Neural Networks with Recurrent Generative Feedback

by   Yujia Huang, et al.

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 the internal generative model and the external environmental. Inspired by this, we enforce consistency in neural networks by incorporating generative recurrent feedback. We instantiate it on convolutional neural networks (CNNs). The proposed framework, termed Convolutional Neural Networks with Feedback (CNN-F), introduces a generative feedback with latent variables into existing CNN architectures, making consistent predictions via alternating MAP inference under a Bayesian framework. CNN-F shows considerably better adversarial robustness over regular feedforward CNNs on standard benchmarks. In addition, With higher V4 and IT neural predictivity, CNN-F produces object representations closer to primate vision than conventional CNNs.



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1 Introduction

Figure 1: An intuitive illustration of recurrent generative feedback in human visual perception system.
Figure 2: Left: CNN, Graphical model for the DGM and the inference network for the DGM.

We use the DGM to as the generative model for the joint distribution of image features

, labels and latent variables . MAP inference for , and is denoted in red, green and blue respectively. and denotes feedforward features and feedback features respectively.
Right: CNN with feedback (CNN-F). CNN-F performs alternating MAP inference via recurrent feedforward and feedback pathways to enforce self-consistency.

Vulnerability in feedforward neural networks Conventional deep neural networks (DNNs) often contain many layers of feedforward connections. With the ever-growing network capacities and representation abilities, they have achieved great success. For example, recent convolutional neural networks (CNNs) have impressive accuracy on large scale image classification benchmarks szegedy_rethinking_2015. However, current CNN models also have significant limitations. For instance, they can suffer significant performance drop from corruptions which barely influence human recognition dodgedistortionperf. Studies also show that CNNs can be misled by imperceptible noise known as adversarial attacks szegedy2013intriguing.

Feedback in the human brain To address the weaknesses of CNNs, we can take inspiration from of how human visual recognition works, and incorporate certain mechanisms into the CNN design. While human visual cortex has hierarchical feedforward connections, backward connections from higher level to lower level cortical areas are something that current artificial networks are lacking felleman_distributed_1991. Studies suggest these backward connections carry out top-down processing which improves the representation of sensory input kok2012less. In addition, evidence suggests recurrent feedback in the human visual cortex is crucial for robust object recognition. For example, humans require recurrent feedback to recognize challenging images kar2019evidence. Obfuscated images can fool humans without recurrent feedback elsayed2018adversarial. Figure 1 shows an intuitive example of recovering a sharpened cat from a blurry cat and achieving consistent predictions after several iterations.

Predictive coding

Computational neuroscientists speculate that Bayesian inference models human perception


. Predictive coding, for example, is a specific formulation of hierarchical Bayesian inference that assumes Gaussian distributions on all variables 

rao_predictive_1999. Predictive coding uses recurrent, feedback pathways to perform Bayesian inference. According to predictive coding theory, feedback pathways encode predictions of lower level inputs. The residual errors are used recurrently to update the predictions.

Summary of results In this paper, we extend the principle of predictive coding to explicitly incorporate Bayesian inference in neural networks via generative feedback connections. We then investigate whether feedback promotes robust object recognition. Our contributions are as follows:

Generative feedback Inspired by the Bayesian brain hypothesis and predictive coding theory, we hypothesize that the feedback process reconstructs the stimulus using an internal generative model of the world. To generate images from a low dimensional label, we also add appropriate auxiliaries that capture variation in the images. Specifically, we adopt a recently proposed model, named the Deconvolutional Generative Model (DGM) NRM, as the generative feedback (Figure 2). The DGM uses hierarchical latent variables to generate images. We show that Bayesian inference in the DGM is achieved by CNN with adaptive nonlinear operators.

