. This contrasts with the exam-style paradigm in which the machine learning community typically applies AI to math. To maximize progress in mathematics using AI, we will need to go beyond simple question answering. With this in mind, we explore the extent to which narrow models trained to solve a fixed mathematical task learn broader mathematical structure that can be extracted by a researcher or other AI system. As a basic test case for this, we use the task of training a neural network to predict a group operation (for example, performing modular arithmetic or composition of permutations). We describe a suite of tests designed to assess whether the model captures significant group-theoretic notions such as the identity element, commutativity, or subgroups. Through extensive experimentation we find evidence that models learn representations capable of capturing abstract algebraic properties. For example, we find hints that models capture the commutativity of modular arithmetic. We are also able to train linear classifiers that reliably distinguish between elements of certain subgroups (even though no labels for these subgroups are included in the data). On the other hand, we are unable to extract notions such as the concept of the identity element. Together, our results suggest that in some cases the representations of even small neural networks can be used to distill interesting abstract structure from new mathematical objects.