The computational design of novel protein structures has the potential to
impact numerous scientific disciplines greatly. Toward this goal, we introduce
$\text{FoldFlow}$ a series of novel generative models of increasing modeling
power based on the flow-matching paradigm over $3\text{D}$ rigid motions --
i.e. the group $\text{SE(3)}$ -- enabling accurate modeling of protein
backbones. We first introduce $\text{FoldFlow-Base}$, a simulation-free
approach to learning deterministic continuous-time dynamics and matching
invariant target distributions on $\text{SE(3)}$. We next accelerate training
by incorporating Riemannian optimal transport to create $\text{FoldFlow-OT}$,
leading to the construction of both more simple and stable flows. Finally, we
design $\text{FoldFlow-SFM}$ coupling both Riemannian OT and simulation-free
training to learn stochastic continuous-time dynamics over $\text{SE(3)}$. Our
family of $\text{FoldFlow}$ generative models offer several key advantages over
previous approaches to the generative modeling of proteins: they are more
stable and faster to train than diffusion-based approaches, and our models
enjoy the ability to map any invariant source distribution to any invariant
target distribution over $\text{SE(3)}$. Empirically, we validate our FoldFlow
models on protein backbone generation of up to $300$ amino acids leading to
high-quality designable, diverse, and novel samples.

FoldFlow 是一系列基于流匹配范式的新型生成模型，通过流动匹配 3D 刚体运动（即群 SE (3)），实现对蛋白质主链的准确建模，具备比其他方法更稳定、更快速的训练能力，并可以将任何不变的源分布映射到 SE (3) 上的任何不变的目标分布。

蛋白质骨架生成的 SE (3)- 随机流匹配

SE(3)-Stochastic Flow Matching for Protein Backbone Generation

$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}}
\newcommand{\eps}{{\varepsilon}} \newcommand{\SetX}{\mathsf{X}}
\newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}}
\newcommand{\Space}{\overline{\mathsf{m}}}
\newcommand{\pth}[2][\!]{#1\left({#2}\right)}$ We revisit the problem of
computing Fr\'echet distance between polygonal curves under $L_1$, $L_2$, and
$L_\infty$ norms, focusing on discrete Fr\'echet distance, where only distance
between vertices is considered. We develop efficient algorithms for two natural
classes of curves. In particular, given two polygonal curves of $n$ vertices
each, a $\eps$-approximation of their discrete Fr\'echet distance can be
computed in roughly $O(n\kappa^3\log n/\eps^3)$ time in three dimensions, if
one of the curves is \emph{$\kappa$-bounded}. Previously, only a
$\kappa$-approximation algorithm was known. If both curves are the so-called
\emph{\backbone~curves}, which are widely used to model protein backbones in
molecular biology, we can $\eps$-approximate their Fr\'echet distance in near
linear time in two dimensions, and in roughly $O(n^{4/3}\log nm)$ time in three
dimensions. In the second part, we propose a pseudo--output-sensitive algorithm
for computing Fr\'echet distance exactly. The complexity of the algorithm is a
function of a quantity we call the \emph{\bwnumber{}}, which is quadratic in
the worst case, but tends to be much smaller in practice.

本文研究计算多边形曲线之间的 Frechet 距离，提出了两种算法来实现离散情况下的逼近和伪输出敏感性算法，并研究应用于蛋白质主干的 Frechet 距离计算。