The Generalized Persistence Diagram (GPD) for multi-parameter persistence naturally extends the classical notion of persistence diagram for one-parameter persistence. However, unlike its classical counterpart, computing the GPD remains a significant challenge. The main hurdle is that, while the GPD is defined as the M\"obius inversion of the Generalized Rank Invariant (GRI), computing the GRI is intractable due to the formidable size of its domain, i.e., the set of all connected and convex subsets in a finite grid in $\mathbb{R}^d$ with $d \geq 2$. This computational intractability suggests seeking alternative approaches to computing the GPD. In order to study the complexity associated to computing the GPD, it is useful to consider its classical one-parameter counterpart, where for a filtration of a simplicial complex with $n$ simplices, its persistence diagram contains at most $n$ points. This observation leads to the question: 'Given a $d$-parameter simplicial filtration, could the cardinality of its GPD (specifically, the support of the GPD) also be bounded by a polynomial in the number of simplices in the filtration?' This is the case for $d=1$, where we compute the persistence diagram directly at the simplicial filtration level. If this were also the case for $d\geq2$, it might be possible to compute the GPD directly and much more efficiently without relying on the GRI. We show that the answer to the question above is negative, demonstrating the inherent difficulty of computing the GPD. More specifically, we construct a sequence of $d$-parameter simplicial filtrations where the cardinalities of their GPDs are not bounded by any polynomial in the the number of simplices. Furthermore, we show that several commonly used methods for constructing multi-parameter filtrations can give rise to such "wild" filtrations.

翻译：广义持久性图是多参数持久性理论中经典单参数持久性图概念的自然推广。然而，与经典情形不同，广义持久性图的计算仍是一个重大挑战。主要障碍在于，虽然广义持久性图被定义为广义秩不变量的默比乌斯反演，但广义秩不变量的计算因其定义域（即$d\geq 2$维空间$\mathbb{R}^d$中有限网格上所有连通凸子集的集合）的规模极其庞大而难以处理。这种计算上的难处理性促使我们寻求计算广义持久性图的替代方法。为研究计算广义持久性图相关的复杂度，考察其经典单参数对应情形是有益的：对于包含$n$个单形的单纯复形滤过，其持久性图最多包含$n$个点。这一观察引出了以下问题："给定$d$参数单纯形滤过，其广义持久性图（具体指广义持久性图的支撑集）的基数是否也能被滤过中单形数量的多项式所界定？"在$d=1$的情形中，我们直接在单纯形滤过层面计算持久性图。若$d\geq 2$时同样成立，则可能无需依赖广义秩不变量而直接、更高效地计算广义持久性图。我们证明上述问题的答案是否定的，这揭示了计算广义持久性图的内在困难。具体而言，我们构造了一系列$d$参数单纯形滤过，其广义持久性图的基数不受任何关于单形数量的多项式函数界定。此外，我们证明多种常用的多参数滤过构造方法都可能产生此类"野生"滤过。