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

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

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

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

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

The claim to be assessed is:

Here, we find that by simply introducing isotropic, symmetric noise centered about zero during training, a layer of artificial neurons will converge to a sparse coding solution.

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

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network. Learning reconstructions of training data with sparse activations is the mainstay of sparse coding (Olshausen & Field, 1997).

Interestingly, the ReLU activation function that is crucial for the results of this paper can be derived as the optimal solution to a simplified sparse coding problem when we assume that the basis vectors are orthogonal (Ba, 2020; Gregor & LeCun, 2010). This results in the optimal solution for activity \(a_{i}\) to be given by \(\operatorname*{ReLU}(u_{i}-\lambda)-\operatorname*{ReLU}(-u_{i}-\lambda)\) or, if we constrain \(a_{i}\) to be non-negative, \(\operatorname*{ReLU}(u_{i}-\lambda)\), where \(u_{i}\) is the value obtained by projecting the data onto basis \(\phi_{i}\).

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 \(\mathbf{x}\).

\[\tilde{\mathbf{x}} =\mathbf{x}+\epsilon,\ \ \epsilon\sim N(0,\sigma I)\] \[\hat{\mathbf{x}} =W_{d}\ \text{ReLU}(W_{e}\tilde{\mathbf{x}}+\mathbf{b}_{e})+ \mathbf{b}_{d}\] \[\text{loss} =\sum_{\mathbf{x}\in X}(\mathbf{x}-\hat{\mathbf{x}})^{2}. \tag{1}\]

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.

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).

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

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

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

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

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse. We train our models in the same way as in Fig. 2 but at epoch 800 linearly anneal each noise level down to 0.0 over the next 800 epochs. Fig. 6 shows that the sparsity levels remain largely unperturbed.

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

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

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

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

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

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

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

Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

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

* 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.

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.

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

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

Input: Figure 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 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 that "simply introducing isotropic, symmetric noise during training causes convergence to a sparse coding solution" is well-supported by comprehensive empirical evidence. The paper demonstrates:

- Consistent sparsification across multiple experimental conditions (datasets, neuron counts, noise distributions, tasks)
- Formation of biologically-plausible receptive fields similar to sparse coding results
- Retention of sparsity even after noise removal
- Clear mechanistic explanation through regularization and noise rejection
- Robust validation through extensive ablation studies
- Quantitative results showing increased sparsity with noise (Fig 2, 7)

The claim avoids overstatement by precisely describing the observation without extrapolating beyond the evidence. The "simply" qualifier is justified given that noise alone, without additional penalties or constraints, produces these effects.

<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:

Here, we find that by simply introducing isotropic, symmetric noise centered about zero during training, a layer of artificial neurons will converge to a sparse coding solution.

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). It was later shown that noise injection is in fact equivalent to having a regularization term that seeks to minimize the \(L_{2}\) norm of the network's Jacobian (Bishop, 1995; Rifai et al., 2011; Alain and Bengio, 2014). From this perspective, it makes sense that sparsity can be used to turn off neurons and minimize this Jacobian, providing a potential explanation for our results.

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network. Learning reconstructions of training data with sparse activations is the mainstay of sparse coding (Olshausen & Field, 1997).

Interestingly, the ReLU activation function that is crucial for the results of this paper can be derived as the optimal solution to a simplified sparse coding problem when we assume that the basis vectors are orthogonal (Ba, 2020; Gregor & LeCun, 2010). This results in the optimal solution for activity \(a_{i}\) to be given by \(\operatorname*{ReLU}(u_{i}-\lambda)-\operatorname*{ReLU}(-u_{i}-\lambda)\) or, if we constrain \(a_{i}\) to be non-negative, \(\operatorname*{ReLU}(u_{i}-\lambda)\), where \(u_{i}\) is the value obtained by projecting the data onto basis \(\phi_{i}\).

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 \(\mathbf{x}\).

\[\tilde{\mathbf{x}} =\mathbf{x}+\epsilon,\ \ \epsilon\sim N(0,\sigma I)\] \[\hat{\mathbf{x}} =W_{d}\ \text{ReLU}(W_{e}\tilde{\mathbf{x}}+\mathbf{b}_{e})+ \mathbf{b}_{d}\] \[\text{loss} =\sum_{\mathbf{x}\in X}(\mathbf{x}-\hat{\mathbf{x}})^{2}. \tag{1}\]

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.

