A key challenge in designing convolutional network models is sizing them appro-priately. Many factors are involved in these decisions, including number of layers, feature maps, kernel sizes, etc. Complicating this further is the fact that each of these influence not only the numbers and dimensions of the activation units, but also the total number of parameters. In this paper we focus on assessing the in-dependent contributions of three of these linked variables: The numbers of layers, feature maps, and parameters. To accomplish this, we employ a recursive con-volutional network whose weights are tied between layers; this allows us to vary each of the three factors in a controlled setting. We find that while increasing the numbers of layers and parameters each have clear benefit, the number of feature maps (and hence dimensionality of the representation) appears ancillary, and finds most of its benefit through the introduction of more weights. Our results (i) empir-ically confirm the notion that adding layers alone increases computational power, within the context of convolutional layers, and (ii) suggest that precise sizing of convolutional feature map dimensions is itself of little concern; more attention should be paid to the number of parameters in these layers instead. 1

Autoencoders are popular feature learning models because they are conceptually simple, easy to train and allow for efficient inference and training. Recent work has shown how certain autoencoders can assign an unnormalized “score ” to data which measures how well the autoencoder can represent the data. Scores are commonly computed by using training criteria that relate the autoencoder to a probabilistic model, such as the Restricted Boltzmann Machine. In this paper we show how an autoencoder can assign meaningful scores to data independently of training procedure and without reference to any probabilistic model, by interpreting it as a dynamical system. We discuss how, and under which conditions, running the dynamical system can be viewed as performing gradient descent in an energy function, which in turn allows us to derive a score via integration. We also show how one can combine multiple, unnormalized scores into a generative classifier. 1.

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Abstract—Autoencoders are popular feature learning models, that are conceptually simple, easy to train and allow for efficient inference and training. Recent work has shown how certain autoencoders can be associated with an energy landscape, akin to negative log-probability in a probabilistic model, which measures how well the autoencoder can represent regions in the input space. The energy landscape has been commonly inferred heuristically, by using a training criterion that relates the autoencoder to a probabilistic model such as a Restricted Boltzmann Machine (RBM). In this paper we show how most common autoencoders are naturally associated with an energy function, independent of the training procedure, and that the energy landscape can be inferred analytically by integrating the reconstruction function of the autoencoder. For autoencoders with sigmoid hidden units, the energy function is identical to the free energy of an RBM, which helps shed light onto the relationship between these two types of model. We also show that the autoencoder energy function allows us to explain common regularization procedures, such as contractive training, from the perspective of dynamical systems. As a practical application of the energy function, a generative classifier based on class-specific autoencoders is presented. Index Terms—Autoencoders, representation learning, unsupervised learning, generative classification F 1