变分混合与多边缘流匹配：推进统计推断及其生物学应用
Variational Mixtures and Multi-Marginal Flow Matching: Advancing Statistical Inference with Biological Applications
- KTH Royal Institute of Technology（KTH皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本论文针对复杂生物系统中的多模态分布，提出从变分推断到多边缘流匹配的系列方法，并推翻混合在变分推断中性能优势的长期误解，应用于三维空间转录组学。
AI中文摘要：
在本论文中，我开发了针对复杂生物系统所产生的分布具有多模态、几何结构且有时仅定义到归一化常数时的统计推断方法。我从变分推断出发，当解析更新方程不可用时，转向黑箱变分推断。为了建立对推断挑战和所提出方法论的理解，我引入了一种新颖的未归一化目标密度（CoLN分布），并将其作为kappa中的受控测试案例。随后，我追踪了一条表达性越来越强的近似轨迹：使用多重重要性采样ELBO评估的集成（论文A）以及自动化组件协作与探索的变分混合（论文B）。由于表达性需要付出代价，我开发了高效的混合学习思想，包括蒙特卡洛目标估计器，以更高效地扩展混合学习（论文C）。作为kappa中的一项新结果，我推翻了一个长达三十年关于在变分推断中使用混合可能带来性能优势的误解。最后，我从变分推断转向流匹配，解决了多边缘设置中对插值器学习进行专门处理的需求（论文D）。通过结合论文A-D的见解，我在第5.5节中推导出一种新方法：具有变分插值器混合的多边缘流匹配。我将这些方法发展联系到生物学应用，特别强调三维空间转录组学，其中堆叠的组织切片在空间上诱导多模态动态。
英文摘要：
In this thesis I develop methods for statistical inference when the distributions arising from complex biological systems are multi-modal, geometrically structured, and sometimes only defined up to a normalizing constant. I start from variational inference and, when analytic update equations are unavailable, move to black-box variational inference. To build intuition regarding inference challenges and the proposed methodologies, I introduce a novel unnormalized target density (the CoLN distribution) and reuse it as a controlled test case in the kappa. I then trace a trajectory of increasingly expressive approximations: ensembles evaluated with the multiple importance sampling ELBO (Paper A) and variational mixtures that automate component cooperation and exploration (Paper B). Because expressivity comes at a cost, I develop efficient mixture learning ideas, including Monte Carlo objective estimators to scale mixture learning more efficiently (Paper C). As a new result in the kappa, I overturn a three decades long misconception regarding the potential performance benefits of using mixtures in variational inference. Finally, I move from variational inference to flow matching, where I address the need for specialized treatment of interpolant learning in multi-marginal settings (Paper D). By combining insights from Papers A-D, I derive in Section 5.5 a new method: multi-marginal flow matching with mixtures of variational interpolants. I connect these methodological developments to biological applications, with special emphasis on three-dimensional spatial transcriptomics, where stacked tissue slices induce multi-modal dynamics across space.
