Paper: Latent Noise Segmentation: How Neural Noise Leads to the Emergence of Segmentation and Grouping (ICLR-2024)

Reviewer comment (Experiments):
The main evaluation is done on a self-made grouping stimulus set, which trains the network quite explicitly to produce the grouping made by human observers, as the intended objects are exactly the groups that vary separately in the stimulus generation, for example by switching color together. While this is clearly not direct supervision, it does create a statistical structure that strongly favours the representation of object-centred dimensions, while there are no variations within objects or over the whole scene. This makes it a lot less surprising that autoencoders find dimensions that create correlations within objects.

Author response:
The motivation behind this choice is a simplification of natural image statistics - the luminance-illuminance-reflectance relationship. For a set of naturalistic images in the same scene, if some surface shares a material, they also share their reflectance, which is the cue of the color in our synthetic dataset. We certainly agree that this relationship does not hold true in all scenarios, but its simplicity lends itself to a more detailed analysis. 


The cue of color is also not strictly necessary (as seen in CelebA): if there was no cue of color, the information revealed by LNS would allow the remaining covariances within objects, such as their position, to dominate. When viewed in segmentation masks, this would certainly make the segmentation masks appear hollow unless there were other simultaneous cues that affected the entire object at once (e.g., 3D rotation), which we do not have in our simplified 2D datasets. To improve the mathematical rigor of the work, and to make the datasets themselves more accessible, we chose to take the simplification. We believe that investigating the more general case where other cues that are less amenable to mathematical exposition are present is a fruitful future direction, but given the extensive theoretical and empirical analysis we show here, outside the scope of this work.

Finally, we added other model comparisons trained and evaluated on the same datasets to show that even when the models are well-trained (Appendix C2), they fail to exhibit the desired grouping results in our datasets (Table 1; we append this table to another comment you raised, Appendix C1).

Reviewer comment (Theory):
The authors emphasise noise strongly in their arguments and explain how iid noise might emphasise the local PCs around the stimulus. While these arguments are not technically incorrect, they seem besides the point. The main step to make this technique work seems to be the shape of the derivative of the decoder around the stimulus. The decoder seems to create correlated changes in the parts that belong to the same object, which yields high similarity for those pixels. It is interesting that this effect is strong enough that few samples are sufficient to separate objects successfully, but in principle any way of estimating this derivative should work. As the noise size here corresponds to the typical delta used to compute the approximate derivative, it is also not surprising that small noise works well. The interesting part is that the derivative creates correlated changes for each object, not that this can be estimated based on noise samples.

Author response:
We agree with the reviewer that any derivative estimation method should work - the interesting part is that something as naive as noise actually estimates this gradient. Second, we agree that the shape of the derivative of the decoder around the stimulus is the relevant part. To briefly clarify this point: at a high level, the reconstruction-oriented training process optimizing ELBO tends to represent local principle components of the training data samples at the derivative of the output with respect to the latents (for more details, please see Rolinek et al. (2019) and Zietlow et al. (2021)). The distributions of the training samples cause the local principal components to exhibit certain patterns. The derivative's components belonging to one object correlate because the distribution of the training samples does not span in the direction where they do not correlate. Likewise, the derivative's components belonging to different objects are not likely to always correlate since the distributions of the training samples span a space characterizing them as independent. We include a full exposition of this in Appendix A. 

To summarize, the distribution of the training samples makes the "shape" of derivatives favorable for our segmentation task (Appendix A). The intriguing fact is that the distribution of the training dataset follows, from the perspective of optics, a simplification of real-world visual scenes, which we discuss in some detail in Appendix E.

Reviewer comment (Theory):
The connection to human or biological vision is weaker than suggested by the authors. In biological vision, segmentation is needed of the internal layer representations, not of the pixels in the image. This requires that the encoder and decoder are related in some way that allows us to connect our noise reconstructions and the encoding elements. Additionally, a biologically plausible clustering algorithm would be required that is based on the found similarities in the noise. Neither of these are present in the model presented here, and substantial revisions would be necessary to transform the proposed method into a biologically plausible method for object segmentation.

Author response:
Thank you for the comment - the connection to biology was also brought up by another reviewer. Our goal is not to propose an implementation-level plausible model of the visual system, but rather a computational model for perceptual grouping.

To make this clear in the manuscript, we have rephrased parts of the paper:

>_**Abstract**: “Finally, we (4) demonstrate the __ecological plausibility__ of the method by analyzing the sensitivity of the DNN to different magnitudes of noise.”_ 

=> _“Finally, we (4) demonstrate the __practical feasibility__ of the method by analyzing the sensitivity of the DNN to different magnitudes of noise.”_

>_**Abstract**: “Together, our results suggest a novel unsupervised segmentation method requiring few assumptions, a new explanation for the formation of perceptual grouping, __and a potential benefit of neural noise in the visual system__”_ 

=> _“Together, our results suggest a novel unsupervised segmentation method requiring few assumptions, a new explanation for the formation of perceptual grouping, **and a novel potential benefit of neural noise.**”_

>_**Introduction**: “Finally, we study how **ecologically plausible LNS is (that is, how in-principle usable the method is by a biological system)**”_ 

=> _“Finally, we study how **practically feasible LNS is (that is, how in-principle usable the method is by any system constrained by limitations like compute time or noise magnitude, such as the primate visual system)**”_

> _**Introduction**: “Our results suggest that an **ecologically plausible** number of time steps” _

=> _“Our results suggest that a **practically feasible** number of time steps”_

> _**Section 3.3**: “To understand how **ecologically plausible** LNS is, we wanted to understand how the amount of noise added in latent space would affect the model’s segmentation performance.”_

=> _“To understand how **practically feasible LNS is for a system constrained by compute time or noise magnitude**, we analyzed how the amount of noise added in latent space would affect the model’s segmentation performance.”_

> _**Conclusion**: “With both learning rules, segmentation performance asymptotes quickly, suggesting that the time scale of the segmentation in our models is **ecologically plausible**.”_

=> _“With both learning rules, segmentation performance asymptotes quickly, suggesting that the time scale of the segmentation in our models is **practically feasible for constrained systems**.”_

>[...] In biological vision we need a segmentation of the internal layer representations, not of the pixels in the image. This requires that the encoder and decoder are in some way related that allows us to connect our noise reconstructions and the encoding elements. [...]

