Eigen-Distortions of Hierarchical Representations

We develop a method for comparing hierarchical image representations in terms of their ability to explain perceptual sensitivity in humans. Specifically, we utilize Fisher information to establish a model-derived prediction of sensitivity to local perturbations of an image. For a given image, we compute the eigenvectors of the Fisher information matrix with largest and smallest eigenvalues, corresponding to the model-predicted mostand least-noticeable image distortions, respectively. For human subjects, we then measure the amount of each distortion that can be reliably detected when added to the image. We use this method to test the ability of a variety of representations to mimic human perceptual sensitivity. We find that the early layers of VGG16, a deep neural network optimized for object recognition, provide a better match to human perception than later layers, and a better match than a 4-stage convolutional neural network (CNN) trained on a database of human ratings of distorted image quality. On the other hand, we find that simple models of early visual processing, incorporating one or more stages of local gain control, trained on the same database of distortion ratings, provide substantially better predictions of human sensitivity than either the CNN, or any combination of layers of VGG16. Human capabilities for recognizing complex visual patterns are believed to arise through a cascade of transformations, implemented by neurons in successive stages in the visual system. Several recent studies have suggested that representations of deep convolutional neural networks trained for object recognition can predict activity in areas of the primate ventral visual stream better than models constructed explicitly for that purpose (Yamins et al. [2014], Khaligh-Razavi and Kriegeskorte [2014]). These results have inspired exploration of deep networks trained on object recognition as models of human perception, explicitly employing their representations as perceptual distortion metrics or loss functions (Hénaff and Simoncelli [2016], Johnson et al. [2016], Dosovitskiy and Brox [2016]). On the other hand, several other studies have used synthesis techniques to generate images that indicate a profound mismatch between the sensitivity of these networks and that of human observers. Specifically, Szegedy et al. [2013] constructed image distortions, imperceptible to humans, that cause their networks to grossly misclassify objects. Similarly, Nguyen and Clune [2015] optimized randomly initialized images to achieve reliable recognition by a network, but found that the resulting ∗Currently at Google, Inc. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. ‘fooling images’ were uninterpretable by human viewers. Simpler networks, designed for texture classification and constrained to mimic the early visual system, do not exhibit such failures (Portilla and Simoncelli [2000]). These results have prompted efforts to understand why generalization failures of this type are so consistent across deep network architectures, and to develop more robust training methods to defend networks against attacks designed to exploit these weaknesses (Goodfellow et al. [2014]). From the perspective of modeling human perception, these synthesis failures suggest that representational spaces within deep neural networks deviate significantly from those of humans, and that methods for comparing representational similarity, based on fixed object classes and discrete sampling of the representational space, are insufficient to expose these deviations. If we are going to use such networks as models for human perception, we need better methods of comparing model representations to human vision. Recent work has taken the first step in this direction, by analyzing deep networks’ robustness to visual distortions on classification tasks, as well as the similarity of classification errors that humans and deep networks make in the presence of the same kind of distortion (Dodge and Karam [2017]). Here, we aim to accomplish something in the same spirit, but rather than testing on a set of handselected examples, we develop a model-constrained synthesis method for generating targeted test stimuli that can be used to compare the layer-wise representational sensitivity of a model to human perceptual sensitivity. Utilizing Fisher information, we isolate the model-predicted most and least noticeable changes to an image. We test these predictions by determining how well human observers can discriminate these same changes. We apply this method to six layers of VGG16 (Simonyan and Zisserman [2015]), a deep convolutional neural network (CNN) trained to classify objects. We also apply the method to several models explicitly trained to predict human sensitivity to image distortions, including both a 4-stage generic CNN, an optimally-weighted version of VGG16, and a family of highly-structured models explicitly constructed to mimic the physiology of the early human visual system. Example images from the paper, as well as additional examples, are available at http://www.cns.nyu.edu/~lcv/eigendistortions/. 1 Predicting discrimination thresholds Suppose we have a model for human visual representation, defined by conditional density p(~r|~x), where ~x is an N -dimensional vector containing the image pixels, and ~r is an M -dimensional random vector representing responses internal to the visual system (e.g., firing rates of a population of neurons). If the image is modified by the addition of a distortion vector, ~x+ αû, where û is a unit vector, and scalar α controls the amplitude of distortion, the model can be used to predict the threshold at which the distorted image can be reliably distinguished from the original image. Specifically, one can express a lower bound on the discrimination threshold in direction û for any observer or model that bases its judgments on ~r (Seriès et al. [2009]): T (û; ~x) ≥ β √ ûTJ−1[~x]û (1) where β is a scale factor that depends on the noise amplitude of the internal representation (as well as experimental conditions, when measuring discrimination thresholds of human observers), and J [~x] is the Fisher information matrix (FIM; Fisher [1925]), a second-order expansion of the log likelihood: J [~x] = E~r|~x [( ∂ ∂~x log p(~r|~x) )( ∂ ∂~x log p(~r|~x) )T] (2) Here, we restrict ourselves to models that can be expressed as a deterministic (and differentiable) mapping from the input pixels to mean output response vector, f(~x), with additive white Gaussian noise in the response space. The log likelihood in this case reduces to a quadratic form: log p(~r|~x) = − 2 ( [~r − f(~x)] [~r − f(~x)] ) + const. Substituting this into Eq. (2) gives: J [~x] = ∂f ∂~x T ∂f ∂~x Thus, for these models, the Fisher information matrix induces a locally adaptive Euclidean metric on the space of images, as specified by the Jacobian matrix, ∂f/∂~x.