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Result · DS-GRADIENT-EFFICIENT-SEMIMETRIC

The profiled gradient defines a semimetric

Proved herepublication: internal

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Meaning

First-order profiled geometry acts through an efficient projection rather than a full positive-definite score metric.

Statement

For regular nonsingular blocks, G_s=I^{-1}-E_λ I_λλ^{-1} E_λ^T is positive semidefinite of rank d_ψ and induces the population first-order efficient-score semimetric.

Relationships

Rests on

The full chain beneath it

Enables

Open next door

Local graph
Local map of Ds first-order gradient is an efficient semimetric2 prerequisites to the left, 3 results it enables or questions it raises to the right, 0 boundary cases below.Infinitesimal cell-transfer identityInfinitesimal cell-tr…Ds objective is the logdet of efficient/profiled Schur informationDs objective is the l…D-style finite exchange-to-geometry theorem fails for DsD-style finite exchan…Safe Ds tangent screeningSafe Ds tangent scree…Population stationary geometry for profiled DsPopulation stationary…Ds first-order gradient is an efficient semimetricDs first-order gradie…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

DS2. Gradient / efficient semimetric

For regular nonsingular blocks,

Gs=I−1−EλIλλ−1Eλ⊤⪰0,\boxed{ G_s = I^{-1} -E_\lambda I_{\lambda\lambda}^{-1}E_\lambda^\top \succeq0, }

of rank dψd_\psi. Population first-order stationarity therefore induces an efficient-score semi-metric / affine-max geometry.

From Profiled Ds-optimality · chapter contents