The Manifold Hypothesis for Gradient-Based Explanations

When do gradient-based explanation algorithms provide meaningful explanations? We propose a necessary criterion: their feature attributions need to be aligned with the tangent space of the data manifold. To provide evidence for this hypothesis, we introduce a framework based on variational autoencoders that allows to estimate and generate image manifolds. Through experiments across a range of different datasets -- MNIST, EMNIST, CIFAR10, X-ray pneumonia and Diabetic Retinopathy detection -- we demonstrate that the more a feature attribution is aligned with the tangent space of the data, the more structured and explanatory it tends to be. In particular, the attributions provided by popular post-hoc methods such as Integrated Gradients, SmoothGrad and Input $\times$ Gradient tend to be more strongly aligned with the data manifold than the raw gradient. As a consequence, we suggest that explanation algorithms should actively strive to align their explanations with the data manifold. In part, this can be achieved by adversarial training, which leads to better alignment across all datasets. Some form of adjustment to the model architecture or training algorithm is necessary, since we show that generalization of neural networks alone does not imply the alignment of model gradients with the data manifold.