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.
翻译:地形数据分析(TDA)是计算数学、连接代数表层和数据科学的一个分支,它提供复杂结构的压缩、噪音-气旋表示;深神经网络(DNNS)学习了与模型结构界定的一系列变异有关的数以百万计的参数,导致输入数据的高维、难于解释的内部表述。随着DNNS在社会多个部门中更加无处不在,人们日益认识到需要数学方法来帮助分析家、研究人员和从业者理解和解释这些模型的内部表述与最后分类的关系。在本文件中,我们从TDA中采用尖端边缘技术,目的是深入了解用于图像分类的进化神经网络的可解释性。我们使用两种通用的TDA方法探索以高维点云的形式模拟隐性启动,并提供实验性证据,证明这些云能收集了关于模型过程的有用结构信息。首先,我们证明基于持续同系的远程测量方法可以用来量化各个层次之间的深层观测结果。我们应用了从深度观测到这些层次结构的深度结构的清晰度技术,我们用这些工具来解释这些图层结构的距离。我们展示了这些图层结构的比较的距离。