  Recent research has shown remarkable success in revealing "steering"
directions in the latent spaces of pre-trained GANs. These directions
correspond to semantically meaningful image transformations e.g., shift, zoom,
color manipulations), and have similar interpretable effects across all
categories that the GAN can generate. Some methods focus on user-specified
transformations, while others discover transformations in an unsupervised
manner. However, all existing techniques rely on an optimization procedure to
expose those directions, and offer no control over the degree of allowed
interaction between different transformations. In this paper, we show that
"steering" trajectories can be computed in closed form directly from the
generator's weights without any form of training or optimization. This applies
to user-prescribed geometric transformations, as well as to unsupervised
discovery of more complex effects. Our approach allows determining both linear
and nonlinear trajectories, and has many advantages over previous methods. In
particular, we can control whether one transformation is allowed to come on the
expense of another (e.g. zoom-in with or without allowing translation to keep
the object centered). Moreover, we can determine the natural end-point of the
trajectory, which corresponds to the largest extent to which a transformation
can be applied without incurring degradation. Finally, we show how transferring
attributes between images can be achieved without optimization, even across
different categories.
