Canonical Correlation Analysis (CCA) is widely used for multimodal data analysis and, more recently, for discriminative tasks such as multi-view learning; however, it makes no use of class labels. Recent CCA methods have started to address this weakness but are limited in that they do not simultaneously optimize the CCA projection for discrimination and the CCA projection itself, or they are linear only. We address these deficiencies by simultaneously optimizing a CCA-based and a task objective in an end-to-end manner. Together, these two objectives learn a non-linear CCA projection to a shared latent space that is highly correlated and discriminative. Our method shows a significant improvement over previous state-of-the-art (including deep supervised approaches) for cross-view classification, regularization with a second view, and semi-supervised learning on real data.READ FULL TEXT VIEW PDF
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CCA is a popular data analysis technique that projects two data sources into a space in which they are maximally correlated (Hotelling1936, ; Bie2005, ). It was initially used for unsupervised data analysis to gain insights into components shared by the two sources (Andrew2013, ; Wang2015_ICML, ; Wang2016, ). CCA is also used to compute a shared latent space for cross-view classification (Kan2015, ; Wang2015_ICML, ; Chandar2016, ; Chang2018, ), for representation learning on multiple views that are then joined for prediction (Sargin2007, ; Dorfer2016, ), and for classification from a single view when a second view is available during training (Arora2012, )
. While some of the correlated CCA features are useful for discriminative tasks, many represent properties that are of no use for classification and obscure correlated information that is beneficial. This problem is magnified with recent non-linear extensions of CCA, implemented via neural networks (NNs), that make significant strides in improving correlation(Andrew2013, ; Wang2015_ICML, ; Wang2016, ; Chang2018, ) but often at the expense of discriminative capability (cf. §4.1
). Therefore, we present a new deep learning technique to project the data from two views to a shared space that is also discriminative.
Most prior work that boosts the discriminative capability of CCA is linear only (Lee2015, ; Singanamalli2014, ; Duan2016, ). More recent work using NNs still remains limited in that it optimizes discriminative capability for an intermediate representation rather than the final CCA projection (Dorfer2016, ), or optimizes the CCA objective only during pre-training, not while training the task objective (Dorfer2018, ). We advocate to jointly optimize CCA and a discriminative objective by computing the CCA projection within a network layer while applying a task-driven operation such as classification. Experimental results show that our method significantly improves upon previous work (Dorfer2016, ; Dorfer2018, ) due to its focus on both the shared latent space and a task-driven objective. The latter is particularly important on small training set sizes.
While alternative approaches to multi-view learning via CCA exist, they typically focus on a reconstruction objective. That is, they transform the input into a shared space such that the input could be reconstructed – either individually, or reconstructing one view from the other. Variations of coupled dictionary learning (Shekhar2014, ; Xu2015, ; Cha2015, ; Bahrampour2015, )
and autoencoders(Wang2015_ICML, ; Bhatt2017, ) have been used in this context. CCA-based objectives, such as the model used in this work, instead learn a transformation to a shared space without the need for reconstructing the input. This task may be easier and sufficient in producing a representation for multi-view classification (Wang2015_ICML, ). We show that the CCA objective can equivalently be expressed as an distance minimization in the shared space plus an orthogonality constraint. Orthogonality constraints help regularize NNs (Huang2018_dbn, ); we present three techniques to accomplish this. While our method is derived from CCA, by manipulating the orthogonality constraints, we obtain deep CCA approaches that compute a shared latent space that is also discriminative.
Overall, our method enables end-to-end training via mini-batches, and we demonstrate the effectiveness of our model for three different tasks: 1) cross-view classification on a variation of MNIST (Lecun1998, ) showing significant improvements in accuracy, 2) regularization when two views are available for training but only one at test time on a cancer imaging and genomic data set with only 1,000 samples, and 3) semi-supervised representation learning to improve speech recognition. In addition, our approach is more robust in the small sample size regime than alternative methods. Our experiments on real data show the effectiveness of our method in learning a shared space that is more discriminative than current state-of-the-art methods for a variety of tasks.
