Identifiability and stability moduli

Use metric spaces $(X,d_X)$ and $(Y,d_Y)$, a nonempty admissible truth class $U\subseteq X$, a forward map $F:U\to Y$, and a metric solution-class space $(Q,d_Q)$ with class map $q:U\to Q$ as follows: $q$ has image all of $Q$, and $q(x)=q(x^{\mathrm{\prime }})$ exactly when $x$ and $x^{\mathrm{\prime }}$ are equivalent for recovery. The library's metric axioms imply nonnegative metric values.

The forward map $F$ is identifiable modulo the recovery equivalence on $U$ if

$x,x^{\mathrm{\prime }}\in U\text{ and }F(x)=F(x^{\mathrm{\prime }})⟹q(x)=q(x^{\mathrm{\prime }})\mathrm{.}$

It is nonidentifiable modulo that equivalence if some two admissible truths have equal data and distinct target classes.

A deterministic data-only estimator is a function $\widehat{q}:Y\to Q$. It is exact on $U$ if $\widehat{q}(F(x))=q(x)$ for every $x\in U$. The diagram below commutes exactly under this condition; it states the defining equality of functions and is not a proof device.

A uniform stability modulus on $U$ is a nondecreasing function $\omega :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$, with $\omega (0)=0$ and ${\lim }_{t\downarrow 0}\omega (t)=0$, such that

$d_Q(q(x),q(x^{\mathrm{\prime }}))\le \omega (d_Y(F(x),F(x^{\mathrm{\prime }})))(x,x^{\mathrm{\prime }}\in U)\mathrm{.}$

Identifiability concerns equality of exact data; the modulus is a separate quantitative bound for all truth pairs. The estimator definition specifies no regularization procedure or convergence guarantee.

Inverse-problem model and solution-class interface

Metric-space axioms

Nonnegative values of metric distances

Christian Clason, Regularization of Inverse Problems, Chapter 3 §§3.1–3.2 and Chapter 4 §4.1

Tristan van Leeuwen and Christoph Brune, 10 Lectures on Inverse Problems and Imaging, Lecture 1 §§1.2–1.3