Misc

In this paper, we present a unified approach to constructing continuous and discrete $\mathrm{PGL}(3)$-invariant integrable systems, formulated in terms of the common dependent variables $z_1,z_2$, from linear spectral problems and their factorisation. Starting from third-order spectral problems, we first provide explicit forms of the differential and difference invariants, generalising the Schwarzian derivative and cross-ratio to the rank-$3$ setting. The factorisation induces dualities among linear spectral problems, underlying the exact discretisation and multi-dimensional consistency of the associated Boussinesq systems. Then, we derive both continuous and discrete $\mathrm{PGL}(3)$-invariant Boussinesq systems, representing natural rank-$3$ generalisations of the Schwarzian KdV and cross-ratio equations. A geometric lifting-decoupling mechanism is developed to explain the reduction of these systems to the $\mathrm{PGL}(2)$-invariant Boussinesq equations. Finally, we derive a ${\mathrm{PGL } }(3)$-invariant system of generating PDEs together with its Lagrangian structure, in which the lattice parameters serve as independent variables, providing the generating PDE system for the Boussinesq hierarchy.

半離散方程式（微分差分方程式ともいう）においては，連続独立変数と離散独立変数が非可換のため，対称性とならない非内在（non-intrinsic）（または不規則（irregular））な変換が生じるという問題があった．本論文では，この問題の解決方法を論じるとともに，半離散方程式における対称性理論について紹介する．具体例として，戸田型，ヴォルテラ型，5 点の伊藤・成田・ボゴヤヴレンスキー方程式などのリー点対称性（Lie point symmetry）を求め，群不変解（group-invariant solution）を導出する．

Diffusion models have emerged as powerful generative models, but their high computation cost in iterative sampling remains a significant bottleneck. In this work, we present an in-depth and insightful study of state-of-the-art acceleration techniques for diffusion models, including caching and quantization, revealing their limitations in computation error and generation quality. To break these limits, this work introduces Modulated Diffusion (MoDiff), an innovative, rigorous, and principled framework that accelerates generative modeling through modulated quantization and error compensation. MoDiff not only inherents the advantages of existing caching and quantization methods but also serves as a general framework to accelerate all diffusion models. The advantages of MoDiff are supported by solid theoretical insight and analysis. In addition, extensive experiments on CIFAR-10 and LSUN demonstrate that MoDiff significant reduces activation quantization from 8 bits to 3 bits without performance degradation in post-training quantization (PTQ). Our code implementation is available at https://github.com/WeizhiGao/MoDiff.