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:

This results in the activity of every neuron existing within a particular range that, when combined with the negative bias term they all agree upon, results in a sparse \(k\) neurons remaining active.

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.

Approaches to introduce activation sparsity have included explicitly using Top-K activation functions (Ranzato et al., 2007; Makhzani and Frey, 2014; Ahmad and Scheinkman, 2019), novel regularization terms (Kurtz et al., 2020; Yang et al., 2020) and other approaches (Schwarz et al., 2021; Molchanov et al., 2017).

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. Top-K activation functions inspired by inhibitory interneurons have been implemented in a number of deep learning models for greater computational efficiency (Ahmad & Scheinkman, 2019), adversarial robustness (Xiao et al., 2020; Tramer et al., 2020), interpretability (Nelson et al., 2022), and continual learning (Shen et al., 2021; Iyer et al., 2022).

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).

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.

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). 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.

These Top-K networks sort the neuron activations and keep only the \(k\) most active. They were either given fixed \(k\) values or could learn the \(k\) values using either a continuous relaxation that is differentiable or the REINFORCE algorithm (see Appendix A for details) (Williams, 2004). These approaches all performed worse than the shared bias (and default ReLU) networks, with the best networks getting similar validation accuracies as the much smaller 100 neuron network and only when they were not sparse, using \(k=3,000\) neurons.

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.

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.

As a result, in the low noise regime, it is optimal for Attention to implement a form of nearest neighbour lookup, activating few keys.

This results in SDM attempting to also perform a nearest neighbour lookup, activating only as many neurons around the noisy query as is necessary to activate those storing the target pattern.

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.

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.

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 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 14: The GELU network sparsifies with noise when trained on CIFAR10. The GELU network closely resembles the ReLU network by sparsifying (but not to the same degree). It also forms biological receptive fields. We use the arbitrary 0.0001 absolute activation threshold to label if a neuron is on or off.

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 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 15: The sigmoid network also sparsifies with noise for CIFAR10. While sigmoid sparsifies in proportion to noise, it does not form biological receptive fields and is less sparse than GELU or ReLU.

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:
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, this claim appears to be well-supported by multiple pieces of evidence. The figures, particularly Figure 2 and Figure 3, demonstrate that neurons do converge to a specific range of activation patterns with negative bias terms, resulting in sparse activation. The evidence shows clear synchronization of bias terms and controlled sparsity levels across different noise conditions. The experimental data directly validates the key components of the claim regarding neuron activity ranges, negative bias terms, and resulting sparsity. The review comments, while noting some limitations in scope, do not specifically challenge this mechanistic claim.

<score>1</score>

Justification:
1. Figure 3 directly shows the convergence of neuron bias terms and resulting activation patterns
2. Figure 2 demonstrates consistent sparsification patterns across noise levels
3. The evidence includes detailed mathematical explanations of how the bias terms and ReLU nonlinearity interact to produce the claimed behavior
4. Multiple experimental variations (different noise levels, architectures) show consistent results
5. The claim's wording accurately reflects the scope and certainty of the findings without overstatement


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:

This results in the activity of every neuron existing within a particular range that, when combined with the negative bias term they all agree upon, results in a sparse \(k\) neurons remaining active.

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.

Approaches to introduce activation sparsity have included explicitly using Top-K activation functions (Ranzato et al., 2007; Makhzani and Frey, 2014; Ahmad and Scheinkman, 2019), novel regularization terms (Kurtz et al., 2020; Yang et al., 2020) and other approaches (Schwarz et al., 2021; Molchanov et al., 2017).

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. Top-K activation functions inspired by inhibitory interneurons have been implemented in a number of deep learning models for greater computational efficiency (Ahmad & Scheinkman, 2019), adversarial robustness (Xiao et al., 2020; Tramer et al., 2020), interpretability (Nelson et al., 2022), and continual learning (Shen et al., 2021; Iyer et al., 2022).

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).

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.

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). 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.

These Top-K networks sort the neuron activations and keep only the \(k\) most active. They were either given fixed \(k\) values or could learn the \(k\) values using either a continuous relaxation that is differentiable or the REINFORCE algorithm (see Appendix A for details) (Williams, 2004). These approaches all performed worse than the shared bias (and default ReLU) networks, with the best networks getting similar validation accuracies as the much smaller 100 neuron network and only when they were not sparse, using \(k=3,000\) neurons.

