Can parameter-free associative lateral connectivity boost generalization performance of CNNs?


Montobbio, Bonnasse-Gahot, Citti, & Sarti (pp2019) present an interesting model of lateral connectivity and its computational function in early visual areas. Lateral connections emanating from each unit drive other units to the degree that they are similar in their receptive profiles. Two units are symmetrically laterally connected if they respond to stimuli in the same region of the visual field with similar selectivity.

More precisely, lateral connectivity in this model implements a diffusion process in a space defined by the similarity of bottom-up filter templates. The similarity of the filters is measured by the inner product of the filter weights. Two filters that do not spatially overlap, thus, are not similar. Two filters are similar to the extent that their filters don’t merely overlap, but have correlated weight templates. Connecting units in proportion to their filter similarity results in a connectivity matrix that defines the paths of diffusion. The diffusion amounts to a multiplication with a convolution matrix. It is the activations (after the ReLU nonlinearity) that form the basis of the linear diffusion process.

The idea is that the lateral connections implement a diffusive spreading of activation among units with similar filters during perceptual inference. The intuitive motivation is that the spreading activation fills in missing information or regularizes the representation. This might make the representation of an image compromised by noise or distortion more like the representation of its uncompromised counterpart.

Instead of performing n iterations of the lateral diffusion at inference, we can equivalently take the convolutional matrix to the n-th power. The recurrent convolutional model is thus equivalent to a feedforward model with the diffusion matrix multiplication inserted after each layer.

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Montobbio’s model for MNIST


In the context of Gabor-like orientation-selective filters, the proposed formula for connectivity results in an anisotropic kernel of lateral connectivity  that looks plausible in that it connects approximately collinear edge filters. This is broadly consistent with anatomical studies showing that V1 neurons selective for oriented edges form long-range (>0.5 mm in tree shrew cortex) horizontal connections that preferentially target neurons selective for collinear oriented edges.


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Figure from Bosking et al. (1997). Long-range lateral connections of oriented-edge-selective neurons in tree-shrew V1 preferentially project to other neurons selective for collinear oriented edges.


Since the similarity between filters is defined in terms of the bottom-up filter templates, it can be computed for arbitrary filters, e.g. filters learned through task training. The lateral connectivity kernel for each filter, thus, does not have to be learned through experience. Adding this type of recurrent lateral connectivity to a convolutional neural network (CNN), thus, does not increase the parameter count.

The authors argue that the proposed connectivity makes CNNs more robust to local perturbations of the image. They tested 2-layer CNNs on MNIST, Kuzushiji-MNIST, Fashion-MNIST, and CIFAR-10. They present evidence that the local anisotropic diffusion of activity improves robustness to noise, occlusions, and adversarial perturbations.

Overall, the authors took inspiration from visual psychophysics (Field et al. 1992; Geisler et al. 2001) and neurobiology (Bosking et al. 1997), abstracted a parsimonious mathematical model of lateral connectivity, and assessed the computational benefits of the model in the context of CNNs that perform visual recognition tasks. The proposed diffusive lateral activation might not be the whole story of lateral and recurrent connectivity in the brain, but it might be part of the story. The idea deserves careful consideration.

The paper is well written and engaging. I’m left with many questions as detailed below. In case the authors chose to revise the paper, it would be great to see some of the questions addressed, a deeper exploration of the functional mechanism underlying the benefits, and some more challenging tests of performance.


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Figure from Geisler et al. (2001). Edge elements tend to be locally approximately collinear in natural images. Given that there is an orientated edge segment (shown as horizontal) in a particular location (shown in the center), the arrangement shows in what direction each orientation (oriented line) is most probable for each distance to the reference location.

Questions and thoughts

1 Can the increase in robustness be attributed to trivial forms of contextual integration?

If the filters were isotropic Gaussian blobs, then the diffusion process would simply blur the image. Blurring can help reduce noise and might reduce susceptibility to adversarial perturbations (especially if the adversary is not enabled to take this into account). Image blurring could be considered the layer-0 version of the proposed model. What is its effect on performance?