We would like to clarify on this point that was also alluded to by another reviewer:
Latent Noise Segmentation is a way to “show” the result of segmentation that the network has already learned during training. Specifically, LNS reveals the shape of the local derivatives around the input sample. This additional information that LNS reveals enables segmentation using a variety of methods. In our paper, we segment by propagating this information through the decoder, and clustering the outputs. Naturally, this segmentation could happen at any downstream layer (thanks to the chain rule: if the clusterability of the representation depends on the shape of the derivatives, which it does, and is present in the last layer, which it is, then it must propagate through the entire decoder and is accessible at any level of representation). 

To make this point clearer, we have changed the text in Section 2.2 as follows:

“[...] This means that we can perturb all units simultaneously with independent noise: training on a generic reconstruction loss encourages the network to code in a manner that allows independent noise to reveal the relevant derivative direction, making the representation clusterable for segmentation (Figure1b).

Indeed, the goal of independent noise is not to perform segmentation itself, but to reveal the local neighborhood of the input stimulus and thus enable segmentation, _from the perspective of the decoder_.”

Reviewer comment (Theory):
I would like to understand how the authors imagine the noise-based segmentation to work in biological vision: How does the autoencoder model map to the brain? And what evidence is there that anything like this might actually happen in the brain?

Author response:
To clarify, our goal is certainly not to propose an implementation-level model of the visual system, but rather, a computational account of perceptual grouping. That being said, our model choice is inspired and motivated by links to biology (VAEs and predictive coding: Marino, 2022; Bouting et al., 2020; VAEs and the Free Energy Principle: Friston, 2010; Mazzaglia, 2022; empirical support for VAEs in macaques: Higgins et al., 2021). While we tried to be as careful as possible when writing the manuscript, we hope that the changes to the text in the manuscript outlined in an earlier reply help clarify this intention.

Reviewer comment (Experiments):
Does this work in any way for natural scenes? Testing this on existing autoencoders for natural scenes could avoid concerns about the training data being very targeted to create the patterns observed in humans.

Author response:
To address the concerns about the training data being very targeted, and about how alternative methods do on these datasets, we now report results for two additional SOTA/near-SOTA models: Genesis and Genesis-v2 (Engelcke et al., 2020; 2022). 

| Model     | Kanizsa           | Closure           | Continuity        | Proximity         | Gradient Occ.     | Illusory Occ.     |
|-----------|-------------------|-------------------|-------------------|-------------------|-------------------|-------------------|
| AE        | 0.859 ± 0.008     | 0.766 ± 0.008     | 0.552 ± 0.008     | **0.996\*** ± 0.001 | 0.926 ± 0.009     | **0.994\*** ± 0.002 |
| VAE       | **0.871** ± 0.005 | **0.795** ± 0.009 | 0.593 ± 0.012     | 0.943 ± 0.010     | 0.918 ± 0.002     | 0.974 ± 0.003     |
| AE Rec.   | 0.246 ± 0.006     | 0.064 ± 0.001     | 0.570 ± 0.006     | 0.141 ± 0.001     | **0.982** ± 0.000 | -0.023 ± 0.001    |
| VAE Rec.  | 0.343 ± 0.004     | 0.084 ± 0.000     | **0.611** ± 0.005 | 0.144 ± 0.001     | 0.977 ± 0.001     | -0.024 ± 0.001    |
| Genesis   | 0.346             | 0.755             | 0.394             | 0.879             | 0.933             | 0.294             |
| Genesis-v2| 0.415             | 0.000             | 0.399             | 0.059             | 0.000             | 0.422             |

*Model results (ARI ± standard error of the mean). AE (Autoencoder) and VAE (Variational Autoencoder) model scores reported for the best noise level and number of samples. AE Rec. and VAE Rec. stand for noisy reconstruction control experiments, and (see Section 2.4 for details). Bold indicates top-performing model. For the AE and the VAE, a star (\*) indicates a statistically significant difference in model performance (Bonferroni corrected).*

These results are now reported in the main text in Table 1. While quantitative results for the Genesis model look impressive, a further investigation in Appendix C1 reveals that the decent quantitative performance is driven by good pixel-level predictions in nearly all cases, and not actual Gestalt grouping phenomena (Figure 10). Indeed, the model appears to primarily follow pixel-level information present in the input.

To verify that this result was not caused by a mistake in our implementation, we followed the official implementation of Genesis and Genesis-v2, and show reconstructions on a separate validation set (Appendix C2).

Thank you once again for your review. We believe that the additional clarifications, experiments and analyses address your comments, and we hope that you consider raising your score.

Reviewer comment (Experiments):
What about comparisons to alternative methods? Are there other methods for segmentation or grouping that get the gestalt tests wrong?

