Explaining the Human Visual Brain Challenge 2019 -- receptive fields and surrogate features

TL;DR

This paper reviews submissions to a challenge focused on aligning neural network features with human brain data from fMRI and MEG, emphasizing receptive fields, surrogate features, and RDM construction.

Contribution

It introduces methods for optimizing receptive field granularity, creating surrogate features with Multidimensional Scaling, and discusses RDM construction nuances.

Findings

Optimal receptive field granularity improves RDM similarity.

Surrogate features via Multidimensional Scaling enhance model-brain alignment.

Attention to RDM construction details is crucial for accurate comparisons.

Abstract

In this paper I review the submission to the Explaining the Human Visual Brain Challenge 2019 in both the fMRI and MEG tracks. The goal was to construct neural network features which generate the so-called representational dissimilarity matrix (RDM) which is most similar to the one extracted from fMRI and MEG data upon viewing a set of images. I review exploring the optimal granularity of the receptive field, a construction of intermediate surrogate features using Multidimensional Scaling and modelling them using neural network features. I also point out some peculiarities of the RDM construction which have to be taken into account.

Equations

$$1-R(x,y)=1-\frac{(x-⟨x⟩)(y-⟨y⟩)}{{\sigma }_x{\sigma }_y}$$

$$x_i=+1+{\epsilon }_i{y}_i=-1+{\tilde{\epsilon }}_i$$

$$1-\frac{x\cdot y}{|x||y|}$$

Full text

1 Introduction

The Explaining the Human Visual Brain Challenge 2019 organized within the Algonauts project [1], has as its aim increasing our understanding of what visual features are processed and encoded by the brain. The challenge specifically concentrates on fMRI data from the early visual cortex (EVC) and inferior temporal cortex (IT) in the fMRI track of the competition and on MEG data from the early and late stages of visual processing in the MEG track of the competition. In the following I will use the above acronyms as well as EARLY and LATE targets for the MEG track.

The competition adopts the Representational Similarity Analysis framework [2] where one first constructs a so-called representational dissimilarity matrix (RDM) out of brain data, whose entries are $1-Pearson\ correlation$ between the neuronal activities/signals corresponding to two images (see Fig. 1).

The goal of the competition is to construct features (e.g. out of Deep Neural Network activations) so that the resulting dissimilarity matrix is closest, as measured by Spearman’s correlation, to the one extracted from the brain averaged over 15 subjects (see Fig. 2). There are two training datasets, with 118 and 92 images respectively and and a test set with 78 images. For more details see [1] and [4].

Below I first discuss some very general ingredients of the approach: some rather unexpected properties of the RDM’s, determining the optimal coarse grained receptive field and the methodology for feature selection. Then I outline the approach using surrogate MDS features and ICA components for the IT and LATE targets, and the models for the EVC and EARLY targets. Since this is a write-up of a competition entry, some clarity of the modelling approach is unfortunately sacrificed for tweaking performance.

2 Peculiarities of the RDM construction

The Representational Dissimilarity Matrices (RDM) by construction have two rather unexpected and somewhat unwelcome features. Firstly, they can miss a very strong discriminative signal as long as it is correlated. Secondly, they are influenced by the presence of irrelevant uninformative features.

The dissimilarity between features $x$ and $y$ representing two images is defined by

[TABLE]

where $R(x,y)$ is Pearson’s correlation coefficient between the two feature vectors. Suppose that the two images are completely dissimilar with the features $x$ being all $+1$, and the features $y$ being all $-1$ (as shown in Fig. 3 (left)). Since the Pearson correlation coefficient subtracts the mean of the feature vector, the difference between the two images will be completely lost. In a more realistic setting when

[TABLE]

with some noise vectors $\varepsilon_{i}$ and $\tilde{\varepsilon}_{i}$, the computed dissimilarity will only be influenced by the correlation of the noise signals and not by the relevant discriminating features.

Suppose that we concatenate to our features a vector of $N$ irrelevant features, which are always zero irrespective of the image (as shown in Fig. 3 (right)). Now the relevant features $+1$ and $-1$ will start to contribute to the dissimilarity. In the limit $N\to\infty$, this corresponds to using a cosine dissimilarity

[TABLE]

for the original features.

For the neural network features, the global signal (e.g. $\left\langle{x}\right\rangle$ above) carries important information and should not be subtracted out. Hence we always add a constant vector to our constructed features11195% overall length. for all entries to the competition. This strongly influences the results. We will refer to this constant as $level$.

It may well be that uninformative voxels may also appear in the brain RDM’s, so the above may not be completely artificial. However the sensitivity of the dissimilarity to such uninformative voxels and the downplaying of correlated signals is rather unwelcome. Fortunately, the two properties work in opposite directions but this adds another degree of freedom to the setup. One way of correcting it would be treat $\left\langle{x}\right\rangle$ as an average over all images for each feature individually. However, the definition of the RDM is of course completely fixed throughout the competition.

