AI 中文总结
该研究提出带拓扑保证的基于传输的嵌入方法，通过定义c-凸势函数的传输映射压缩数据，在COIL和SYMSOL数据集实验中验证其能保留拓扑且避免虚假同调。
AI 中文摘要
来自图像集合的点云、马尔可夫链蒙特卡洛的样本或随机游走的状态，通常具有简单的底层几何结构，但会被噪声、高环境维度以及欧氏距离无法反映相似性的问题所掩盖。UMAP和t-SNE等方法可将此类数据压缩为可用形式，但依赖启发式选择，且无法保证输出能反映输入的拓扑结构。我们提出一种带有此类保证的压缩方法：将数据编码为正的m×n随机矩阵Q=(q_ij)，例如点云上随机游走的转移矩阵，在概率单纯形Δ_n上定义势函数ψ(p)=log∑_i exp(-KL(p,q_i•))，其中KL(p,q)为Kullback-Leibler散度，且证明对于代价函数c(p,q)=KL(p,q)，ψ在最优传输理论意义下是c-凸的。相关的传输映射会坍缩噪声方向，同时可证保留拓扑：ψ的超水平集与c-共轭函数的超水平集同伦等价，其像为可重采样的压缩拓扑空间族，可解释为alpha形状的连续类似物。我们通过两个实验验证该方法：一是从COIL图像数据集中恢复相机角度的圆，而标准PCA流程会产生虚假同调；二是从SYMSOL位姿估计基准中45000个四面体视角恢复商空间SO(3)/A_4。
英文摘要
Point clouds arising in image collections, samples from Markov chain Monte Carlo, or states of a random walk, often have a simple underlying geometry which is obscured by noise, high ambient dimension, and the failure of Euclidean distance to reflect similarity. Methods such as UMAP and t-SNE condense such data into usable form, but rely on heuristic choices and provide no guarantee that the output reflects the topology of the input. We introduce a condensation method that comes with such a guarantee. Encoding the data as a positive $m\times n$ stochastic matrix $Q=(q_{ij})$, for instance the transition matrix of a random walk on the point cloud, we define a potential function $ψ(p)=\log \sum_{i} \exp(-KL(p,q_{i\bullet}))$ on the probability simplex $Δ_n$, where $KL(p,q)$ is the Kullback-Leibler divergence, and prove that $ψ$ is $c$-convex in the sense of Optimal Transport Theory for the cost function $c(p,q)=KL(p,q)$. The associated transport map collapses noisy directions while provably preserving topology: the super-level sets of $ψ$ are homotopy equivalent to those of a $c$-conjugate function, whose image is a condensed, resampleable family of topological spaces which can be interpreted as a continuous analog of an alpha shape. We demonstrate the method by recovering the circle of camera angles from the COIL image dataset, where a standard PCA pipeline produces spurious homology, and the quotient $SO(3)/A_4$ from $45{,}000$ views of a tetrahedron in the SYMSOL pose-estimation benchmark.
Comments27 pages, 6 figures
