Weak limit theorems for univariate k-mean clustering under a nonregular condition
Principal points and self-consistency (31 Aug 2026)
The DS17 study (DS-STABLE-BASINS) registered the principal-points /
self-consistency branch of this cluster. Flury introduces population
-means points and the elliptical theme; Tarpey–Flury define
self-consistency , the vocabulary DS17's tilt-consistent
strip rules specialize; Tarpey–Li–Flury prove the forced principal-subspace
alignment of self-consistent points under elliptical symmetry — the closest
published structural cousin of DS17's conditional-centering obstruction
(both are "self-consistency plus a symmetry/centering hypothesis forces a
degenerate configuration" results), with no nuisance/profiled block, no
self-regenerating tilt coefficient, and no conditional-mean-orthogonality
hypothesis. Serinko–Babu is the nearest 1-D interval-partition asymptotics
precedent: singular-Hessian degeneracy at a unique global k-means optimum
with nonstandard rates — a per-law description of degeneracy when it occurs,
never a class-wide forcing theorem. Triangulation:
LITERATURE/audits/DS-STABLE-BASINS-31-August-2026.md.
J. Multivariate Anal. 41(2):273-296; primary PDF read verbatim. 1-D interval-partition (split-function) formulation of population k-means; the unique global optimizer with singular Hessian yields nonstandard n^{1/4} rates (double exponential, k=2). Nearest precedent for degeneracy AT a 1-D self-consistent partition optimum; describes what happens when degeneracy occurs for one law, never a class-wide forcing theorem like DS17.
Cited by
- Exact off-class global basin transfers to empirical exchange-stable optimaDS-NONCENTERED-GLOBAL-BASIN-TRANSFER
- Inhabitation reduces to a scalar root equation per lawDS-STABLE-BASINS-FIXED-POINT-GATE
- Measured gate scans: eight (L)-laws empty, mix3 inhabited free, sign-split family at v_2DS-STABLE-BASINS-GATE-SCANS
Relevant to
Claims this source bears on without being cited in their proofs.