As neural networks have become more powerful, an increasing number of studies have sought to decipher their internal representations [26, 16, 4, 2, 11, 25, 21]. Most of these have focused on the role of individual units in the computations performed by individual networks. Comparing population representations across networks has proven especially difficult, largely because networks converge to apparently distinct solutions in which it is difficult to find one-to-one mappings of units .
Recently,  applied Canonical Correlation Analysis (CCA) as a tool to compare representations across networks. CCA had previously been used for tasks such as computing the similarity between modeled and measured brain activity , and training multi-lingual word embeddings in language models . Because CCA is invariant to linear transforms, it is capable of finding shared structure across representations which are superficially dissimilar, making CCA an ideal tool for comparing the representations across groups of networks and for comparing representations across time in RNNs.
Using CCA to investigate the representations of neural networks, we make three main contributions:
Building off of , we demonstrate that groups of networks which generalize converge to more similar solutions than those which memorize (Section 3.1), that wider networks converge to more similar solutions than narrower networks (Section 3.2), and that networks with identical topology but distinct learning rates converge to a small set of diverse solutions (Section 3.3).
2 Canonical Correlation Analysis on Neural Network Representations
Canonical Correlation Analysis , is a statistical technique for relating two sets of observations arising from an underlying process. It identifies the ’best’ (maximizing correlation) linear relationships (under mutual orthogonality and norm constraints) between two sets of multidimensional variates.
Concretely, in our setting, the underlying process is a neural network being trained on some task. The multidimensional variates are neuron activation vectors over some dataset . As in has on . If , then the neuron outputs scalars , which can be stacked to form a vector.111This is different than the vector of all neuron outputs on a single input: , which is also sometimes referred to as an activation vector.
A single neuron activation vector is one multidimensional variate, and a layer of neurons gives us a set of multidimensional variates. In particular, we can consider two layers, , of a neural network as two sets of observations, to which we can then apply CCA, to determine the similarity between two layers. Crucially, this similarity measure is invariant to (invertible) affine transforms of either layer, which makes it especially apt for neural networks, where the representation at each layer typically goes through an affine transform before later use. Most importantly, it also enables comparisons between different neural networks,222Including those with different topologies such that and have different sizes. which is not naively possible due to a lack of any kind of neuron to neuron alignment.
2.1 Mathematical Details of Canonical Correlation
Here we overview the formal mathematical interpretation of CCA, as well as the optimization problem to compute it. Let be and dimensional matrices respectively, with representing multidimensional variates, and representing multidimensional variates. We wish to find vectors in respectively, such that the dot product
is maximized. Assuming the variates in are centered, and letting denote the by covariance of , denote the by covariance of , and the cross covariance:
We can change basis, to and to get
which can be solved with a singular value decomposition:
with in (*) being the first left and right singular vectors, and the top singular value of corresponding to the canonical correlation coefficient , which tells us how well correlated the vectors and (both vectors in ) are.
In fact, are really the first in a series, and can be denoted . Next in the series are , the second left and right singular vectors, and the corresponding second highest singular value of . denotes the correlation between and , which is the next highest possible correlation under the constraint that and .
The output of CCA is a series of singular vectors which are pairwise orthogonal, their corresponding vectors in : and , and finally their correlation coefficient , with . Letting , we end up with non-zero .
Note that the orthogonality of also results in the orthogonality of , as
and so our CCA directions are also orthogonal.
2.2 Beyond Mean CCA Similarity
To determine the representational similarity between two layers ,  prunes neurons with a preprocessing SVD step, and then applies CCA to . They then represent the similarity of by the mean correlation coefficient. Adapting this to make a distance measure, :
One drawback with this measure is that it implicitly assumes that all CCA vectors are equally important to the representations at layer . However, there has been ample evidence that DNNs do not rely on the full dimensionality of a layer to represent high performance solutions [12, 6, 1, 20, 15, 21, 14]. As a result, the mean correlation coefficient will typically underestimate the degree of similarity.
To investigate this further, we first asked whether, over the course of training, all CCA vectors converge to their final representations before the network’s performance converges. To test this, we computed the CCA similarity between layer at times throughout training with layer at the final timestep . Viewing the sorted CCA coefficients , we can see that many of the coefficients continue to change well after the network’s performance has converged (Figure 1a-c, Figure A1). This result suggests that the unconverged coefficients and their corresponding vectors may represent “noise” which is unnecessary for high network performance.
