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.

翻译：我们提出一种基于训练神经网络实现幂等性的生成建模新方法。幂等算子是指连续应用不会改变初始应用结果的操作，即$f(f(z))=f(z)$。所提出的模型$f$通过以下目标函数训练，将源分布（如高斯噪声）映射到目标分布（如真实图像）：(1) 目标分布中的实例应映射到自身，即$f(x)=x$。我们将目标流形定义为所有被$f$映射到自身的实例集合。(2) 构成源分布的实例应映射到已定义的目标流形上。这通过优化幂等项$f(f(z)) = f(z)$实现，该项鼓励$f(z)$的值域位于目标流形上。在理想假设下，该过程可证明收敛到目标分布。该策略使模型能够单步生成输出，保持一致的潜在空间，同时允许通过连续应用进行优化。此外，我们发现通过处理来自目标分布和源分布的输入，该模型能巧妙地将损坏或修改后的数据投影回目标流形。这项工作是迈向"全局投影器"的第一步，该投影器能将任意输入投影到目标数据分布中。