We first introduce CCA and present our task-driven approach in §3. Linear and non-linear CCA are unsupervised and find the shared signal between a pair of data sources, by maximizing the sum correlation between corresponding projections. Let and be mean-centered input data from two different views with samples and , features, respectively.
CCA. The objective is to maximize the correlation between and , where and
are projection vectors(Hotelling1936, ). The first canonical directions are found via
and subsequent projections are found by maximizing the same correlation but in orthogonal directions. Combining the projection vectors into matrices and (), CCA can be reformulated as a trace maximization under orthonormality constraints on the projections, i.e.,
for covariance matrices , , and cross-covariance matrix . Let
and its singular value decomposition (SVD) bewith singular values in descending order. and are computed from the top singular vectors of as and where denotes the first columns of matrix . The sum correlation in the projection space is equivalent to
i.e., the sum of the top singular values. A regularized variation of CCA (RCCA) ensures that the covariance matrices are positive definite by computing the covariance matrices as and , for regularization parameter
and identity matrix(Bilenko2016, ).
. Deep CCA adds non-linear projections to CCA by non-linearly mapping the input via a multilayer perceptron (MLP). In particular, inputsand are mapped via non-linear functions and , parameterized by and , resulting in activations and (assumed to be mean centered) (Andrew2013, ). When implemented by a NN, and are the output activations of the final layer with features. Fig. 1(a) shows the network structure.
DCCA optimizes the same objective as CCA, see Eq. (1), but using activations and . Regularized covariance matrices are computed accordingly and the solution for and can be computed using SVD just as with linear CCA. When (i.e., the number of CCA components is equal to the number of features in and ), optimizing the sum correlation in the projection space, as in Eq. (2), is equivalent to optimizing the following matrix trace norm objective (TNO)
where as in case of CCA (Andrew2013, ). DCCA optimizes this objective directly, without a need to compute the CCA projection within the network. The TNO is optimized first, followed by a linear CCA operation before downstream tasks like classification are performed.
SoftCCA. While DCCA enforces orthogonality constraints on projections and , SoftCCA relaxes them using regularization (Chang2018, ). Final projection matrices and are integrated into and as the top network layer. The trace objective for DCCA in Eq. (1) can be rewritten as minimizing the distance between the projections when each feature in and is normalized to a unit variance (Li2003, ), leading to111We use this distance objective in our formulation. Regularization in SoftCCA penalizes the off-diagonal elements of the covariance matrix , using a running average computed over batches as and a loss of . Overall, the SoftCCA loss takes the form
Supervised CCA methods. CCA, DCCA, and SoftCCA are all unsupervised methods to learn a projection to a shared space in which the data is maximally correlated. Although these methods have shown utility for discriminative tasks, a CCA decomposition may not be optimal for classification because features that are correlated may not be discriminative. Our experiments will show that maximizing the correlation objective too much can degrade performance on discriminative tasks.
CCA has previously been extended to supervised settings by maximizing the total correlation between each view and the training labels in addition to each pair of views (Lee2015, ; Singanamalli2014, ), and by maximizing the separation of classes (Kan2015, ; Dorfer2016, ). Although these methods incorporate the class labels, they do not directly optimize for classification. Dorfer et. al’s CCA Layer (CCAL) is the closest to our method. It optimizes a task loss operating on a CCA projection; however, the CCA objective itself is only optimized during pre-training, not in an end-to-end manner (Dorfer2018, ). Other supervised CCA methods are linear only (Singanamalli2014, ; Lee2015, ; Kan2015, ; Duan2016, ). Instead of computing the CCA projection within the network, as in CCAL, we optimize the non-linear mapping into the shared space together with the CCA part.
To compute a shared latent space that is also discriminative, we start with the DCCA formulation and add a task-driven term to the optimization objective. The CCA component finds features that are correlated between views, while the task component ensures that they are also discriminative. This model can be used for representation learning on multiple views before joining representations for prediction (Sargin2007, ; Dorfer2016, ) and for classification when two views are available for training but only one at test time (Arora2012, ). In §4, we demonstrate both use cases on real data. Our methods and related NN models from the literature are summarized in Tab. S1 (suppl. material); Fig. 1 shows schematic diagrams.
