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This paper introduces a new method to build linear flows, by taking theexponential of a linear transformation. This linear transformation does notneed to be invertible itself, and the exponential has the following desirableproperties: it is guaranteed to be invertible, its inverse is straightforwardto compute and the log Jacobian determinant is equal to the trace of the lineartransformation. An important insight is that the exponential can be computedimplicitly, which allows the use of convolutional layers. Using this insight,we develop new invertible transformations named convolution exponentials andgraph convolution exponentials, which retain the equivariance of theirunderlying transformations. In addition, we generalize Sylvester Flows andpropose Convolutional Sylvester Flows which are based on the generalization andthe convolution exponential as basis change. Empirically, we show that theconvolution exponential outperforms other linear transformations in generativeflows on CIFAR10 and the graph convolution exponential improves the performanceof graph normalizing flows. In addition, we show that Convolutional SylvesterFlows improve performance over residual flows as a generative flow modelmeasured in log-likelihood.

The Convolution Exponential and Generalized Sylvester Flows