We next asked whether the CCA vectors which stabilize early in training remain stable. To test this, we computed the CCA vectors between layer at timestep in training and timestep . We then computed the similarity between the top vectors (those which stabilized early) and the bottom vectors (those which had not stabilized) with the representation at all other training times. Consistent with our intuition, we found that those vectors which stabilized early remained stable, while the unstable vectors continued to vary, and therefore likely represent noise.
These results suggest that task-critical representations are learned by midway through training, while the noise only approaches its final value towards the end. We therefore suggest a simple and easy to compute variation that takes this into account. We also discuss an alternate approach in Section A.2.
One way to address this issue is to replace the mean by a weighted mean, in which canonical correlations which are more important to the underlying representation have higher weight. We propose a simple method, projection weighting, to determine these weights. We base our proposition on the hypothesis that CCA vectors that account for (loosely speaking) a larger proportion of the original outputs are likely to be more important to the underlying representation.
More formally, let layer , have neuron activation vectors , and CCA vectors . We know from (**) that are orthogonal. Because computing CCA can result in the accrual of small numerical errors , we first explicitly orthonormalize via Gram-Schmidt. We then identify how much of the original output is accounted for by each :
Normalizing this to get weights , with , we can compute the projection weighted CCA distance333We note that this is technically a pseudo-distance rather than a distance as it is non-symmetric.:
As a simple test of the benefits of projection weighting, we constructed a toy case in which we used CCA to compare the representations of two networks with common (signal) and uncommon (noise) structure, each of a fixed dimensionality. We then used the naive mean and projected weighted mean to measure the CCA distance between these two networks as a function of the ratio of signal dimensions to noise dimensions. As expected we found that while the naive mean was extremely sensitive to this ratio, the projection weighted mean was largely robust (Figure 2).
3 Using CCA to measure the similarity of converged solutions
Because CCA measures the distance between two representations independent of linear transforms, it enables formerly difficult comparisons between the representations of different networks. Here, we use this property of CCA to evaluate whether groups of networks trained on CIFAR-10 with different random initializations converge to similar solutions under the following conditions:
3.1 Generalizing networks converge to more similar solutions than memorizing networks
It has recently been observed that DNNs are capable of solving image classification tasks even when the labels have been randomly permuted . Such networks must, by definition, memorize the training data, and therefore cannot generalize beyond the training set. However, the representational properties which distinguish networks which memorize from those which generalize remain unclear.
In particular, we hypothesize that the representational similarity in a group of generalizing networks (networks trained on the true labels) should differ from the representational similarity of memorizing networks (networks trained on random labels.)
To test this hypothesis, we trained groups of five networks with identical topology on either unmodified CIFAR-10 or CIFAR-10 with random labels (the same set of random labels was used for all networks), all of which were trained to near-zero training loss444Details of the architectures and training procedures for this and following experiments can be found in Appendix A.4.. Critically, the randomization of CIFAR-10 labels was consistent for all networks. To evaluate the similarity of converged solutions, we then measured the pairwise projection weighted CCA distance for each layer among networks trained on unmodified CIFAR-10 (“Generalizing”), among networks trained on randomized label CIFAR-10 (“Memorizing”) and between each pair of networks trained on unmodified and random label CIFAR-10 (“Inter”). For all analyses, the representation in a given layer was obtained by averaging across all spatial locations within each filter.
Remarkably, we found that not only do generalizing networks converge to more similar solutions than memorizing networks (to be expected, since generalizing networks are more constrained), but memorizing networks are as similar to each other as they are to a generalizing network. This result suggests that the solutions found by memorizing networks were as diverse as those found across entirely different dataset labellings.
We also found that at early layers, all networks converged to equally similar solutions, regardless of whether they generalize or memorize (Figure 3). Intuitively, this makes sense as the feature detectors found in early layers of CNNs are likely required regardless of the dataset labelling. In contrast, however, at later layers, groups of generalizing networks converged to substantially more similar solutions than groups of memorizing networks (Figure 3
). Even among networks which generalize, the CCA distance between solutions found in later layers was well above zero, suggesting that the solutions found were quite diverse. At the softmax layer, sets of both generalizing and memorizing networks converged to highly similar solutions when CCA distance was computed based on training data; when test data was used, however, only generalizing networks converged to similar softmax outputs (FigureA10), again reflecting that each memorizing network memorizes the training data using a different strategy.
