A Sparse Coding Interpretation of Neural Networks and Theoretical Implications

Neural networks, specifically deep convolutional neural networks, have achieved unprecedented performance in various computer vision tasks, but the rationale for the computations and structures of successful neural networks is not fully understood. Theories abound for the aptitude of convolutional neural networks for image classification, but less is understood about why such models would be capable of complex visual tasks such as inference and anomaly identification. Here, we propose a sparse coding interpretation of neural networks that have ReLU activation and of convolutional neural networks in particular. In sparse coding, when the model's basis functions are assumed to be orthogonal, the optimal coefficients are given by the soft-threshold function of the basis functions projected onto the input image. In a non-negative variant of sparse coding, the soft-threshold function becomes a ReLU. Here, we derive these solutions via sparse coding with orthogonal-assumed basis functions, then we derive the convolutional neural network forward transformation from a modified non-negative orthogonal sparse coding model with an exponential prior parameter for each sparse coding coefficient. Next, we derive a complete convolutional neural network without normalization and pooling by adding logistic regression to a hierarchical sparse coding model. Finally we motivate potentially more robust forward transformations by maintaining sparse priors in convolutional neural networks as well performing a stronger nonlinear transformation.