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.

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.

As a result, in the low noise regime, it is optimal for Attention to implement a form of nearest neighbour lookup, activating few keys.

This results in SDM attempting to also perform a nearest neighbour lookup, activating only as many neurons around the noisy query as is necessary to activate those storing the target pattern.

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.

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.

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 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 14: The GELU network sparsifies with noise when trained on CIFAR10. The GELU network closely resembles the ReLU network by sparsifying (but not to the same degree). It also forms biological receptive fields. We use the arbitrary 0.0001 absolute activation threshold to label if a neuron is on or off.

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 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 15: The sigmoid network also sparsifies with noise for CIFAR10. While sigmoid sparsifies in proportion to noise, it does not form biological receptive fields and is less sparse than GELU or ReLU.

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:
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 that neurons converge to a particular range and agree on a negative bias term resulting in k-sparse activations is well supported by multiple pieces of evidence:

1. Figure 3 clearly shows neuron bias terms converging to synchronized negative values as training progresses
2. Figure 2 demonstrates increasing sparsity (fewer active neurons) with higher noise levels
3. The evidence discusses how the bias terms become "more negative to raise the activation threshold and increase sparsity"
4. Table 1 and multiple figures show consistent sparsification across different architectures and conditions

The claim accurately reflects what is demonstrated in the data without overstating or exaggerating the findings. The mechanism described aligns well with the empirical evidence presented.

<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:

This results in the activity of every neuron existing within a particular range that, when combined with the negative bias term they all agree upon, results in a sparse \(k\) neurons remaining active.

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.

Approaches to introduce activation sparsity have included explicitly using Top-K activation functions (Ranzato et al., 2007; Makhzani and Frey, 2014; Ahmad and Scheinkman, 2019), novel regularization terms (Kurtz et al., 2020; Yang et al., 2020) and other approaches (Schwarz et al., 2021; Molchanov et al., 2017).

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. Top-K activation functions inspired by inhibitory interneurons have been implemented in a number of deep learning models for greater computational efficiency (Ahmad & Scheinkman, 2019), adversarial robustness (Xiao et al., 2020; Tramer et al., 2020), interpretability (Nelson et al., 2022), and continual learning (Shen et al., 2021; Iyer et al., 2022).

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).

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.

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). 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.

These Top-K networks sort the neuron activations and keep only the \(k\) most active. They were either given fixed \(k\) values or could learn the \(k\) values using either a continuous relaxation that is differentiable or the REINFORCE algorithm (see Appendix A for details) (Williams, 2004). These approaches all performed worse than the shared bias (and default ReLU) networks, with the best networks getting similar validation accuracies as the much smaller 100 neuron network and only when they were not sparse, using \(k=3,000\) neurons.

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.

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.

As a result, in the low noise regime, it is optimal for Attention to implement a form of nearest neighbour lookup, activating few keys.

This results in SDM attempting to also perform a nearest neighbour lookup, activating only as many neurons around the noisy query as is necessary to activate those storing the target pattern.

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.

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.

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 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 14: The GELU network sparsifies with noise when trained on CIFAR10. The GELU network closely resembles the ReLU network by sparsifying (but not to the same degree). It also forms biological receptive fields. We use the arbitrary 0.0001 absolute activation threshold to label if a neuron is on or off.

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 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 15: The sigmoid network also sparsifies with noise for CIFAR10. While sigmoid sparsifies in proportion to noise, it does not form biological receptive fields and is less sparse than GELU or ReLU.

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:
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 assess the claim: "This results in the activity of every neuron existing within a particular range that, when combined with the negative bias term they all agree upon, results in a sparse k neurons remaining active."

Summary of evaluation:
The evidence strongly supports this claim. Figure 3 clearly shows how neuron bias terms synchronize to negative values as training progresses, especially at higher noise levels. The evidence demonstrates that this leads to sparse activation patterns where only a subset of neurons remain active, as shown in Figures 2, 14, and 15 across different activation functions. The relationship between bias terms, noise levels, and resulting sparsity is well-documented in the data and matches the claim's description without overstatement.

<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:

This results in the activity of every neuron existing within a particular range that, when combined with the negative bias term they all agree upon, results in a sparse \(k\) neurons remaining active.