Importantly, because each network learned a different linear transform of a similar solution, traditional distance metrics, such as cosine or Euclidean distance, were insufficient to reveal this difference (Figure A5). Additionally, while unweighted CCA revealed the same broad pattern, it does not reveal that generalizing networks get more similar in the final two layers (Figure A9).
3.2 Wider networks converge to more similar solutions
In the model compression literature, it has been repeatedly noted that while networks are robust to the removal of a large fraction of their parameters (in some cases, as many as 90%), networks initialized and trained from the start with fewer parameters converge to poorer solutions than those derived from pruning a large networks [8, 9, 6, 1, 20, 15]. Recently,  proposed the “lottery ticket hypothesis,” which hypothesizes that larger networks are more likely to converge to good solutions because they are more likely to contain a sub-network with a “lucky” initialization. If this were true, we might expect that groups of larger networks are more likely to contain the same “lottery ticket” sub-network and are therefore more likely to converge to similar solutions than smaller networks.
To test this intuition, we trained groups of convolutional networks with increasing numbers of filters at each layer. We then used projection weighted CCA to measure the pairwise similarity between each group of networks of the same size. Consistent with our intuition, we found that larger networks converged to much more similar solutions than smaller networks (Figure 4).555To control for variability in CCA distance due to comparisons across representations of different sizes, a random subset of 128 filters from the final layer were used for all network comparisons. This bias should, if anything, lead to an overestimate of the distance between groups of larger networks, as they are more heavily subsampled. This is also consistent with the equivalence of deep networks to Gaussian processes (GPs) in the limit of infinite width [13, 17]. If each unit in a layer corresponds to a draw from a GP, then as the number of units increases the CCA distance will go to zero.
Interestingly, we also found that networks which converged to more similar solutions also achieved noticeably higher test accuracy. In fact, we found that across pairs of networks, the correlation between test accuracy and the pairwise CCA distance was -0.96 (Figure 4), suggesting that the CCA distance between groups of identical networks with different random initializations (computed using the train data) may serve as a strong predictor of test accuracy. It may therefore enable accurate prediction of test performance without requiring the use of a validation set.
3.3 Across many initializations and learning rates, networks converge to discriminable clusters of solutions
Here, we ask whether networks trained on the same data with different initializations and learning rates converge to the same solutions. To test this, we measured the pairwise CCA distance between networks trained on unmodified CIFAR-10. Interestingly, when we plotted the pairwise distance matrix (Figure 5), we observed a block diagonal structure consistent with five clusters of converged network solutions, with one cluster highly dissimilar to the other four clusters. Despite the fact that these networks all achieved similar train loss (and many reached similar test accuracy as well), these clusters corresponded with the learning rate used to train each network. This result suggests that there exist multiple minima in the optimization landscape to which networks may converge which are largely specified by the optimization parameters.
In , the authors also observed clusters of network solutions using the relationship between networks’ robustness to cumulative deletion or “ablation” of filters and generalization error. To test whether the same clusters are found via these distinct approaches, we assigned a color to each cluster found using CCA (see bars on left and top in Figure 5), and used these colors to identify the same networks in a plot of ablation robustness vs. generalization error (Figure 5). Surprisingly, the clusters found using CCA aligned nearly perfectly with those observed using ablation robustness.
This result suggests not only that networks with different learning rates converge to distinct clusters of solutions, but also that these clusters can be uncovered independently using multiple methods, each of which measures a different property of the learned solution. Moreover, analyzing these networks using traditional metrics, such as generalization error, would obscure the differences between many of these networks.
4 CCA on Recurrent Neural Networks
So far, CCA has been used to study feedforward networks. We now use CCA to investigate RNNs. Our RNNs are LSTMs used for the Penn Treebank (PTB) and WikiText-2 (WT2) language modelling tasks, following the implementation in [18, 19].
