Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Here, we show that by simply injecting symmetric, random, noise during training in reconstruction or classification tasks, artificial neural networks with ReLU activation functions eliminate this difference; the neurons converge to a sparse coding solution where only a small fraction are active for any input.

The review comment to be evaluated is:

Strengths:
* The paper tackles the important problem of bridging the gap between biological plausible learning and current artificial neural networks.

Weaknesses:
* The paper mostly focuses on 1-hidden layer networks. It is not too clear why and how these results are relevant for practice, where typically much deeper networks are considered. For classification tasks, the observed sparsity seems to be much less pronounced.
* Unlike claimed in the paper, using the L1-norm can lead to biologically plausible and sparse features even for classification tasks, see e.g. Figure 4 in "Towards Learning Convolutions from Scratch" (https://papers.nips.cc/paper/2020/file/5c528e25e1fdeaf9d8160dc24dbf4d60-Paper.pdf)
* There are some issues regarding novelty/clarity (see next section)

The evidence to be evaluated is:Gaussian noise injection has been interpreted as a form of model regularization, to help avoid overfitting (Zur et al., 2009) and improve generalization (Sietsma and Dow, 1991; Matsuoka, 1992). It was later shown that noise injection is in fact equivalent to having a regularization term that seeks to minimize the \(L_{2}\) norm of the network's Jacobian (Bishop, 1995; Rifai et al., 2011; Alain and Bengio, 2014). From this perspective, it makes sense that sparsity can be used to turn off neurons and minimize this Jacobian, providing a potential explanation for our results.

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network. Learning reconstructions of training data with sparse activations is the mainstay of sparse coding (Olshausen & Field, 1997). Sparse coding uses an overcomplete basis to learn a reconstruction of data with an \(L_{1}\) regularization penalty on neuron activity.

The objective function aims to minimize the mean squared-error between the reconstructed image and the original image while having sparse neuron activity: \(\operatorname*{arg\,min}_{\Phi,\mathbf{n}}(\mathbf{x}-\hat{\mathbf{x}})^{2}+ \lambda\sum_{i}^{N}|a_{i}|\). This sparse coding task is an example of a statistical model converging with neuroscience because, in addition to producing sparse representations, the basis functions learn to become Gabor filters like those found in area V1 of the visual system (Olshausen & Field, 1997). Interestingly, the ReLU activation function that is crucial for the results of this paper can be derived as the optimal solution to a simplified sparse coding problem when we assume that the basis vectors are orthogonal (Ba, 2020; Gregor & LeCun, 2010). This results in the optimal solution for activity \(a_{i}\) to be given by \(\operatorname*{ReLU}(u_{i}-\lambda)-\operatorname*{ReLU}(-u_{i}-\lambda)\) or, if we constrain \(a_{i}\) to be non-negative, \(\operatorname*{ReLU}(u_{i}-\lambda)\), where \(u_{i}\) is the value obtained by projecting the data onto basis \(\phi_{i}\). Intuitively, if the values of our target image \(\mathbf{x}\) reconstructed by \(a_{i}\phi_{i}\) are too small, it is better for \(a_{i}=0\) than to predict \(\mathbf{x}\), where the threshold for this tradeoff is given by \(\lambda\), which behaves as an activation threshold.

These Top-K networks, when trained with noise, are also closely related to associative memory models such as Sparse Distributed Memory (SDM) and Modern Hopfield Networks that activate a sparse subset of neurons and were developed to handle noisy queries (Kanerva, 1988; Krotov & Hopfield, 2016).

By default, we train a single hidden layer ReLU network with different amounts of random Gaussian noise applied to the input. The objective is to remove the noise and reconstruct the original input. Formally, noise is applied from an isotropic Gaussian with variance \(\sigma\) to the input \(\mathbf{x}\). The corrupted image is then put through our encoder and decoder.

\[\tilde{\mathbf{x}} =\mathbf{x}+\epsilon,\ \ \epsilon\sim N(0,\sigma I)\] \[\hat{\mathbf{x}} =W_{d}\ \text{ReLU}(W_{e}\tilde{\mathbf{x}}+\mathbf{b}_{e})+ \mathbf{b}_{d}\] \[\text{loss} =\sum_{\mathbf{x}\in X}(\mathbf{x}-\hat{\mathbf{x}})^{2}. \tag{1}\] We use Kaiming randomly initialized weights (He et al., 2015) and train until the fraction of active neurons converges.

Figure 2 shows that a single hidden layer ReLU network with 10,000 neurons learns to become a Top-K network with smaller \(k\) values as the amount of noise increases. This network was trained on the CIFAR10 embeddings to mimick training within a deeper network layer and where the manifold is smoother than when training on raw image pixels. While Fig. 2 only shows the mean number of neurons that are active across all 50,000 CIFAR10 inputs per epoch, Figure 3 shows the fraction of active neurons for every input.

It is clear that these values all synchronize to a value decided on by the network and are responsible for the sparse \(k\) neurons remaining active after the ReLU nonlinearity is applied. As noise increases, the bias terms become more negative to raise the activation threshold and increase sparsity. The \(L_{2}\) norms of the key vectors also become smaller. We hypothesize this is a counter-response to the \(L_{2}\) norms of the data increasing with noise and explains why in high noise settings (\(\sigma\geq 0.8\)) neuron activations linger around \(\sim\)50% sparsity before rapidly becoming very sparse (see Fig. 2). Before the key vector \(L_{2}\) norms are sufficiently small, noisy data with large \(L_{2}\) norms produce very large neuron activations (both positive and negative) and the bias terms initialized close to 0 have no real effect on which neurons are active. Because the key vectors are randomly initialized from a symmetric distribution around 0, in expectation half of the neurons are active after the ReLU nonlinearity is applied. This means the network must wait to first shrink its key vector \(L_{2}\) norms before the bias terms can affect activation sparsity.

Looking more closely at the sparse solutions found by our ReLU networks, we discovered that for \(\sigma\geq 0.3\), neurons begin to form receptive fields reminiscent of V1 Gabor filters and on/off center surround retinal ganglion cells (Sterling & Laughlin, 2015; Olshausen & Field, 1997).

This is a similar result to that of sparse coding (Olshausen & Field, 1997), however, here we use noise with the ReLU activation function instead of a sparsity penalty and train on full images instead of 8x8 patches.

When we tried training the network without any noise but instead with an \(L_{1}\) activation penalty motivated by the original sparse coding setup (Olshausen & Field, 1997), no biological receptive fields were observed. In addition, this approach resulted in unstable training, more dead neurons, less sparsity, and no synchronization of key vector norms or bias terms.1 Footnote 1: We did not implement the full sparse coding framework, only the \(L_{1}\) penalty, however, this shows that it is insufficient to implement sparse coding while noisy training is.

With these models we find that our networks become even more sparse and do so more quickly, as we would expect due to the explaining away of redundant neurons (see Appendix C).

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse.

Fig. 6 shows that the sparsity levels remain largely unperturbed.

In fact, for classification \(\sigma\in\{0.1,0.3\}\) both get 94.3% validation accuracy instead of the 93.4% that the baseline gets, showing slightly better generalization while also being more sparse (66% for baseline versus 55% for \(\sigma=0.1\) and 45% for \(\sigma=0.3\)).

**Model Ablations -** Figure 7 shows the mean neuron sparsity at network convergence across a number of different experimental conditions: (i) three datasets: CIFAR10 pixels, CIFAR10 latent embeddings, MNIST; (ii) two noise distributions: Gaussian and Laplace; (iii) four numbers of neurons: 100, 1,000, 10,000 and 100,000; (iv) three training tasks: reconstruction, classification, and Transformer next token prediction.

We found that every optimizer with a high enough learning rate converged to a sparse Top-K solution without creating dead neurons.2

We found for the latent CIFAR10 dataset, both sigmoid and GELU (Hendrycks and Gimpel, 2016) had no sparsity but while sigmoid performs worse, GELU does just as well as ReLU. If we raised our arbitrary activation threshold to \(<0.01\) we found that GELU does become sparse with respect to noise but still not as sparse as ReLU while sigmoid remains fully dense (see Appendix E).

While still neither activation function becomes as sparse as ReLU, the GELU networks implement the same Top-K convergence as ReLU shown in Fig. 3 and form biologically plausible receptive fields like those of Fig. 5.

The correlation between noise and activation sparsity remains and is again retained even after noise is removed.

Figure 7(d) shows the final sparsity amounts of the first model layer with each of the other layers giving similar results with \(\sim\)3x to \(\sim\)9x more sparsity when \(\sigma=3.0\), compared to \(\sigma=0.0\).

An increase in noise will create more erroneous low-level activations in neurons that must be ignored to maximize signal to noise ratio. Therefore, during noisy training, the model must learn weight vectors to encode information about the data distribution that can be separated from the noise using only ReLU functions.

Similarly, we hypothesize that the encoding layer in our network is learning such projections, so that thresholding performs noise rejection while also producing sparse activations.

