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How to force unsupervised neural networks to discover the right representation of images

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One appealing way to design an object recognition system is to define objects recursively in terms of their parts and the required spatial relationships between the parts and the whole. These relationships can be represented by the coordinate transformation between an intrinsic frame of reference embedded in the part and an intrinsic frame embedded in the whole. This transformation is unaffected by the viewpoint so this form of knowledge about the shape of an object is viewpoint invariant. A natural way for a neural network to implement this knowledge is by using a matrix of weights to represent each part-whole relationship and a vector of neural activities to represent the pose of each part or whole relative to the viewer. The pose of the whole can then be predicted from the poses of the parts and, if the predictions agree, the whole is present. This leads to neural networks that can recognize objects over a wide range of viewpoints using neural activities that are ``equivariant’’ rather than invariant: as the viewpoint varies the neural activities all vary even though the knowledge is viewpoint-invariant. The ``capsules’’ that implement the lowest-level parts in the shape hierarchy need to extract explicit pose parameters from pixel intensities and these pose parameters need to have the right form to allow coordinate transformations to be implemented by matrix multiplies. These capsules are quite easy to learn from pairs of transformed images if the neural net has direct, non-visual access to the transformations, as it would if it controlled them. (Joint work with Sida Wang and Alex Krizhevsky)

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