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).

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

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

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

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

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse. We train our models in the same way as in Fig. 2 but at epoch 800 linearly anneal each noise level down to 0.0 over the next 800 epochs. Fig. 6 shows that the sparsity levels remain largely unperturbed.

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

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

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

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

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

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

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

Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

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

* 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.

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.

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

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

Input: Figure 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 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've carefully reviewed the claim "Here, we find that by simply introducing isotropic, symmetric noise centered about zero during training, a layer of artificial neurons will converge to a sparse coding solution" against the provided evidence and reviewer comments.

Summary: The evidence strongly supports this claim through extensive experimental validation across multiple conditions (datasets, neuron counts, noise distributions, tasks). The figures (especially 2, 7, 14, 15) demonstrate consistent sparsification effects. The authors appropriately qualify their claim with "a layer" rather than overgeneralizing. The reviewer notes the sparsity effects are "relatively large and robust." While the reviewer points out limitations regarding practicality and biological relevance, these don't directly challenge the core claim about noise inducing sparsity.

<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:

Here, we find that by simply introducing isotropic, symmetric noise centered about zero during training, a layer of artificial neurons will converge to a sparse coding solution.

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). It was later shown that noise injection is in fact equivalent to having a regularization term that seeks to minimize the \(L_{2}\) norm of the network's Jacobian (Bishop, 1995; Rifai et al., 2011; Alain and Bengio, 2014). From this perspective, it makes sense that sparsity can be used to turn off neurons and minimize this Jacobian, providing a potential explanation for our results.

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network. Learning reconstructions of training data with sparse activations is the mainstay of sparse coding (Olshausen & Field, 1997).

Interestingly, the ReLU activation function that is crucial for the results of this paper can be derived as the optimal solution to a simplified sparse coding problem when we assume that the basis vectors are orthogonal (Ba, 2020; Gregor & LeCun, 2010). This results in the optimal solution for activity \(a_{i}\) to be given by \(\operatorname*{ReLU}(u_{i}-\lambda)-\operatorname*{ReLU}(-u_{i}-\lambda)\) or, if we constrain \(a_{i}\) to be non-negative, \(\operatorname*{ReLU}(u_{i}-\lambda)\), where \(u_{i}\) is the value obtained by projecting the data onto basis \(\phi_{i}\).

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 \(\mathbf{x}\).

\[\tilde{\mathbf{x}} =\mathbf{x}+\epsilon,\ \ \epsilon\sim N(0,\sigma I)\] \[\hat{\mathbf{x}} =W_{d}\ \text{ReLU}(W_{e}\tilde{\mathbf{x}}+\mathbf{b}_{e})+ \mathbf{b}_{d}\] \[\text{loss} =\sum_{\mathbf{x}\in X}(\mathbf{x}-\hat{\mathbf{x}})^{2}. \tag{1}\]

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.

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).

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

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

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

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

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse. We train our models in the same way as in Fig. 2 but at epoch 800 linearly anneal each noise level down to 0.0 over the next 800 epochs. Fig. 6 shows that the sparsity levels remain largely unperturbed.

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

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

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

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

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

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

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

Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

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

* 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.

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.

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

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

Input: Figure 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 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, this claim appears to be well-stated and accurately reflects the evidence presented. The paper provides extensive experimental validation across multiple conditions (datasets, noise distributions, neuron counts, tasks) showing that introducing isotropic, symmetric noise during training consistently leads to sparse coding solutions. The evidence includes:
- Detailed empirical results showing increased sparsity with noise (Figs 2,3,7)
- Formation of biologically plausible receptive fields (Figs 5,8)
- Retained sparsity after noise removal (Fig 6)
- Theoretical connections to sparse coding and noise handling
- Robustness checks across various experimental conditions

The review comments acknowledge this as a valid finding, with the main criticism being about the advance being minor rather than questioning the claim's validity.

<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:

Here, we find that by simply introducing isotropic, symmetric noise centered about zero during training, a layer of artificial neurons will converge to a sparse coding solution.

The review comment to be evaluated is:

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

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

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

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network. Learning reconstructions of training data with sparse activations is the mainstay of sparse coding (Olshausen & Field, 1997).