Our initial attempt at doing so did not produce performance gains, though it did increase sparsity.

Using deep learning to give insights into neuroscience, this result suggests that in the presence of noise, the biological sparse coding solution is optimal. Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

**Conclusion -** We have shown that the realistic assumption of noisy training data is sufficient for feed-forward layers to organically become highly sparse and retain their sparsity, even after noise is removed. This result is particularly striking because of the way the networks converge with sparse coding, producing biological receptive fields and approximating an inhibitory interneuron that implements a Top-K operation. Combining noisy training with other approaches to induce sparsity, including taking additional ideas from sparse coding, may result in even higher degrees of sparsity. More broadly, this work builds a new bridge between artificial and biological neural networks by showing how they can be more similar than they may otherwise appear.

Here we show additional images of the receptive fields for networks trained with different amounts of noise (Fig. 8), the queries used to activate these neurons (Fig. 9) and how the receptive fields look more biological as the network is trained for longer (Fig. 10).

Training the same networks as before but using Eq. 2 with \(t^{*}=1\) and \(\gamma=1\) we present our results in Table 1. We show the sparsity amounts of the networks in Fig. 11.

Removing the bias terms and relying just on the inhibitory interneuron, we found that it performs well (Table 1) and becomes sparse as a function of noise but results in a number of dead neurons and is not as sparse as the baseline ReLU or Explaining Away models.

Neither of these networks are as sparse as the ReLU network. Also neither network creates sparsity through dead neurons, but given that all neurons have non-zero absolute activity, having dead neurons is impossible. While we just show the mean active neurons here, both solutions look like those of Fig. 3 with the neurons for all inputs converging to a Top-K solution where the bias terms become negative and synchronize along with the key vector \(L_{2}\) norms.

Input: Table 1: Reconstruction Validation Accuracy - The baseline ReLU model varying the number of neurons, compared to the shared bias, explaining away, and inhibitory interneuron networks. We only show noise up to σ = 0.8 because the higher noise levels present trivial solutions that all networks perform equally well on. Due to very high convergence between random seeds, we just present a single run for each model.

Input: Figure 7: Training ablations. The positive correlation between noise and sparsity is robust to (a) neuron counts, (b) datasets, (c) noise distributions, and (d) training tasks.

Input: Figure 2: There is a positive relationship between activation sparsity and noise. The average fraction of neurons active during each CIFAR10 latent input over the course of training for 800 epochs. Each line corresponds to training randomly initialized networks with different noise levels σ denoted by different colors. We show the average of three different random seeds and their standard error of the mean (not visible as the variance is so low). Interestingly, the higher noise levels take longer to sparsify but upon convergence become the sparsest.

Input: Figure 3: Noise induces the formation of Top-K networks. Shown column-wise are the fraction of neurons that are active for each training input (Left), the values of each of the 10,000 neurons bias terms (Middle) and the L2 norms of their key vectors (Right), as a function of number of training epochs. Each is shown for three different representative noise levels (rows). These plots show each input to the network and use the density of blue to represent how many times a particular value occurs.

Input: Figure 5: Biological receptive fields form with noise. Networks trained on CIFAR10 pixels with σ = 0.8 Gaussian noise. The most active 125 neuron receptive fields are reshaped into 32x32x3 and re-scaled so their values span the range of pixel values [0, 255]. Neurons are sorted by their activity levels (top left is the most active, going across rows then down columns) for a random noisy input of a car. Starting at σ ≥ 0.1 and particularly for σ = 0.8 (shown here) we observe the formation of many Gabor filters and on/off center surround receptive fields that are unique to noisy training.

Input: Figure 6: Noisy pretraining retains network sparsity. The vertical purple lines denote the start of noise annealing where each noise level is linearly decayed down to σ = 0. For both reconstruction (left) and classification (right) tasks the networks remain highly sparse even after all noise is removed. Moreover, there is no noticeable performance difference between any of the networks while some of those trained with moderate noise perform slightly better than the noise free baseline. We truncate the x-axis to start at epoch 600 for the sake of clarity. The classification networks continue to sparsify even when noise annealing has started because unlike in the reconstruction task, the sparsity level did not converge within the first 800 epochs.

Input: Figure 8: Receptive fields across noise levels. We show for all noise levels the 625 most active neurons for the truck query (middle of Fig. 9). For 0.8 ≥ σ which activate fewer neurons, we also show the cat query below it. Biological like receptive fields start to appear for σ = 0.1 and can be seen most clearly for σ ∈ {0.3, 0.8, 1.5}.

Input: Figure 12: Dead Neurons during the reconstruction task. During noise annealing, the higher noise levels σ ≥ 0.8 see a number of dead neurons. However, this does not full explain their sparsity levels and on this trivial task it does not hurt their reconstruction performance.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
I'll assess the claim: "Here, we show that by simply injecting symmetric, random, noise during training in reconstruction or classification tasks, artificial neural networks with ReLU activation functions eliminate this difference; the neurons converge to a sparse coding solution where only a small fraction are active for any input."

Summary of evaluation:
The evidence strongly supports the main aspects of this claim through extensive experimental results. Figures 2, 3, and 7 demonstrate clear convergence to sparse solutions across multiple conditions. The sparsity is retained even after noise removal (Fig. 6). However, the claim slightly overstates by saying it "eliminates this difference" when the reviewer points out the sparsity is less pronounced in classification tasks, and results focus heavily on single-layer networks. The evidence shows the effect is strong but not complete elimination.

<score>2</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Here, we show that by simply injecting symmetric, random, noise during training in reconstruction or classification tasks, artificial neural networks with ReLU activation functions eliminate this difference; the neurons converge to a sparse coding solution where only a small fraction are active for any input.

The review comment to be evaluated is:

In this empirical paper, the authors observe the correlation between the sparsity of the learned representations and the level of noise injected into the input of Neural Networks with the ReLU activation function. Experiments have been carried out on CIFAR10 both on pixel data and on a neural representation, MNIST, and on the WikiText-103 dataset. They report a positive correlation between the magnitude of the injected noise and the sparsity of the representation. They observe Gabor-like filters when reconstructing with noise on pixel data. For the classification task, the performance of the model did not decrease with sparsity. Also, compared with using only the top-K activations of a model learned without noise, the models learned with noise gave better validation accuracies.

The evidence to be evaluated is:Gaussian noise injection has been interpreted as a form of model regularization, to help avoid overfitting (Zur et al., 2009) and improve generalization (Sietsma and Dow, 1991; Matsuoka, 1992). It was later shown that noise injection is in fact equivalent to having a regularization term that seeks to minimize the \(L_{2}\) norm of the network's Jacobian (Bishop, 1995; Rifai et al., 2011; Alain and Bengio, 2014). From this perspective, it makes sense that sparsity can be used to turn off neurons and minimize this Jacobian, providing a potential explanation for our results.

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network. Learning reconstructions of training data with sparse activations is the mainstay of sparse coding (Olshausen & Field, 1997). Sparse coding uses an overcomplete basis to learn a reconstruction of data with an \(L_{1}\) regularization penalty on neuron activity.

The objective function aims to minimize the mean squared-error between the reconstructed image and the original image while having sparse neuron activity: \(\operatorname*{arg\,min}_{\Phi,\mathbf{n}}(\mathbf{x}-\hat{\mathbf{x}})^{2}+ \lambda\sum_{i}^{N}|a_{i}|\). This sparse coding task is an example of a statistical model converging with neuroscience because, in addition to producing sparse representations, the basis functions learn to become Gabor filters like those found in area V1 of the visual system (Olshausen & Field, 1997). Interestingly, the ReLU activation function that is crucial for the results of this paper can be derived as the optimal solution to a simplified sparse coding problem when we assume that the basis vectors are orthogonal (Ba, 2020; Gregor & LeCun, 2010). This results in the optimal solution for activity \(a_{i}\) to be given by \(\operatorname*{ReLU}(u_{i}-\lambda)-\operatorname*{ReLU}(-u_{i}-\lambda)\) or, if we constrain \(a_{i}\) to be non-negative, \(\operatorname*{ReLU}(u_{i}-\lambda)\), where \(u_{i}\) is the value obtained by projecting the data onto basis \(\phi_{i}\). Intuitively, if the values of our target image \(\mathbf{x}\) reconstructed by \(a_{i}\phi_{i}\) are too small, it is better for \(a_{i}=0\) than to predict \(\mathbf{x}\), where the threshold for this tradeoff is given by \(\lambda\), which behaves as an activation threshold.

These Top-K networks, when trained with noise, are also closely related to associative memory models such as Sparse Distributed Memory (SDM) and Modern Hopfield Networks that activate a sparse subset of neurons and were developed to handle noisy queries (Kanerva, 1988; Krotov & Hopfield, 2016).

