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Result · D-POP-VORONOI

Stationary population D partitions are geometric

Proved herepublication: internal

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Meaning

The population first-order condition gives nearest-centre geometry under its measure and regularity assumptions.

Statement

A regular atomless stationary D quantizer assigns almost every score to a nearest centroid in the common metric I_q^{-1}; pairwise boundaries are affine hyperplanes.

Assumptions

  • I_q positive definite

  • regular first variation

  • no relevant boundary mass for pointwise statement

deployable population geometry

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Relationships

Rests on

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Local graph
Local map of Population stationary D geometry1 prerequisites to the left, 0 results it enables or questions it raises to the right, 0 boundary cases below.Infinitesimal cell-transfer identityInfinitesimal cell-tr…Population stationary D geometryPopulation stationary…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

D1. Population stationary geometry

A regular atomless stationary D quantizer satisfies

q(s)∈arg⁡min⁡b(s−μb)⊤Iq−1(s−μb)a.e.\boxed{ q(s)\in\arg\min_b (s-\mu_b)^\top I_q^{-1}(s-\mu_b) \quad\text{a.e.} }

Thus cells form a self-consistent common-metric Mahalanobis Voronoi / affine-max partition.

This is a first-order stationarity result, not a global-optimality theorem.

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