Volterra Networks：面向像素空间扩散的多项式去噪架构
Polynomial Denoisers on the Image Manifold: Volterra Networks for Self-Contained Pixel Diffusion
做扩散模型架构的值得看这篇：作者证明了 softmax attention 表达不了多项式去噪算子，于是干脆用 Volterra 层全替换，不靠 attention、卷积和激活函数也能在 ImageNet 上打平同预算的像素扩散模型。想理解 attention 在去噪里到底缺什么、以及流形维度如何决定逼近速率的，这篇把理论和架构对得很齐。
一篇 arXiv 论文提出用 Volterra 层替代 softmax attention 来构建像素空间扩散模型。理论出发点是：对高斯噪声而言，最优去噪器可展开为观测信号的 Wiener–Volterra 多项式泛函级数，而论文证明 softmax attention 以及包括 linear attention 在内的任何归一化混合规则都无法实现超越仿射映射的多项式映射。
基于这一结论，作者提出的架构为：原始图像 patch 经过激进的瓶颈嵌入，再串联低秩 Volterra 层——每层本身即是非线性算子，因此整个模型不含激活函数，也不使用 attention、卷积或任何预训练组件。在数据流形上，其逼近速率由流形维度而非像素空间的环境维度决定。
在 ImageNet 256×256 和 512×512 上的实验显示，该模型在匹配参数量与 FLOP 预算下，与自足式（self-contained）像素空间扩散模型表现相当。这是首个混合算子全部采用 Volterra 层的像素扩散架构，为去噪算子的理论最优性与网络结构之间建立了直接对应。
Polynomial Denoisers on the Image Manifold: Volterra Networks for Self-Contained Pixel Diffusion
Natural images are often conjectured to lie on a low-dimensional manifold within the high-dimensional pixel space. Recent work leverages this to build self-contained pixel-space diffusion models without specialized machinery or pretrained components. A plain Transformer on raw image patches suffices, provided it predicts the clean image rather than the noise and compresses each patch through an aggressive bottleneck. The mixing operator, however, has remained softmax attention, inherited from sequence modeling rather than derived from denoising. For Gaussian noise, the optimal denoiser admits a canonical expansion in polynomial functionals of the observation, the Wiener--Volterra series. We prove that softmax attention, and any normalized mixing rule (including linear attention), cannot realize any polynomial map beyond an affine one. In contrast, Volterra layers realize this class directly and, on a data manifold, achieve approximation rates governed by the manifold dimension rather than the ambient pixel dimension. We therefore propose the architecture these results prescribe; raw image patches passed through a bottleneck embedding, followed by cascaded low-rank Volterra layers, which carry no activation function because each layer is nonlinear in its own right. To our knowledge this is the first pixel-space diffusion architecture whose mixing operators are Volterra layers throughout, with no attention, no convolution, no pretrained components, and no activation function on the residual path. On ImageNet $256 \times 256$ and $512 \times 512$, the resulting model is competitive with self-contained pixel-space diffusion at matched parameter and FLOP budgets.
This report is categorized under AITOP's "Papers" desk. Within our cross-regional analytical framework, it directly maps to AI developer toolchains, coding agents, and engineering workflows. Key entities and technical topics under continuous tracking include Volterra Networks, 扩散模型, softmax attention, 像素空间.
Source attribution: curated from arXiv cs.AI (@Siddharth Roheda, Rohit Chowdhury, Amit Satish Unde, Aniruddha Bala).