By default, we train a single hidden layer ReLU network with different amounts of random Gaussian noise applied to the input. The objective is to remove the noise and reconstruct the original input. Formally, noise is applied from an isotropic Gaussian with variance \(\sigma\) to the input \(\mathbf{x}\). The corrupted image is then put through our encoder and decoder.

\[\tilde{\mathbf{x}} =\mathbf{x}+\epsilon,\ \ \epsilon\sim N(0,\sigma I)\] \[\hat{\mathbf{x}} =W_{d}\ \text{ReLU}(W_{e}\tilde{\mathbf{x}}+\mathbf{b}_{e})+ \mathbf{b}_{d}\] \[\text{loss} =\sum_{\mathbf{x}\in X}(\mathbf{x}-\hat{\mathbf{x}})^{2}. \tag{1}\] We use Kaiming randomly initialized weights (He et al., 2015) and train until the fraction of active neurons converges.

Figure 2 shows that a single hidden layer ReLU network with 10,000 neurons learns to become a Top-K network with smaller \(k\) values as the amount of noise increases. This network was trained on the CIFAR10 embeddings to mimick training within a deeper network layer and where the manifold is smoother than when training on raw image pixels. While Fig. 2 only shows the mean number of neurons that are active across all 50,000 CIFAR10 inputs per epoch, Figure 3 shows the fraction of active neurons for every input.

It is clear that these values all synchronize to a value decided on by the network and are responsible for the sparse \(k\) neurons remaining active after the ReLU nonlinearity is applied. As noise increases, the bias terms become more negative to raise the activation threshold and increase sparsity. The \(L_{2}\) norms of the key vectors also become smaller. We hypothesize this is a counter-response to the \(L_{2}\) norms of the data increasing with noise and explains why in high noise settings (\(\sigma\geq 0.8\)) neuron activations linger around \(\sim\)50% sparsity before rapidly becoming very sparse (see Fig. 2). Before the key vector \(L_{2}\) norms are sufficiently small, noisy data with large \(L_{2}\) norms produce very large neuron activations (both positive and negative) and the bias terms initialized close to 0 have no real effect on which neurons are active. Because the key vectors are randomly initialized from a symmetric distribution around 0, in expectation half of the neurons are active after the ReLU nonlinearity is applied. This means the network must wait to first shrink its key vector \(L_{2}\) norms before the bias terms can affect activation sparsity.

Looking more closely at the sparse solutions found by our ReLU networks, we discovered that for \(\sigma\geq 0.3\), neurons begin to form receptive fields reminiscent of V1 Gabor filters and on/off center surround retinal ganglion cells (Sterling & Laughlin, 2015; Olshausen & Field, 1997).

This is a similar result to that of sparse coding (Olshausen & Field, 1997), however, here we use noise with the ReLU activation function instead of a sparsity penalty and train on full images instead of 8x8 patches.

When we tried training the network without any noise but instead with an \(L_{1}\) activation penalty motivated by the original sparse coding setup (Olshausen & Field, 1997), no biological receptive fields were observed. In addition, this approach resulted in unstable training, more dead neurons, less sparsity, and no synchronization of key vector norms or bias terms.1 Footnote 1: We did not implement the full sparse coding framework, only the \(L_{1}\) penalty, however, this shows that it is insufficient to implement sparse coding while noisy training is.

With these models we find that our networks become even more sparse and do so more quickly, as we would expect due to the explaining away of redundant neurons (see Appendix C).

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse.

Fig. 6 shows that the sparsity levels remain largely unperturbed.

In fact, for classification \(\sigma\in\{0.1,0.3\}\) both get 94.3% validation accuracy instead of the 93.4% that the baseline gets, showing slightly better generalization while also being more sparse (66% for baseline versus 55% for \(\sigma=0.1\) and 45% for \(\sigma=0.3\)).

**Model Ablations -** Figure 7 shows the mean neuron sparsity at network convergence across a number of different experimental conditions: (i) three datasets: CIFAR10 pixels, CIFAR10 latent embeddings, MNIST; (ii) two noise distributions: Gaussian and Laplace; (iii) four numbers of neurons: 100, 1,000, 10,000 and 100,000; (iv) three training tasks: reconstruction, classification, and Transformer next token prediction.

We found that every optimizer with a high enough learning rate converged to a sparse Top-K solution without creating dead neurons.2

We found for the latent CIFAR10 dataset, both sigmoid and GELU (Hendrycks and Gimpel, 2016) had no sparsity but while sigmoid performs worse, GELU does just as well as ReLU. If we raised our arbitrary activation threshold to \(<0.01\) we found that GELU does become sparse with respect to noise but still not as sparse as ReLU while sigmoid remains fully dense (see Appendix E).

While still neither activation function becomes as sparse as ReLU, the GELU networks implement the same Top-K convergence as ReLU shown in Fig. 3 and form biologically plausible receptive fields like those of Fig. 5.

The correlation between noise and activation sparsity remains and is again retained even after noise is removed.

Figure 7(d) shows the final sparsity amounts of the first model layer with each of the other layers giving similar results with \(\sim\)3x to \(\sim\)9x more sparsity when \(\sigma=3.0\), compared to \(\sigma=0.0\).

An increase in noise will create more erroneous low-level activations in neurons that must be ignored to maximize signal to noise ratio. Therefore, during noisy training, the model must learn weight vectors to encode information about the data distribution that can be separated from the noise using only ReLU functions.

Similarly, we hypothesize that the encoding layer in our network is learning such projections, so that thresholding performs noise rejection while also producing sparse activations.

Our initial attempt at doing so did not produce performance gains, though it did increase sparsity.

Using deep learning to give insights into neuroscience, this result suggests that in the presence of noise, the biological sparse coding solution is optimal. Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

**Conclusion -** We have shown that the realistic assumption of noisy training data is sufficient for feed-forward layers to organically become highly sparse and retain their sparsity, even after noise is removed. This result is particularly striking because of the way the networks converge with sparse coding, producing biological receptive fields and approximating an inhibitory interneuron that implements a Top-K operation. Combining noisy training with other approaches to induce sparsity, including taking additional ideas from sparse coding, may result in even higher degrees of sparsity. More broadly, this work builds a new bridge between artificial and biological neural networks by showing how they can be more similar than they may otherwise appear.

Here we show additional images of the receptive fields for networks trained with different amounts of noise (Fig. 8), the queries used to activate these neurons (Fig. 9) and how the receptive fields look more biological as the network is trained for longer (Fig. 10).

Training the same networks as before but using Eq. 2 with \(t^{*}=1\) and \(\gamma=1\) we present our results in Table 1. We show the sparsity amounts of the networks in Fig. 11.

Removing the bias terms and relying just on the inhibitory interneuron, we found that it performs well (Table 1) and becomes sparse as a function of noise but results in a number of dead neurons and is not as sparse as the baseline ReLU or Explaining Away models.

Neither of these networks are as sparse as the ReLU network. Also neither network creates sparsity through dead neurons, but given that all neurons have non-zero absolute activity, having dead neurons is impossible. While we just show the mean active neurons here, both solutions look like those of Fig. 3 with the neurons for all inputs converging to a Top-K solution where the bias terms become negative and synchronize along with the key vector \(L_{2}\) norms.

Input: Table 1: Reconstruction Validation Accuracy - The baseline ReLU model varying the number of neurons, compared to the shared bias, explaining away, and inhibitory interneuron networks. We only show noise up to σ = 0.8 because the higher noise levels present trivial solutions that all networks perform equally well on. Due to very high convergence between random seeds, we just present a single run for each model.

Input: Figure 7: Training ablations. The positive correlation between noise and sparsity is robust to (a) neuron counts, (b) datasets, (c) noise distributions, and (d) training tasks.

Input: Figure 2: There is a positive relationship between activation sparsity and noise. The average fraction of neurons active during each CIFAR10 latent input over the course of training for 800 epochs. Each line corresponds to training randomly initialized networks with different noise levels σ denoted by different colors. We show the average of three different random seeds and their standard error of the mean (not visible as the variance is so low). Interestingly, the higher noise levels take longer to sparsify but upon convergence become the sparsest.

Input: Figure 3: Noise induces the formation of Top-K networks. Shown column-wise are the fraction of neurons that are active for each training input (Left), the values of each of the 10,000 neurons bias terms (Middle) and the L2 norms of their key vectors (Right), as a function of number of training epochs. Each is shown for three different representative noise levels (rows). These plots show each input to the network and use the density of blue to represent how many times a particular value occurs.

Input: Figure 5: Biological receptive fields form with noise. Networks trained on CIFAR10 pixels with σ = 0.8 Gaussian noise. The most active 125 neuron receptive fields are reshaped into 32x32x3 and re-scaled so their values span the range of pixel values [0, 255]. Neurons are sorted by their activity levels (top left is the most active, going across rows then down columns) for a random noisy input of a car. Starting at σ ≥ 0.1 and particularly for σ = 0.8 (shown here) we observe the formation of many Gabor filters and on/off center surround receptive fields that are unique to noisy training.