While DCCA optimizes the sum correlation through an equivalent loss function (TNO), the CCA projection itself is computed onlyafter optimization. Hence, the projections cannot be used to optimize another task simultaneously. The main challenge in developing a task-optimal form of deep CCA that discriminates based on the CCA projection is in computing this projection within the network – a necessary step to enable simultaneous training of both objectives. We tackle this by focusing on the two components of DCCA: maximizing the sum correlation between activations and and enforcing orthonormality constraints within and . We achieve both by transforming the CCA objective and present three methods that progressively relax the orthogonality constraints.
We further improve upon DCCA by enabling mini-batch computations for improved flexibility and test performance. DCCA was developed for large batches because correlation is not separable across batches. While large batch implementations of stochastic gradient optimization can increase computational efficiency via parallelism, small batch training provides more up-to-date gradient calculations, allowing a wider range of learning rates and improving test accuracy (Masters2018, ). We reformulate the correlation objective as the
distance (following SoftCCA), enabling separability across batches. We ensure a normalization to one via batch normalization without the scale and shift parameters(Ioffe2015, ).
Task-driven objective. First, we apply non-linear functions and (via MLPs) to each view and , i.e., and . Second, a task-specific function operates on the outputs and . In particular, and are optimized so that the distance between and is minimized; therefore, can be trained to operate on both inputs and
. We combine CCA and task-driven objectives as a weighted sum with a hyperparameter for tuning. This model is flexible, in that the task-driven goal can be used for classification(Krizhevsky2012, ; Dorfer2016_lda, ), regression (Katzman2016, ), clustering (Caron2018, ), or any other task. See Tab. S1 (suppl. material) for an overview.
Orthogonality constraints. The remaining complications for mini-batch optimization are the orthogonality constraints, for which we propose three solutions, each handling the orthogonality constraints of CCA in a different way: whitening, soft decorrelation, and no decorrelation.
1) Whitening (TOCCA-W). CCA applies orthogonality constraints to and . We accomplish this with a linear whitening transformation that transforms the activations such that their covariance becomes the identity matrix, i.e., features are uncorrelated. Decorrelated Batch Normalization (DBN) has previously been used to regularize deep models by decorrelating features (Huang2018_dbn, ) and inspired our solution. In particular, we apply a transformation to make orthonormal, i.e., .
We use a Zero-phase Component Analysis (ZCA) whitening transform composed of three steps: rotate the data to decorrelate it, rescale each axis, and rotate back to the original space. Each of these transformations is learned from the data. Any matrix satisfying whitens the data, where denotes the covariance matrix of . As is only defined up to a rotation, it is not unique. PCA whitening follows the first two steps and uses the eigendecomposition of : for and , where
. As PCA whitening suffers from stochastic axis swapping, neurons are not stable between batches(Huang2018_dbn, ). ZCA whitening uses the transformation in which PCA whitening is first applied, followed by a rotation back to the original space. Adding the rotation brings the whitened data as close as possible to the original data (Kessy2015, ).
Computation of is clearly depend on . While Huang et al. (Huang2018_dbn, ) used a running average of over batches, we apply this stochastic approximation to for each view using the update for batch where is the covariance matrix for the current batch and is the momentum. We then compute the ZCA transformation from to do whitening as . At test time, from the last training batch is used. Algorithm 1 (suppl. material) describes ZCA whitening in greater detail. In summary, TOCCA-W integrates both the correlation and task-driven objectives, with decorrelation performed by whitening, into
where and are whitened outputs of and , respectively.
2) Soft decorrelation (TOCCA-SD). While fully independent components may be beneficial in regularizing NNs on some data sets, a softer decorrelation may be more suitable on others. In this second formulation we relax the orthogonality constraints using regularization, following the Decorr loss of SoftCCA (Chang2018, ). The loss function for this formulation is
3) No decorrelation (TOCCA-ND). When CCA is used in an unsupervised manner, some form of orthogonality constraint or decorrelation is necessary to ensure that and do not simply produce multiple copies of the same feature. While this result could maximize the sum correlation, it is not helpful in capturing useful projections. In the task-driven setting, the discriminative term ensures that the features in and are not replicates of the same information. TOCCA-ND therefore removes the decorrelation term entirely, forming the simpler objective
These three models allow testing whether whitening or soft decorrelation benefit a task-driven model.
