  The dynamics of a turbulent flow tend to occupy only a portion of the phase
space at a statistically stationary regime. From a dynamical systems point of
view, this portion is the attractor. The knowledge of the turbulent attractor
is useful for two purposes, at least: (i) We can gain physical insight into
turbulence (what is the shape and geometry of the attractor?), and (ii) it
provides the minimal number of degrees of freedom to accurately describe the
turbulent dynamics. Autoencoders enable the computation of an optimal latent
space, which is a low-order representation of the dynamics. If properly trained
and correctly designed, autoencoders can learn an approximation of the
turbulent attractor, as shown by Doan, Racca and Magri (2022). In this paper,
we theoretically interpret the transformations of an autoencoder. First, we
remark that the latent space is a curved manifold with curvilinear coordinates,
which can be analyzed with simple tools from Riemann geometry. Second, we
characterize the geometrical properties of the latent space. We mathematically
derive the metric tensor, which provides a mathematical description of the
manifold. Third, we propose a method -- proper latent decomposition (PLD) --
that generalizes proper orthogonal decomposition of turbulent flows on the
autoencoder latent space. This decomposition finds the dominant directions in
the curved latent space. This theoretical work opens up computational
opportunities for interpreting autoencoders and creating reduced-order models
of turbulent flows.