3 Effective receptive fields

The spatial size of the outputs of the different convolutional layers vary significantly (e.g. for resnet50 we have $56\times 56$, $28\times 28$, $14\times 14$ and $7\times 7$). This corresponds to looking at the images at different ”granularities”. Of course, the features (channels) for each layer also differ in their ”complexity” going intuitively from low-level features to high-level ones. It is interesting to investigate what is the natural granularity (and whether something like this exists) for brain regions/processing stages appearing in the competition, disentangled from the low-level/high-level feature characteristics.

To this end, we can transform the outputs from the convolutional layers through a max-pooling operation leading to a specified $k\times k$ output size222This can be done using the adaptive_max_pool2d operation in PyTorch.. The results for the average score on the 118 image dataset of the transformed resnet family convolutional layers for different modalities is shown in Fig. 4. Hence we pick out a $5\times 5$ grid for the EVC, EARLY and LATE targets, while IT favours a very coarse $2\times 2$ grid. Some random trials on the test set also support this conclusion.

It is interesting to note that if one would use average pooling with the same settings, the results would be much worse – single digit scores with the same settings as in Fig. 4. Thus the brain seems to pick out just the strongest signals in appropriate parts of the image.

In the following we always preprocessed the convolutional layers in this way, normalizing the resulting outputs so that the mean of nonzero elements would be equal to unity.

4 Methods for feature selection

In the competition entries we frequently screened features for their relevance and eliminated the irrelevant ones. Due to the fact that the dissimilarity matrix is a very global way of summarizing features, it seems rather difficult to define a relevant feature without a particular context of other features. Hence, we always define relevance in the context of some prescribed set of features. We evaluate the score on this reference set, and compute the difference w.r.t this score when we erase a feature from the set333By replacing it with the constant $level$ of section 2., or when we add an external feature to this set444In the context of convolutional features, we always erase or add an output channel as a whole..

The rather complicated prescriptions outlined below are motivated by trying to isolate genuine relevant or irrelevant features, so that they would have a good chance of generalizing to new data. However this is by no means guaranteed to work, and only works to some extent.

We used two possible ways of scoring features:

1. A

For each of the 15 subjects we individually evaluate the reference score and the modified score (with an added or erased feature). We then take the mean, as well as the $z$-score of the 15 differences. 2. B

We randomly choose 30 subsets of around $1/4$ images and use these for the reference and modified scores. Similarly we take the mean and $z$-score of the 30 differences555For some inessential historical reason, we did that within an overall 5-fold cross-validation..

For features constructed from some fitting process, as in section 5, we use option A on the CV test folds, as well as on predictions on the other dataset (e.g. 118 if the model was trained on 92 images and vice-versa). We will refer to this procedure later as MDS pruning.

For adding features we used a slightly more complicated process. First we used 10 different splits of the dataset into 5 parts. For each part we computed the gain in the score resulting from adding a feature. From these 50 differences we computed the mean. As a set of candidate features to add we picked those which had positive mean for both the 118 and 92 image dataset. Then we added successively the best features and kept those which still had a positive mean. We will refer to this procedure later as Feature Adding.

An interesting outcome of quantifying feature importance is the relevance of the $5\times 5$ spatial grid when modelling EARLY, LATE (and to some extent EVC). Applying algorithm A by dropping all channels in a particular grid position, we get a $5\times 5$ matrix of $z$-scores (shown in Fig.5 for the LATE and EARLY processing stages on the 118 image dataset modelled by the maxpool2 layer of vgg19). We henceforth drop the corners of the EARLY stage features. We would also drop the top and bottom row of the LATE stage features, but in the end we adopted a different approach in this case. Both conclusions were confirmed on the test set. We also found that dropping corners on the EVC also gave a perfomance boost.

5 Surrogate MDS features and the IT and LATE features

In order to model the dissimilarity matrices for the IT and LATE part of the challenge, we decided to construct first completely abstractly features which could reproduce well the dissimilarity matrix. These features do not have any a-priori interpretation and are defined only for a given training dataset, hence we call them surrogate features. Then we fit various models to reproduce these abstract features from the neural network features. These models can then be applied to the test dataset to yield the predicted features.

The idea of the construction is to use Multidimensional Scaling (MDS) embedding constructed out of the average (across subjects) dissimilarity matrix666More precisely we find that the $4^{th}$ power of the mean RDM works best with the MDS algorithm.. MDS aims to find a low dimensional embedding which preserves as well as possible the distances (related here to dissimilarities) between data points (images). Of course, it does not follow that the resulting coordinates will lead to the original dissimilarity matrix, but experiments on the 118 dataset gave sufficiently high scores so that trying to develop a custom embedding algorithm was not worthwhile during the competition.

Since the MDS algorithm is probabilistic, we fix 10 random seeds and construct 10-dimensional MDS embeddings giving in total 100 features. For the 118 image dataset, these features allow theoretically to obtain a score of around 77%. In practice the score will be lower as we have to express (approximately) these features in terms of DNN features. Due to the small number of images and huge number of DNN features we have to impose strong regularization or sparsity constraints. Also, in order to reduce somewhat the number of DNN features, we fit the model to each layer in turn and then combine the results using ridge regression.

