In deep dictionary learning multiple dictionaries are learned based on information at various levels of abstraction. We propose a novel hierarchical discriminative dictionary learning layer embedded within a neural network with an image classification objective. Discrimination is induced in the learned synthesis dictionaries at multiple hierarchical levels in a simple way by a one-hot-code representation of the class labels during the training backward pass. In addition, local sparse representation objectives are approximated during the forward pass, introducing local regularization. We evaluate our proposal on five known datasets and we either outperform state-of-the-art methods or achieve competitive classification results. INDEX TERMS Dictionary learning, machine learning, image classification, neural network.

Grid cells in rat medial entorhinal cortex are widely thought to play a major role in spatial behavior. However, the exact computational role of the population of grid cells is not known. Here we provide a descriptive model, which nonetheless considers biologically feasible mechanisms, whereby the grid cells are viewed as a two-dimensional Fourier basis set, in hexagonal coordinates, with restricted availability of basis functions. With known properties imposed in the model parameters, we demonstrate how various empirical benchmark findings are straight-forward to understand in this model. We also explain how complex computations, inherent in a Fourier model, are feasible in the medial entorhinal cortex with simple mechanisms. We further suggest, based on model experiments, that grid cells may support a form of lossy compression of contextual information, enabling its representation in an efficient manner. In sum, this hexagonal Fourier model suggests how the entire population of grid cells may be modeled in a principled way, incorporates biologically feasible mechanisms and provides a potentially powerful interpretation of the relationship between grid-cell activity and contextual information beyond spatial knowledge. This enables various phenomena to be modeled with relatively simple mechanisms, and leads to novel and testable predictions.

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