This paper introduces a new method to build linear flows, by taking the
exponential of a linear transformation. This linear transformation does not
need to be invertible itself, and the exponential has the following desirable
properties: it is guaranteed to be invertible, its inverse is straightforward
to compute and the log Jacobian determinant is equal to the trace of the linear
transformation. An important insight is that the exponential can be computed
implicitly, which allows the use of convolutional layers. Using this insight,
we develop new invertible transformations named convolution exponentials and
graph convolution exponentials, which retain the equivariance of their
underlying transformations. In addition, we generalize Sylvester Flows and
propose Convolutional Sylvester Flows which are based on the generalization and
the convolution exponential as basis change. Empirically, we show that the
convolution exponential outperforms other linear transformations in generative
flows on CIFAR10 and the graph convolution exponential improves the performance
of graph normalizing flows. In addition, we show that Convolutional Sylvester
Flows improve performance over residual flows as a generative flow model
measured in log-likelihood