Input: Figure 6: Noisy pretraining retains network sparsity. The vertical purple lines denote the start of noise annealing where each noise level is linearly decayed down to σ = 0. For both reconstruction (left) and classification (right) tasks the networks remain highly sparse even after all noise is removed. Moreover, there is no noticeable performance difference between any of the networks while some of those trained with moderate noise perform slightly better than the noise free baseline. We truncate the x-axis to start at epoch 600 for the sake of clarity. The classification networks continue to sparsify even when noise annealing has started because unlike in the reconstruction task, the sparsity level did not converge within the first 800 epochs.

Input: Figure 8: Receptive fields across noise levels. We show for all noise levels the 625 most active neurons for the truck query (middle of Fig. 9). For 0.8 ≥ σ which activate fewer neurons, we also show the cat query below it. Biological like receptive fields start to appear for σ = 0.1 and can be seen most clearly for σ ∈ {0.3, 0.8, 1.5}.

Input: Figure 12: Dead Neurons during the reconstruction task. During noise annealing, the higher noise levels σ ≥ 0.8 see a number of dead neurons. However, this does not full explain their sparsity levels and on this trivial task it does not hurt their reconstruction performance.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, the claim appears to be well-supported. The paper provides extensive experimental evidence demonstrating that injecting random symmetric noise during training leads ReLU networks to converge to sparse coding solutions. The evidence includes:
- Comprehensive ablation studies showing robustness across datasets, architectures, and conditions
- Clear visualization of neuron activation patterns and receptive fields
- Quantitative measurements of sparsity levels
- Theoretical connections to sparse coding literature
- Validation that sparsity persists after noise removal
- Performance metrics showing maintained or improved accuracy

The claim language appropriately reflects the scope and certainty of the findings without overreaching.

<score>1</score>

The claim is well-stated and accurately reflects the empirical evidence presented in the paper.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

The resulting network learns receptive fields like those of primary visual cortex and remains sparse even when noise is removed in later stages of learning.

The evidence to be evaluated is:From this perspective, it makes sense that sparsity can be used to turn off neurons and minimize this Jacobian, providing a potential explanation for our results.

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network.

This sparse coding task is an example of a statistical model converging with neuroscience because, in addition to producing sparse representations, the basis functions learn to become Gabor filters like those found in area V1 of the visual system (Olshausen & Field, 1997).

Figure 2 shows that a single hidden layer ReLU network with 10,000 neurons learns to become a Top-K network with smaller \(k\) values as the amount of noise increases. This network was trained on the CIFAR10 embeddings to mimick training within a deeper network layer and where the manifold is smoother than when training on raw image pixels. While Fig. 2 only shows the mean number of neurons that are active across all 50,000 CIFAR10 inputs per epoch, Figure 3 shows the fraction of active neurons for every input.

It is clear that these values all synchronize to a value decided on by the network and are responsible for the sparse \(k\) neurons remaining active after the ReLU nonlinearity is applied. As noise increases, the bias terms become more negative to raise the activation threshold and increase sparsity.

We hypothesize this is a counter-response to the \(L_{2}\) norms of the data increasing with noise and explains why in high noise settings (\(\sigma\geq 0.8\)) neuron activations linger around \(\sim\)50% sparsity before rapidly becoming very sparse (see Fig. 2).

Because the key vectors are randomly initialized from a symmetric distribution around 0, in expectation half of the neurons are active after the ReLU nonlinearity is applied.

Looking more closely at the sparse solutions found by our ReLU networks, we discovered that for \(\sigma\geq 0.3\), neurons begin to form receptive fields reminiscent of V1 Gabor filters and on/off center surround retinal ganglion cells (Sterling & Laughlin, 2015; Olshausen & Field, 1997).

This is a similar result to that of sparse coding (Olshausen & Field, 1997), however, here we use noise with the ReLU activation function instead of a sparsity penalty and train on full images instead of 8x8 patches.

The longer training progressed, the more similar the receptive fields appeared to those found in biology.

When we tried training the network without any noise but instead with an \(L_{1}\) activation penalty motivated by the original sparse coding setup (Olshausen & Field, 1997), no biological receptive fields were observed.

With these models we find that our networks become even more sparse and do so more quickly, as we would expect due to the explaining away of redundant neurons (see Appendix C).

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse.

Fig. 6 shows that the sparsity levels remain largely unperturbed.

In fact, for classification \(\sigma\in\{0.1,0.3\}\) both get 94.3% validation accuracy instead of the 93.4% that the baseline gets, showing slightly better generalization while also being more sparse (66% for baseline versus 55% for \(\sigma=0.1\) and 45% for \(\sigma=0.3\)).

**Model Ablations -** Figure 7 shows the mean neuron sparsity at network convergence across a number of different experimental conditions: (i) three datasets: CIFAR10 pixels, CIFAR10 latent embeddings, MNIST; (ii) two noise distributions: Gaussian and Laplace; (iii) four numbers of neurons: 100, 1,000, 10,000 and 100,000; (iv) three training tasks: reconstruction, classification, and Transformer next token prediction.

While still neither activation function becomes as sparse as ReLU, the GELU networks implement the same Top-K convergence as ReLU shown in Fig. 3 and form biologically plausible receptive fields like those of Fig. 5.

The correlation between noise and activation sparsity remains and is again retained even after noise is removed.

Figure 7(d) shows the final sparsity amounts of the first model layer with each of the other layers giving similar results with \(\sim\)3x to \(\sim\)9x more sparsity when \(\sigma=3.0\), compared to \(\sigma=0.0\).

In natural images, for example, certain projections (e.g., with Gabor filters) have high kurtosis which allows them to be easily distinguished from noise via thresholding or "coring" (Simoncelli and Adelson, 1996). Similarly, we hypothesize that the encoding layer in our network is learning such projections, so that thresholding performs noise rejection while also producing sparse activations.

Using deep learning to give insights into neuroscience, this result suggests that in the presence of noise, the biological sparse coding solution is optimal. Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

**Conclusion -** We have shown that the realistic assumption of noisy training data is sufficient for feed-forward layers to organically become highly sparse and retain their sparsity, even after noise is removed. This result is particularly striking because of the way the networks converge with sparse coding, producing biological receptive fields and approximating an inhibitory interneuron that implements a Top-K operation.

More broadly, this work builds a new bridge between artificial and biological neural networks by showing how they can be more similar than they may otherwise appear.

Here we show additional images of the receptive fields for networks trained with different amounts of noise (Fig. 8), the queries used to activate these neurons (Fig. 9) and how the receptive fields look more biological as the network is trained for longer (Fig. 10).

We show the sparsity amounts of the networks in Fig. 11.

Removing the bias terms and relying just on the inhibitory interneuron, we found that it performs well (Table 1) and becomes sparse as a function of noise but results in a number of dead neurons and is not as sparse as the baseline ReLU or Explaining Away models.

Input: Table 1: Reconstruction Validation Accuracy - The baseline ReLU model varying the number of neurons, compared to the shared bias, explaining away, and inhibitory interneuron networks. We only show noise up to σ = 0.8 because the higher noise levels present trivial solutions that all networks perform equally well on. Due to very high convergence between random seeds, we just present a single run for each model.

Input: Figure 7: Training ablations. The positive correlation between noise and sparsity is robust to (a) neuron counts, (b) datasets, (c) noise distributions, and (d) training tasks.

Input: Figure 2: There is a positive relationship between activation sparsity and noise. The average fraction of neurons active during each CIFAR10 latent input over the course of training for 800 epochs. Each line corresponds to training randomly initialized networks with different noise levels σ denoted by different colors. We show the average of three different random seeds and their standard error of the mean (not visible as the variance is so low). Interestingly, the higher noise levels take longer to sparsify but upon convergence become the sparsest.

Input: Figure 3: Noise induces the formation of Top-K networks. Shown column-wise are the fraction of neurons that are active for each training input (Left), the values of each of the 10,000 neurons bias terms (Middle) and the L2 norms of their key vectors (Right), as a function of number of training epochs. Each is shown for three different representative noise levels (rows). These plots show each input to the network and use the density of blue to represent how many times a particular value occurs.

Input: Figure 5: Biological receptive fields form with noise. Networks trained on CIFAR10 pixels with σ = 0.8 Gaussian noise. The most active 125 neuron receptive fields are reshaped into 32x32x3 and re-scaled so their values span the range of pixel values [0, 255]. Neurons are sorted by their activity levels (top left is the most active, going across rows then down columns) for a random noisy input of a car. Starting at σ ≥ 0.1 and particularly for σ = 0.8 (shown here) we observe the formation of many Gabor filters and on/off center surround receptive fields that are unique to noisy training.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
I'll evaluate the claim: "The resulting network learns receptive fields like those of primary visual cortex and remains sparse even when noise is removed in later stages of learning."

