المفاهيم الأساسية
The Morse theory framework provides insights into critical points and homology changes in the k-NN distance function.
الملخص
The content explores Morse theory applied to the k-th nearest neighbor distance function in Rd. It presents a combinatorial-geometric characterization of critical points, their indices, and homological effects. The study aims to analyze persistent homology in order-k Delaunay mosaics and random k-fold coverage. Key highlights include:
- Introduction to the k-nearest neighbor distance function.
- Application of Morse theory to non-differentiable functions.
- Combinatorial-geometric description of critical points.
- Detailed analysis of homological connectivity for random k-fold cover.
- Relationship between Morse theory and order-k Delaunay mosaics.
الإحصائيات
Passing the threshold r = ((log n + (i − 1) log log n)/n)1/d, the i-th homology of Br(Pn) will remain unchanged if we further increase r.
For i = d, this analysis describes the exact moment at which Br(Pn) covers the manifold, with critical points of index d corresponding to the last uncovered connected components.
A homogeneous Poisson process on a d-dimensional compact manifold with rate n was considered for analysis purposes.
اقتباسات
"Key reason for interest in d(k)P comes from its sub-level sets being k-fold covers."
"Our results provide new means to analyze homology and persistent homology of Delaunay mosaics."
"The behavior of d(k)P is similar to classical Morse theory but can have several simultaneous changes at each critical level."