让神经网络在特定空间下变线性，实现单步扩散采样和投影生成。
Who Said Neural Networks Aren't Linear?
- 通过可逆网络构造新向量空间，使神经网络变为线性算子。
- 扩散模型采样从数百步压缩为一步，且保持生成质量。
- 适用于需要投影、风格迁移的生成任务，可解释性强。
尽管神经网络以非线性著称，但线性是相对于向量空间而言的。本文提出一种基于结构传输的框架，通过可逆神经网络 $g_x$ 和 $g_y$ 定义新的加法与数乘操作，构造出新的向量空间 $X$ 与 $Y$，使得神经网络 $f(x) = g_y^{-1}(A g_x(x))$ 在其中表现为线性算子。这类架构称为 Linearizer，使其可直接应用线性代数工具（如 SVD、伪逆、正交投影等）。进一步证明：共享一个神经网络的两个 Linearizer 的复合仍为 Linearizer。利用该性质，本文实现了将扩散模型的数百次采样步骤压缩为单步；同时通过强制幂等性（$f(f(x))=f(x)$），构建全局投影生成模型，并成功实现模块化风格迁移。
原文摘要 · Abstract (English)
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.
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