Crate spectrafit_varpro 

spectrafit-varpro — Variable Projection (VarPro) solver for separable models.

A graph is separable when every model node has exactly one linear parameter (amplitude) and the remaining parameters are nonlinear shape parameters. Examples: Gaussian, Lorentzian, Voigt, step functions, Fano.

For such graphs varpro eliminates the linear amplitude dimensions from the optimisation, reducing it to a pure nonlinear problem over the shape params. This gives faster convergence and better numerical conditioning vs vanilla LM.

Bounds

varpro has no native bounds support. When any nonlinear parameter has finite bounds and options.solver == "varpro", we fall back to LM and log a warning. With options.solver == "auto" we transparently fall back.

Jacobian variant: Kaufman approximation, not full Golub-Pereyra

The projected-residual Jacobian that drives the nonlinear optimisation step is supplied by the varpro crate (the [dependencies] varpro entry in this crate’s Cargo.toml), not by this crate’s own code. That crate implements the Kaufman (1975) approximation to the classic Golub-Pereyra (1973) projected Jacobian — confirmed by its own module doc comment (varpro v0.14.0, src/lib.rs, line 99): “The VarPro algorithm implemented here follows (O’Leary2013), but uses the Kaufman approximation to calculate the Jacobian.” Concretely, the full Golub-Pereyra derivative of the projected residual has two terms; the Kaufman approximation drops the second one (the term involving the derivative of the projector P = I - Φ Φ⁺ itself), keeping only the cheaper first term. This is the standard trade-off documented in the literature: fewer FLOPs per Jacobian evaluation, at the cost of a Gauss-Newton-only convergence-rate guarantee near the solution (full Golub-Pereyra retains second-order terms that can matter on ill-conditioned or rank-deficient projections) and a covariance that is only an approximation of the true joint covariance over both linear and nonlinear parameters. tests/parity/test_varpro_equivalence.py checks empirically that, on a well-posed separable problem, this approximation still converges to the same minimum as full joint LM — the approximation changes the path and the covariance, not the optimum.

References:

• Golub, G., Pereyra, V. (1973). “The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate.” SIAM J. Numer. Anal. 10(2), 413-432.
• Kaufman, L. (1975). “A variable projection method for solving separable nonlinear least squares problems.” BIT 15, 49-57.

Structs

GraphSeparableModel

Implements varpro’s SeparableNonlinearModel trait for an arbitrary separable spectrafit graph.

Functions

is_separable

Returns true when every node in the graph is either separable (has an amplitude linear param + nonlinear shape params) or invariant (all params linear). Returns false for unknown or non-conforming node types.

solve_varpro

Run the varpro solver on one or more datasets sharing the same separable model.