Summary: The evidence strongly supports this claim through multiple angles:
1. Figure 5 directly shows the formation of Gabor filters and center-surround receptive fields, which are characteristic of V1 neurons
2. Figures 2 and 3 demonstrate the emergence and maintenance of sparsity
3. The paper explicitly tests and confirms that sparsity remains after noise removal (Fig. 6)
4. Multiple ablation studies (Fig. 7) show the robustness of these findings across different conditions

The claim is well-stated and accurately reflects the evidence without exaggeration. It makes specific assertions that are directly supported by experimental results and visualizations.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

The resulting network learns receptive fields like those of primary visual cortex and remains sparse even when noise is removed in later stages of learning.

The review comment to be evaluated is:

Strengths

The paper is simple and easy to follow.

The sparsity effects shown appear relatively large and robust.

Adding noise is a simple manipulation that can have benefits beyond sparsity, and so there is some appeal to using this manipulation as a way to generate sparsity.

Weaknesses

It is odd that the authors focus so much on a one-layer model. For this finding to be broadly relevant, it is important to test whether this effect occurs in large-scale DNNs and whether networks can generate sparsity while maintaining good performance on real-world tasks. The only thing that is done along these lines is with a single, small GPT2 model.

It appears you need fairly high levels of noise to generate substantial sparsity (e.g., sigma=0.3 to get below 20% activation in Figure 2, which based on Figure 4 is quite a lot of noise). It seems this might limit the practical utility of the method, though it is hard to say in part because experiments are mostly limited to the single-layer training regime.

The link with biological receptive fields is tenuous. There are multiple ways of generating center-surround and oriented/bandpass receptive fields (e.g., whitening for center-surround RFs, sparsity for oriented RFs, CNNs often exhibit similar structure at the first layer). It is not clear this approach provides a better fit to biological data than these alternatives and there is no quantification of the match to biology or model comparison. Moreover, there are so many differences between the models trained here and biological networks, that it is hard to know the relevance of this finding, particularly given that the results can vary with the activation function (e.g., the sigmoid activation function does not generate sparsity) and the optimization algorithm (e.g., SGD with a low learning rate apparently does not generate sparsity).

The evidence to be evaluated is:From this perspective, it makes sense that sparsity can be used to turn off neurons and minimize this Jacobian, providing a potential explanation for our results.

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network.

This sparse coding task is an example of a statistical model converging with neuroscience because, in addition to producing sparse representations, the basis functions learn to become Gabor filters like those found in area V1 of the visual system (Olshausen & Field, 1997).

Figure 2 shows that a single hidden layer ReLU network with 10,000 neurons learns to become a Top-K network with smaller \(k\) values as the amount of noise increases. This network was trained on the CIFAR10 embeddings to mimick training within a deeper network layer and where the manifold is smoother than when training on raw image pixels. While Fig. 2 only shows the mean number of neurons that are active across all 50,000 CIFAR10 inputs per epoch, Figure 3 shows the fraction of active neurons for every input.

It is clear that these values all synchronize to a value decided on by the network and are responsible for the sparse \(k\) neurons remaining active after the ReLU nonlinearity is applied. As noise increases, the bias terms become more negative to raise the activation threshold and increase sparsity.

We hypothesize this is a counter-response to the \(L_{2}\) norms of the data increasing with noise and explains why in high noise settings (\(\sigma\geq 0.8\)) neuron activations linger around \(\sim\)50% sparsity before rapidly becoming very sparse (see Fig. 2).

Because the key vectors are randomly initialized from a symmetric distribution around 0, in expectation half of the neurons are active after the ReLU nonlinearity is applied.

Looking more closely at the sparse solutions found by our ReLU networks, we discovered that for \(\sigma\geq 0.3\), neurons begin to form receptive fields reminiscent of V1 Gabor filters and on/off center surround retinal ganglion cells (Sterling & Laughlin, 2015; Olshausen & Field, 1997).

This is a similar result to that of sparse coding (Olshausen & Field, 1997), however, here we use noise with the ReLU activation function instead of a sparsity penalty and train on full images instead of 8x8 patches.

The longer training progressed, the more similar the receptive fields appeared to those found in biology.

When we tried training the network without any noise but instead with an \(L_{1}\) activation penalty motivated by the original sparse coding setup (Olshausen & Field, 1997), no biological receptive fields were observed.

With these models we find that our networks become even more sparse and do so more quickly, as we would expect due to the explaining away of redundant neurons (see Appendix C).

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse.

Fig. 6 shows that the sparsity levels remain largely unperturbed.

In fact, for classification \(\sigma\in\{0.1,0.3\}\) both get 94.3% validation accuracy instead of the 93.4% that the baseline gets, showing slightly better generalization while also being more sparse (66% for baseline versus 55% for \(\sigma=0.1\) and 45% for \(\sigma=0.3\)).

**Model Ablations -** Figure 7 shows the mean neuron sparsity at network convergence across a number of different experimental conditions: (i) three datasets: CIFAR10 pixels, CIFAR10 latent embeddings, MNIST; (ii) two noise distributions: Gaussian and Laplace; (iii) four numbers of neurons: 100, 1,000, 10,000 and 100,000; (iv) three training tasks: reconstruction, classification, and Transformer next token prediction.

While still neither activation function becomes as sparse as ReLU, the GELU networks implement the same Top-K convergence as ReLU shown in Fig. 3 and form biologically plausible receptive fields like those of Fig. 5.

The correlation between noise and activation sparsity remains and is again retained even after noise is removed.

Figure 7(d) shows the final sparsity amounts of the first model layer with each of the other layers giving similar results with \(\sim\)3x to \(\sim\)9x more sparsity when \(\sigma=3.0\), compared to \(\sigma=0.0\).

In natural images, for example, certain projections (e.g., with Gabor filters) have high kurtosis which allows them to be easily distinguished from noise via thresholding or "coring" (Simoncelli and Adelson, 1996). Similarly, we hypothesize that the encoding layer in our network is learning such projections, so that thresholding performs noise rejection while also producing sparse activations.

Using deep learning to give insights into neuroscience, this result suggests that in the presence of noise, the biological sparse coding solution is optimal. Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

**Conclusion -** We have shown that the realistic assumption of noisy training data is sufficient for feed-forward layers to organically become highly sparse and retain their sparsity, even after noise is removed. This result is particularly striking because of the way the networks converge with sparse coding, producing biological receptive fields and approximating an inhibitory interneuron that implements a Top-K operation.

More broadly, this work builds a new bridge between artificial and biological neural networks by showing how they can be more similar than they may otherwise appear.

Here we show additional images of the receptive fields for networks trained with different amounts of noise (Fig. 8), the queries used to activate these neurons (Fig. 9) and how the receptive fields look more biological as the network is trained for longer (Fig. 10).

We show the sparsity amounts of the networks in Fig. 11.

Removing the bias terms and relying just on the inhibitory interneuron, we found that it performs well (Table 1) and becomes sparse as a function of noise but results in a number of dead neurons and is not as sparse as the baseline ReLU or Explaining Away models.

Input: Table 1: Reconstruction Validation Accuracy - The baseline ReLU model varying the number of neurons, compared to the shared bias, explaining away, and inhibitory interneuron networks. We only show noise up to σ = 0.8 because the higher noise levels present trivial solutions that all networks perform equally well on. Due to very high convergence between random seeds, we just present a single run for each model.

Input: Figure 7: Training ablations. The positive correlation between noise and sparsity is robust to (a) neuron counts, (b) datasets, (c) noise distributions, and (d) training tasks.

Input: Figure 2: There is a positive relationship between activation sparsity and noise. The average fraction of neurons active during each CIFAR10 latent input over the course of training for 800 epochs. Each line corresponds to training randomly initialized networks with different noise levels σ denoted by different colors. We show the average of three different random seeds and their standard error of the mean (not visible as the variance is so low). Interestingly, the higher noise levels take longer to sparsify but upon convergence become the sparsest.

Input: Figure 3: Noise induces the formation of Top-K networks. Shown column-wise are the fraction of neurons that are active for each training input (Left), the values of each of the 10,000 neurons bias terms (Middle) and the L2 norms of their key vectors (Right), as a function of number of training epochs. Each is shown for three different representative noise levels (rows). These plots show each input to the network and use the density of blue to represent how many times a particular value occurs.

Input: Figure 5: Biological receptive fields form with noise. Networks trained on CIFAR10 pixels with σ = 0.8 Gaussian noise. The most active 125 neuron receptive fields are reshaped into 32x32x3 and re-scaled so their values span the range of pixel values [0, 255]. Neurons are sorted by their activity levels (top left is the most active, going across rows then down columns) for a random noisy input of a car. Starting at σ ≥ 0.1 and particularly for σ = 0.8 (shown here) we observe the formation of many Gabor filters and on/off center surround receptive fields that are unique to noisy training.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim and evidence, I find this claim to be moderately overstated. While there is clear evidence that the network develops receptive fields similar to those in primary visual cortex (as shown in Figure 5) and maintains sparsity after noise removal (demonstrated in multiple experiments), there are important limitations noted in the reviewer comments that aren't acknowledged in the claim. The biological similarity is "tenuous" since there are multiple ways to generate such receptive fields, and no quantitative comparison to biological data is provided. The high noise levels required (σ≥0.3) for these effects also limit the practical implications.

