[1912.11460v2] Characterizing the Decision Boundary of Deep Neural Networks
This is fairly unexplored in the machine learning literature, and thus in this work, we embarked upon a research inquiry to study the decision boundary of DNNs and investigated their behaviors through the lens of their decision boundaries

\begin{abstract}
Deep neural networks and in particular, deep neural classifiers have become an integral part of many modern applications. Despite their practical success, we still have limited knowledge of how they work and the demand for such an understanding is evergrowing. In this regard, one crucial aspect of deep neural network classifiers that can help us deepen our knowledge about their decision-making behavior is to investigate their decision boundaries. Nevertheless, this is contingent upon having access to samples populating the areas near the decision boundary. To achieve this, we propose a novel approach we call \textbf{Deep} \textbf{D}ecision boundary \textbf{I}nstance \textbf{G}eneration (\textbf{DeepDIG}). DeepDIG utilizes a method based on adversarial example generation as an effective way of generating samples near the decision boundary of any deep neural network model. Then, we introduce a set of important principled characteristics that take advantage of the generated instances near the decision boundary to provide multifaceted understandings of deep neural networks. We have performed extensive experiments on multiple representative datasets across various deep neural network models and characterized their decision boundaries.
\end{abstract}
‹Figure 1: A high-level illustration of Deep Decision boundary Instance Generation (DeepDIG). For a given pre-trained deep neural network model and two classes s and t, DeepDIG tries to find instances as close as possible to the decision boundary between the two classes s and t. (Introduction)Figure 2: The proposed framework Deep Decision boundary Instance Generation (DeepDIG). It consists of three components. In component (I), targeted adversarial examples of source instances are generated (x̂t ). In component (II), from adversarial examples of component (I), a new set of adversarial examples are generated (x̂s ) which are classified as s. Finally, in component (III), a binary search based algorithm is employed to refine and identify the borderline instances near the decision boundary. (Proposed Framework (DeepDIG))Figure 3: An illustration of capturing geometrical complexity of a decision boundary through measuring the oscillation between two decision (classification) regions rs and rt for samples on a borderline trajectory. Decision boundary in case (I) is geometrically more complex than that of (II). (Decision Boundary Complexity in the Input Space)Figure 4: MNISTFCN Figure 5: MNISTCNN Figure 6: Projection of embeddings of training and test samples as well as borderline instances onto a 2D space (MNIST) (Inter-model Decision Boundary Characterization)Figure 7: FashionMNISTFCN Figure 8: FashionMNISTCNN Figure 9: Projection of embeddings of training and test samples as well as borderline instances onto a 2D space (FashionMNIST) (Inter-model Decision Boundary Characterization)Figure 10: CIFAR10ResNet Figure 11: CIFAR10GoogleNet Figure 12: Projection of embeddings of training and test samples as well as borderline instances onto a 2D space (CIFAR10) (Inter-model Decision Boundary Characterization)Figure 13: Input space Decision boundary Complexity (IDC) of model MNISTCNN according to the characteristic measure IDC discussed in Section ?? (Intra-model Decision Boundary Characterization)Figure 14: Embedding space Decision boundary Complexity 1 (EDC1) i.e., the distance from the separating hyperplane for all class pairs in model MNISTCNN according to the characteristic measure EDC1 discussed in Section ?? (Intra-model Decision Boundary Characterization)Figure 15: Embedding space Decision boundary Complexity 2 (EDC2) i.e., the accuracy against the linear SVM for all class pairs in model MNISTCNN according to the characteristic measure EDC2 discussed in Section ?? (Intra-model Decision Boundary Characterization)›