For the IT target, we use ridge777RidgeCV from sklearn. regression, as well as orthogonal matching pursuit OMP(6) (with 6 nonzero components per layer) trained on the 118 dataset and OMP(7) trained on the 92 image dataset. All these models are trained on layers of the resnet50 network (with convolutional layers reduced to $2\times 2$). We concatenate the features giving a vector of length 300 for each image. We then use the MDS pruning procedure by imposing positivity on the 118 dataset. The final answer is extended with a constant level of $1$ of section 2 (the precise value is not important as long as it is away from zero). This yields a score of 19.42 on the test set.

From some earlier attempts we found that ICA features work rather well on the IT (but worse than above). Hence, we concatenate the previous 300 features with 75+75 ICA components of block1 and block3 respectively888Since ICA works worse for the IT target, we shrink them by multiplying by 0.25 and 0.5 respectively., from resnet34. ICA is fitted on the 118 and 92 datasets concatenated together. This raises the score to 20.77 for the IT target on the test set.

For the LATE target, a cross validation study of ridge and OMP does not give good results. However, rather surprisingly a nonlinear model – gradient boosting regression works rather well. Since we expect the data to be very noisy and prone to overfitting, we use only very small trees (with $n_{estimators}=5$) and Huber loss. We concatenate GBR(5) fitted on the 118 dataset MDS and GBR(5) fitted on the 92 dataset MDS (again on the resnet50 network with convolutional layers reduced now to $5\times 5$)). We then use the MDS pruning procedure by eliminating features which are negative on both datasets. Again we extend the set with a constant level of $1$. This yields a score of 57.10. Eliminating a slightly smaller number of features by raising the elimination threshold to 0.05 gives the final score 57.38 for the LATE target on the test set.

6 The EARLY and EVC features

The models for EARLY and EVC are quite unsophisticated and involve a choice of DNN layer, reduction to $5\times 5$, eliminating corners of the receptive field, eliminating some features using some pruning procedure and then enhancing the most relevant ones. One can make a very simple model for the LATE target along these lines which gives 33.24 on the test set (just after pruning). However, for our final model we adopted the approach using MDS features. The details for the EARLY and EVC targets are given below.

For the EARLY target, we use the maxpool2 layer of vgg19. For the pruning step we use algorithm A of section 4 eliminating features with negative $z$-scores bigger than 0.15 on either dataset. We than used the Feature Adding procedure on the same layer, and concatenated these features with a multiplier $4$. The constant level was here set to 0.5. This yields a score of 46.91 for the EARLY target on the test set.

For the EVC target, we use the block2 layer of resnet18. For the pruning step we use algorithm B of section 4 eliminating 1/4 of the worst features (according to their $z$-scores). We than used the Feature Adding procedure on the same layer, and concatenated these features with a multiplier $2$. The constant level was here set to 0. This yields a score of 28.29. Performing Feature Adding with the maxpool2 layer of vgg19 and concatenating with multiplier 0.5 gives a slight boost to 28.40 for the EVC target on the test set.

7 Conclusions

The key difficulty in the challenge is the relatively small number of images in comparison with the enormous number of potential DNN features. In addition, the three datasets are quite distinct, far from satisfying an i.i.d. assumption, so that there is a huge potential for overfitting. E.g. direct optimization of the score as a function of feature weights using some gradient free optimizer did not generalize well when doing cross-validation. The same happened for some approaches which had the potential of working for much larger datasets like a Siamese network on top of the DNN features. In view of the above we concentrated on rather simple models.

In addition, unfortunately cross-validation was not always a reliable indicator of performance on the test set. An extreme example was a submission which (erronously) added worst features from the remaining layers of resnet18 in the Feature Adding step to EVC, which gave 32.68 on the test set.

Some properties seemed quite robust, however, like the optimal effective granularity of the receptive field is not very large (around $5\times 5$ or $2\times 2$), as well as the fact that roughly second level layers (like block2 or maxpool2) seemed to be the best starting point for the early stages EVC and EARLY targets.

The insensitivity of the RDM construction to correlated features as well as the fact that it is influenced by completely uninformative features suggests that it would be better to modify its definition e.g. by subtracting out the featurewise average over the dataset.

The approach which worked best for the late stages (IT and LATE targets), used surrogate features constructed out of MDS embedding of the (transformed) dissimilarity matrix and then various models for fitting these features. If one would want to seriously proceed along this route, then one should develop a different embedding algorithm which would be directly optimized for reproducing the given RDM. Of course, given the original brain fMRI and MEG data, one could directly try to fit these data (after some appropriate preprocessing) with the neural network features. This would have the additional benefit of explaining some concrete real brain feature. We intend to investigate this approach in the future.

Acknowledgments. This work was done in preparation for the Foundation for Polish Science (FNP) project Biologically inspired Artificial Neural Networks POIR.04.04.00-00-14DE/18-00. I would like to thank the organizers for a very stimulating challenge.