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Result · E-SUPERGRADIENT

The E criterion has explicit supergradients

Establishedpublication: published

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Meaning

Classical spectral analysis describes the supergradient when the smallest eigenvalue is simple or repeated.

Statement

For F_E(I)=lambda_min(I), a simple minimum eigenvalue has gradient vv^T; with multiplicity r the superdifferential is {VHV^T:H>=0,tr H=1}.

Relationships

Enables

Raises

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Local graph
Local map of E-optimality supergradient structure0 prerequisites to the left, 4 results it enables or questions it raises to the right, 0 boundary cases below.Repeated minimum eigenvalues make one-transfer first-order E geometry non-identifyingRepeated minimum eige…Safe E supergradient screeningSafe E supergradient…Population E common-supergradient geometryPopulation E common-s…Quantitative E necessity bound under spectral gapQuantitative E necess…E-optimality supergradient structureE-optimality supergra…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

E1. Supergradient structure

If the minimum eigenvalue is simple with unit eigenvector vv, one gradient is vv⊤vv^\top. At multiplicity rr with basis VV,

∂+λmin⁡(I)={VHV⊤:H⪰0, tr⁡H=1}.\partial^+\lambda_{\min}(I) =\{VHV^\top:H\succeq0,\ \operatorname{tr}H=1\}.

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