RemarkRemark: AI-adaptedProof: Not applicablePipeline-generated

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The Gauss–Newton information term omits residual curvature from the full Hessian

Remark

For a $C^{2}$ residual map $r : U \to R^{m}$ on an open $U \subseteq R^{n}$ , with $m, n \geq 1$ , consider the unweighted nonlinear least-squares objective $ϕ (θ) = \frac{1}{2} ∥ r (θ) ∥_{2}^{2}$ . Then the matrix $J (θ)^{T} J (θ)$ is the Gauss–Newton term of the Hessian. The exact identity is $\nabla^{2} ϕ (θ) = J (θ)^{T} J (θ) + \sum_{i} r_{i} (θ) \nabla^{2} r_{i} (θ)$ by The nonlinear least-squares gradient and Hessian formulas. Thus the two matrices agree at zero residual, but the information matrix defined in Finite nonlinear observation designs, local identifiability, and weighted information is not asserted to be the full Hessian at a nonzero-residual point.

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Sources

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, second edition, Ch. 10, §§10.1 and 10.3 (standard reference, not scraped)