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.

Approaches to introduce activation sparsity have included explicitly using Top-K activation functions (Ranzato et al., 2007; Makhzani and Frey, 2014; Ahmad and Scheinkman, 2019), novel regularization terms (Kurtz et al., 2020; Yang et al., 2020) and other approaches (Schwarz et al., 2021; Molchanov et al., 2017).

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. Top-K activation functions inspired by inhibitory interneurons have been implemented in a number of deep learning models for greater computational efficiency (Ahmad & Scheinkman, 2019), adversarial robustness (Xiao et al., 2020; Tramer et al., 2020), interpretability (Nelson et al., 2022), and continual learning (Shen et al., 2021; Iyer et al., 2022).

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).

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.

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). 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.

These Top-K networks sort the neuron activations and keep only the \(k\) most active. They were either given fixed \(k\) values or could learn the \(k\) values using either a continuous relaxation that is differentiable or the REINFORCE algorithm (see Appendix A for details) (Williams, 2004). These approaches all performed worse than the shared bias (and default ReLU) networks, with the best networks getting similar validation accuracies as the much smaller 100 neuron network and only when they were not sparse, using \(k=3,000\) neurons.

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.

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.

As a result, in the low noise regime, it is optimal for Attention to implement a form of nearest neighbour lookup, activating few keys.

This results in SDM attempting to also perform a nearest neighbour lookup, activating only as many neurons around the noisy query as is necessary to activate those storing the target pattern.

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.

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.

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 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 14: The GELU network sparsifies with noise when trained on CIFAR10. The GELU network closely resembles the ReLU network by sparsifying (but not to the same degree). It also forms biological receptive fields. We use the arbitrary 0.0001 absolute activation threshold to label if a neuron is on or off.

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 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 15: The sigmoid network also sparsifies with noise for CIFAR10. While sigmoid sparsifies in proportion to noise, it does not form biological receptive fields and is less sparse than GELU or ReLU.

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:
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-stated based on extensive empirical support. The evidence shows that:

1. Networks consistently develop synchronized negative bias terms that work with ReLU activation to maintain sparse activations (Fig 2, 3)
2. This sparsification occurs robustly across different neuron counts, datasets, noise types, and tasks (Fig 7)
3. The sparsity remains even after noise is removed (Fig 6)
4. Both GELU and sigmoid networks demonstrate similar sparsification patterns (Fig 14, 15)

The evidence directly demonstrates how neurons converge to synchronized negative bias values that result in sparse k-active neurons, validating the claim's specific mechanism. The claim doesn't overstate or generalize beyond what is shown.

<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 degree to which our network becomes a sparse coding, Top-K network is proportional to the amount of noise applied, up to a noise limit.

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).

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.

Top-K activation functions inspired by inhibitory interneurons have been implemented in a number of deep learning models for greater computational efficiency (Ahmad & Scheinkman, 2019), adversarial robustness (Xiao et al., 2020; Tramer et al., 2020), interpretability (Nelson et al., 2022), and continual learning (Shen et al., 2021; Iyer et al., 2022).

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 \(\mathbf{x}\).

To ensure our results are robust, we vary a number of settings, finding that in every case the relationship between noise and network sparsity holds.

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.

This reveals that as noise increases, the variance for how many neurons are active shrinks, so that the mean is an accurate summary statistic.

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).

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

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.

For classification, it is clear that while the correlation between noise and sparsity holds, it is not quite as strong.

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 \(\sigma=0.3\)). Note that for the networks with \(\sigma\geq 0.8\) for reconstruction and \(\sigma\geq 3.0\) for classification, annealing the noise results in dead neurons, presumably because the data manifold no longer spans as large a space.

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. 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.

The positive correlation between noise and sparsity is opposite to Transformer Attention and associative memory models such as SDM and Hopfield Networks (Bricken and Pehlevan, 2021).

As a result, in the low noise regime, it is optimal for Attention to implement a form of nearest neighbour lookup, activating few keys. When there is more noise, especially noise that moves the query so far from its target key that it is no longer the nearest, it is optimal to activate more keys and average over them, losing accuracy but giving a solution in the correct neighbourhood.

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.

