The day in conversation for 2026-10-19

The discussion: Tom: That localized region map for controlling family-wise error in persistence diagrams seems powerful, but I’m curious how robust that localized structure is when you shift from additive landmark responses to, say, multiplicative ones. Jane: The robustness depends entirely on the smoothness of the underlying density structure within that neighborhood; if the zero-exclusion intervals vary wildly across radii, the map becomes noisy quickly. What specific metric did you use to quantify that structural variation in your simulation? Lu: We relied on a kernel density estimator applied to the expected feature mass within those regions, which we compared against standard Gaussian multiplier bootstrap estimates. Did you find that the performance gain from using the localized map outweighed the computational overhead of calculating those specific local densities? Meng: I found that while it was faster for high-dimensional data sets due to sparsity assumptions, applying it to persistence diagrams with very dense point clouds required significantly more samples to maintain similar coverage guarantees compared to earlier methods. How did you normalize that sample requirement across different radii? Lalam: That normalization is tricky; we initially tested a fixed budget allocation based on the expected complexity of the neighborhood, but I wonder if that assumption holds when the underlying persistence diagrams exhibit multi-modal clustering, which complicates defining a single center point.