Computational complexity. Due to the eigendecomposition, TOCCA-W has a complexity of compared to for TOCCA-SD, with respect to output dimension . However, is typically small () and this extra computation is only performed once per batch. The difference in runtime is less than 6.5% for a batch size of 100 or 9.4% for a batch size of 30 (Table S3, suppl. material).
In summary, all three variants are motivated by adding a task-driven component to deep CCA. TOCCA-ND is the most relaxed and directly attempts to obtain identical latent representations. Experiments will show that whitening (TOCCA-W) and soft decorrelation (TOCCA-SD) provide a beneficial regularization. Further, since the distance that we optimize was shown to be equivalent to the sum correlation (cf. §2 SoftCCA paragraph), all three TOCCA models maintain the goals of CCA just with different relaxations of the orthogonality constraints. See Tab. S1 (suppl. material) for an overview.
We validated our methods on three different data sets: MNIST handwritten digits, the Carolina Breast Cancer Study (CBCS) using imaging and genomic features, and speech data from the Wisconsin X-ray Microbeam Database (XRMB). Our experiments show the utility of our methods for (1) cross-view classification, (2) regularization with a second view during training when only one view is available at test time, and (3) representation learning on multiple views that are joined for prediction.
Implementation.222Code will be available on GitHub soon.
Each layer of our network consists of a fully connected layer, followed by a ReLU activation and batch normalization(Ioffe2015, ). We used the Nadam optimizer and tuned hyperparameters on a validation set via random search; settings and ranges are specified in Table S2Dorfer2016, ) also use ReLu activation and batch normalization. We modified CCAL- (Dorfer2018, ) to use a softmax function and cross-entropy loss for classification, instead of a pairwise ranking loss for retrieval, referring to this modification as CCAL-.
We formed a multi-view data set from the MNIST handwritten digit image data set (Lecun1998, ). Following Andrew et al. (Andrew2013, ), we split each image in half horizontally, creating left and right views that are each pixels. All images were flattened into a vector with 392 features. The full data set consists of 60k training images and 10k test images. We used a random set of up to 50k for training and the remaining training images for validation. We used the full 10k image test set.
We evaluated cross-view classification accuracy by first computing the projection for each view, then we trained a linear SVM on one view’s projection, and finally we used the other view’s projection at test time. While the task-driven methods presented in this work learn a classifier within the model, this test setup enables a fair comparison with the unsupervised CCA variants and validates the discriminativity of the features learned. Notably, using the built-in softmax classifier performed similarly to the SVM (not shown), as much of the power of our methods comes from the representation learning part. We do not compare with a simple supervised NN because this setup does not learn the shared space necessary for cross-view classification. We report results averaged over five randomly selected training/validation sets; the test set always remained the same.
Correlation vs. classification accuracy We first demonstrate the importance of adding a task-driven component to DCCA by showing that maximizing the sum correlation between views is not sufficient. Fig. 2 (left) shows the sum correlation vs. cross-view classification accuracy across many different hyperparameter settings for DCCA (Andrew2013, ), SoftCCA (Chang2018, ), and TOCCA. We used 50 components for each; thus, the maximum sum correlation was 50. The sum correlation was measured after applying linear CCA to ensure that components were independent. With DCCA a larger correlation tended to produce a larger classification accuracy, but there was still a large variance in classification accuracy amongst hyperparameter settings that produced a similar sum correlation. For example, with the two farthest right points in the plot (colored red), their classification accuracy differs by 10%, and they are not even the points with the best classification accuracy (colored purple). The pattern is different for SoftCCA. There was an increase in classification accuracy as sum correlation increased but only up to a point. For higher sum correlations, the classification accuracy varied even more from 20% to 80%. Further experiments (not shown) have indicated that when the sole objective is correlation, some of the projection directions are simply not discriminative, particularly when there are a large number of classes. Hence, optimizing for sum correlation alone does not guarantee a discriminative model. TOCCA-W and TOCCA-SD show a much greater classification accuracy across a wide range of correlations and, overall, the best accuracy when correlation is greatest.