* Zur et al. (2009) Richard M. Zur, Yulei Jiang, Lorenzo L. Pesce, and Karen Drukker. Noise injection for training artificial neural networks: a comparison with weight decay and early stopping. _Medical physics_, 36 10:4810-8, 2009.

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.

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 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 14: The GELU network sparsifies with noise when trained on CIFAR10. The GELU network closely resembles the ReLU network by sparsifying (but not to the same degree). It also forms biological receptive fields. We use the arbitrary 0.0001 absolute activation threshold to label if a neuron is on or off.

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 15: The sigmoid network also sparsifies with noise for CIFAR10. While sigmoid sparsifies in proportion to noise, it does not form biological receptive fields and is less sparse than GELU or ReLU.

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: 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:
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 careful review, the claim "The degree to which our network becomes a sparse coding, Top-K network is proportional to the amount of noise applied, up to a noise limit" appears to be well-supported by extensive empirical evidence. The figures (particularly 2, 3, 6, 7, 14, and 15) consistently demonstrate this relationship across different architectures, datasets, noise distributions, and training tasks. The evidence shows clear proportionality between noise levels and sparsity, with proper acknowledgment of limitations (e.g., very high noise levels leading to dead neurons). The claim is carefully qualified with "up to a noise limit" rather than making absolute statements.

<score>1</score>

Justification:
- Multiple figures demonstrate consistent positive correlation between noise and sparsity
- Results are validated across different:
  * Network sizes (100 to 100,000 neurons)
  * Datasets (CIFAR10, MNIST)
  * Noise distributions (Gaussian, Laplace)
  * Tasks (reconstruction, classification, language modeling)
  * Architectures (ReLU, GELU, Sigmoid)
- Limitations are explicitly acknowledged (e.g., dead neurons at high noise levels)
- Claim uses measured language ("proportional", "up to a limit") rather than absolute statements
- Quantitative evidence is provided through graphs and tables
- Results show robustness across multiple experimental conditions


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 degree to which our network becomes a sparse coding, Top-K network is proportional to the amount of noise applied, up to a noise limit.

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: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).

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.

Top-K activation functions inspired by inhibitory interneurons have been implemented in a number of deep learning models for greater computational efficiency (Ahmad & Scheinkman, 2019), adversarial robustness (Xiao et al., 2020; Tramer et al., 2020), interpretability (Nelson et al., 2022), and continual learning (Shen et al., 2021; Iyer et al., 2022).

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 \(\mathbf{x}\).

To ensure our results are robust, we vary a number of settings, finding that in every case the relationship between noise and network sparsity holds.

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.

This reveals that as noise increases, the variance for how many neurons are active shrinks, so that the mean is an accurate summary statistic.

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).

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

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.

For classification, it is clear that while the correlation between noise and sparsity holds, it is not quite as strong.

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 \(\sigma=0.3\)). Note that for the networks with \(\sigma\geq 0.8\) for reconstruction and \(\sigma\geq 3.0\) for classification, annealing the noise results in dead neurons, presumably because the data manifold no longer spans as large a space.

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. 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.

The positive correlation between noise and sparsity is opposite to Transformer Attention and associative memory models such as SDM and Hopfield Networks (Bricken and Pehlevan, 2021).

As a result, in the low noise regime, it is optimal for Attention to implement a form of nearest neighbour lookup, activating few keys. When there is more noise, especially noise that moves the query so far from its target key that it is no longer the nearest, it is optimal to activate more keys and average over them, losing accuracy but giving a solution in the correct neighbourhood.

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.

* Zur et al. (2009) Richard M. Zur, Yulei Jiang, Lorenzo L. Pesce, and Karen Drukker. Noise injection for training artificial neural networks: a comparison with weight decay and early stopping. _Medical physics_, 36 10:4810-8, 2009.

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.

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 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 14: The GELU network sparsifies with noise when trained on CIFAR10. The GELU network closely resembles the ReLU network by sparsifying (but not to the same degree). It also forms biological receptive fields. We use the arbitrary 0.0001 absolute activation threshold to label if a neuron is on or off.

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 15: The sigmoid network also sparsifies with noise for CIFAR10. While sigmoid sparsifies in proportion to noise, it does not form biological receptive fields and is less sparse than GELU or ReLU.

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: 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:
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 degree to which our network becomes a sparse coding, Top-K network is proportional to the amount of noise applied, up to a noise limit."

