Convolutions Through the Lens of Tensor Networks
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Despite their simple intuition, convolutions are more tedious to analyze than dense layers, which complicates the generalization of theoretical and algorithmic ideas. We provide a new perspective onto convolutions through tensor networks (TNs) which allow reasoning about the underlying tensor multiplications by drawing diagrams, and manipulating them to perform function transformations, sub-tensor access, and fusion. We demonstrate this expressive power by deriving the diagrams of various autodiff operations and popular approximations of second-order information with full hyper-parameter support, batching, channel groups, and generalization to arbitrary convolution dimensions. Further, we provide convolution-specific transformations based on the connectivity pattern which allow to re-wire and simplify diagrams before evaluation. Finally, we probe computational performance, relying on established machinery for efficient TN contraction. Our TN implementation speeds up a recently-proposed KFAC variant up to 4.5x and enables new hardware-efficient tensor dropout for approximate backpropagation.
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