Author response:
To address the concerns about the training data being very targeted, and about how alternative methods do on these datasets, we now report results for two additional SOTA/near-SOTA models: Genesis and Genesis-v2 (Engelcke et al., 2020; 2022). 

| Model     | Kanizsa           | Closure           | Continuity        | Proximity         | Gradient Occ.     | Illusory Occ.     |
|-----------|-------------------|-------------------|-------------------|-------------------|-------------------|-------------------|
| AE        | 0.859 ± 0.008     | 0.766 ± 0.008     | 0.552 ± 0.008     | **0.996\*** ± 0.001 | 0.926 ± 0.009     | **0.994\*** ± 0.002 |
| VAE       | **0.871** ± 0.005 | **0.795** ± 0.009 | 0.593 ± 0.012     | 0.943 ± 0.010     | 0.918 ± 0.002     | 0.974 ± 0.003     |
| AE Rec.   | 0.246 ± 0.006     | 0.064 ± 0.001     | 0.570 ± 0.006     | 0.141 ± 0.001     | **0.982** ± 0.000 | -0.023 ± 0.001    |
| VAE Rec.  | 0.343 ± 0.004     | 0.084 ± 0.000     | **0.611** ± 0.005 | 0.144 ± 0.001     | 0.977 ± 0.001     | -0.024 ± 0.001    |
| Genesis   | 0.346             | 0.755             | 0.394             | 0.879             | 0.933             | 0.294             |
| Genesis-v2| 0.415             | 0.000             | 0.399             | 0.059             | 0.000             | 0.422             |

*Model results (ARI ± standard error of the mean). AE (Autoencoder) and VAE (Variational Autoencoder) model scores reported for the best noise level and number of samples. AE Rec. and VAE Rec. stand for noisy reconstruction control experiments, and (see Section 2.4 for details). Bold indicates top-performing model. For the AE and the VAE, a star (\*) indicates a statistically significant difference in model performance (Bonferroni corrected).*

These results are now reported in the main text in Table 1. While quantitative results for the Genesis model look impressive, a further investigation in Appendix C1 reveals that the decent quantitative performance is driven by good pixel-level predictions in nearly all cases, and not actual Gestalt grouping phenomena (Figure 10). Indeed, the model appears to primarily follow pixel-level information present in the input.

To verify that this result was not caused by a mistake in our implementation, we followed the official implementation of Genesis and Genesis-v2, and show reconstructions on a separate validation set (Appendix C2).

Thank you once again for your review. We believe that the additional clarifications, experiments and analyses address your comments, and we hope that you consider raising your score.

Reviewer comment (Experiments):
Can the authors please comment on whether they experimented with harder versions of GG? Here are a few potential options: (1) Change the size of the square/circle between train and test splits for Kanisza Squares, Closure, Continuity, (2) Use different rotations (without overlap between train and test splits) of the Kanisza squares / Closure squares, (3) Use different colors / textures as backgrounds in all tasks. Any modification of the dataset in the spirit of the modifications suggested here will further strengthen the message that simple low-level statistics don't drive the emergence of perceptual grouping.

Author response:
Although we don’t have quantitative results to report on this, during the design of the GG datasets we did have a variant with different sizes for Illusory Occlusion. Qualitatively, the results did not differ. This is not surprising because Latent Noise Segmentation can be viewed as a form of “simultaneous random latent traversal” (Kingma & Welling, 2013; Higgins et al., 2017). Specifically, in latent traversal you change the activity of one latent unit at a time to understand how that changes the output image. Thus, when adding noise to the latents in the presence of a generative factor of size in the dataset (and thus the latents), size merely provides an additional cue for grouping the contours of the image. A similar effect already exists in the data, induced by the generative factor of position. 

We attempted to avoid making the GG datasets difficult for reasons that did not directly contribute to testing for grouping capabilities of the model. This was done to make the datasets, and training on them, as accessible as possible as low-cost tests of specifically grouping. We believe our additional baseline results suggest that we somewhat succeeded in hitting the sweet spot - it is possible to train other models to convergence on the datasets - but together the datasets discriminate between models that perform grouping, and those that do not. Indeed, there is still upwards room on our datasets quantitatively, as even though our approach solves the necessary perceptual grouping conditions, it is clear that models that do so more accurately could improve quantitative scores on these benchmarks further.

Reviewer comment (Theory):
The paper claims that the results show “potential benefit of neural noise in the visual system” and repeatedly claims to study the ecological plausibility of the method, but there is not a substantial connection between this method and biology. Noise is prevalent in biological vision, but the method relies on small-amplitude noise, whereas spiking noise is closer to Poisson.

Author response:
Thank you for your detailed review. We are glad you found our datasets interesting, and approach novel and creative. We have made several improvements to our manuscript as a result of your comments, and believe it now more carefully represents the primary goal of our work: that of a computational approach to perceptual grouping and segmentation, rather than an implementation-level explanation of specific neural processes in the primate visual system.

>The abstract claims that the results show, “potential benefit of neural noise in the visual system” and the paper repeatedly claims to study the ecologically plausibility of the method. Noise is certainly prevalent in biological vision, but I'm not convinced that there is a substantial connection between this method and biology.