Self-consistency We introduce generative feedback to NN and impose self-consistency to enable robustness. Intuitively, self-consistency means that the label, auxillary information and images should be consistent with each other. Furthermore, our internal model of the world should be consistent with the external stimuli. Mathematically, we use a generative model to describe the joint distribution of labels, latent variables and features of input images. The predicted label, latent variables and features are maximum a posteriori (MAP) estimates conditioned on the other two elements. They are self-consistent if the MAP estimates are consistent with each other (Figure 4).

CNN with Feedback (CNN-F) We incorporate generative recurrent feedback into CNN and term this model as CNN-F. We impose self-consistency in CNN-F by performing alternating MAP inference for the label, latent variables and image features. MAP inference for the label and image features given latent variables is straightforward from classification and generation networks. MAP inference for the hierarchical latent variables is approximated by iterated conditional modes (ICM) besag1986statistical, where we iteratively compute MAP estimates for latent variables at one layer conditioning on all other layers. This leads to recurrent generative feedback to feedforward layers in CNN-F (Figure 2).

Adversarial robustness We evaluate the adversarial robustness of CNN-F on MNIST and Fashion-MNIST datasets. CNN-F achieves significantly better adversarial accuracy than CNN under both standard training and adversarial training. Furthermore, training and evaluating CNN-F with more iterations both help improve robustness, indicating that recurrent feedback is crucial for recognizing challenging images.

Biological plausibility

Given the lateral and backward connections in the primate brain, we investigate whether the CNN-F’s generative feedback produces more biologically similar neural networks. Trained on ImageNet-12, we show that the CNN-F has higher V4 and IT neural predictivity compared to its corresponding CNN. This demonstrates that the CNN-F models human vision significantly closer compared to CNNs.

2 Approach

Figure 3: Feedforward and feedback pathway in CNN-F. and are computed by the feedforward pathway and is computed from the feedback pathway.

In this section, we first formally define self-consistency. Then we give a specific form of generative feedback in CNN and impose self-consistency on it. We term this model as CNN-F. Finally we show the training and testing procedure in CNN-F. Throughout, we use the following notations:

Let be the input of a neural network and be the output. In image classification problems, is image and

is one-hot encoded label.

is the total number of classes. is usually much less than . We use to denote the total number of layers of the network, and index the input layer to the feedforward network as layer . Let be encoded feature of at layer of the feedforward pathway. Feedforward pathway computes feature map from layer to layer , and feedback pathway generates from layer to . and have the same dimensions. To generate from , we introduce latent variables for each layer of CNN. Let be latent variables at layer , where are the number of channels, height and width for the corresponding feature map. Finally, we use to denote the joint distribution parameterized by . includes the weight of convolutional and fully connected layers and the bias term . We use , and to denote the MAP estimates of conditioning on the other two variables.

2.1 Generative feedback and Self-consistency

Figure 4: Self-consistency among and consistency between and .

Human brain and NN are similar in having a hierarchical structure. In human visual perception, external stimuli are first preprocessed by lateral geniculate nucleus (LGN) and then sent to be processed by V1, V2, V4 and Inferior Temporal (IT) cortex in the ventral cortical visual system. Conventional NN use feedforward layers to model this process and learn a one-direction mapping from input to output. However, numerous studies suggest that in addition to the feedforward connections from V1 to IT, there are feedback connections among these cortical areas felleman_distributed_1991.

Inspired by the Bayesian brain hypothesis and the predictive coding theory, we propose to add generative feedback connections to NN. Since is usually of much higher dimension than , we introduce latent variables to account for the information loss in the feedforward process. We then propose to model the feedback connections as MAP estimation from an internal generative model that describes the joint distribution of and . Furthermore, we realize recurrent feedback by imposing self-consistency (Definition 2.1).

Definition 2.1.

