inv_gft_newton: Inverse GFT by Full Newton

In GFT: Generalized Fisher Transformation of Correlation Matrices

Inverse GFT by Full Newton

Description

Reconstructs the correlation matrix C with \mathrm{vecl}(\log C) = z by a full Newton method with the exact O(n^4) Hessian recomputed at every iteration, Armijo backtracking on the objective, and an optional fixed-point warm start. Reference implementation for benchmarking against inv_gft.

Usage

inv_gft_newton(z, x0 = NULL, tol = 1e-13, maxit = 500, warm = 1,
               safeguard = TRUE)

Arguments

z	numeric vector of length n(n-1)/2.
x0	optional starting value of length n; defaults to zero.
tol	convergence tolerance on \|\mathrm{diag}(e^A) - 1\|_\infty.
maxit	maximum number of iterations.
warm	number of initial fixed-point steps.
safeguard	logical. If TRUE (the default), apply the two rounding-floor safeguards described below. If FALSE, reproduce the published comparator with only the Armijo line search added, which may stagnate short of tol.

Details

Each iteration forms the exact Hessian in O(n^4) time and solves the Newton system by Cholesky factorization; if the factorization fails or the line search cannot make progress, a fixed-point step is substituted. inv_gft obtains the same fast local convergence at O(n^3) cost per iteration by solving the Newton system inexactly and matrix-free.

Near the solution the objective f = \mathrm{tr}(e^A) - 1'x is flat to within cancellation, so the Armijo condition carries no information: the line search exhausts all of its halvings, the fixed-point fallback is taken, and the iteration can lock at an error several orders of magnitude above the attainable floor. With safeguard = TRUE two further rules prevent this. The full Newton step is taken untested once the predicted decrease falls below the floating-point resolution of f, and a persistent lack of progress below 10^{-9} hands the iteration to the contractive fixed point.

safeguard = FALSE disables both and reproduces the comparator benchmarked in the reference below, namely the method of Chen, Fei and Yu (2025) with only the Armijo line search added. In that mode converged can be FALSE on a small fraction of otherwise well conditioned problems; this is the documented behaviour of that variant, not a defect of the implementation.

Value

An object of class "gft_inv"; see inv_gft for the components.

References

Archakov, I. and Hansen, P. R. (2026). A variational approach to the generalized Fisher transformation of correlation matrices. Working paper.

See Also

inv_gft.

Examples

z <- gft(0.9^abs(outer(1:5, 1:5, "-")))
r <- inv_gft_newton(z)
r$converged

GFT documentation built on Aug. 23, 2026, 5:10 p.m.