Domain adaptation techniques address the problem of reducing the sensitivity of machine learning methods to the so-called domain shift, namely the difference between source (training) and target (test) data distributions. In particular, unsupervised domain adaptation assumes no labels are available in the target domain. To this end, aligning second order statistics (covariances) of target and source domains have proven to be an effective approach ti fill the gap between the domains. However, covariance matrices do not form a subspace of the Euclidean space, but live in a Riemannian manifold with non-positive curvature, making the usual Euclidean metric suboptimal to measure distances. In this paper, we extend the idea of training a neural network with a constraint on the covariances of the hidden layer features, by rigorously accounting for the curved structure of the manifold of symmetric positive definite matrices. The resulting loss function exploits a theoretically sound geodesic distance on such manifold. Results show indeed the suboptimal nature of the Euclidean distance. This makes us able to perform better than previous approaches on the standard Office dataset, a benchmark for domain adaptation techniques.
Last worth to mention is Batch Normalization @cite_26 , recently revisited under a specific domain adaptation perspective @cite_14 . The idea is quite similar in spirit to the idea of aligning second order statistics. However Batch Normalization tries to compensate for the source's internal covariance shift by normalizing each mini-batch to be zero-mean and unit variance (whitening), while not taking into account target's statistics: layer's activation are decorrelated for the source data, but not for target points.
@cite_14 @cite_26
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While the authors of Batch Normalization (BN) identify and address an important problem involved in training deep networks-- Internal Covariate Shift-- the current solution has certain drawbacks. Specifically, BN depends on batch statistics for layerwise input normalization during training which makes the estimates of mean and standard deviation of input (distribution) to hidden layers inaccurate for validation due to shifting parameter values (especially during initial training epochs). Also, BN cannot be used with batch-size 1 during training. We address these drawbacks by proposing a non-adaptive normalization technique for removing internal covariate shift, that we call Normalization Propagation. Our approach does not depend on batch statistics, but rather uses a data-independent parametric estimate of mean and standard-deviation in every layer thus being computationally faster compared with BN. We exploit the observation that the pre-activation before Rectified Linear Units follow Gaussian distribution in deep networks, and that once the first and second order statistics of any given dataset are normalized, we can forward propagate this normalization without the need for recalculating the approximate statistics for hidden layers. While the authors of Batch Normalization (BN) identify and address an important problem involved in training deep networks- Internal Covariate Shift- the current solution has certain drawbacks. For instance, BN depends on batch statistics for layerwise input normalization during training which makes the estimates of mean and standard deviation of input (distribution) to hidden layers inaccurate due to shifting parameter values (especially during initial training epochs). Another fundamental problem with BN is that it cannot be used with batch-size 1 during training. We address these drawbacks of BN by proposing a non-adaptive normalization technique for removing covariate shift, that we call Normalization Propagation. Our approach does not depend on batch statistics, but rather uses a data-independent parametric estimate of mean and standard-deviation in every layer thus being computationally faster compared with BN. We exploit the observation that the pre-activation before Rectified Linear Units follow Gaussian distribution in deep networks, and that once the first and second order statistics of any given dataset are normalized, we can forward propagate this normalization without the need for recalculating the approximate statistics for hidden layers. Batch Normalization (BN) is capable of accelerating the training of deep models by centering and scaling activations within mini-batches. In this work, we propose Decorrelated Batch Normalization (DBN), which not just centers and scales activations but whitens them. We explore multiple whitening techniques, and find that PCA whitening causes a problem we call stochastic axis swapping, which is detrimental to learning. We show that ZCA whitening does not suffer from this problem, permitting successful learning. DBN retains the desirable qualities of BN and further improves BN's optimization efficiency and generalization ability. We design comprehensive experiments to show that DBN can improve the performance of BN on multilayer perceptrons and convolutional neural networks. Furthermore, we consistently improve the accuracy of residual networks on CIFAR-10, CIFAR-100, and ImageNet. Batch normalization is a ubiquitous deep learning technique that normalizes activations in intermediate layers. It is associated with improved accuracy and faster learning, but despite its enormous success there is little consensus regarding why it works. We aim to rectify this and take an empirical approach to understanding batch normalization. Our primary observation is that the higher learning rates that batch normalization enables have a regularizing effect that dramatically improves generalization of normalized networks, which is both demonstrated empirically and motivated theoretically. We show how both activations and gradient information becomes less input dependent across spatial dimensions and examples within a mini-batch for deep unnormalized networks, and show how this limit possible learning rates. Motivated by recent results in random matrix theory, we argue that this ill-conditioning is due to fluctuations in random initialization, shedding new light on classical initialization schemes and their consequences.