In this section, we provide proofs for Proposition 2.1.B. Inthe proof, we inherit the notations that weuseforprovingTheorem2.1. The instance normalization that we incorporate into the DGM is not the same as the instance normalization that is typically used in image stylization [35]. CNN-F-5 significantly improves the robustness of CNN. CNN-F achieves higher accuracy on MNIST than CNN for under both standard training and adversarial training.

A.1 Generative model We choose the deconvolutional generative model (DGM) [25] as the generative feedback in CNN-F. The graphical model of the DGM is shown in Figure 2 (middle). In this section, we provide proofs for Theorem 2.1. Without loss of generality, we consider a DGM that has the following architecture. Lemma A.1 shows that logits output from the corresponding CNN of the DGM is proportional to the inner product of generated image and input image plus Lemma A.1 to show that CNN performs Bayesian inference in the DGM.

In contrast, human perception is much more robust to such perturbations.

We thank the reviewers for their thoughtful feedback. Due to space limit, we address the main concerns here. R4 How do losses affect performance? We included the ablation study in Appendix (line 522). Empirically, we find that 5 iterations lead to a stable solution.

Neural networks are vulnerable to input perturbations such as additive noise and adversarial attacks. In contrast, human perception is much more robust to such perturbations. The Bayesian brain hypothesis states that human brains use an internal generative model to update the posterior beliefs of the sensory input. This mechanism can be interpreted as a form of self-consistency between the maximum a posteriori (MAP) estimation of an internal generative model and the external environment. Inspired by such hypothesis, we enforce self-consistency in neural networks by incorporating generative recurrent feedback. We instantiate this design on convolutional neural networks (CNNs). The proposed framework, termed Convolutional Neural Networks with Feedback (CNN-F), introduces a generative feedback with latent variables to existing CNN architectures, where consistent predictions are made through alternating MAP inference under a Bayesian framework. In the experiments, CNN-F shows considerably improved adversarial robustness over conventional feedforward CNNs on standard benchmarks.

In this paper, we present a method aimed at integrating domain knowledge abstracted as logic rules into the predictive behaviour of a neural network using feature extracting functions for visual sentiment analysis. We combine the declarative first-order logic rules which represent the human knowledge in a logically-structured format making use of feature-extracting functions. These functions are embodied as programming functions which can represent, in a straightforward manner, the applicable domain knowledge as a set of logical instructions and provide a cumulative set of probability distributions of the input data. These distributions can then be used during the training process in a mini-batch strategy. In contrast with other neural logic approaches, the programmatic nature in practice of these functions do not require any kind of special mathematical encoding, which makes our method very general in nature. We also illustrate the utility of our method for sentiment analysis and compare our results to those obtained using a number of alternatives elsewhere in the literature.