Consider another simplified scenario: If the network were linear, then the lateral connectivity would modify the effective filters, but each filter would still be a linear combination of the input. The model with lateral connectivity could thus be replaced by an equivalent feedforward model with larger kernels. Larger kernels might yield responses that are more robust to noise. Here the activation function is nonlinear, but the benefits might work similarly. It would be good to assess whether larger kernels in a feedforward network bring similar benefits to generalization performance.


2 Were the adversarial perturbations targeted at the tested model?

Robustness to adversarial attack should be tested using adversarial examples targeting each particular model with a given combination of numbers of iterations of lateral diffusion in layers 1 and 2. Was this the case?


3 Is the lateral diffusion process invertible?

The lateral diffusion is a linear transform that maps to a space of equal dimension (like Gaussian blurring of an image).

If the transform were invertible, then it would constitute the simplest possible change (linear, information preserving) to the representational geometry (as characterized by the Euclidean representational distance matrix for a set of stimuli). To better understand why this transform helps, then, it would be interesting to investigate how it changes the representational geometry for a suitable set of stimuli.

If lateral diffusion were not invertible, then it is perhaps best thought of as an intelligent type of pooling (despite the output dimension being equal to the input dimension).


4 Do the lateral connections make representations of corrupted images more similar to representations of uncorrupted versions of the same images?

The authors offer an intuitive explanation of the benefits to performance: Lateral diffusion restores the missing parts or repairs what has been corrupted (presumably using accurate prior information about the distribution of natural images). One could directly assess whether this is the case by assessing whether lateral diffusion moves the representation of a corrupted image closer to the representation of its uncorrupted variant.


5 Do correlated filter templates imply correlated filter responses under natural stimulation?

Learned filters reflect features that occur in the training images. If each image is composed of a mosaic of overlapping features, it is intuitive that filters whose templates overlap and are correlated will tend to co-occur and hence yield correlated responses across natural images. The authors seem to assume that this is true. But is there a way to prove that the correlations between filter templates really imply correlation of the filter outputs under natural stimulation? For independent noise images, filters with correlated templates will surely produce correlated outputs. However, it’s easy to imagine stimuli for which filters with correlated templates yield uncorrelated or anticorrelated outputs.


6 Does lateral connectivity reflecting the correlational structure of filter responses under natural stimulation work even better than the proposed approach?

Would the performance gains be larger or smaller if lateral connectivity were determined by filter-output correlation under natural stimulation, rather than by filter-template similarity?

Is filter-template similarity just a useful approximation to filter-output correlation under natural stimulation, or is there a more fundamental computational motivation for using it?


7 How does the proposed lateral connectivity compare to learned lateral connectivity when the number of connections (instead of the number of parameters) is matched?

It would be good to compare CNNs with lateral diffusive connectivity to recurrent convolutional neural networks (RCNNs) for matched sizes of bottom-up and lateral filters (and matched numbers of connections, not parameters). In addition, it would then be interesting to initialize the RCNNs with diffusive lateral connectivity according to the proposed model (after initial training without lateral connections). Lateral connections could precede (as in typical RCNNs) or follow (as in KerCNNs) the nonlinear activation function.


8 Does the proposed mechanism have a motivation in terms of a normative model of visual inference?

Can the intuition that lateral connections implement shrinkage to a prior about natural image statistics be more explicitly justified?

If the filters serve to infer features of a linear generative model of the image, then features with correlated templates are anti-correlated given the image (competing to explain the same variance). This suggests that inhibitory connections are needed to implement the dynamics for inference. Cortex does rely on local inhibition. How does local inhibitory connectivity fit into the picture?

Can associative filling in and competitive explaining away be reconciled and combined?



  • A mathematical model of lateral connectivity, motivated by human visual contour integration and studies on V1 long-range lateral connectivity, is tested in terms of the computational benefits it brings in the context of CNNs that recognize images.
  • The model is intuitive, elegant, and parsimonious in that it does not require learning of additional parameters.
  • The paper presents initial evidence for improved generalization performance in the context of deep convolutional neural networks.



