  Finding latent structures in data is drawing increasing attention in diverse
fields such as image and signal processing, fluid dynamics, and machine
learning. In this work we examine the problem of finding the main modes of
gradient flows. Gradient descent is a fundamental process in optimization where
its stochastic version is prominent in training of neural networks. Here our
aim is to establish a consistent theory for gradient flows $\psi_t = P(\psi)$,
where $P$ is a nonlinear homogeneous operator. Our proposed framework stems
from analytic solutions of homogeneous flows, previously formalized by
Cohen-Gilboa, where the initial condition $\psi_0$ admits the nonlinear
eigenvalue problem $P(\psi_0)=\lambda \psi_0 $. We first present an analytic
solution for \ac{DMD} in such cases. We show an inherent flaw of \ac{DMD},
which is unable to recover the essential dynamics of the flow. It is evident
that \ac{DMD} is best suited for homogeneous flows of degree one. We propose an
adaptive time sampling scheme and show its dynamics are analogue to homogeneous
flows of degree one with a fixed step size. Moreover, we adapt \ac{DMD} to
yield a real spectrum, using symmetric matrices. Our analytic solution of the
proposed scheme recovers the dynamics perfectly and yields zero error. We then
proceed to show that in the general case the orthogonal modes $\{ \phi_i \}$
are approximately nonlinear eigenfunctions $P(\phi_i) \approx\lambda_i \phi_i
$. We formulate Orthogonal Nonlinear Spectral decomposition (\emph{OrthoNS}),
which recovers the essential latent structures of the gradient descent process.
Definitions for spectrum and filtering are given, and a Parseval-type identity
is shown.
