Cliff Joslyn

Emilie Purvine, Davis Brown, Brett Jefferson et al.

Topological data analysis (TDA) is a branch of computational mathematics, bridging algebraic topology and data science, that provides compact, noise-robust representations of complex structures. Deep neural networks (DNNs) learn millions of parameters associated with a series of transformations defined by the model architecture, resulting in high-dimensional, difficult-to-interpret internal representations of input data. As DNNs become more ubiquitous across multiple sectors of our society, there is increasing recognition that mathematical methods are needed to aid analysts, researchers, and practitioners in understanding and interpreting how these models' internal representations relate to the final classification. In this paper, we apply cutting edge techniques from TDA with the goal of gaining insight into the interpretability of convolutional neural networks used for image classification. We use two common TDA approaches to explore several methods for modeling hidden-layer activations as high-dimensional point clouds, and provide experimental evidence that these point clouds capture valuable structural information about the model's process. First, we demonstrate that a distance metric based on persistent homology can be used to quantify meaningful differences between layers, and we discuss these distances in the broader context of existing representational similarity metrics for neural network interpretability. Second, we show that a mapper graph can provide semantic insight into how these models organize hierarchical class knowledge at each layer. These observations demonstrate that TDA is a useful tool to help deep learning practitioners unlock the hidden structures of their models.

Eric Yeats, John Buckheit, Sarah Scullen et al.

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.

Henry Kvinge, Brett Jefferson, Cliff Joslyn et al.

As data grows in size and complexity, finding frameworks which aid in interpretation and analysis has become critical. This is particularly true when data comes from complex systems where extensive structure is available, but must be drawn from peripheral sources. In this paper we argue that in such situations, sheaves can provide a natural framework to analyze how well a statistical model fits at the local level (that is, on subsets of related datapoints) vs the global level (on all the data). The sheaf-based approach that we propose is suitably general enough to be useful in a range of applications, from analyzing sensor networks to understanding the feature space of a deep learning model.