(Self-consistency) Given a joint distribution parameterized by , are self-consistent if they satisfy the following constraints:


In words, self-consistency means that MAP estimates from an internal generative model are consistent with each other. In addition to self-consistency, we also impose the consistency constraint between and the external input features (Figure 4). We hypothesize that for easy images (familiar images to human, clean images in the training dataset for NN), the from the first feedforward pass should automatically satisfy the self-consistent constraints. Therefore, feedback need not be triggered. For challenging images (unfamiliar images to human, unseen perturbed images for NN), recurrent feedback is needed to obtain self-consistent and to match with . This recurrence accounts for the dynamics in neural circuits kietzmann2019recurrence and the longer time that people need to process challenging images kar2019evidence.

2.2 Generative Feedback in CNN-F

CNN have been used to model the hierarchical structure of human retinatopic fields eickenberg2017seeing; horikawa2017hierarchical, and have achieved state-of-the-art performance in image classification. Therefore, we introduce generative feedback to CNN and impose self-consistency on it. We term the resulting model as CNN-F.

We choose to use the DGM NRM as generative feedback in the CNN-F. The DGM introduces hierarchical binary latent variables and generates images from coarse to fine details. The generation process in the DGM is shown in Figure 3. First, is sampled from the label distribution. Then each entry of

is sampled from a Bernoulli distribution parameterized by

and a bias term . and are then used to generate the layer below:


In this paper, we assume to be uniform, which is realistic under the balanced label scenario. We assume that follows Gaussian distribution centered at

with standard deviation


2.3 Recurrence in CNN-F

In this section, we show that self-consistent in the DGM can be obtained via alternately propagating along feedforward and feedback pathway in CNN-F.

Feedforward and feedback pathway in CNN-F

The feedback pathway in CNN-F takes the same form as the generation process in the DGM (Equation 2). The feedforward pathway in CNN-F takes the same form as CNN except for the nonlinear operators. In conventional CNN, nonlinear operators are and , where is the dimension of the pooling region in the feature map (typically equals to or ). In contrast, we use and given in Equation 14 in the feedforward pathway of CNN-F. These operators adaptively choose how to activate the feedforward feature map based on the sign of the feedback feature map. The feedforward pathway computes using the recursion 111 takes the form of or ..

MAP inference in the DGM

We present MAP inference for in the DGM in Theorem 2.1

. First, we define generative classifier as a neural network that outputs the MAP distribution of

in a generative model. A well known example is that logistic regression is the generative classifier derived from Gaussian Naive Bayes model, where

is Boolean variable modeled by a Bernoulli distribution and is assumed to follow Gaussian distribution. We use and to denote latent variables that are at a layer followed by and respectively. denotes indicator function. To ease MAP inference in the DGM, we have the following assumptions:

Assumption 2.1.
  • [ nosep,font=, leftmargin=3em,itemindent=-1em,align=left]

  • The generated image at layer from the DGM has a constant norm:

  • Prior distribution on the label is a uniform distribution:

  • Normalization factor in for each category is constant:

Theorem 2.1 (MAP inference in the DGM).

Under Assumption 2.1, the followings hold:

  • [ nosep,font=, leftmargin=3em,itemindent=-1em,align=left]

  • Let be the feature at layer , then .

  • CNN with and is the generative classifier derived from the DGM.

  • MAP estimate of conditioned on and in the DGM is:


For part A, we have . The second equality is obtained because is a deterministic function of and . The third equality is obtained because . For part B and C, please refer to Appendix A. ∎


Theorem 2.1.A and B show that is the generated feature map at bottom level in CNN-F and is the output from the feedfoward pathway.
Theorem 2.1.C states that if the sign of the feedforward feature map matches with that of the feedback feature map. at locations that satisfy one of these two requirements: 1) the value in the feedback feature map is non-negative and it is the maximum value within the local pooling region or 2) the value in the feedback feature map is negative and it is the minimum value within the local pooling region. Using Theorem 2.1.C, we approximate by greedily finding the MAP estimate of conditioning on all other layers.