One specific question we explore is whether the learning dynamics of RNNs mirror the “bottom up” convergence observed in the feedforward case in , as well as investigating whether CCA produces qualitatively better outputs than other metrics. However, in the case of RNNs, there are two possible notions of “time”. There is the training timestep, which affects the values of the weights, but also a ‘sequence timestep’ – the number of tokens of the sequence that have been fed into the recurrent net. This latter notion of time does not explicitly change the weights, but results in updated values of the cell state and hidden state of the network, which of course affect the representations of the network.
In this work, we primarily focus on the training notion of time; however, we perform a preliminary investigation of the sequence notion of time as well, demonstrating that CCA is capable of finding similarity across sequence timesteps which are missed by traditional metrics (Figures A2, A4), but also that even CCA often fails to find similarity in the hidden state across sequence timesteps, suggesting that representations over sequence timesteps are often not linearly similar (Figure A3).
4.1 Learning Dynamics Through Training Time
To measure the convergence of representations through training time, we computed the projection weighted mean CCA value for each layer’s representation throughout training to its final representation. We observed bottom-up convergence in both Penn Treebank and WikiText-2 (Figure 6a-b). We repeated these experiments with cosine and Euclidean distance (Figure A8), finding that while these other metrics also reveal a bottom up convergence, the results with CCA highlight this phenomena much more clearly.
5 Discussion and future work
In this study, we developed CCA as a tool to gain insights on many representational properties of deep neural networks. We found that the representations in hidden layers of a neural network contain both “signal” components, which are stable over training and correspond to performance curves, and an unstable “noise” component. Using this insight, we proposed projection weighted CCA, adapting 
. Leveraging the ability of CCA to compare across different networks, we investigated the properties of converged solutions of convolutional neural networks (Section3), finding that networks which generalize converge to more similar solutions than those which memorize (Section 3.1), that wider networks converge to more similar solutions than narrow networks (Section 3.2), and that across otherwise identical networks with different random initializations and learning rates, networks converge to diverse clusters of solutions (Section 3.3). We also used projection weighted CCA to study the dynamics (both across training time and sequence steps) of RNNs, (Section 4), finding that RNNs exhibit bottom-up convergence over the course of training (Section 4.1), and that across sequence timesteps, RNN representations vary nonlinearly (Section A.3).
One interesting direction for future work is to examine what is unique about directions which are preserved across networks trained with different initializations. Previous work has demonstrated that these directions are sufficient for the network computation , but the properties that make these directions special remain unknown. Furthermore, the attributes which specifically distinguish the diverse solutions found in Figure 5 remain unclear. We also observed that networks which converge to similar solutions exhibit higher generalization performance (Figure 4). In future work, it would be interesting to explore whether this insight could be used as a regularizer to improve network performance. Additionally, it would be useful to explore whether this result is consistent in RNNs as well as CNNs. Another interesting direction would be to investigate which aspects of the representation present in RNNs is stable over time and which aspects vary. Additionally, in previous work , it was observed that fixing layers in CNNs over the course of training led to better test performance (“freeze training”). An interesting open question would be to investigate whether a similar training protocol could be adapted for RNNs.
We would like to thank Jascha Sohl-Dickstein for critical feedback on the manuscript, and Jason Yosinski, Jon Kleinberg, Martin Wattenberg, Neil Rabinowitz, Justin Gilmer, and Avraham Ruderman for helpful discussion.
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a.1 Performance Plots for Models
We include the train/test curves for models trained in Figure 1. Comparing the curves to Figure 1, we can see that for all the models, there is a train time where performance is almost equivalent to final performance, but most CCA coefficients still haven’t converged. This suggests that the vectors associated with these are noise in the representation, which is not necessary for doing well at the task.
a.2 Additional reduction methods for CCA
Another potential method to reduce across CCA vectors of varying importnace is to estimate the number of important CCA vectors
, and perform an average over this. A statistical hypothesis test, proposed by Bartlett, and known as Bartlett’s test, attempts to identify the number of statistically significant canonical correlations. Key to the test is the computation of Bartlett’s statistic:
where, in the same notation as previously, is the number of datapoints, and are the number of neurons in , with
. The null hypothesisis that there are statistically significant canonical correlations with the remaining
are generated randomly via a normal distribution. Under the null, the distribution of becomes chi-squared with degrees of freedom. We can then compute the value of and determine if satisfactorily explains the data.