  • The computational benefits of the proposed lateral connectivity is tested only in the context of toy tasks and two-layer neural networks.
  • Some trivial explanations for the performance benefits have not been ruled out yet.
  • It’s unclear how to choose the number of iterations of lateral diffusion for each of the the two layers, and choosing the best combination might positively bias the estimate of the gain in accuracy.


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Figure from Boutin et al. (pp2019) showing how feedback from layer 2 to layer 1 in a sparse deep predictive coding model trained on natural images can give rise to collinear “association fields” (a concept suggested by Field et al. (1993) on the basis of psychophysical experiments). Montobbio et al. plausibly suggest that direct lateral connections may contribute to this function.

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Figure from Montobbio et al. showing the kinds of perturbations that lateral connectivity rendered the networks more robust to.


Minor point

“associated to” -> “associated with” (in several places)

Is a cow-mug a cow to the ventral stream, and a mug to a deep neural network?


An elegant new study by Bracci, Kalfas & Op de Beeck (pp2018) suggests that the prominent division between animate and inanimate things in the human ventral stream’s representational space is based on a superficial analysis of visual appearance, rather than on a deeper analysis of whether the thing before us is a living thing or a lifeless object.

Bracci et al. assembled a beautiful set of stimuli divided into 9 equivalent triads (Figure 1). Each triad consists of an animal, a manmade object, and a kind of hybrid of the two: an artefact of the same category and function as the object, designed to resemble the animal in the triad.

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Figure 1: The entire set of 9 triads = 27 stimuli. Detail from Figure 1 of the paper.


Bracci et al. measured response patterns to each of the 27 stimuli (stimulus duration: 1.5 s) using functional magnetic resonance imaging (fMRI) with blood-oxygen-level-dependent (BOLD) contrast and voxels of 3-mm width in each dimension. Sixteen subjects viewed the images in the scanner while performing each of two tasks: categorizing the images as depicting something that looks like an animal or not (task 1) and categorizing the images as depicting a real living animal or a lifeless artefact (task 2).

The authors performed representational similarity analysis, computing representational dissimilarity matrices (RDMs) using the correlation distance (1 – Pearson correlation between spatial response patterns). They averaged representational dissimilarities of the same kind (e.g. between the animal and the corresponding hybrid) across the 9 triads. To compare different kinds of representational distance, they used ANOVAs and t tests to perform inference (treating the subject variable as a random effect). They also studied the representations of the stimuli in the last fully connected layers of two deep neural networks (DNNs; VGG-19, GoogLeNet) trained to classify objects, and in human similarity judgments. For the DNNs and human judgements, they used stimulus bootstrapping (treating the stimulus variable as a random effect) to perform inference.

Results of a series of well-motivated analyses are summarized in Figure 2 below (not in the paper). The most striking finding is that while human judgments and DNN last-layer representations are dominated by the living/nonliving distinction, human ventral temporal cortex (VTC) appears to care more about appearance: the hybrid animal-lookalike objects, despite being lifeless artefacts, fall closer to the animals than to the objects. In addition, the authors find:

  • Clusters of animals, hybrids, and objects: In VTC, animals, hybrids, and objects form significantly distinct clusters (average within-cluster dissimilarity < average between-cluster dissimilarity for all three pairs of categories). In DNNs and behavioral judgments, by contrast, the hybrids and the objects do not form significantly distinct clusters (but animals form a separate cluster from hybrids and from objects).
  • Matching of animals to corresponding hybrids: In VTC, the distance between a hybrid animal-lookalike and the corresponding animal is significantly smaller than that between a hydrid animal-lookalike and a non-matching animal. This indicates that VTC discriminates the animals and animal-lookalikes and (at least to some extent) matches the lookalikes to the correct animals. This effect was also present in the similarity judgments and DNNs. However, the latter two similarly matched the hybrids up with their corresponding objects, which was not a significant effect in VTC.


