论文标题
多参数持续模块的局部等效性
Local Equivalence of Metrics for Multiparameter Persistence Modules
论文作者
论文摘要
多参数持久性的理想不变将是歧视性，可计算和稳定的。在这项工作中，我们分析了稳定的，可计算的多功能器持久性模块的判别能力：光纤条形码。纤维条代码等效于等级不变，并编码多组模块的1参数子模块的条形码。众所周知，这种不变的是全球不完整的。但是，在这项工作中，我们表明，通过在交织距离之间显示局部等效度（在有限释放的模块上完成）和在光纤条形码上的匹配距离之间，对于有限呈现的模块，纤维条代码已在本地完成。更准确地说，我们表明：对于有限的多参数模块$ m $，有一个$ m $的社区，在交织的距离$ d_i $中，匹配距离（$ d_0 $）满足以下Bi-lipschitz不平等现象$ \ frac $ \ frac {1} {1} {1} {34} d_i（M，n）n） D_i（m，n）$在该社区中的所有$ n $大约$ m $。因此，该社区中没有其他模块具有与$ M $相同的光纤条形码。
An ideal invariant for multiparameter persistence would be discriminative, computable and stable. In this work we analyse the discriminative power of a stable, computable invariant of multiparameter persistence modules: the fibered bar code. The fibered bar code is equivalent to the rank invariant and encodes the bar codes of the 1-parameter submodules of a multiparameter module. This invariant is well known to be globally incomplete. However in this work we show that the fibered bar code is locally complete for finitely presented modules by showing a local equivalence of metrics between the interleaving distance (which is complete on finitely-presented modules) and the matching distance on fibered bar codes. More precisely, we show that: for a finitely-presented multiparameter module $M$ there is a neighbourhood of $M$, in the interleaving distance $d_I$, for which the matching distance, $d_0$, satisfies the following bi-Lipschitz inequalities $\frac{1}{34}d_I(M,N) \leq d_0(M,N) \leq d_I(M,N)$ for all $N$ in this neighbourhood about $M$. As a consequence no other module in this neighbourhood has the same fibered bar code as $M$.
