发表机构
University of Cambridge; University of California（剑桥大学; 加利福尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于高斯混合潜分布的拉回几何，用于多模态数据，通过责任加权期望精度定义度量，在归一化流中自适应学习分量，实验证明能减少传输变形并提高插值真实性。
AI 中文摘要
数据驱动的黎曼几何提供了高维数据的非线性插值和几何表示。为了使这些操作在统计上有意义，观测值之间的路径应优先穿过高似然区域。现有的可扩展拉回构造通常使用单峰高斯潜分布，假设数据靠近单一流形。对于多模态数据，将分离的模态或局部结构映射到单个高斯区域可能需要大量的传输变形，并损害由此产生的几何。我们引入了一种适用于混合流形上数据的拉回几何。使用潜高斯混合，我们将其黎曼度量定义为责任加权期望分量精度的矩阵平方。该度量光滑且正定，并在单分量极限下恢复现有的高斯构造。对于结构化重叠混合，我们建立了对数密度沿测地线为凹的条件，为所提出的几何与穿过高似然区域的路径之间提供了形式化联系，并推导了相应的局部曲率关系。我们在具有自适应混合学习的归一化流中实例化此几何，使活动分量的数量从数据中涌现，并支持逐分量重建和局部有效维度估计。在合成几何数据、具有已知参考轨迹的受控多视图图像设置以及MNIST上的实验表明，传输变形减少、路径支持具有竞争力、参考轨迹恢复接近，且插值真实性提高。这些结果将可扩展拉回几何扩展到靠近单一流形的数据集之外，同时保留可处理且可解释的局部结构。
英文摘要
Data-driven Riemannian geometry provides nonlinear interpolation and geometric representations of high-dimensional data. For these operations to be statistically meaningful, paths between observations should preferentially traverse high-likelihood regions. Existing scalable pullback constructions typically use a unimodal Gaussian latent distribution, assuming that the data reside close to a single manifold. For multimodal data, mapping separated modes or local structures into one Gaussian region can require substantial transport deformation and compromise the resulting geometry. We introduce a pullback geometry for data supported on mixtures of manifolds. Using a latent Gaussian mixture, we define its Riemannian metric as the matrix square of the responsibility-weighted expected component precision. The metric is smooth and positive definite and recovers the existing Gaussian construction in the single-component limit. For structured overlapping mixtures, we establish conditions under which the log-density is concave along geodesics, providing a formal connection between the proposed geometry and paths through high-likelihood regions, and derive the corresponding local curvature relations. We instantiate this geometry in a normalizing flow with adaptive mixture learning, allowing the number of active components to emerge from the data and supporting component-wise reconstruction and local effective-dimension estimation. Experiments on synthetic geometric data, a controlled multi-view image setting with a known reference trajectory, and MNIST show reduced transport distortion, competitive path support, close reference-trajectory recovery, and improved interpolation realism. These results extend scalable pullback geometry beyond datasets that reside close to a single manifold while retaining tractable and interpretable local structure.