We attempted to be extraordinarily careful with our phrasing in the paper, not making any direct claims about human/primate vision. Indeed, in the discussion, we explicitly state:
_“Finally, while we do not make any claims about the primate visual system, our results suggest a potential benefit of independent noise in the visual system.“_

That being said, we apologize that there remains a lack of clarity about the objectives of the work in its framing. In the work, we study our approach of perceptual grouping and segmentation from a computational perspective, looking for a computational solution (rather than an implementation-level solution) to the well-known Gestalt principles of grouping. Our main goal is not to provide an exhaustive explanation of segmentation in primates, but rather to take a substantial and theoretically-grounded step towards an explanation of the Gestalt grouping laws in a computational framework. Our work is certainly inspired by biology, and our model choice was specifically driven by their computational and experimental links to biology (_Predictive coding: Marino, 2022; Bouting et al., 2020; Free Energy Principle: Friston, 2010; Mazzaglia, 2022; empirical support in macaques: Higgins et al., 2021_), but we agree that this does not constitute a link to biology. 

As such, we have rephrased our terminology in parts of the paper, specifically removing mentions of ecological plausibility:

>_**Abstract**: “Finally, we (4) demonstrate the __ecological plausibility__ of the method by analyzing the sensitivity of the DNN to different magnitudes of noise.”_ 

=> _“Finally, we (4) demonstrate the __practical feasibility__ of the method by analyzing the sensitivity of the DNN to different magnitudes of noise.”_

>_**Abstract**: “Together, our results suggest a novel unsupervised segmentation method requiring few assumptions, a new explanation for the formation of perceptual grouping, __and a potential benefit of neural noise in the visual system__”_ 

=> _“Together, our results suggest a novel unsupervised segmentation method requiring few assumptions, a new explanation for the formation of perceptual grouping, **and a novel potential benefit of neural noise.**”_

>_**Introduction**: “Finally, we study how **ecologically plausible LNS is (that is, how in-principle usable the method is by a biological system)**”_ 

=> _“Finally, we study how **practically feasible LNS is (that is, how in-principle usable the method is by any system constrained by limitations like compute time or noise magnitude, such as the primate visual system)**”_

> _**Introduction**: “Our results suggest that an **ecologically plausible** number of time steps” _

=> _“Our results suggest that a **practically feasible** number of time steps”_

> _**Section 3.3**: “To understand how **ecologically plausible** LNS is, we wanted to understand how the amount of noise added in latent space would affect the model’s segmentation performance.”_

=> _“To understand how **practically feasible LNS is for a system constrained by compute time or noise magnitude**, we analyzed how the amount of noise added in latent space would affect the model’s segmentation performance.”_

> _**Conclusion**: “With both learning rules, segmentation performance asymptotes quickly, suggesting that the time scale of the segmentation in our models is **ecologically plausible**.”_

=> _“With both learning rules, segmentation performance asymptotes quickly, suggesting that the time scale of the segmentation in our models is **practically feasible for constrained systems**.”_

Reviewer comment (Theory):
The method uses agglomerative clustering, which raises questions about its biological plausibility.

Author response:
>  Issues include: 1) reliance of the method on small-amplitude noise, whereas spiking noise is closer to Poisson

Our analyses show that for Latent Noise Segmentation, the noise scale is only meaningful in a comparative, not an absolute sense. What we believe matters is the ratio of noise scale to latent traversal scale (as mentioned by reviewer zkrR). As such, without knowing the scale of the true signal vs the scale of independent noise, a direct implementation-level comparison (especially given how much we do not know about neural coding and noise in cortex) seems difficult in principle. Indeed, we show that for the AE model, various different levels of noise result in good performance, and that if we fix our model coding scheme (in the case of the VAE, where we fix the latent prior to be normally distributed), we consistently find a small range of optimal noise values (Figures 5 and 6).

>Issues include: 2) use of agglomerative clustering

We have made changes in the manuscript to further clarify that the point of LNS is not that we apply a specific clustering algorithm, but that the noise added in latent space reveals a clusterable representation - one that was not clusterable prior to adding noise:

Section 2.2: “[...] This means that we can perturb all units simultaneously with independent noise: training on a generic reconstruction loss encourages the network to code in a manner that allows independent noise to reveal the relevant derivative direction, making the representation clusterable for segmentation (Figure1b).
Indeed, the goal of independent noise is not to perform segmentation itself, but to reveal the local neighborhood of the input stimulus and thus enable segmentation, _from the perspective of the decoder_.”

To further demonstrate that the specific clustering algorithm is not critical to the method, we have added a new set of experiments in Appendix D1.

| Model      | Kanizsa                 | Closure                 | Continuity              | Proximity               | Gradient Occ.           | Illusory Occ.           |
|------------|-------------------------|-------------------------|-------------------------|-------------------------|-------------------------|-------------------------|
| AE+Agg     | 0.859 ± 0.008           | 0.766 ± 0.008           | 0.552 ± 0.008           | **0.996 ± 0.001**       | **0.926 ± 0.009**       | 0.994 ± 0.002           |
| VAE+Agg    | **0.871 ± 0.005**       | 0.795 ± 0.009           | **0.593 ± 0.012**       | 0.943 ± 0.010           | 0.918 ± 0.002           | 0.974 ± 0.003           |
| AE+KMeans  | 0.854 ± 0.003           | 0.803 ± 0.001           | 0.553 ± 0.004           | 0.995 ± 0.000           | 0.922 ± 0.006           | **0.998 ± 0.000**       |
| VAE+KMeans | 0.866 ± 0.003           | **0.852 ± 0.006**       | 0.589 ± 0.006           | 0.921 ± 0.005           | 0.924 ± 0.003           | 0.986 ± 0.001           |
| AE+Agg'    | 0.557 ± 0.007           | 0.296 ± 0.006           | 0.322 ± 0.004           | 0.929 ± 0.004           | 0.702 ± 0.004           | 0.520 ± 0.027           |
| VAE+Agg'   | 0.457 ± 0.008           | 0.324 ± 0.004           | 0.302 ± 0.004           | 0.752 ± 0.012           | 0.594 ± 0.009           | 0.438 ± 0.006           |

