fitting

Sampling-kind primitives for the geometry-generic Gauss-Newton fitting model.

The Gauss-Newton model m(p) = c + gᵀp + ½ pᵀ(JᵀJ)p of a least-squares objective factors over two independent choices, and the fitting layer is generic in both:

• the sampling kind (apply / entries / probe) – the bare single-sample Jacobian 𝒥 / its transpose 𝒥ᵀ (from t3toolbox.backend.probing), plus the kind-specific ‖·‖² reduction over the sample stack; and

• the geometry (manifold / corewise) – the gauge projection Π (geometry.project), with the Riemannian forward J = 𝒥∘Π and gradient Jᵀr = Π∘𝒥ᵀr.

This module supplies the kind half: a SamplingKind descriptor bundling the bare probing primitives + the reduction per kind (APPLY / ENTRIES / PROBE). The geometry half lives in the geometry (t3toolbox.manifold.MANIFOLD / COREWISE); the frontend t3toolbox.fitting.GaussNewtonModel composes the two. The §6.3 corewise substitution (U,O,P,Q) -> (U,G,G,G) is now nothing but CorewiseGeometry.frame – there is no separate corewise backend.

probe is vector-valued (one free mode per probe), so its residual / forward output is a sequence of d arrays and its reduction sums the free mode too; apply / entries are the scalar all-modes special case. The objective constant c = ½‖r‖² is just ½ · sumsq(r) (the same reduction as the model’s quadratic term ½‖𝒥Πp‖²), so the kind needs no separate objective function.

Attributes

APPLY
ENTRIES
PROBE
probe_kind
apply_derivatives_kind
entries_derivatives_kind
probe_derivatives_kind

Classes

SamplingKind	A sampling kind: the bare J / J^T for one measurement operator, plus its reductions.
ScalarOutputKind	Kinds whose output is a scalar per measurement (apply / entries): every mode is contracted, so
ProbeOutputKind	Kinds whose output is a vector per measurement per mode (probe): one free mode each, so the
ApplyKind	The all-modes apply: one scalar per measurement. Unweighted (no mode or order axis).
EntriesKind	The all-modes entries: one scalar per multi-index. Unweighted.
ProbeKind	The vector-valued probe (one free mode per measurement), optionally per-mode weighted.
ApplyDerivativesKind	Symmetric directional derivatives of the all-modes apply, orders 0..order, in direction P.
EntriesDerivativesKind	ApplyDerivativesKind at integer grid points. sample = (index, pp). Order-only weight.
ProbeDerivativesKind	Vector-valued derivative probing: the residual is a list of d order-leading arrays.

Functions

sumsq_over_samples(out, n_w)	The ‖·‖² reduction for the scalar-output (apply / entries) kinds: sum out**2 over the
sumsq_over_probes(zz, n_w)	The ‖·‖² reduction for the vector-output probe kind: sum over the leading n_w sample
block_sumsq_over_samples(out, n_w, has_order)	Per-(mode, order) sum-of-squares for the scalar-output apply/entries kinds -> a 2-D
block_sumsq_over_probes(zz, n_w, has_order)	Per-(mode, order) sum-of-squares for the vector-output probe kinds -> a 2-D (d, n_order)

Module Contents

APPLY

ENTRIES

PROBE

probe_kind

apply_derivatives_kind

entries_derivatives_kind

probe_derivatives_kind