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Open question · OPEN-ATOMIC-RANDOMIZATION-GAP

Can randomization help on atomic laws?

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Meaning

Determine the possible advantage of randomized labels when atomless purification is unavailable.

Scientific context

Randomized and deterministic rules attain the same population optimum when the score law is atomless (SOFT-HARD-ATOMLESS-EQUIVALENCE). Real samples are atomic. Whether splitting a single atom between labels can strictly beat every deterministic rule is open.

Statement

For atomic/finite score laws, determine whether splitting an atom among labels can strictly improve the objective over every deterministic hard score quantizer.

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Local map of Atomic randomization gap2 prerequisites to the left, 0 results it enables or questions it raises to the right, 0 boundary cases below.Soft assignments are an actual randomized quantizerSoft assignments are…Randomized and deterministic population optima coincide for moment-based criteria under atomlessnessRandomized and determ…Atomic randomization gapAtomic randomization…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The question in full

Atomic randomization gap

For atomic score laws, can splitting an atom among labels strictly improve D or DsD_s? Smallest exact counterexample, or conditions for no gap.

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