*Table 2: Model results for different clustering algorithms (ARI ± standard error of the mean).*
*Agg = the agglomerative clustering algorithm from `sklearn` with `metric` being "euclidean" and `linkage` being "ward".*
*Agg' = the agglomerative clustering algorithm from `sklearn` with `metric` being "euclidean" and `linkage` being "complete".*
*KMeans = the K-means algorithm from `sklearn`.*
*These scores are obtained for the optimal noise level and number of samples found for Agg.*

In summary, while clustering algorithm can change the results (agglomerative clustering with the Ward linkage does better than Agglomerative Clustering with the Complete linkage), the results do not crucially depend on the clustering algorithm (K-means clustering performs as well as, or better than, Agglomerative Clustering with the Ward linkage).

In addition, we now report UMAP visualizations for the clusterability of the examples shown in Figure 4. This additional section can be found in Appendix F2, and further bolsters the claim that the specific clustering method is not important, but rather, that clustering is only a part of the algorithm that allows us to convert already-existing clusters into segmentation masks for visualization.

Reviewer comment (Experiments):
There is a lack of comparison with neural data, which weakens the claim of biological relevance.

Author response:
> Issues include: 3) lack of comparison with neural data

While we agree that comparing to neural data would be a nice addition, we wanted to focus on a behavioral result in combination with an in-depth theoretical account of the phenomenon. A substantial reason for this is a recent movement in the community of modeling human vision (Bowers et al., 2022) that argues that predicting neural activity has gained a large amount of traction in the literature, while targeted modeling of specific phenomena in human vision with controlled experiments has fallen by the wayside. 

Here, we aim to explicitly address the latter: we take some of the most classical results in human vision with over a century of history, and model them directly with control experiments, and provide a detailed exposition of why and how our methodology works. Specifically, because of the relative simplicity of our approach we are able to formulate links between basic image statistics, and segmentation using our method (illuminance-luminance-reflectance, Appendix A1 & E1). We thus strongly believe this is a timely contribution from _also_ this perspective (in addition to the contribution of the methodology itself), even if it slightly deviates from the mainstream of neural predictivity analysis.

Reviewer comment (Evaluation):
The method shows poor performance on natural images. Although tested on celebrity faces in addition to Gestalt dataset images, the results are generally poor. According to Table 1, even in controlled circumstances, the model only outperforms the control (same model with noise applied to output instead of latent representation) in 4 out of 6 cases.

Author response:
>To elaborate on point 5 above, the method is not meant to be practical, but its low practicality is also a concern for biological plausibility. It is not extensively tested on natural or otherwise practical images but in addition to the Gestalt dataset images, it was tested on celebrity faces and the paper claims that the method “often finds a semantically meaningful segmentation of face-hair-background”. However, examination of the results (Appendix A.3) shows that the results are generally poor. According to Table 1, even in these artificial circumstances, the model only outperforms the control in 4/6 cases. The control is the same model with noise applied to the output rather than to the latent representation. It could be interesting to also contrast with established strong segmentation methods, particularly if this method agreed with humans in conditions where others don’t. However, some promise of strong performance would be needed in an convincing model of human vision.

Thank you for the suggestion - we think your suggestion of evaluating other models on our datasets is relevant and important. We now report results for two additional SOTA/near-SOTA models: Genesis and Genesis-v2 (Engelcke et al., 2020; 2022). 

| Model     | Kanizsa           | Closure           | Continuity        | Proximity         | Gradient Occ.     | Illusory Occ.     |
|-----------|-------------------|-------------------|-------------------|-------------------|-------------------|-------------------|
| AE        | 0.859 ± 0.008     | 0.766 ± 0.008     | 0.552 ± 0.008     | **0.996\*** ± 0.001 | 0.926 ± 0.009     | **0.994\*** ± 0.002 |
| VAE       | **0.871** ± 0.005 | **0.795** ± 0.009 | 0.593 ± 0.012     | 0.943 ± 0.010     | 0.918 ± 0.002     | 0.974 ± 0.003     |
| AE Rec.   | 0.246 ± 0.006     | 0.064 ± 0.001     | 0.570 ± 0.006     | 0.141 ± 0.001     | **0.982** ± 0.000 | -0.023 ± 0.001    |
| VAE Rec.  | 0.343 ± 0.004     | 0.084 ± 0.000     | **0.611** ± 0.005 | 0.144 ± 0.001     | 0.977 ± 0.001     | -0.024 ± 0.001    |
| Genesis   | 0.346             | 0.755             | 0.394             | 0.879             | 0.933             | 0.294             |
| Genesis-v2| 0.415             | 0.000             | 0.399             | 0.059             | 0.000             | 0.422             |

*Model results (ARI ± standard error of the mean). AE (Autoencoder) and VAE (Variational Autoencoder) model scores reported for the best noise level and number of samples. AE Rec. and VAE Rec. stand for noisy reconstruction control experiments, and (see Section 2.4 for details). Bold indicates top-performing model. For the AE and the VAE, a star (\*) indicates a statistically significant difference in model performance (Bonferroni corrected).*

These results are now reported in the main text in Table 1. While quantitative results for the Genesis model look impressive, a further investigation in Appendix C1 reveals that the decent quantitative performance is driven by good pixel-level predictions in nearly all cases, and not actual Gestalt grouping phenomena (Figure 10). Indeed, the model appears to primarily follow pixel-level information present in the input.

