arXiv：cs.LG（机器学习，全量分类）· Honglei Brinkmann, Lucas Ng, Georgios Batzolis, Mark Girolami, Carola-Bibiane Sch\"onlieb, Willem Diepeveen·· 14 小时前AI 评分32
超越单模态基：面向多模态数据的拉回几何
Beyond Unimodal Bases: Pullback Geometry for Multimodal Data
AI 导读
研究者提出一种面向多流形混合支撑数据的拉回几何，用潜变量高斯混合将黎曼度量定义为责任加权期望分量精度的矩阵平方，在单分量极限下可退化为已有高斯构造。该度量光滑且正定，并在结构化重叠混合下给出测地线上对数密度凹性的条件及局部曲率关系。方法在归一化流中实现自适应混合学习，合成几何数据、受控多视角图像与 MNIST 实验显示传输畸变降低、参考轨迹恢复更接近且插值更真实。
正文
Abstract: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.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.00708 [cs.LG] |
| (or arXiv:2610.00708v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.00708 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Honglei Brinkmann [view email]
[v1]
Wed, 30 Sep 2026 20:55:56 UTC (17,350 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org