  We propose a new approach for generative modeling based on training a neural
network to be idempotent. An idempotent operator is one that can be applied
sequentially without changing the result beyond the initial application, namely
$f(f(z))=f(z)$. The proposed model $f$ is trained to map a source distribution
(e.g, Gaussian noise) to a target distribution (e.g. realistic images) using
the following objectives: (1) Instances from the target distribution should map
to themselves, namely $f(x)=x$. We define the target manifold as the set of all
instances that $f$ maps to themselves. (2) Instances that form the source
distribution should map onto the defined target manifold. This is achieved by
optimizing the idempotence term, $f(f(z))=f(z)$ which encourages the range of
$f(z)$ to be on the target manifold. Under ideal assumptions such a process
provably converges to the target distribution. This strategy results in a model
capable of generating an output in one step, maintaining a consistent latent
space, while also allowing sequential applications for refinement.
Additionally, we find that by processing inputs from both target and source
distributions, the model adeptly projects corrupted or modified data back to
the target manifold. This work is a first step towards a ``global projector''
that enables projecting any input into a target data distribution.