Interestingly, the ReLU activation function that is crucial for the results of this paper can be derived as the optimal solution to a simplified sparse coding problem when we assume that the basis vectors are orthogonal (Ba, 2020; Gregor & LeCun, 2010). This results in the optimal solution for activity \(a_{i}\) to be given by \(\operatorname*{ReLU}(u_{i}-\lambda)-\operatorname*{ReLU}(-u_{i}-\lambda)\) or, if we constrain \(a_{i}\) to be non-negative, \(\operatorname*{ReLU}(u_{i}-\lambda)\), where \(u_{i}\) is the value obtained by projecting the data onto basis \(\phi_{i}\).

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 \(\mathbf{x}\).

\[\tilde{\mathbf{x}} =\mathbf{x}+\epsilon,\ \ \epsilon\sim N(0,\sigma I)\] \[\hat{\mathbf{x}} =W_{d}\ \text{ReLU}(W_{e}\tilde{\mathbf{x}}+\mathbf{b}_{e})+ \mathbf{b}_{d}\] \[\text{loss} =\sum_{\mathbf{x}\in X}(\mathbf{x}-\hat{\mathbf{x}})^{2}. \tag{1}\]

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.

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).

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

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

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

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

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse. We train our models in the same way as in Fig. 2 but at epoch 800 linearly anneal each noise level down to 0.0 over the next 800 epochs. Fig. 6 shows that the sparsity levels remain largely unperturbed.

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

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

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

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

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

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

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

Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

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

* 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.

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.

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

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

Input: Figure 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 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:
Here is my evaluation:

The claim "by simply introducing isotropic, symmetric noise centered about zero during training, a layer of artificial neurons will converge to a sparse coding solution" is largely well-supported by the comprehensive evidence provided. The paper demonstrates this through extensive experiments showing:

1. Clear correlation between noise and sparsity across multiple settings (Fig 7)
2. Formation of biologically plausible receptive fields (Fig 5, 8)
3. Retention of sparsity after noise removal (Fig 6)
4. Robustness across architectures, datasets, and tasks

While the reviewer notes limitations regarding deeper networks, the core claim about noise inducing sparse coding is carefully qualified and supported by thorough empirical evidence.

<score>1</score>

Justification details:
- The claim makes a focused statement about noise inducing sparsity and uses measured language ("will converge")
- Extensive ablation studies validate the robustness
- Results are demonstrated across multiple experimental conditions
- Biological receptive field emergence provides additional validation
- The paper acknowledges limitations and provides clear evidence for each component of the claim


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

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

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

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

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

The claim to be assessed is:

Here, we find that by simply introducing isotropic, symmetric noise centered about zero during training, a layer of artificial neurons will converge to a sparse coding solution.

The review comment to be evaluated is:

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

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

However, we are unaware of any existing work which finds that noisy training with a ReLU activation function not only results in sparsity but also converges to a sparse coding, Top-K network. Learning reconstructions of training data with sparse activations is the mainstay of sparse coding (Olshausen & Field, 1997).

Interestingly, the ReLU activation function that is crucial for the results of this paper can be derived as the optimal solution to a simplified sparse coding problem when we assume that the basis vectors are orthogonal (Ba, 2020; Gregor & LeCun, 2010). This results in the optimal solution for activity \(a_{i}\) to be given by \(\operatorname*{ReLU}(u_{i}-\lambda)-\operatorname*{ReLU}(-u_{i}-\lambda)\) or, if we constrain \(a_{i}\) to be non-negative, \(\operatorname*{ReLU}(u_{i}-\lambda)\), where \(u_{i}\) is the value obtained by projecting the data onto basis \(\phi_{i}\).

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 \(\mathbf{x}\).

\[\tilde{\mathbf{x}} =\mathbf{x}+\epsilon,\ \ \epsilon\sim N(0,\sigma I)\] \[\hat{\mathbf{x}} =W_{d}\ \text{ReLU}(W_{e}\tilde{\mathbf{x}}+\mathbf{b}_{e})+ \mathbf{b}_{d}\] \[\text{loss} =\sum_{\mathbf{x}\in X}(\mathbf{x}-\hat{\mathbf{x}})^{2}. \tag{1}\]

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.

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).

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

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

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

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

Investigating our noise trained sparse networks, we find that even after the noise is removed, they remain sparse. We train our models in the same way as in Fig. 2 but at epoch 800 linearly anneal each noise level down to 0.0 over the next 800 epochs. Fig. 6 shows that the sparsity levels remain largely unperturbed.

