  Neural-network-based compressors have proven to be remarkably effective at
compressing sources, such as images, that are nominally high-dimensional but
presumed to be concentrated on a low-dimensional manifold. We consider a
continuous-time random process that models an extreme version of such a source,
wherein the realizations fall along a one-dimensional "curve" in function space
that has infinite-dimensional linear span. We precisely characterize the
optimal entropy-distortion tradeoff for this source and show numerically that
it is achieved by neural-network-based compressors trained via stochastic
gradient descent. In contrast, we show both analytically and experimentally
that compressors based on the classical Karhunen-Lo\`{e}ve transform are highly
suboptimal at high rates.