Iterative inference in CNN-F

For self-consistency on , we solve the problem in 1 via alternating optimization. The resulting iterative inference in CNN-F is as follows (Figure 2, right). In the initialization step, image is passed through a standard CNN, and latent variables are initialized using Equation 18 and 19 with conventional and . The feedback generative network then uses and to generate intermediate features

, where the subscript denotes the number of iterations. In practice, we use logits instead of one-hot encoded label in the generative feedback to maintain uncertainty in each category. We assign

to the input features for the next iteration , where is the layer that we reconstruct to in the feedback pathway. is then fed back to the feedforward pathway for the next iteration. Starting from the first iteration, we use and instead of and and in the feedforward pathway to infer (Theroem 2.1). This iterative inference procedure is described in Algorithm 1.

Input :  Input image , index of the target layer to reconstruct, maximum number of iterations N.
Initialize by standard the CNN;
while t < N do
       Feedback pathway: generate from and , .;
       Feedforward pathway: predict from and compute , .;
end while
Algorithm 1 Iterative inference in CNN-F

2.4 Training and testing in CNN-F

During training, we have three goals: 1) train a generative model to model the data distribution, 2) train a generative classifier and 3) enforce self-consistency in the model. We first approximate self-consistent and then update model parameters based on the losses listed in Table 1. Each loss term is computed for every iteration. Minimizing the reconstruction loss and conditional latent likelihood loss is equivalent to maximizing the log likelihood of . Minimizing the reconstruction loss also improves the consistency between and . Minimizing the Cross entropy loss helps training the generative classifier. During testing time, CNN-F finds self-consistent given the input image using iterative inference described in Algorithm 1.

Form Purpose
Cross entropy loss classification
Reconstruction loss generation, self-consistency
Conditional latent likelihood loss 222According to the distribution specified by DGM, , see Appendix A. generation
Table 1: Training losses used to train CNN-F.

3 CNN-F models correspond more to primate vision compared to a CNN

Figure 5: The generative feedback in CNN-F models predict a significant portion of primate vision neural responses. Left The Brain-Score’s experimental paradigm presents visual image stimuli both to primates and a neural network. The neural similarity is measured by the correlation between the neural network’s activations and the primate’s neural responses. Middle With the VGG-16 architecture, the CNN-F’s generative feedback increases the correspondence with the primate’s V4 and IT neural responses. Right The CNN-F’s generative feedback produces a drop in accuracy on ILSVRC-12 compared to the CNN.

A CNN has a substantial correspondence with the primate visual cortex SchrimpfKubilius2018BrainScore

. To measure this correspondence, neural predictivity is a benchmark that quantifies the similarity between an artificial response—such as a CNN’s hidden layer activations—and a biological response—such as a neuronal activations. To measure similarity, neural predictivity linearly maps an CNN’s hidden layer activations to the primate visual cortex’s neural activations using a PLS regression model with 25 components 

yamins_using_2016. The Pearson’s from this regression quantifies the neural predictivity.

Given the prevalence of backwards connections in the visual cortex, we investigated whether generative feedback increases the neural predictivity of a CNN with respect to the primate visual system. To investigate whether generative feedback produces object representations closer to primate vision, we trained a CNN-F and a CNN with the VGG-16 architecture on the ILSVRC-12 (ImageNet-2012) dataset simonyan_very_2015. We compared the biological correspondence of the CNN-F and CNN through the Brain-Score, which contains neural similarity benchmarks on V4 and IT (Figure 5).

Figure 5 shows how the models perform on the Brain-Score and ImageNet. In both V4 and IT neural predictivity, the CNN-F has a greater correspondence with the primate brain compared to the CNN (V4: , ; IT: , ) with a decrease in Top-1 ILSVRC-12 classification accuracy. This demonstrates that the CNN-F’s generative feedback—with a CNN’s convolutional layers—corresponds to a significant portion of primate vision neural responses.

