论文标题
UMAP的稳定性
Stability for UMAP
论文作者
论文摘要
本文显示了Healy-Mcinnes Umap Construction $ V（X，N）$，作为与扩展的伪式度量空间（EP-Metric Space）相关的越野式覆盖物体的迭代推动力，该对象由有限套装$ x $的元素的选择所定义。在另一个有限套件中的包含$ x \子集y $定义了UMAP Systems $ v（x，n）\ to V（y，n'）$的地图，在兼容的社区系统$ n'$ for $ y $的情况下。还有一张诱发的ep-metric空间$（x，d）\ to（y，d'）$的地图，其中$ d $和$ d'$是$ x $ and $ y $定义的指标的colimits（全球平均值）。我们证明了将此Ep-Metric空间图限制为全局组件的稳定性结果。该稳定性结果通过切除路径成分转化为UMAP系统全局组件的稳定性结果。
This paper displays the Healy-McInnes UMAP construction $V(X,N)$ as an iterated pushout of Vietoris-Rips objects associated to extended pseudo metric spaces (ep-metric spaces) defined by choices of neighbourhoods of the elements of a finite set $X$. An inclusion $X \subset Y$ in another finite set defines a map of UMAP systems $V(X,N) \to V(Y,N')$ in the presence of a compatible system of neighbourhoods $N'$ for $Y$. There is also an induced map of ep-metric spaces $(X,D) \to (Y,D')$, where $D$ and $D'$ are colimits (global averages) of the metrics defined by the neighbourhood systems for $X$ and $Y$. We prove a stablity result for the restriction of this ep-metric space map to global components. This stability result translates, via excision for path components, to a stability result for global components of the UMAP systems.