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Figure 2: A qualitative visual summary of the results. Connection lines indicate different kinds of representational dissimilarity, illustrated for two triads although estimates and tests are based on averages across all 9 triads. Gray underlays indicate clusters (average within-cluster dissimilarity < average between-cluster dissimilarity, significant). Arcs indicate significantly different representational dissimilarities. It would be great if the authors added a figure like this in the revision of the paper. However, unlike the mock-up above, it should be a quantitatively accurate multidimensional scaling (MDS, metric stress) arrangement, ideally based on unbiased crossvalidated representational dissimilarity estimates.


The effect of the categorization task on the VTC representation was subtle or absent, consistent with other recent studies (cf. Nastase et al. 2017, open review). The representation appears to be mostly stimulus driven.

The results of Bracci et al. are consistent with the idea that the ventral stream transforms images into a semantic representation by computing features that are grounded in visual appearance, but correlated with categories (Jozwik et al. 2015). VTC might be 5-10 nonlinear transformations removed from the image. While it may emphasize visual features that help with categorization, it might not be the stage where all the evidence is put together for our final assessment of what we’re looking at. VTC, thus, is fooled by these fun artefacts, and that might be what makes them so charming.

Although this interpretation is plausible enough and straightforward, I am left with some lingering thoughts to the contrary.

What if things were the other way round? Instead of DNNs judging correctly where VTC is fooled, what if VTC had a special ability that the DNNs lack: to see the analogy between the cow and the cow-mug, to map the mug onto the cow? The “visual appearance” interpretation is based on the deceptively obvious assumption that the cow-mug (for example) “looks like” a cow. One might, equally compellingly, argue that it looks like a mug: it’s glossy, it’s conical, it has a handle. VTC, then, does not fail to see the difference between the fake animal and the real animal (in fact these categories do cluster in VTC). Rather it succeeds at making the analogy, at mapping that handle onto the tail of a cow, which is perhaps an example of a cognitive feat beyond current AI.

Bracci et al.’s results are thought-provoking and the study looks set to inspire computational and empirical follow-up research that links vision to cognition and brain representations to deep neural network models.



  • addresses an important question
  • elegant design with beautiful stimulus set
  • well-motivated and comprehensive analyses
  • interesting and thought-provoking results
  • two categorization tasks, promoting either the living/nonliving or the animal-appearance/non-animal appearance division
  • behavioral similarity judgment data
  • information-based searchlight mapping, providing a broader view of the effects
  • new data set to be shared with the community



  • representational geometry analyses, though reasonable, are suboptimal
  • no detailed analyses of DNN representations (only the last fully connected layers shown, which are not expected to best model the ventral stream) or the degree to which they can explain the VTC representation
  • only three ROIs (V1, posterior VTC, anterior VTC)
  • correlation distance used to measure representational distances (making it difficult to assess which individual representational distances are significantly different from zero, which appears important here)


Suggestions for improvement

The analyses are effective and support most of the claims made. However, to push this study from good to excellent, I suggest the following improvements.


Major points

Improved representational-geometry analysis

The key representational dissimilarities needed to address the questions of this study are labeled a-g in Figure 2. It would be great to see these seven quantities estimated, tested for deviation from 0, and all 7 choose 2 = 21 pairwise comparisons tested. This would address which distinctions are significant and enable addressing all the questions with a consistent approach, rather than combining many qualitatively different statistics (including clustering index, identity index, and model RDM correlation).

With the correlation distance, this would require a split-data RDM approach, consistent with the present approach, but using the repeated response measurements to the same stimulus to estimate and remove the positive bias of the correlation-distance estimates. However, a better approach would be to use a crossvalidated distance estimator (more details below).