Summary of Evidence Review:
The evidence strongly supports this claim through multiple experiments and visualizations. Figures 2, 3, 7, and others consistently show that increasing noise levels lead to greater sparsity across different architectures, datasets, and training tasks. The relationship holds up to certain noise thresholds (σ≥0.8 for reconstruction, σ≥3.0 for classification), beyond which networks develop dead neurons. The evidence demonstrates this relationship is robust across varying neuron counts, datasets, noise distributions, and tasks. The reviewer comments acknowledge the "sparsity effects shown appear relatively large and robust."

Score Justification:
This claim is well-stated (score of 1) because it:
- Accurately reflects the dose-dependent relationship between noise and sparsity
- Acknowledges the existence of a noise limit
- Is supported by comprehensive empirical evidence
- Avoids overextending beyond what the data shows
- Has been validated across multiple experimental conditions

<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 degree to which our network becomes a sparse coding, Top-K network is proportional to the amount of noise applied, up to a noise limit.

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: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).

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.

Top-K activation functions inspired by inhibitory interneurons have been implemented in a number of deep learning models for greater computational efficiency (Ahmad & Scheinkman, 2019), adversarial robustness (Xiao et al., 2020; Tramer et al., 2020), interpretability (Nelson et al., 2022), and continual learning (Shen et al., 2021; Iyer et al., 2022).

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 \(\mathbf{x}\).

To ensure our results are robust, we vary a number of settings, finding that in every case the relationship between noise and network sparsity holds.

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.

This reveals that as noise increases, the variance for how many neurons are active shrinks, so that the mean is an accurate summary statistic.

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).

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

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.

For classification, it is clear that while the correlation between noise and sparsity holds, it is not quite as strong.

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 \(\sigma=0.3\)). Note that for the networks with \(\sigma\geq 0.8\) for reconstruction and \(\sigma\geq 3.0\) for classification, annealing the noise results in dead neurons, presumably because the data manifold no longer spans as large a space.

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. 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.

The positive correlation between noise and sparsity is opposite to Transformer Attention and associative memory models such as SDM and Hopfield Networks (Bricken and Pehlevan, 2021).

As a result, in the low noise regime, it is optimal for Attention to implement a form of nearest neighbour lookup, activating few keys. When there is more noise, especially noise that moves the query so far from its target key that it is no longer the nearest, it is optimal to activate more keys and average over them, losing accuracy but giving a solution in the correct neighbourhood.

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.

* Zur et al. (2009) Richard M. Zur, Yulei Jiang, Lorenzo L. Pesce, and Karen Drukker. Noise injection for training artificial neural networks: a comparison with weight decay and early stopping. _Medical physics_, 36 10:4810-8, 2009.

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.

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 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 14: The GELU network sparsifies with noise when trained on CIFAR10. The GELU network closely resembles the ReLU network by sparsifying (but not to the same degree). It also forms biological receptive fields. We use the arbitrary 0.0001 absolute activation threshold to label if a neuron is on or off.

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 15: The sigmoid network also sparsifies with noise for CIFAR10. While sigmoid sparsifies in proportion to noise, it does not form biological receptive fields and is less sparse than GELU or ReLU.

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: 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:
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 that "the degree to which our network becomes a sparse coding, Top-K network is proportional to the amount of noise applied, up to a noise limit" is well-supported by extensive experimental data. The figures (particularly 2, 7, and 3) demonstrate clear positive correlation between noise levels and network sparsity across multiple conditions, including varying neuron counts, datasets, noise distributions, and tasks. The evidence shows this relationship holds up to certain noise thresholds (σ≥0.8 for reconstruction, σ≥3.0 for classification), beyond which dead neurons emerge. The claim carefully acknowledges this limit rather than overstating the relationship.

<score>1</score>

I justify this score of 1 (well-stated) because:

1. Multiple experimental validations show consistent noise-sparsity relationship
2. Clear evidence from comprehensive ablation studies
3. Authors acknowledge limitations and thresholds
4. Supporting visuals (Figures 2,3,6,7) directly demonstrate the claimed relationship
5. The review comment acknowledges this as a key strength, noting it was "tested in several different circumstances"
6. Careful qualification of "up to a noise limit" prevents overstatement

The claim represents the evidence accurately without exaggeration.