网格距离测度下的持久同调：Betti 0分量合并规则问询

Great question—this is a common point of confusion when moving from high-level TDA concepts to practical grid-based data, so let’s break this down step by step.

First, forget the idea of a fixed "distance threshold" for merging—this isn’t about geometric proximity alone. Instead, "close enough" is defined by your filtration function and the progression of the simplicial complex as you increase (or decrease) the filtration value.

Here’s the concrete breakdown for lower-level sets (the most common scenario for grid data):

• Your n×n grid is a set of 0-dimensional simplices (vertices), each assigned a value from the grid (this is your filtration function f).
• A Betti-0 component is just a connected set of vertices in the simplicial complex at a given filtration value.
• Two components merge when a 1-dimensional simplex (edge) connecting a vertex from each component is added to the complex. The filtration value at which this happens is the maximum of the two vertices’ values (for lower-level sets—we include all simplices where f(vertex) ≤ current filtration value).

In short: components merge when the filtration scale reaches the point where the "bridge" (edge) between them is included in the complex. It’s about topological connectivity under the filtration, not a fixed geometric distance.

You’re right that cross (4-neighbor) and square (8-neighbor) are starting points, but the deeper piece is how we convert the grid into a simplicial complex—this is where triangulation comes in.

For an n×n grid:

• 0-simplices: Every grid cell is a vertex, with value equal to its grid entry.
• 1-simplices: Edges connect vertices based on your chosen neighborhood (4 or 8), but the key is each edge’s filtration value is set to the maximum of its two vertices’ values.
• 2-simplices: To turn the grid’s square cells into a simplicial complex, we triangulate each square—usually by splitting it along one diagonal (e.g., top-left to bottom-right). Each resulting triangle is a 2-simplex, with its filtration value set to the maximum of its three vertices’ values.

This triangulation matters because it allows the complex to capture higher-dimensional topological features, but for Betti-0, the critical part is the 1-simplices (edges) that connect vertices. Choosing 4 vs 8 neighbors changes which edges exist, which in turn changes when components merge (8-neighbor will lead to earlier merges for diagonal vertices).

The TDA package’s functions like gridDiag are built specifically for grid data, and they handle the triangulation automatically behind the scenes. Here’s what you need to know:

• When you pass a grid to gridDiag, it first triangulates each square cell into two triangles (as we discussed) to form a simplicial complex.
• The filtration parameter controls whether you’re using lower or upper level sets—this determines how the filtration values are applied to simplices.
• For Betti-0, the package tracks when edges are added to the complex (at their max vertex value) and uses that to record component merges in the persistence diagram.

If you want to customize the neighborhood, you can pre-process your grid into a point cloud and use ripsDiag instead, but gridDiag is optimized for grid structures and uses the triangulated complex by default.

内容的提问来源于stack exchange，提问作者darren86