Multidimensional scaling (MDS) to visualize representational geometries

This study has 27 unique stimuli, a number well suited for visualization of the representational geometries by MDS. To appreciate the differences between the triads (each of which has unique features), it would be great to see an MDS of all 27 objects and perhaps also MDS arrangements of subsets, e.g. each triad or pairs of triads (so as to reduce distortions due to dimensionality reduction).

Most importantly, the key representational dissimilarities a-g can be visualized in a single MDS as shown in Figure 2 above, using two triads to illustrate the triad-averaged representational geometry (showing average within- and between-triad distances among the three types of object). The MDS could use 2 or 3 dimensions, depending on which variant better visually conveys the actual dissimilarity estimates.


Crossvalidated distance estimators

The correlation distance is not an ideal dissimilarity measure because a large correlation distance does not indicate that two stimuli are distinctly represented. If a region does not respond to either stimulus, for example, the correlation of the two patterns (due to noise) will be close to 0 and the correlation distance will be close to 1, a high value that can be mistaken as indicating a decodable stimulus pair.

Crossvalidated distances such as the linear-discriminant t value (LD-t; Kriegeskorte et al. 2007, Nili et al. 2014) or the crossnobis distance (also known as the linear discriminant contrast, LDC; Walther et al. 2016) would be preferable. Like decoding accuracy, they use crossvalidation to remove bias (due to overfitting) and indicate that the two stimuli are distinctly encoded. Unlike decoding accuracy, they are continuous and nonsaturating, which makes them more sensitive and a better way to characterize representational geometries.

Since the LD-t and the crossnobis distance estimators are symmetrically distributed about 0 under the null hypothesis (H0: response patterns drawn from the same distribution), it would be straightforward to test these distances (and averages over sets of them) for deviation from 0, treating subjects and/or stimuli as random effects, and using t tests, ANOVAs, or nonparametric alternatives. Comparing different dissimilarities or set-average dissimilarities is similarly straightforward.


Linear crossdecoding with generalization across triads

An additional analysis that would give complementary information is linear decoding of categorical divisions with generalization across stimuli. A good approach would be leave-one-triad-out linear classification of:

  • living versus nonliving
  • things that look like animals versus other things
  • animal-lookalikes versus other things
  • animals versus animal-lookalikes
  • animals versus objects
  • animal-lookalikes versus objects

This might work for devisions that do not show clustering (within dissimilarity < between dissimilarity), which would indicate linear separability in the absence of compact clusters.

For the living/nonliving destinction, for example, the linear discriminant would select responses that are not confounded by animal-like appearance (as most VTC responses seem to be), responses that distinguish living things from animal-lookalike objects. This analysis would provide a good test of the existence of such responses in VTC.


More layers of the two DNNs

To assess the hypothesis that VTC computes features that are more visual than semantic with DNNs, it would be useful to include an analysis of all the layers of each of the two DNNs, and to test whether weighted combinations of layers can explain the VTC representational geometry (cf. Khaligh-Razavi & Kriegeskorte 2014).


More ROIs

How do these effects look in V2, V4, LOC, FFA, EBA, and PPA?


Minor points

The use of the term “bias” in the abstract and main text is nonstandard and didn’t make sense to me. Bias only makes sense when we have some definition of what the absence of bias would mean. Similarly the use of “veridical” in the abstract doesn’t make sense. There is no norm against which to judge veridicality.


The polar plots are entirely unmotivated. There is no cyclic structure or even meaningful order to the the 9 triads.


“DNNs are very good, and even better than than human visual cortex, at identifying a cow-mug as being a mug — not a cow.” This is not a defensible claim for several reasons, each of which by itself suffices to invalidate this.

  • fMRI does not reveal all the information in cortex.
  • VTC is not all of visual cortex.
  • VTC does cluster animals separately from animal-lookalikes and from objects.
  • Linear readout of animacy (cross-validated across triads) might further reveal that the distinction is present (even if it is not dominant in the representational geometry.



Grammar, typos

“how an object looks like” -> ‘how an object looks” or “what an object looks like”

“as oppose to” -> “as opposed to”

“where observed” -> “were observed”