To verify that this result was not caused by a mistake in our implementation, we followed the official implementation of Genesis and Genesis-v2, and show reconstructions on a separate validation set (Appendix C2).

Finally, we would like to emphasize out that coping with full-scale naturalistic images and revealing computational principles of perceptual grouping are fundamentally different objectives of research. Here, we mainly focused on the latter.

> What size are the GG datasets?

Training: 30,000 images, validation: 300 images that are not present in the training set, test: 100 images. We have also added this information in Appendix E1.

Thank you again for your comments, which have helped us improve our manuscript. We believe our substantial additional experiments and analyses, combined with a careful rephrasing of critical parts in the paper address your comments, and we hope that you consider increasing your score as a result of these changes.

Reviewer comment (Experiments):
The method is not extensively tested on natural or practical images, and its low practicality is also a concern for biological plausibility. It would be interesting to contrast results with established strong segmentation methods, especially if this method agreed with humans where others don’t, but strong performance evidence would be needed for a convincing model of human vision.

Author response:
>To elaborate on point 5 above, the method is not meant to be practical, but its low practicality is also a concern for biological plausibility. It is not extensively tested on natural or otherwise practical images but in addition to the Gestalt dataset images, it was tested on celebrity faces and the paper claims that the method “often finds a semantically meaningful segmentation of face-hair-background”. However, examination of the results (Appendix A.3) shows that the results are generally poor. According to Table 1, even in these artificial circumstances, the model only outperforms the control in 4/6 cases. The control is the same model with noise applied to the output rather than to the latent representation. It could be interesting to also contrast with established strong segmentation methods, particularly if this method agreed with humans in conditions where others don’t. However, some promise of strong performance would be needed in an convincing model of human vision.

Thank you for the suggestion - we think your suggestion of evaluating other models on our datasets is relevant and important. We now report results for two additional SOTA/near-SOTA models: Genesis and Genesis-v2 (Engelcke et al., 2020; 2022). 

| Model     | Kanizsa           | Closure           | Continuity        | Proximity         | Gradient Occ.     | Illusory Occ.     |
|-----------|-------------------|-------------------|-------------------|-------------------|-------------------|-------------------|
| AE        | 0.859 ± 0.008     | 0.766 ± 0.008     | 0.552 ± 0.008     | **0.996\*** ± 0.001 | 0.926 ± 0.009     | **0.994\*** ± 0.002 |
| VAE       | **0.871** ± 0.005 | **0.795** ± 0.009 | 0.593 ± 0.012     | 0.943 ± 0.010     | 0.918 ± 0.002     | 0.974 ± 0.003     |
| AE Rec.   | 0.246 ± 0.006     | 0.064 ± 0.001     | 0.570 ± 0.006     | 0.141 ± 0.001     | **0.982** ± 0.000 | -0.023 ± 0.001    |
| VAE Rec.  | 0.343 ± 0.004     | 0.084 ± 0.000     | **0.611** ± 0.005 | 0.144 ± 0.001     | 0.977 ± 0.001     | -0.024 ± 0.001    |
| Genesis   | 0.346             | 0.755             | 0.394             | 0.879             | 0.933             | 0.294             |
| Genesis-v2| 0.415             | 0.000             | 0.399             | 0.059             | 0.000             | 0.422             |

*Model results (ARI ± standard error of the mean). AE (Autoencoder) and VAE (Variational Autoencoder) model scores reported for the best noise level and number of samples. AE Rec. and VAE Rec. stand for noisy reconstruction control experiments, and (see Section 2.4 for details). Bold indicates top-performing model. For the AE and the VAE, a star (\*) indicates a statistically significant difference in model performance (Bonferroni corrected).*

These results are now reported in the main text in Table 1. While quantitative results for the Genesis model look impressive, a further investigation in Appendix C1 reveals that the decent quantitative performance is driven by good pixel-level predictions in nearly all cases, and not actual Gestalt grouping phenomena (Figure 10). Indeed, the model appears to primarily follow pixel-level information present in the input.

To verify that this result was not caused by a mistake in our implementation, we followed the official implementation of Genesis and Genesis-v2, and show reconstructions on a separate validation set (Appendix C2).

Finally, we would like to emphasize out that coping with full-scale naturalistic images and revealing computational principles of perceptual grouping are fundamentally different objectives of research. Here, we mainly focused on the latter.

> What size are the GG datasets?

Training: 30,000 images, validation: 300 images that are not present in the training set, test: 100 images. We have also added this information in Appendix E1.

Thank you again for your comments, which have helped us improve our manuscript. We believe our substantial additional experiments and analyses, combined with a careful rephrasing of critical parts in the paper address your comments, and we hope that you consider increasing your score as a result of these changes.

Reviewer comment (Experiments):
The role of the post-processing step is not evaluated: does agglomerative clustering play a big role? There are other standard clustering methods that could be tested.

Author response:
Thank you for your review, and your positive assessment of our paper.

>The role of the post-processing step is not evaluated : does agglomerative clustering play a big role ? There are other standard clustering methods that could be tested.

