Neural networks are famously nonlinear. However, linearity is defined relative to a pair of vector spaces, $f:X \to Y$. Leveraging the algebraic concept of transport of structure, we propose a method to explicitly identify non-standard vector spaces where a neural network acts as a linear operator. When sandwiching a linear operator $A$ between two invertible neural networks, $f(x)=g_y^{-1}(A g_x(x))$, the corresponding vector spaces $X$ and $Y$ are induced by newly defined addition and scaling actions derived from $g_x$ and $g_y$. We term this kind of architecture a Linearizer. This framework makes the entire arsenal of linear algebra, including SVD, pseudo-inverse, orthogonal projection and more, applicable to nonlinear mappings. Furthermore, we show that the composition of two Linearizers that share a neural network is also a Linearizer. We leverage this property and demonstrate that training diffusion models using our architecture makes the hundreds of sampling steps collapse into a single step. We further utilize our framework to enforce idempotency (i.e. $f(f(x))=f(x)$) on networks leading to a globally projective generative model and to demonstrate modular style transfer.

翻译：神经网络以非线性著称。然而，线性性是相对于一对向量空间 $f:X \to Y$ 定义的。利用代数中的结构迁移概念，我们提出了一种方法，可以显式地识别出非标准的向量空间，使得神经网络在其中表现为线性算子。当将一个线性算子 $A$ 夹在两个可逆神经网络之间，即 $f(x)=g_y^{-1}(A g_x(x))$ 时，相应的向量空间 $X$ 和 $Y$ 由从 $g_x$ 和 $g_y$ 导出的新定义的加法与数乘运算所诱导。我们将此类架构称为线性化器。该框架使得包括奇异值分解、伪逆、正交投影等在内的整个线性代数工具库可应用于非线性映射。此外，我们证明了共享一个神经网络的两个线性化器的复合也是一个线性化器。我们利用这一性质，并展示了使用我们的架构训练扩散模型，可以将数百个采样步骤坍缩为单一步骤。我们进一步利用该框架在网络中强制幂等性（即 $f(f(x))=f(x)$），从而得到一个全局投影生成模型，并演示了模块化的风格迁移。