4 CNN-F produces more robust object recognition

4.1 CNN-F is robust against adversarial attacks

(a) Standard training. Testing w/ FGSM.
(b) Standard training. Testing w/ PGD-40.
(c) Train CNN-F with different number of iterations. Testing w/ PGD-40.
(d) Evaluate a CNN-F- model with various number of iterations against PGD-40 attack.
Figure 6: Adversarial robustness of CNN-F with standard training. CNN-F achieves higher accuracy on Fashion-MNIST than CNN under standard training on Fashion-MNIST. More iterations are needed for larger adversarial perturbation magnitude. CNN-F- stands for CNN-F trained with iterations; PGD-40 stands for a PGD attack with 40 steps.
Attack methods

We explore various ways to attack the CNN-F. First, we attempt to perform an end-to-end BPDA (athalye2018obfuscated) attack on CNN-F. Due to the approximation of non-differentiable activation operators and the depth of the unrolled CNN-F, the effectiveness of this attack degrades. Second, we attack the first feedforward pass of CNN-F. This is equivalent to attacking a CNN with the same parameters and attempting to transfer the attack from the CNN over to the CNN-F. We call this method transfer attack for short. Transfer attack overcomes the obfuscated gradient issue, and is more effective than end-to-end attack. Therefore, the adversarial accuracy we present here is against transfer attack on the cross entropy loss. We use the Fast Gradient Sign Attack Method (FGSM) goodfellow2014explaining Projected Gradient Descent (PGD) method to attack. For PGD attack, we generate adversarial samples within -norm constraint, and denote the maximum -norm between adversarial images and clean images as .

Standard Training

We train a CNN-F model with two convolutional layers followed by two fully-connected layers. For training details and results on MNIST dataset, please refer to Appendix B. For all the figures in the paper, the reported accuracy is averaged over 5 runs and the error bar indicates standard deviation. We test the robustness of CNN-F under two settings: standard training on clean images and adversarial training (madry2017towards), and show improvement of CNN-F compared to CNN in both settings. The results for standard training are shown in Figure 6. We see that CNN-F has considerably better robustness than CNN. Furthermore, training and evaluating CNN-F with more iterations both improve robustness, and we see larger improvements for higher (Figure 5(c), 5(d)). This indicates that recurrent feedback is crucial for recognizing challenging images.

Adversarial Training

Instead of training CNN-F solely on adversarial images as conventional adversarial training madry2017towards, we train CNN-F with both clean images and adversarial images in a data augmentation manner. We use cross entropy loss on both clean images and adversarial images, and we let the CNN-F to reconstruct adversarial samples to the corresponding clean images. We find that training CNN-F in this way mitigate the overfitting problem in conventional adversarial training. As shown in Figure 7, CNN-F-5 trained with both clean images and adversarial images achieves high accuracy on both clean images and adversarial images.

Furthermore, CNN-F generalizes better to unseen attacks compared to CNN. Figure 7(a) shows that CNN-F trained with FGSM adversarial images generalizes better to PGD-40 attacks compared to CNN. Figure 7(b) shows that CNN-F trained with PGD-40 attack suffers less from accuracy drop against adversaries with larger strength.

(a) Adv. training w/ FGSM . Testing w/ FGSM.
(b) Adv. training w/ PGD-40 . Testing w/ PGD-40.
Figure 7: Adversarial robustness of CNN-F with adversarial training. CNN-F achieves higher adversarial accuracy and natural accuracy than CNN when trained with both clean and adversarial images on Fashion-MNIST.
(a) Adv. training w/ FGSM . Testing w/ PGD-40.
(b) Adv. training w/ PGD-40 . Testing w/ PGD-40.
Figure 8: CNN-F generalize better to unseen attacks than CNN. (a) Performance of CNN-F trained with FGSM adversarial samples against PGD-40 adversaries. (b) Performance of CNN-F trained with PGD-40 adversarial samples with against adversaries with different strength.