We have conducted additional experiments to evaluate how critical the specific clustering algorithm is for our results. We provide results below and in Appendix D1.

| Model      | Kanizsa                 | Closure                 | Continuity              | Proximity               | Gradient Occ.           | Illusory Occ.           |
|------------|-------------------------|-------------------------|-------------------------|-------------------------|-------------------------|-------------------------|
| AE+Agg     | 0.859 ± 0.008           | 0.766 ± 0.008           | 0.552 ± 0.008           | **0.996 ± 0.001**       | **0.926 ± 0.009**       | 0.994 ± 0.002           |
| VAE+Agg    | **0.871 ± 0.005**       | 0.795 ± 0.009           | **0.593 ± 0.012**       | 0.943 ± 0.010           | 0.918 ± 0.002           | 0.974 ± 0.003           |
| AE+KMeans  | 0.854 ± 0.003           | 0.803 ± 0.001           | 0.553 ± 0.004           | 0.995 ± 0.000           | 0.922 ± 0.006           | **0.998 ± 0.000**       |
| VAE+KMeans | 0.866 ± 0.003           | **0.852 ± 0.006**       | 0.589 ± 0.006           | 0.921 ± 0.005           | 0.924 ± 0.003           | 0.986 ± 0.001           |
| AE+Agg'    | 0.557 ± 0.007           | 0.296 ± 0.006           | 0.322 ± 0.004           | 0.929 ± 0.004           | 0.702 ± 0.004           | 0.520 ± 0.027           |
| VAE+Agg'   | 0.457 ± 0.008           | 0.324 ± 0.004           | 0.302 ± 0.004           | 0.752 ± 0.012           | 0.594 ± 0.009           | 0.438 ± 0.006           |

*Table 2: Model results for different clustering algorithms (ARI ± standard error of the mean).*
*Agg = the agglomerative clustering algorithm from `sklearn` with `metric` being "euclidean" and `linkage` being "ward".*
*Agg' = the agglomerative clustering algorithm from `sklearn` with `metric` being "euclidean" and `linkage` being "complete".*
*KMeans = the K-means algorithm from `sklearn`.*
*These scores are obtained for the optimal noise level and number of samples found for Agg.*

In summary, while clustering algorithm can change the results (agglomerative clustering with the Ward linkage does better than Agglomerative Clustering with the Complete linkage), the results do not crucially depend on the clustering algorithm (K-means clustering performs as well as, or better than, Agglomerative Clustering with the Ward linkage).

Furthermore, to clarify that the reason LNS works is not because of the clustering algorithm but because it makes the underlying representation clusterable, we have changed the text in Section 2.2:

“[...] This means that we can perturb all units simultaneously with independent noise: training on a generic reconstruction loss encourages the network to code in a manner that allows independent noise to reveal the relevant derivative direction, making the representation clusterable for segmentation (Figure1b).
Indeed, the goal of independent noise is not to perform segmentation itself, but to reveal the local neighborhood of the input stimulus and thus enable segmentation, _from the perspective of the decoder_.”

In addition, we now report UMAP visualizations for the clusterability of the examples shown in Figure 4. This additional section can be found in Appendix F2.

Reviewer comment (Experiments):
Even if it's not designed for the segmentation of natural images and if it's likely to not perform very well compared to SOTA algorithms, it is worth testing it. I have in mind a recent paper (Vacher et al 2022) in which deep neural network features are evaluated for segmenting natural images.

Author response:
>Even if it's not designed for the segmentation of natural images and if it's likely to not perform very well compared to SOTA algorithms it is worth testing it. I have in mind a recent paper (Vacher et al 2022) in which deep neural network features are evaluated for segmenting natural images.

We believe that dealing with naturalistic images and understanding the perceptual biases that lead to grouping are different directions of research and chose to focus on the latter in this work. To address a more naturalistic case where real image statistics play a role, we evaluated our models on the CelebA dataset (Appendix B). We think your suggestion is interesting and would have loved to do it, but we also believe that training generative models on meaningfully large datasets is quite an engineering challenge due to the changes in scale and architecture required (e.g., Child, 2021; Vahdat & Kautz, 2020).

Instead, as suggested by other reviewers, to contextualize our model in the extant literature, we have instead conducted control experiments on other SOTA/near-SOTA models on our datasets (Genesis and Genesis-v2; Engelcke et al, 2020; 2022), results for which we now report in the main text, Table 1.

| Model     | Kanizsa           | Closure           | Continuity        | Proximity         | Gradient Occ.     | Illusory Occ.     |
|-----------|-------------------|-------------------|-------------------|-------------------|-------------------|-------------------|
| AE        | 0.859 ± 0.008     | 0.766 ± 0.008     | 0.552 ± 0.008     | **0.996\*** ± 0.001 | 0.926 ± 0.009     | **0.994\*** ± 0.002 |
| VAE       | **0.871** ± 0.005 | **0.795** ± 0.009 | 0.593 ± 0.012     | 0.943 ± 0.010     | 0.918 ± 0.002     | 0.974 ± 0.003     |
| AE Rec.   | 0.246 ± 0.006     | 0.064 ± 0.001     | 0.570 ± 0.006     | 0.141 ± 0.001     | **0.982** ± 0.000 | -0.023 ± 0.001    |
| VAE Rec.  | 0.343 ± 0.004     | 0.084 ± 0.000     | **0.611** ± 0.005 | 0.144 ± 0.001     | 0.977 ± 0.001     | -0.024 ± 0.001    |
| Genesis   | 0.346             | 0.755             | 0.394             | 0.879             | 0.933             | 0.294             |
| Genesis-v2| 0.415             | 0.000             | 0.399             | 0.059             | 0.000             | 0.422             |

*Model results (ARI ± standard error of the mean). AE (Autoencoder) and VAE (Variational Autoencoder) model scores reported for the best noise level and number of samples. AE Rec. and VAE Rec. stand for noisy reconstruction control experiments, and (see Section 2.4 for details). Bold indicates top-performing model. For the AE and the VAE, a star (\*) indicates a statistically significant difference in model performance (Bonferroni corrected).*

While quantitative results for the Genesis model look impressive, a further investigation in Appendix C1 reveals that the decent quantitative performance is driven by good pixel value predictions in nearly all cases, and not actual Gestalt grouping phenomena (Figure 10). Indeed, the model appears to primarily follow pixel value information present in the input.