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

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

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

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

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

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

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

Moreover, the fact that noise alone results in sparse coding rather than having a sparsity penalty in the loss function suggests that the primary driver behind sparse coding may be the handling of noise with metabolic efficiency as an added benefit, rather than the other way around.

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

* 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.

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.

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

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

Input: Figure 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 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:
Summary: The claim appears well-stated (level 1) based on extensive empirical evidence. The paper demonstrates through multiple experiments, visualizations, and ablation studies that introducing isotropic Gaussian noise during training consistently leads to sparse coding solutions. Key evidence includes: (1) Robust sparsification across different datasets, architectures, and noise distributions (Fig. 7), (2) Formation of biologically plausible Gabor-like receptive fields (Fig. 5, 8), (3) Retention of sparsity after noise removal (Fig. 6), and (4) Maintained or improved performance on tasks. The claim is measured and accurately reflects the strength of the supporting evidence without exaggeration.

<score>1</score>

Justification:
1. Strong experimental validation:
- Comprehensive ablation studies across datasets, noise distributions, neuron counts, and tasks (Fig. 7)
- Consistent results with low variance across random seeds (Fig. 2)
- Clear quantitative metrics for sparsity levels and performance

2. Clear mechanistic understanding:
- Detailed analysis of bias terms and key vector norms (Fig. 3)
- Theoretical connections to sparse coding literature
- Well-documented failure cases (L1 penalty comparison)

3. Conservative phrasing:
- Uses "will converge" rather than stronger claims about optimality
- Specifies "isotropic, symmetric noise centered about zero"
- Does not overstate generality beyond tested conditions

4. Review comment aligns with claim's scope and findings

The evidence comprehensively supports the specific claim without extending beyond what is demonstrated.


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 solution mimicks a simplified version of the biological inhibitory interneuron circuit.

The evidence to be evaluated is: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). However, in all of these instances there are explicit mechanisms built into the network to enforce Top-K and handle repercussions such as creating dead neurons. 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).

This network was trained on the CIFAR10 embeddings to mimick training within a deeper network layer and where the manifold is smoother than when training on raw image pixels.

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

An increase in noise will create more erroneous low-level activations in neurons that must be ignored to maximize signal to noise ratio.

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.

We have shown the extent to which a feedforward ReLU MLP is capable of converging with a biological inhibitory interneuron circuit.

This result is particularly striking because of the way the networks converge with sparse coding, producing biological receptive fields and approximating an inhibitory interneuron that implements a Top-K operation.

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

We can model explaining away and facilitation with an additional weight matrix \(G\in\mathbb{R}^{m\times m}\) (recall \(m\) is the number of neurons) that we run in a recurrent inner loop. Our one hidden layer network becomes: \[\mu_{0} =W_{e}\tilde{\mathbf{x}}+\mathbf{b}_{e}\] \[\mathbf{z}_{t} =\text{ReLU}(\mu_{t})\] \[\Delta\mu =\big{(}\mu_{0}-G\mathbf{z}_{t}\big{)}-\mu_{t}\] \[\mu_{t+1} =\mu_{t}+\gamma\Delta\mu\] \[\hat{\mathbf{x}} =W_{d}\text{ ReLU}(\mu_{t^{*}})+\mathbf{b}_{d}, \tag{2}\]

We can also replace \(G\mathbf{z}_{t}\) with \(G_{d}G_{e}\mathbf{z}_{t}\) where \(G_{e}\in\mathbb{R}^{1\times m},G_{d}\in\mathbb{R}^{m\times 1}\) implement a single inhibitory interneuron that uses \(2m\) rather than \(m^{2}\) parameters and must learn each of its afferent and efferent connections. When the neurons still have their bias terms this solution has no effect. 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. However, this shows that the full inhibitory interneuron solution is at least plausible.

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

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

Output:
I'll evaluate the claim "This solution mimicks a simplified version of the biological inhibitory interneuron circuit."

Summary:
The evidence shows that while the model implements certain biological-like properties (sparse activation, Top-K convergence, and formation of biologically plausible receptive fields), it's a significant simplification. The paper acknowledges this by showing that replacing G with GdGe creates a "single inhibitory interneuron" that performs "well" but has limitations like dead neurons and less sparsity than baseline models. The claim carefully uses the word "mimicks" and "simplified version," which accurately reflects the evidence's scope without overstating the biological similarity.