However, the iterative nature of this metric makes it expensive to compute. We therefore focus on projection weighting in this work, and leave further exploration of Bartlett’s test for a future study.
a.3 Representation Dynamics in RNNs Through Sequence (Time) Steps
Here, we investigate the utility of CCA for analyzing representations of RNNs unrolled across sequence time steps. As a toy example of CCA’s benefit in this case, we first initialize a linear vanilla RNN with a unitary recurrent matrix (such that it simply rotates the hidden representation on each timestep). We then use cosine distance, Euclidean distance, and CCA to compare the hidden representation at each timestep to the representation at the final timestep (FigureA2a-c). While both cosine and Euclidean distance fail to realize the similarity between timesteps, CCA, because of its invariance to linear transforms, immediately recognizes that the representations at all timesteps are linearly equivalent.
However, as linear networks are limited in their representational capabilities, we next examine a toy case of a network involving both a linear and non-linear component. We again initialize a simple RNN with the following update rule:
where is the hidden state at time ,
represents the sigmoid nonlinearity,is a random rotation matrix, ), and is a scale factor between the linear and non-linear components. For values of as high as (suggesting that the nonlinear component has 100 times the magnitude of the linear component), we again find that, in contrast to CCA, cosine and Euclidean distance fail to recognize the similarity between timesteps (Figure A2d-f).
However, both of the above cases are toy examples. We next analyze the application of CCA to the more realistic situation of LSTM networks trained on PTB and WikiText-2. To do this, we unroll the RNN for sequence steps, and collect the activations of each neuron in the hidden state over the appropriate sequence tokens for each of the timesteps. More precisely, we can represent our output by a matrix with dimensions where is the number of neurons and is the total sequence length. Our per sequence step matrices would then be , with consisting of all the outputs corresponding to sequence tokens with index equal to modulo , and our matrix would have dimensions . We can then compare to analogous with the comparison to the final timestep. We then apply CCA, Cosine and Euclidean distance as above. To our surprise, the hidden state varies significantly from sequence timestep to sequence timestep, Figure A4.
The above result demonstrates that the hidden state varies nonlinearly in the presence of unique inputs. However, this nonlinearity could be caused by the recurrent dynamics or novel inputs. To disambiguate these two cases, we asked how the hidden state changes when the same input is repeated. We therefore repeat the same input for timesteps, beginning the repetition after some percentage of previous steps containing unique inputs (e.g., through the input sequence tokens). When the repeating inputs were presented early in the sequence, CCA recognized that the hidden state was highly similar, while cosine and Euclidean distance remained insensitive to this similarity (Figure A4, light blue lines). This result appears to suggest that the recurrent dynamics are approximately linear in nature.
However, when the same set of repeating inputs was presented late in the sequence (Figure A4, dark blue lines), we found that the CCA distance increased markedly, suggesting that the nonlinearity of the recurrent dynamics depends not only on the (fixed) recurrent matrix, but also on the sequence history of the network.
a.4 Experimental details
CIFAR-10 ConvNet Architecture:
The convolutional networks trained on CIFAR-10 were identical to those used in 
. All CIFAR-10 networks were trained for 100 epochs using the Adam optimizer with default parameters, unless otherwise specified (learning rate: 0.001, beta1: 0.9, beta2: 0.999). Default layer sizes were: 64, 64, 128, 128, 128, 256, 256, 256, 512, 512, 512, with strides of 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, respectively. All kernels were 3x3 and a batch size of 32. Batch normalization layers were present after each convolutional layer. For the experiments in Section 3.2, all layers were scaled equally by a constant factor0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0.
for training the word level Penn TreeBank and WikiText-2 LSTM models, (without finetuning or continuous cache pointer augmentation). All hyperparameters were left unmodified, so experiments can be reproduced by training LSTM models using the command to runmain.py, and then applying CCA to the hidden states, via the open source implementation777https://github.com/google/svcca/.
Generate vectors in of ‘signal’ (iid standard normal), for and concatenate this matrix with a noise matrix: to. (Note that the noise being lower magnitude than the signal is something that we see in typical neural networks – work on network compression has showing that pruning low magnitude weights is an effective compression strategy.) Putting together gives matrix , (neurons) by (datapoints). Apply a randomly sampled orthonormal transform to the by subset of to get a new by matrix, and again add iid noise of dimensions by to get matrix . Apply CCA based methods to detect similarity between . Of particular interest are cases (low dim. signal in noise).