<score>3</score>

Justification:
1. The evidence does support formation of Gabor-like filters (Fig 5) and sustained sparsity (Fig 2, Fig 6)
2. However, reviewer highlights valid concerns about:
- Lack of quantitative comparison to biological data
- Multiple alternative ways to generate similar receptive fields
- High noise requirements potentially limiting practical utility
3. The claim would be better stated with qualifiers acknowledging these limitations and the specific conditions required.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

The resulting network learns receptive fields like those of primary visual cortex and remains sparse even when noise is removed in later stages of learning.

The review comment to be evaluated is:

Strengths

1. The authors find a simple, implicit manner to create top-k networks that compares favorably to explicit methods

2. The method is tested in several different circumstances, with different tasks, numbers of neurons, and noise levels. Simple characteristics of the network, such as the fraction of active neurons and their biases, are well described.

Weaknesses

Major

1. As the authors observe, the results are recapitulated using a global negative bias term, acting as common inhibition that requires greater input values to activate neurons and resulting in top-k networks.  Thus, while such an observation may indeed be useful for those interested in such networks, the advance it comprises is a minor one.

Minor

1. While the presented results may bear on biological considerations, as suggested therein, caution is advised, as noise in biological networks is dynamic and thus not perfectly analogous to simply adding noise to inputs of a feedforward network.

The evidence to be evaluated is:From this perspective, it makes sense that sparsity can be used to turn off neurons and minimize this Jacobian, providing a potential explanation for our results.

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network.

This sparse coding task is an example of a statistical model converging with neuroscience because, in addition to producing sparse representations, the basis functions learn to become Gabor filters like those found in area V1 of the visual system (Olshausen & Field, 1997).

Figure 2 shows that a single hidden layer ReLU network with 10,000 neurons learns to become a Top-K network with smaller \(k\) values as the amount of noise increases. This network was trained on the CIFAR10 embeddings to mimick training within a deeper network layer and where the manifold is smoother than when training on raw image pixels. While Fig. 2 only shows the mean number of neurons that are active across all 50,000 CIFAR10 inputs per epoch, Figure 3 shows the fraction of active neurons for every input.

It is clear that these values all synchronize to a value decided on by the network and are responsible for the sparse \(k\) neurons remaining active after the ReLU nonlinearity is applied. As noise increases, the bias terms become more negative to raise the activation threshold and increase sparsity.

We hypothesize this is a counter-response to the \(L_{2}\) norms of the data increasing with noise and explains why in high noise settings (\(\sigma\geq 0.8\)) neuron activations linger around \(\sim\)50% sparsity before rapidly becoming very sparse (see Fig. 2).

Because the key vectors are randomly initialized from a symmetric distribution around 0, in expectation half of the neurons are active after the ReLU nonlinearity is applied.

Looking more closely at the sparse solutions found by our ReLU networks, we discovered that for \(\sigma\geq 0.3\), neurons begin to form receptive fields reminiscent of V1 Gabor filters and on/off center surround retinal ganglion cells (Sterling & Laughlin, 2015; Olshausen & Field, 1997).

This is a similar result to that of sparse coding (Olshausen & Field, 1997), however, here we use noise with the ReLU activation function instead of a sparsity penalty and train on full images instead of 8x8 patches.

The longer training progressed, the more similar the receptive fields appeared to those found in biology.

When we tried training the network without any noise but instead with an \(L_{1}\) activation penalty motivated by the original sparse coding setup (Olshausen & Field, 1997), no biological receptive fields were observed.

With these models we find that our networks become even more sparse and do so more quickly, as we would expect due to the explaining away of redundant neurons (see Appendix C).

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse.

Fig. 6 shows that the sparsity levels remain largely unperturbed.

In fact, for classification \(\sigma\in\{0.1,0.3\}\) both get 94.3% validation accuracy instead of the 93.4% that the baseline gets, showing slightly better generalization while also being more sparse (66% for baseline versus 55% for \(\sigma=0.1\) and 45% for \(\sigma=0.3\)).

**Model Ablations -** Figure 7 shows the mean neuron sparsity at network convergence across a number of different experimental conditions: (i) three datasets: CIFAR10 pixels, CIFAR10 latent embeddings, MNIST; (ii) two noise distributions: Gaussian and Laplace; (iii) four numbers of neurons: 100, 1,000, 10,000 and 100,000; (iv) three training tasks: reconstruction, classification, and Transformer next token prediction.

While still neither activation function becomes as sparse as ReLU, the GELU networks implement the same Top-K convergence as ReLU shown in Fig. 3 and form biologically plausible receptive fields like those of Fig. 5.

The correlation between noise and activation sparsity remains and is again retained even after noise is removed.

Figure 7(d) shows the final sparsity amounts of the first model layer with each of the other layers giving similar results with \(\sim\)3x to \(\sim\)9x more sparsity when \(\sigma=3.0\), compared to \(\sigma=0.0\).

In natural images, for example, certain projections (e.g., with Gabor filters) have high kurtosis which allows them to be easily distinguished from noise via thresholding or "coring" (Simoncelli and Adelson, 1996). Similarly, we hypothesize that the encoding layer in our network is learning such projections, so that thresholding performs noise rejection while also producing sparse activations.

Using deep learning to give insights into neuroscience, this result suggests that in the presence of noise, the biological sparse coding solution is optimal. Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

**Conclusion -** We have shown that the realistic assumption of noisy training data is sufficient for feed-forward layers to organically become highly sparse and retain their sparsity, even after noise is removed. This result is particularly striking because of the way the networks converge with sparse coding, producing biological receptive fields and approximating an inhibitory interneuron that implements a Top-K operation.

More broadly, this work builds a new bridge between artificial and biological neural networks by showing how they can be more similar than they may otherwise appear.

Here we show additional images of the receptive fields for networks trained with different amounts of noise (Fig. 8), the queries used to activate these neurons (Fig. 9) and how the receptive fields look more biological as the network is trained for longer (Fig. 10).

We show the sparsity amounts of the networks in Fig. 11.

Removing the bias terms and relying just on the inhibitory interneuron, we found that it performs well (Table 1) and becomes sparse as a function of noise but results in a number of dead neurons and is not as sparse as the baseline ReLU or Explaining Away models.

Input: Table 1: Reconstruction Validation Accuracy - The baseline ReLU model varying the number of neurons, compared to the shared bias, explaining away, and inhibitory interneuron networks. We only show noise up to σ = 0.8 because the higher noise levels present trivial solutions that all networks perform equally well on. Due to very high convergence between random seeds, we just present a single run for each model.

Input: Figure 7: Training ablations. The positive correlation between noise and sparsity is robust to (a) neuron counts, (b) datasets, (c) noise distributions, and (d) training tasks.

Input: Figure 2: There is a positive relationship between activation sparsity and noise. The average fraction of neurons active during each CIFAR10 latent input over the course of training for 800 epochs. Each line corresponds to training randomly initialized networks with different noise levels σ denoted by different colors. We show the average of three different random seeds and their standard error of the mean (not visible as the variance is so low). Interestingly, the higher noise levels take longer to sparsify but upon convergence become the sparsest.

Input: Figure 3: Noise induces the formation of Top-K networks. Shown column-wise are the fraction of neurons that are active for each training input (Left), the values of each of the 10,000 neurons bias terms (Middle) and the L2 norms of their key vectors (Right), as a function of number of training epochs. Each is shown for three different representative noise levels (rows). These plots show each input to the network and use the density of blue to represent how many times a particular value occurs.

Input: Figure 5: Biological receptive fields form with noise. Networks trained on CIFAR10 pixels with σ = 0.8 Gaussian noise. The most active 125 neuron receptive fields are reshaped into 32x32x3 and re-scaled so their values span the range of pixel values [0, 255]. Neurons are sorted by their activity levels (top left is the most active, going across rows then down columns) for a random noisy input of a car. Starting at σ ≥ 0.1 and particularly for σ = 0.8 (shown here) we observe the formation of many Gabor filters and on/off center surround receptive fields that are unique to noisy training.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Let me evaluate the claim: "The resulting network learns receptive fields like those of primary visual cortex and remains sparse even when noise is removed in later stages of learning."

This claim appears well-stated based on the extensive evidence provided:

1. Figure 5 clearly shows the formation of Gabor filters and center-surround receptive fields similar to V1, particularly at σ ≥ 0.8
2. Multiple figures (2, 3, 7) demonstrate consistent sparsity across different conditions
3. The evidence explicitly states that "sparsity levels remain largely unperturbed" after noise removal (Fig. 6)
4. The results are robust across multiple ablation studies shown in Figure 7
5. The review comments don't dispute these specific findings

The claim is precise and measured, avoiding overstatement while accurately reflecting the demonstrated results.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

The resulting network learns receptive fields like those of primary visual cortex and remains sparse even when noise is removed in later stages of learning.