4.2 The generative feedback in CNN-F models restores perturbed images

Figure 9: The generative feedback in CNN-F models restores perturbed images. a, The decision cell cross-sections for a CNN-F trained on Fashion-MNIST. Arrows visualize the feedback direction on the cross-section. b, Fashion-MNIST classification accuracy on PGD adversarial examples; Grad-CAM activations visualize the CNN-F model’s attention from incorrect (iter. 1) to correct predictions (iter. 2). c, Grad-CAM activations across different feedback iterations in the CNN-F. d, From left to right: clean images, corrupted images, and images restored by the CNN-F’s feedback.

Given that CNN-F models are robust to adversarial attacks, we examine the models’ mechanism for robustness. Studies suggest that feedback in the visual cortex is crucial to robust object recognition (kar2019evidence; elsayed2018adversarial)

. We investigate this principle with generative feedback in CNN-F models. We train a CNN-F model on Fashion-MNIST. A validation image is then selected from Fashion-MNIST. Using the image’s two largest principal components, a two-dimensional hyperplane 

intersects the image with the image at the center. Vector arrows visualize the generative feedback’s perturbation on the hyperplane’s position. In Figure 

9 (a), we find that generative feedback perturbs samples across decision boundaries toward the validation image. This demonstrates that the CNN-F’s generative feedback can restore perturbed, distorted images to their uncorrupted objects.

We further explore this principle with regard to adversarial examples. The CNN-F model can correct initially wrong predictions. Figure 9 (b) Grad-CAM activations visualize the network’s attention from an incorrect prediction to a correct prediction on PGD-40 adversarial samples (selvaraju2017grad). To correct predictions, the CNN-F model does not initially focus on specific features. Rather, it either identifies the entire object or the entire image. With generative feedback, the CNN-F begins to focus on specific features. This is reproduced in clean images as well as images corrupted by blurring and additive noise 9 (c). Furthermore, with these perceptible corruptions, the CNN-F model can reconstruct the clean image with generative feedback 9 (d). This demonstrates that the generative feedback is one mechanism that restores perturbed images.

5 Related work

Robust neural networks with latent variables

Latent variable models are a unifying theme in robust neural networks. The consciousness prior bengio_consciousness_2019 postulates that natural representations—such as language—operate in a low-dimensional space, which may restrict expressivity but also may facilitate rapid learning. If adversarial attack introduce examples outside this low-dimensional manifold, latent variable models can map these samples back to the manifold. A related mechanism for robustness is state reification lamb_state-reification_2019

. Similar to self-consistency, state reification models the distribution of hidden states over the training data. It then maps less likely states to more likely states. MagNet and Denoising Feature Matching introduce similar mechanisms: using autoencoders on the input space to detect adversarial examples and restore them in the input space 

meng_magnet_2017; warde-farley+al-2017-denoisegan-iclr

. Lastly, Defense-GAN proposes a generative adversarial network to approximate the data manifold 

samangouei2018defense. CNN-F generalizes these themes into a Bayesian framework. Intuitively, CNN-F can be viewed as an autoencoder. In contrast to standard autoencoders, CNN-F requires stronger constraints through Bayes rule. CNN-F—through self-consistency—constrains the generated image to satisfy the maximum a posteriori on the predicted output.

Computational models of human vision

Recurrent models and Bayesian inference have been two prevalent concepts in computational visual neuroscience. Recently, kubilius_cornet_2018 proposed CORnet as a more accurate model of human vision by modeling recurrent cortical pathways. Like CNN-F, they show CORnet has a larger V4 and IT neural similarity compared to a CNN with similar weights. linsley_learning_2018 suggests hGRU as another recurrent model of vision. Distinct from other models, hGRU models lateral pathways in the visual cortex to global contextual information. While Bayesian inference is a candidate for visual perception, a Bayesian framework is absent in these models. The recursive cortical network (RCN) proposes a hierarchal conditional random field as a model for visual perception george_generative_2017. In contrast to neural networks, RCN uses belief propagation for both training and inference. With the representational ability of neural networks, we propose CNN-F to approximate Bayesian inference with recurrent circuits in neural networks.