Effect of batch size. Fig. 2 (right) plots the batch size vs. classification accuracy for a training set size of . We tested batch sizes from to ; a batch size of 10 or 30 was best for all three variations of TOCCA. This is in line with previous work that found the best performance with a batch size between 2 and 32 (Masters2018, ). We used a batch size of 32 in the remaining experiments on MNIST.
Effect of training set size. We manipulated the training set size in order to study the robustness of our methods. In particular, Fig. 2 (right) shows the cross-view classification accuracy for training set sizes from to . While we expected that performance would decrease for smaller training set sizes, some methods were more susceptible to this degradation than others. The classification accuracy with CCA dropped significantly for and , due to overfitting and instability issues related to the covariance and cross-covariance matrices. SoftCCA shows similar behavior (prior work (Chang2018, ) on this method did not test such small training set sizes).
Across all training set sizes, our TOCCA variations consistently exhibited good performance, e.g., increasing classification accuracy from 78.3% to 86.7% for with TOCCA-SD. Increases in accuracy over TOCCA-ND were small, indicating that the different decorrelation schemes have only a small effect on this data set; the task-driven component is the main reason for the success of our method. In particular, the classification accuracy with did better than the unsupervised DCCA method on . Further, TOCCA with did better than linear methods on , clearly showing the benefits of the proposed formulation. We also examined the CCA projections qualitatively via a 2D -SNE embedding (VanDerMaaten2008, ). Fig. 3 shows the CCA projection of the left view for each method. As expected, the task-driven variant produced more clearly separated classes.
In this experiment, we address the following question: Given two views available for training but only one at test time, does the additional view help to regularize the model?
We study this question using 1,003 patient samples with image and genomic data from CBCS333http://cbcs.web.unc.edu/for-researchers/ (Troester2018, ). Images consisted of four cores per patient from a tissue microarray that was stained with hematoxylin and eosin. Image features were extracted using a VGG16 backbone (Simonyan2015, )
, pre-trained on ImageNet, by taking the mean of the 512D output of the fourth set of conv. layers across the tissue region and further averaging across all core images for the same patient. For gene expression (GE), we used the set of 50 genes in the PAM50 array(Parker2009, ). The data set was randomly split into half for training and one quarter for validation/testing; we report the mean over eight cross-validation runs. Classification tasks included predicting (1) Basal vs. non-Basal genomic subtype using images, which is typically done from GE, and (2) predicting grade 1 vs. 3 from GE, typically done from images. This is not a multi-task classification setup; it is a means for one view to stabilize the representation of the other.
We tested different classifier training methods when only one view was available at test time: a) a linear SVM trained on one view, b) a deep NN trained on one view using the same architecture as the lower layers of TOCCA, c) CCAL- trained on both views, d) TOCCA trained on both views. Table 1 lists the classification accuracy for each method and task. When predicting genomic subtype Basal from images, all our methods showed an improvement in classification accuracy; the best result was with TOCCA-W, which produced a 2.2% improvement. For predicting grade from GE, all our methods again improved the accuracy – by up to 3.2% with TOCCA-W. These results show that having additional information during training can boost performance at test time.
Notably, this experiment used a static set of pre-trained VGG16 image features in order to assess the utility of the method. The network itself could be fine-tuned end-to-end with our TOCCA model, providing an easy opportunity for data augmentation and likely further improvements in classification accuracy.
Our final experiments use speech data from XRMB, consisting of simultaneously recorded acoustic and articulatory measurements. Prior work has shown that CCA-based algorithms can improve phonetic recognition (Wang2015_ICASSP, ; Wang2015_ICML, ; Wang2016, ; Dorfer2016, ). The 45 speakers were split into 35 for training, 2 for validation, and 8 for testing – a total of 1,429,236 samples for training, 85,297 for validation, and 111,314 for testing.444http://ttic.uchicago.edu/~klivescu/XRMB_data/full/README The acoustic features are 112D and the articulatory ones are 273D. We removed the per-speaker mean & variance for both views. Samples are annotated with one of 38 phonetic labels.