The review comment to be evaluated is:

Strengths:
* The paper tackles the important problem of bridging the gap between biological plausible learning and current artificial neural networks.

Weaknesses:
* The paper mostly focuses on 1-hidden layer networks. It is not too clear why and how these results are relevant for practice, where typically much deeper networks are considered. For classification tasks, the observed sparsity seems to be much less pronounced.
* Unlike claimed in the paper, using the L1-norm can lead to biologically plausible and sparse features even for classification tasks, see e.g. Figure 4 in "Towards Learning Convolutions from Scratch" (https://papers.nips.cc/paper/2020/file/5c528e25e1fdeaf9d8160dc24dbf4d60-Paper.pdf)
* There are some issues regarding novelty/clarity (see next section)

The evidence to be evaluated is:From this perspective, it makes sense that sparsity can be used to turn off neurons and minimize this Jacobian, providing a potential explanation for our results.

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network.

This sparse coding task is an example of a statistical model converging with neuroscience because, in addition to producing sparse representations, the basis functions learn to become Gabor filters like those found in area V1 of the visual system (Olshausen & Field, 1997).

Figure 2 shows that a single hidden layer ReLU network with 10,000 neurons learns to become a Top-K network with smaller \(k\) values as the amount of noise increases. This network was trained on the CIFAR10 embeddings to mimick training within a deeper network layer and where the manifold is smoother than when training on raw image pixels. While Fig. 2 only shows the mean number of neurons that are active across all 50,000 CIFAR10 inputs per epoch, Figure 3 shows the fraction of active neurons for every input.

It is clear that these values all synchronize to a value decided on by the network and are responsible for the sparse \(k\) neurons remaining active after the ReLU nonlinearity is applied. As noise increases, the bias terms become more negative to raise the activation threshold and increase sparsity.

We hypothesize this is a counter-response to the \(L_{2}\) norms of the data increasing with noise and explains why in high noise settings (\(\sigma\geq 0.8\)) neuron activations linger around \(\sim\)50% sparsity before rapidly becoming very sparse (see Fig. 2).

Because the key vectors are randomly initialized from a symmetric distribution around 0, in expectation half of the neurons are active after the ReLU nonlinearity is applied.

Looking more closely at the sparse solutions found by our ReLU networks, we discovered that for \(\sigma\geq 0.3\), neurons begin to form receptive fields reminiscent of V1 Gabor filters and on/off center surround retinal ganglion cells (Sterling & Laughlin, 2015; Olshausen & Field, 1997).

This is a similar result to that of sparse coding (Olshausen & Field, 1997), however, here we use noise with the ReLU activation function instead of a sparsity penalty and train on full images instead of 8x8 patches.

The longer training progressed, the more similar the receptive fields appeared to those found in biology.

When we tried training the network without any noise but instead with an \(L_{1}\) activation penalty motivated by the original sparse coding setup (Olshausen & Field, 1997), no biological receptive fields were observed.

With these models we find that our networks become even more sparse and do so more quickly, as we would expect due to the explaining away of redundant neurons (see Appendix C).

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse.

Fig. 6 shows that the sparsity levels remain largely unperturbed.

In fact, for classification \(\sigma\in\{0.1,0.3\}\) both get 94.3% validation accuracy instead of the 93.4% that the baseline gets, showing slightly better generalization while also being more sparse (66% for baseline versus 55% for \(\sigma=0.1\) and 45% for \(\sigma=0.3\)).

**Model Ablations -** Figure 7 shows the mean neuron sparsity at network convergence across a number of different experimental conditions: (i) three datasets: CIFAR10 pixels, CIFAR10 latent embeddings, MNIST; (ii) two noise distributions: Gaussian and Laplace; (iii) four numbers of neurons: 100, 1,000, 10,000 and 100,000; (iv) three training tasks: reconstruction, classification, and Transformer next token prediction.

While still neither activation function becomes as sparse as ReLU, the GELU networks implement the same Top-K convergence as ReLU shown in Fig. 3 and form biologically plausible receptive fields like those of Fig. 5.

The correlation between noise and activation sparsity remains and is again retained even after noise is removed.

Figure 7(d) shows the final sparsity amounts of the first model layer with each of the other layers giving similar results with \(\sim\)3x to \(\sim\)9x more sparsity when \(\sigma=3.0\), compared to \(\sigma=0.0\).

In natural images, for example, certain projections (e.g., with Gabor filters) have high kurtosis which allows them to be easily distinguished from noise via thresholding or "coring" (Simoncelli and Adelson, 1996). Similarly, we hypothesize that the encoding layer in our network is learning such projections, so that thresholding performs noise rejection while also producing sparse activations.

Using deep learning to give insights into neuroscience, this result suggests that in the presence of noise, the biological sparse coding solution is optimal. Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

**Conclusion -** We have shown that the realistic assumption of noisy training data is sufficient for feed-forward layers to organically become highly sparse and retain their sparsity, even after noise is removed. This result is particularly striking because of the way the networks converge with sparse coding, producing biological receptive fields and approximating an inhibitory interneuron that implements a Top-K operation.

More broadly, this work builds a new bridge between artificial and biological neural networks by showing how they can be more similar than they may otherwise appear.

Here we show additional images of the receptive fields for networks trained with different amounts of noise (Fig. 8), the queries used to activate these neurons (Fig. 9) and how the receptive fields look more biological as the network is trained for longer (Fig. 10).

We show the sparsity amounts of the networks in Fig. 11.

Removing the bias terms and relying just on the inhibitory interneuron, we found that it performs well (Table 1) and becomes sparse as a function of noise but results in a number of dead neurons and is not as sparse as the baseline ReLU or Explaining Away models.

Input: Table 1: Reconstruction Validation Accuracy - The baseline ReLU model varying the number of neurons, compared to the shared bias, explaining away, and inhibitory interneuron networks. We only show noise up to σ = 0.8 because the higher noise levels present trivial solutions that all networks perform equally well on. Due to very high convergence between random seeds, we just present a single run for each model.

Input: Figure 7: Training ablations. The positive correlation between noise and sparsity is robust to (a) neuron counts, (b) datasets, (c) noise distributions, and (d) training tasks.

Input: Figure 2: There is a positive relationship between activation sparsity and noise. The average fraction of neurons active during each CIFAR10 latent input over the course of training for 800 epochs. Each line corresponds to training randomly initialized networks with different noise levels σ denoted by different colors. We show the average of three different random seeds and their standard error of the mean (not visible as the variance is so low). Interestingly, the higher noise levels take longer to sparsify but upon convergence become the sparsest.

Input: Figure 3: Noise induces the formation of Top-K networks. Shown column-wise are the fraction of neurons that are active for each training input (Left), the values of each of the 10,000 neurons bias terms (Middle) and the L2 norms of their key vectors (Right), as a function of number of training epochs. Each is shown for three different representative noise levels (rows). These plots show each input to the network and use the density of blue to represent how many times a particular value occurs.

Input: Figure 5: Biological receptive fields form with noise. Networks trained on CIFAR10 pixels with σ = 0.8 Gaussian noise. The most active 125 neuron receptive fields are reshaped into 32x32x3 and re-scaled so their values span the range of pixel values [0, 255]. Neurons are sorted by their activity levels (top left is the most active, going across rows then down columns) for a random noisy input of a car. Starting at σ ≥ 0.1 and particularly for σ = 0.8 (shown here) we observe the formation of many Gabor filters and on/off center surround receptive fields that are unique to noisy training.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Let me evaluate the claim: "The resulting network learns receptive fields like those of primary visual cortex and remains sparse even when noise is removed in later stages of learning."

Brief Assessment:
The evidence strongly supports this claim. Figure 5 clearly shows Gabor-like filters and center-surround receptive fields similar to V1, particularly at σ ≥ 0.3. Multiple figures (2, 3, 7) demonstrate consistent sparsity patterns across different conditions. Figure 6 specifically confirms sparsity remains after noise removal. The claim is measured in its wording, acknowledging similarity to biological receptive fields rather than claiming exact replication, and is well-supported by comprehensive experimental evidence across multiple conditions and ablations.

<score>1</score>

Detailed Justification:
1. Biological Receptive Fields:
- Direct evidence from Figure 5 shows Gabor-like filters and center-surround patterns
- Text confirms these emerge at σ ≥ 0.3 and become more biologically plausible with training
- Multiple citations to established biological parallels (Olshausen & Field, 1997; Sterling & Laughlin, 2015)

2. Maintained Sparsity:
- Figure 2 and 3 demonstrate clear sparsification during training
- Figure 7 shows robustness across different conditions
- Evidence shows sparsity remains after noise removal
- Validated across multiple architectures, datasets, and conditions

3. Review Comments:
The reviewer's concerns about depth and L1-norm alternatives don't directly challenge the specific claim about receptive fields and sparsity maintenance, which is well-supported by the evidence presented.

