Brett Jefferson

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.

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.

Brett Jefferson, Carlos Ortiz Marrero

We explore rigorous, systematic, and controlled experimental evaluation of adversarial examples in the real world and propose a testing regimen for evaluation of real world adversarial objects. We show that for small scene/ environmental perturbations, large adversarial performance differences exist. Current state of adversarial reporting exists largely as a frequency count over a dynamic collections of scenes. Our work underscores the need for either a more complete report or a score that incorporates scene changes and baseline performance for models and environments tested by adversarial developers. We put forth a score that attempts to address the above issues in a straight-forward exemplar application for multiple generated adversary examples. We contribute the following: 1. a testbed for adversarial assessment, 2. a score for adversarial examples, and 3. a collection of additional evaluations on testbed data.