论文标题
持续模块卷积
Convolution of Persistence Modules
论文作者
论文摘要
我们将实价的多参数持久性模块作为滑轮和Cosheaves进行研究。使用有关持久模块的同源代数的最新工作，我们定义了持久模块的衍生复合物之间的两个不同的卷积操作。我们表明，这些操作之一在典型上是对分级模块的衍生张量产物的同构。我们提供了用于计算单参数间隔可分解模块之间卷积的公式。我们的卷积操作类似于Schapira和Kashiwara引入的$ \ Mathbb {r}^n $上衍生的可构造式滑轮的复合物的卷积。在我们的环境中，$ \ mathbb {r}^n $具有非标准拓扑。我们显示我们的卷积操作满足了与标准拓扑结合$ \ mathbb {r}^n $在$ \ mathbb {r}^n $上的卷积的类似属性。我们为持久模块的衍生复合物定义了一个卷积距离，并表明它扩展了经典的交织距离。我们还从捆捆和Cosheaf的角度证明了稳定性。
We conduct a study of real-valued multi-parameter persistence modules as sheaves and cosheaves. Using the recent work on the homological algebra for persistence modules, we define two different convolution operations between derived complexes of persistence modules. We show that one of these operations is canonically isomorphic to the derived tensor product of graded modules. We give formulas for computing convolutions between single-parameter interval decomposable modules. Our convolution operations are analogous to the convolution of derived complexes of constructible sheaves on $\mathbb{R}^n$ introduced by Schapira and Kashiwara. In our setting, $\mathbb{R}^n$ has a non-standard topology. We show our convolution operation satisfies analogous properties to the convolution of constructible sheaves on $\mathbb{R}^n$ with the standard topology. We define a convolution distance for derived complexes of persistence modules and show that it extends the classical interleaving distance. We also prove stability results from the sheaf and cosheaf points of view.
