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Result · INFO-RETENTION-SPECTRUM

Every retention eigenvalue lies between zero and one

Derived from the literaturepublication: derived from published

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Meaning

Whitening against the full information reveals the retained fraction in each information direction.

Scientific context

The binned information matrix, normalized by the unbinned one, has every eigenvalue between zero and one. This gives the natural scale for reporting how much information a partition retains — a spectrum of numbers in the unit interval rather than an unbounded quantity.

Statement

R=I_full^{-1/2} I_q I_full^{-1/2} satisfies 0<=R<=I, so every retention eigenvalue lies in [0,1].

Relationships

Rests on

The full chain beneath it

Enables

Open next door

In the library

Carried by InformationReport.retained_matrix, retained_eigenvalues. Reference.

Every object and refusal

Local graph
Local map of Normalized retained-information spectrum1 prerequisites to the left, 5 results it enables or questions it raises to the right, 0 boundary cases below.Exact Fisher-information loss decompositionExact Fisher-informat…D-efficiency relative to unbinned inferenceD-efficiency relative…Direction-resolved retained-information diagnosticsDirection-resolved re…Finite-sample uncertainty for retention estimatesFinite-sample uncerta…Reported versus true retention of a frozen label map under a score error: an exact two-term expansion in the Fisher-whitened score errorReported versus true…Applying a frozen rule to the proxy instead of the truth: mislabel mass under a margin condition and an exact Loewner sandwich on the retained informationApplying a frozen rul…Normalized retained-information spectrumNormalized retained-i…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

U7. Normalized retained-information spectrum
R=Ifull−1/2IqIfull−1/2,0⪯R⪯I.R=I_{\rm full}^{-1/2}I_qI_{\rm full}^{-1/2}, \qquad 0\preceq R\preceq I.

Thus every retention eigenvalue lies in [0,1][0,1]. Natural summaries are

1dtr⁡R,(det⁡R)1/d,λmin⁡(R).\frac1d\operatorname{tr}R, \qquad (\det R)^{1/d}, \qquad \lambda_{\min}(R).

The middle quantity is D-efficiency.

From Universal information structure · chapter contents