ScoreQuant

Result · TRACE-WHITENED-KMEANS

Trace retention is whitened k-means

Derived from the literaturepublication: derived from published

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Meaning

Maximizing normalized trace is equivalent to minimizing within-cell scatter in Fisher-whitened score space.

Scientific context

Once the score coordinates are put into Fisher-whitened units, maximizing the retained trace is exactly weighted k-means — the same objective, the same optimum, the same algorithm. This explains why running k-means on whitened scores is a principled answer, and it pins down precisely which question that answer is answering.

This is a corollary of the information-loss decomposition (FI-LOSS-DECOMPOSITION) and is explicitly not presented as a theorem in its own right: that normalized trace after Fisher whitening is equivalent to weighted k-means was already established.

Statement

d-tr(I_full^{-1}I_q)=E||z-E[z|Z]||^2 for z=I_full^{-1/2}s; maximizing retained normalized trace is weighted k-means.

Where it stops

This control case is known in substance and is not a novelty claim.

Closest prior work

Nearest prior work

Relationships

Rests on

Enables

Raises

Open next door

In the library

Carried by KMeansConfig, NormalizedTrace. Reference.

Every object and refusal

Local graph
Local map of Normalized trace equals Fisher-whitened k-means1 prerequisites to the left, 1 results it enables or questions it raises to the right, 0 boundary cases below.Quantized Fisher-information identityQuantized Fisher-info…High-rate D quantization asymptoticsHigh-rate D quantizat…Normalized trace equals Fisher-whitened k-meansNormalized trace equa…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

T1. Fisher-whitened normalized trace equals weighted k-means

With

z=Ifull−1/2s,z=I_{\rm full}^{-1/2}s, d−tr⁡(Ifull−1Iq)=E∥z−E[z∣Z]∥2.\boxed{ d-\operatorname{tr}(I_{\rm full}^{-1}I_q) =E\|z-E[z\mid Z]\|^2. }

Therefore maximizing normalized retained trace is exactly squared-error vector quantization / weighted k-means in whitened score space.

This is useful as a baseline, not a headline novelty claim.

From The trace criterion · chapter contents