<score>1</score>

Justification:
1. The claim is well-stated because:
- It explicitly acknowledges simplification rather than claiming exact replication
- The evidence supports basic biological similarities:
  - Formation of biological receptive fields (Fig. 5)
  - Implementation of Top-K operations similar to inhibitory circuits
  - Sparse activation patterns
2. The paper's own discussion acknowledges limitations while demonstrating partial success: "performs well...but results in a number of dead neurons and is not as sparse as the baseline"
3. The mathematical formulation (Equation 2) shows a simplified implementation that captures key biological circuit properties without overclaiming complete biological fidelity

The modest wording of the claim ("mimicks", "simplified") aligns well with both the demonstrated capabilities and acknowledged limitations in the evidence.


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

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

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

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

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

The claim to be assessed is:

This solution mimicks a simplified version of the biological inhibitory interneuron circuit.

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: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). However, in all of these instances there are explicit mechanisms built into the network to enforce Top-K and handle repercussions such as creating dead neurons. 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).

This network was trained on the CIFAR10 embeddings to mimick training within a deeper network layer and where the manifold is smoother than when training on raw image pixels.

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

An increase in noise will create more erroneous low-level activations in neurons that must be ignored to maximize signal to noise ratio.

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.

We have shown the extent to which a feedforward ReLU MLP is capable of converging with a biological inhibitory interneuron circuit.

This result is particularly striking because of the way the networks converge with sparse coding, producing biological receptive fields and approximating an inhibitory interneuron that implements a Top-K operation.

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

We can model explaining away and facilitation with an additional weight matrix \(G\in\mathbb{R}^{m\times m}\) (recall \(m\) is the number of neurons) that we run in a recurrent inner loop. Our one hidden layer network becomes: \[\mu_{0} =W_{e}\tilde{\mathbf{x}}+\mathbf{b}_{e}\] \[\mathbf{z}_{t} =\text{ReLU}(\mu_{t})\] \[\Delta\mu =\big{(}\mu_{0}-G\mathbf{z}_{t}\big{)}-\mu_{t}\] \[\mu_{t+1} =\mu_{t}+\gamma\Delta\mu\] \[\hat{\mathbf{x}} =W_{d}\text{ ReLU}(\mu_{t^{*}})+\mathbf{b}_{d}, \tag{2}\]

We can also replace \(G\mathbf{z}_{t}\) with \(G_{d}G_{e}\mathbf{z}_{t}\) where \(G_{e}\in\mathbb{R}^{1\times m},G_{d}\in\mathbb{R}^{m\times 1}\) implement a single inhibitory interneuron that uses \(2m\) rather than \(m^{2}\) parameters and must learn each of its afferent and efferent connections. When the neurons still have their bias terms this solution has no effect. 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. However, this shows that the full inhibitory interneuron solution is at least plausible.

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

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

Output:
After reviewing the evidence, the claim that "This solution mimicks a simplified version of the biological inhibitory interneuron circuit" appears overstated. While the evidence shows some biological-like features (Gabor filters and center-surround receptive fields) emerging under noisy conditions, and demonstrates an implementation of inhibitory connections, the reviewers correctly point out that multiple mechanisms can generate such patterns. The evidence lacks direct quantitative comparisons to biological circuits, and the model's behavior varies significantly with different activation functions and optimization methods. The biological relevance is limited by numerous differences between this simplified model and actual neural circuits.

<score>4</score>

Justification:
1. The evidence shows only superficial similarities rather than true biological mimicry:
- The paper shows emergence of Gabor-like filters and center-surround patterns, but as the reviewer notes, these can arise from multiple mechanisms
- The mathematical implementation (equation 2) shows a simplified inhibitory circuit, but doesn't demonstrate biological fidelity

2. Key limitations not acknowledged in the claim:
- No quantitative comparison to biological data
- Results depend heavily on specific activation functions and optimization parameters
- The reviewer points out that biological networks differ in many important ways

3. Overstatement in language:
- Claims to "mimic" biological circuits without sufficient validation
- The evidence mainly shows some emergent visual properties that resemble biological patterns, rather than true circuit mimicry
- As noted in the review, "the link with biological receptive fields is tenuous"

The claim would be better stated as suggesting inspiration from or loose similarity to biological circuits, rather than claiming mimicry.