The claim is precise and well-qualified, avoiding overstatement while being thoroughly supported by the evidence.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

The resulting network learns receptive fields like those of primary visual cortex and remains sparse even when noise is removed in later stages of learning.

The review comment to be evaluated is:

In this empirical paper, the authors observe the correlation between the sparsity of the learned representations and the level of noise injected into the input of Neural Networks with the ReLU activation function. Experiments have been carried out on CIFAR10 both on pixel data and on a neural representation, MNIST, and on the WikiText-103 dataset. They report a positive correlation between the magnitude of the injected noise and the sparsity of the representation. They observe Gabor-like filters when reconstructing with noise on pixel data. For the classification task, the performance of the model did not decrease with sparsity. Also, compared with using only the top-K activations of a model learned without noise, the models learned with noise gave better validation accuracies.

The evidence to be evaluated is:From this perspective, it makes sense that sparsity can be used to turn off neurons and minimize this Jacobian, providing a potential explanation for our results.

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network.

This sparse coding task is an example of a statistical model converging with neuroscience because, in addition to producing sparse representations, the basis functions learn to become Gabor filters like those found in area V1 of the visual system (Olshausen & Field, 1997).

Figure 2 shows that a single hidden layer ReLU network with 10,000 neurons learns to become a Top-K network with smaller \(k\) values as the amount of noise increases. This network was trained on the CIFAR10 embeddings to mimick training within a deeper network layer and where the manifold is smoother than when training on raw image pixels. While Fig. 2 only shows the mean number of neurons that are active across all 50,000 CIFAR10 inputs per epoch, Figure 3 shows the fraction of active neurons for every input.

It is clear that these values all synchronize to a value decided on by the network and are responsible for the sparse \(k\) neurons remaining active after the ReLU nonlinearity is applied. As noise increases, the bias terms become more negative to raise the activation threshold and increase sparsity.

We hypothesize this is a counter-response to the \(L_{2}\) norms of the data increasing with noise and explains why in high noise settings (\(\sigma\geq 0.8\)) neuron activations linger around \(\sim\)50% sparsity before rapidly becoming very sparse (see Fig. 2).

Because the key vectors are randomly initialized from a symmetric distribution around 0, in expectation half of the neurons are active after the ReLU nonlinearity is applied.

Looking more closely at the sparse solutions found by our ReLU networks, we discovered that for \(\sigma\geq 0.3\), neurons begin to form receptive fields reminiscent of V1 Gabor filters and on/off center surround retinal ganglion cells (Sterling & Laughlin, 2015; Olshausen & Field, 1997).

This is a similar result to that of sparse coding (Olshausen & Field, 1997), however, here we use noise with the ReLU activation function instead of a sparsity penalty and train on full images instead of 8x8 patches.

The longer training progressed, the more similar the receptive fields appeared to those found in biology.

When we tried training the network without any noise but instead with an \(L_{1}\) activation penalty motivated by the original sparse coding setup (Olshausen & Field, 1997), no biological receptive fields were observed.

With these models we find that our networks become even more sparse and do so more quickly, as we would expect due to the explaining away of redundant neurons (see Appendix C).

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse.

Fig. 6 shows that the sparsity levels remain largely unperturbed.

In fact, for classification \(\sigma\in\{0.1,0.3\}\) both get 94.3% validation accuracy instead of the 93.4% that the baseline gets, showing slightly better generalization while also being more sparse (66% for baseline versus 55% for \(\sigma=0.1\) and 45% for \(\sigma=0.3\)).

**Model Ablations -** Figure 7 shows the mean neuron sparsity at network convergence across a number of different experimental conditions: (i) three datasets: CIFAR10 pixels, CIFAR10 latent embeddings, MNIST; (ii) two noise distributions: Gaussian and Laplace; (iii) four numbers of neurons: 100, 1,000, 10,000 and 100,000; (iv) three training tasks: reconstruction, classification, and Transformer next token prediction.

While still neither activation function becomes as sparse as ReLU, the GELU networks implement the same Top-K convergence as ReLU shown in Fig. 3 and form biologically plausible receptive fields like those of Fig. 5.

The correlation between noise and activation sparsity remains and is again retained even after noise is removed.

Figure 7(d) shows the final sparsity amounts of the first model layer with each of the other layers giving similar results with \(\sim\)3x to \(\sim\)9x more sparsity when \(\sigma=3.0\), compared to \(\sigma=0.0\).

In natural images, for example, certain projections (e.g., with Gabor filters) have high kurtosis which allows them to be easily distinguished from noise via thresholding or "coring" (Simoncelli and Adelson, 1996). Similarly, we hypothesize that the encoding layer in our network is learning such projections, so that thresholding performs noise rejection while also producing sparse activations.

Using deep learning to give insights into neuroscience, this result suggests that in the presence of noise, the biological sparse coding solution is optimal. Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

**Conclusion -** We have shown that the realistic assumption of noisy training data is sufficient for feed-forward layers to organically become highly sparse and retain their sparsity, even after noise is removed. This result is particularly striking because of the way the networks converge with sparse coding, producing biological receptive fields and approximating an inhibitory interneuron that implements a Top-K operation.

More broadly, this work builds a new bridge between artificial and biological neural networks by showing how they can be more similar than they may otherwise appear.

Here we show additional images of the receptive fields for networks trained with different amounts of noise (Fig. 8), the queries used to activate these neurons (Fig. 9) and how the receptive fields look more biological as the network is trained for longer (Fig. 10).

We show the sparsity amounts of the networks in Fig. 11.

Removing the bias terms and relying just on the inhibitory interneuron, we found that it performs well (Table 1) and becomes sparse as a function of noise but results in a number of dead neurons and is not as sparse as the baseline ReLU or Explaining Away models.

Input: Table 1: Reconstruction Validation Accuracy - The baseline ReLU model varying the number of neurons, compared to the shared bias, explaining away, and inhibitory interneuron networks. We only show noise up to σ = 0.8 because the higher noise levels present trivial solutions that all networks perform equally well on. Due to very high convergence between random seeds, we just present a single run for each model.

Input: Figure 7: Training ablations. The positive correlation between noise and sparsity is robust to (a) neuron counts, (b) datasets, (c) noise distributions, and (d) training tasks.

Input: Figure 2: There is a positive relationship between activation sparsity and noise. The average fraction of neurons active during each CIFAR10 latent input over the course of training for 800 epochs. Each line corresponds to training randomly initialized networks with different noise levels σ denoted by different colors. We show the average of three different random seeds and their standard error of the mean (not visible as the variance is so low). Interestingly, the higher noise levels take longer to sparsify but upon convergence become the sparsest.

Input: Figure 3: Noise induces the formation of Top-K networks. Shown column-wise are the fraction of neurons that are active for each training input (Left), the values of each of the 10,000 neurons bias terms (Middle) and the L2 norms of their key vectors (Right), as a function of number of training epochs. Each is shown for three different representative noise levels (rows). These plots show each input to the network and use the density of blue to represent how many times a particular value occurs.

Input: Figure 5: Biological receptive fields form with noise. Networks trained on CIFAR10 pixels with σ = 0.8 Gaussian noise. The most active 125 neuron receptive fields are reshaped into 32x32x3 and re-scaled so their values span the range of pixel values [0, 255]. Neurons are sorted by their activity levels (top left is the most active, going across rows then down columns) for a random noisy input of a car. Starting at σ ≥ 0.1 and particularly for σ = 0.8 (shown here) we observe the formation of many Gabor filters and on/off center surround receptive fields that are unique to noisy training.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim and evidence, I find that it is generally well-stated but has minor overstatement. The evidence strongly supports that the network learns receptive fields similar to those in primary visual cortex (V1), with clear demonstrations of Gabor-like filters in Figure 5 when noise ≥ 0.3. The claim about remaining sparse after noise removal is also supported by multiple experiments across different conditions (shown in Figures 2, 3, 6, and 7). However, the claim could be more precise by specifying the noise threshold (σ ≥ 0.3) needed for biological-like receptive fields to emerge.

<score>2</score>

Justification:
1. Strong evidence supports V1-like receptive fields:
- Figure 5 shows clear Gabor filters and center-surround patterns
- Multiple citations to biological similarity (Olshausen & Field, 1997)
- Results replicated across different conditions (Figure 7)

2. Sparsity retention is well-documented:
- Figure 6 shows sparsity levels remain after noise removal
- Validated across multiple architectures and datasets (Figure 7)
- Quantitative results show maintained sparsity (e.g., 45% for σ=0.3)

3. Minor overstatement:
- Could specify noise threshold (σ ≥ 0.3) needed for biological receptive fields
- Could acknowledge variation in sparsity levels across different conditions

The claim earns a 2 as it is largely well-stated with strong supporting evidence but could be more precise in its conditions.