Feedback networks

Feedback Network zamir2017feedback uses convLSTM as building blocks and adds skip connections between different time steps. This architecture enables early prediction and enforces hierarchical structure in the label space. nayebi2018task uses architecture search to design local recurrent cells and long range feedback to boost classification accuracy. wen2018deep

designs a bi-directional recurrent neural network by recursively performing bottom up and top down computations. The model achieves more accurate and definitive image classification. Despite the promising progress on using feedback to improve classification accuracy on clean images, none of the these works aim to improve classification robustness.

Combining top-down and bottom-up signals in RNNs

mittal2020learning proposes combining attention and modularity mechanisms to route bottom-up (feedforward) and top-down (feedback) signals. They extend the Recurrent Independent Mechanisms (RIMs) goyal2019recurrent framework to a bidirectional structure such that each layer of the hierarchy can send information in both bottom-up direction and top-down direction. Our approach uses approximate Bayesian inference to provide top-down communication, which is more consistent with the Bayesian brain framework and predictive coding.

6 Conclusion

Inspired by the feedback connections in the brain, we propose to introduce recurrent generative feedback to neural networks. We instantiate the framework on CNN and term the model as CNN-F. We then demonstrate that the proposed feedback mechanism significantly improves the adversarial robustness in CNN. We visualized the dynamical behavior of CNN-F and shows its capability of restoring corrupted images. Furthermore, we find that the generative feedback of CNN-F predicts a significant portion of primate vision neural responses.


We thank Francisco Luongo and Haotao Wang for useful discussions. Y. Huang is supported by DARPA LwLL grants. J. Gornet is supported by supported by the NIH Predoctoral Training in Quantitative Neuroscience 1T32NS105595-01A1. D. Y. Tsao is supported by Howard Hughes Medical Institute and Tianqiao and Chrissy Chen Institute for Neuroscience. A. Anandkumar is supported in part by Bren endowed chair, DARPA LwLL grants, Tianqiao and Chrissy Chen Institute for Neuroscience, Microsoft, Google, and Adobe faculty fellowships.



Appendix A Deconvolutional Generative Model

a.1 Generative model

We choose the deconvolutional generative model (DGM) NRM as the generative feedback in CNN-F. The graphical model of the DGM is shown in Figure 2 (middle). The DGM has the same architecture as CNN and generates images from high level to low level. Since low level features usually have higher dimension than high level features, the DGM introduces latent variables at each level to account for uncertainty in the generation process.
Let be label, is the number of classes. Let be image and be encoded features of after convolutional layers. In a DGM with layers in total, denotes generated feature map at layer , and denotes latent variables at layer . We use and to denote latent variables at a layer followed by and respectively. In addition, we use to denote the

th entry in a tensor. Let

and be the weight parameters and bias parameters at layer in the DGM. We use to denote deconvolutional transpose in deconvolutional layers and to denote matrix transpose in fully connected layers. In addition, we use and to denote upsampling and downsampling respectively. The generation process in the DGM is as follows:


In the above generation process, we generate all the way to the image level. If we choose to stop at layer to generate image features , the final generation step is instead of 12.

The joint distribution of latent variables from layer to conditioning on is


where .

a.2 Proof for Theorem 2.1.B

In this section, we provide proofs for 2.1.B. In the proofs, we use to denote the feedforward feature map after convolutional layer in the CNN of the same architecture as the DGM, and use to denote layers after nonlinear operators. Let be the logits output from fully-connected layer of the CNN. Without loss of generality, we consider a DGM that has the following architecture. We list the corresponding feedforward feature maps on the left column:

We prove Theorem 2.1.B which states that CNN with and is the generative classifier derived from the DGM by proving Lemma A.1 first.

Definition A.1.

and are nonlinear operators that adaptively choose how to activate the feedforward feature map based on the sign of the feedback feature map.

Definition A.2 (generative classifier).

Let be the logits output of a CNN, and be the joint distribution specified by a generative model. A CNN is a generative classifier of a generative model if .