To verify that this result was not caused by a mistake in our implementation (e.g. total failure of model training), we followed the official implementation of Genesis and Genesis-v2, and show reconstructions on a separate validation set (Appendix C2).

Reviewer comment (Evaluation):
Latent space of VAE are known to enable appealing morphing between natural images (by linearly interpolating the latent code) so I am wondering what would be the segmentation related uncertainty that could be obtained with this method. I guess this would require a more involved post-processing step.

Author response:
>Latent space of VAE are known to enable appealing morphing between natural images (by linearly interpolating the latent code) so I am wondering what would be the segmentation related uncertainty that could be obtained with this method ... I guess this would require a more involved post-processing step.

This is quite an interesting question and something we only briefly mention in the paper (Section 2.2, first paragraph). We believe that Latent Noise Segmentation can be thought of as simultaneous random latent traversal of all the latent nodes. As such, Figure 6a may help answer your question: we know that in VAEs, the prior is set to $N(0, 1)$, and as such have a meaningful reference point for the “size” of the latent traversal. What is interesting is that in the case of the VAE, performance decreases substantially at higher levels of noise (i.e., larger “random latent traversals”). This suggests that the level of noise (magnitude of the “random latent traversal”) must be large enough to get a reliable signal, but small enough to not meaningfully cause uncertainty about the objects in the input image.

Reviewer comment (Theory):
I'm unclear on the extent to which Latent Noise Segmentation is a method to reveal the perceptual groups already formed inside the network through some mechanism or another, versus a mechanism that gives rise to perceptual groups in its own right. Is the hypothesis that LNS is a way of "doing" segmentation or is it a way to "show" the result of segmentation, if that makes sense?

Author response:
In this phrasing, LNS is a way to “show” the result of segmentation that the network has already learned during training. Specifically, LNS reveals the shape of the local derivatives around the input sample. This additional information that LNS reveals enables segmentation using a variety of methods. In our paper, we segment by propagating this information through the decoder, and clustering the outputs. 

To make this point clearer, we have changed the text in Section 2.2 as follows:
“[...] This means that we can perturb all units simultaneously with independent noise: training on a generic reconstruction loss encourages the network to code in a manner that allows independent noise to reveal the relevant derivative direction, making the representation clusterable for segmentation (Figure 1b).
Indeed, the goal of independent noise is not to perform segmentation itself, but to reveal the local neighborhood of the input stimulus and thus enable segmentation, _from the perspective of the decoder_.”

Reviewer comment (Experiments):
In Section 2, does pretraining refer to training on GG before doing the noise injection, or was there any pretraining on natural images? If so, it would be interesting to see segmentation maps for GG test images before training on GG train images. Would it group by continuity just by learning from natural statistics, for example?

Author response:
Pretraining refers to normal AE/VAE training on any desired dataset of interest before a noise injection. We did not use any naturalistic images in pre-training, and contained our training and analysis only to the GG and CelebA datasets. That being said, we agree that it would be interesting to see whether naturalistic image statistics would, in practice, result in good segmentation performance on our datasets. Unfortunately, training generative models on large datasets is a substantial and costly engineering effort (see e.g., Child, 2021; Vahdat & Kautz, 2020), and we hoped to demonstrate the scalability of the approach using the CelebA dataset. Due to the substantially different architecture of other architectures and the limited size of naturalistic datasets for generative models, we believe that exploring LNS in faithfully naturalistic pre-trained models would be a meaningfully large independent project - here, we have focused solely on the empirical and theoretical underpinnings of the novel method.

Reviewer comment (Experiments):
I think it might be worthwhile to show examples of VAE outputs without noise (i.e., the actual reconstructions, not the segmentation maps). If it groups the pixels of the Kanizsa squares together supposedly "perceives" the square, does it output its illusionary contours?

Author response:
Thank you for the suggestion. We now plot the model reconstructions in Appendix F3, and report:

“To support our control experiment, we also visualize the model output reconstructions (Figure 14). In accordance with the control experiments, we find that the outputs in some cases contain some information about the identity of the objects, even when it does not veridically exist in the input image. However, as reported in Table 1, the amount of information in the reconstructions is not as high as it is when using Latent Noise Segmentation. This is because of the tension between reconstructing an image well (which leads to poor segmentation without LNS in test samples), and segmenting the image components based on their color values. In other words, the fact that the model is still able to segment in some cases can be understood as an artefact caused by the small size of our model and imperfect reconstruction performance. The training process implicitly optimizes toward the success of LNS segmentation, but against the success of direct reconstruction segmentation.”

Reviewer comment (Writing):
"Time steps" is used to refer to the number of noisy samples needed for a segmentation map. Does it bear any relation with actual time in the human visual system?

Author response:
While one could interpret it that way, there is a clear degree of separation in our model and the human visual system (leaving aside the more philosophical considerations): our models deal with continuous activation values, while neurons spike. While it is possible to model deep neural networks as spiking networks (for example, by matching the activation values of a continuous network with the firing rates of spiking networks; Rueckauer et al., 2017), we have not considered it here to get an estimate of how the time steps should be implemented in the corresponding implementation of spiking networks.