Our task on this data set was representation learning for multi-view prediction – that is, using both views of data to learn a shared discriminative representation. We trained each model using both views and their labels. To test each CCA model, we followed prior work and concatenated the original input features from both views with the projections from both views. Due to the large training set size, we used a Linear Discriminant Analysis (LDA) classifier for efficiency. The same construction was used at test time. This setup was used to assess whether a task-optimal DCCA model can improve discriminative power. We tested TOCCA with a task-driven loss of LDA (Dorfer2016_lda, ) or softmax to demonstrate the flexibility of our model.
We compared the discriminability of a variety of methods to learn a shared latent representation. Table 4 lists the classification results with a baseline that used only the original input features for LDA. Although deep methods, i.e., DCCA and SoftCCA, improved upon the linear methods, all TOCCA variations significantly outperformed previous state-of-the-art techniques. Using softmax consistently beat LDA by a large margin. TOCCA-SD and TOCCA-ND produced equivalent results as a weight of on the decorrelation term performed best. However, TOCCA-W showed the best result with an improvement of 15% over the best alternative method.
TOCCA can also be used in a semi-supervised manner when labels are available for only some samples. Table 5 lists the results for TOCCA-W in this setting. With 0% labeled data, the result would be similar to DCCA. Notably, a large improvement over the unsupervised results in Table 4 is seen even with labels for only 10% of the training samples.
We proposed a method to find a shared latent space that is also discriminative by adding a task-driven component to deep CCA while enabling end-to-end training. This was accomplished by replacing the CCA projection with distance minimization and orthogonality constraints on the activations, and was implemented in three different ways. TOCCA-W or TOCCA-SD performed the best, dependent on the data set – both of which include some means of decorrelation to provide an extra regularizing effect to the model and thereby outperforming TOCCA-ND.
TOCCA showed large improvements over state-of-the-art in cross-view classification accuracy on MNIST and significantly increased robustness when the training set size was small. On CBCS, TOCCA provided a regularizing effect when both views were available for training but only one at test time. TOCCA also produced a large increase over state-of-the-art for multi-view representation learning on a much larger data set, XRMB. On this data set we also demonstrated a semi-supervised approach to get a large increase in classification accuracy with only a small proportion of the labels. Using a similar technique, our method could be applied when some samples are missing a second view.
Classification tasks using a softmax operation or LDA were explored in this work; however, the formulation presented can also be used with other tasks such as regression or clustering. Another possible avenue for future work entails extracting components shared by both views as well as individual components. This approach has been developed for dictionary learning (Lock2013, ; Ray2014_bioinformatics, ; Feng2018, ) but could be extended to deep CCA-based methods. Finally, we have yet to apply data augmentation to the proposed framework; this could provide a significant benefit for small training sets.
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This supplementary material includes additional details on our TOCCA algorithm and experiments, including 1) a comparison of our formulation with other related CCA approaches, 2) pseudocode for the ZCA whitening algorithm used by TOCCA-W, 3) details on hyperparameter selection, and 4) training runtime experiments.
Table S1 compares our three TOCCA formulations with other related linear and deep CCA methods.
|DCCA||where (TNO, equivalent to CCA objective)|
|computed after optimization complete|
|CCAL-||where , is pairwise ranking loss|
|Decorr||where is running mean across batches of|
|Task||where can be cross-entropy or any other task-driven loss|
Pseudocode for ZCA whitening used to achieve orthogonality in our TOCCA-W implementation is shown in Algorithm 1.
A random search over hyperparameters was used to train our methods. The hyperparameter settings and ranges for each data set are provided in Table S2.
|Hidden layer size||500||200||1,000|
|Output layer size||50||50||112|
|Loss function weight|
|regularizer||, 0||, 0||, 0|
|Soft decorrelation regularizer|
The computational complexity of TOCCA-W is greater than that of TOCCA-SD due to the eigendecomposition operation (see §3 in the main article); however, this extra computation is only carried out once per batch. A runtime comparison of the two methods on all three data sets is provided in Table S3.
|Data set||Batch size||Epochs||TOCCA-W||TOCCA-SD|
|MNIST||100||200||488 s||418 s|
|MNIST||30||200||1071 s||1036 s|
|CBCS||100||400||103 s||104 s|
|XRMB||50,000||100||3056 s||3446 s|