Open question · OPEN-D-UNRESTRICTED-CONSISTENCY

Does finite optimality survive the population limit?

Openpublication: open

Meaning

Establish unrestricted global D consistency beyond compact rule families with imposed margins.

Scientific context

Consistency is proved only for a restricted class of rules (CONSISTENCY-RESTRICTED-AFFINE). Whether unrestricted empirical global optima converge, in value or in decision, to population optima is open. The structural theorem makes the determinant case look favourable — global optima are geometrically realizable, so the unrestricted and restricted problems have the same optimal value — but that observation falls short of a proof.

Statement

Determine whether unrestricted empirical global D optima converge in value/decision to population global hard quantizers.

Closest prior work

Nearest prior work

Scott, Symons (1971)Clustering Methods Based on Likelihood Ratio Criteria

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Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The question in full

Unrestricted D global consistency

Restricted compact affine-max consistency is established. Open: do unrestricted empirical global D optima converge in value and decision to population global D quantizers under natural assumptions? Finite geometric realizability may reduce this to a controlled geometric class, but the metric and centroids are data-dependent and singular boundaries must be controlled.

From OPEN_PROBLEMS.md