Lemma A.1.

Let be the label and be the image. is the logits output of the CNN that has the same architecture and parameters as the DGM. is the generated image from the DGM. is a constant. . Then we have



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 . Recall from 12, since the DGM assumes to follow a Gaussian distribution centered at , the inner product between and is related to . Recall from equation 13 that conditionoal distribution of latent variables in the DGM is parameterized by . Using these insights, we can use Lemma A.1 to show that CNN performs Bayesian inference in the DGM.

In the proof, the fully-connected layer applies a linear transformation to the input without any bias added. For fully-connected layer with bias term, we modify

to :

The logits are computed by

Following a very similar proof as of Lemma A.1, we can show that


With Lemma A.1, we can prove Theorem 2.1.B.

Assumption A.1.

The generated image from the DGM has a constant norm:

Assumption A.2.

Prior distribution on the label is a uniform distribution:

Assumption A.3.

Normalization factor in for each category is constant:

Theorem (Theorem 2.1.B).

Under assumptions A.1, A.2 and A.3, CNN with and is the generative classifier derived from the DGM.


We use to denote the joint distribution specified by the DGM. In addition, we use to denote the Softmax output from the CNN, i.e. To simplify the notation, we use instead of to denote latent variables across layers.

(Assumption A.1)
(Assumption A.3)
(Lemma A.1)

We obtain line for the following reasons: according to Assumption A.2, and because only is variable, and are given. We obtained line because given and , the logits output are fixed. Therefore, Take exponential on both sides of the above equation, we have:


where is a scale factor.
Since both and are distributions, we have and . Summing over on both sides of equation 17, we have Therefore, we have . ∎

We have proved that CNN with and is the generative classifier derived from the DGM that generates to layer . In fact, we can extend the results to all intermediate layers in the DGM.

Assumption A.4.

Each generated layer in the DGM has a constant norm:

Assumption A.5.

Normalization factor in up to each layer is constant:

Corollary A.1.1.

Under assumptions A.4, A.2 and A.5, CNN with and starting with an intermediate layer is the generative classifier derived from the DGM that generates to the same intermediate layer:

a.3 Proof for Theorem 2.1.C

In this section, we provide proofs for 2.1.C. In the proofs, we inherit the notations that we use for proving 2.1.B. Without loss of generality, we consider a DGM that has the same architecture as the one we use to prove 2.1.B.

Theorem (Theorem 2.1.C).

Under assumptions A.1, A.2 and A.3, MAP estimate of conditioned on and in the DGM is:

(Assumption A.3 and A.2)
(Assumption A.1)

Using Lemma A.1, the MAP estimate of is:

The MAP estimate of is:

a.4 Incorporating instance normalization in the DGM

Inspired by the constant norm assumptions (Assumption A.1 and A.4), we incorporate instance normalization into the DGM. We use to denote instance normalization, and to denote layers after instance normalization. In this section, we prove that with instance normalization, CNN is still the generative classifier derived from the DGM. Without loss of generality, we consider a DGM that has the following architecture. We list the corresponding feedforward feature maps on the left column:

Assumption A.6.

Feedforward feature maps and feedback feature maps have the same norm:

Lemma A.2.

Let be the label and be the image. is the logits output of the CNN that has the same architecture and parameters as the DGM. is the generated image from the DGM, and is normalized by norm. is a constant. . Then we have

(Assumption A.6)
(Assumption A.6)
(Assumption A.6)

Theorem A.3.

Under assumptions A.6 and A.2, CNN with and and instance normalization is the generative classifier derived from the DGM with instance normalization.


The proof of Theorem A.3 is very similar to that of Theorem 2.1 using Lemma A.2. Therefore, we omit the detailed proof here. ∎


The instance normalization that we incorporate into the DGM is not the same as the instance normalization that people typically used for image stylization ulyanov2016instance. The conventional instance normalization computes output from input as , where