DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Inverse reconstruction risk under train and test laws

Definition

Let $(Y, Y)$ and $(Q, Q)$ be measurable data and target spaces, with $Q$ the Borel sigma-algebra of the target metric space. Let $P$ be a probability measure on the product space $(Y \times Q, Y \otimes Q)$ , and let $ℓ : Q \times Q \to [0, + \infty]$ be product-measurable, where $ℓ (t, a)$ is the loss for target $t$ and action $a$ . For a point reconstruction rule $g : (Y, Y) \to (Q, Q)$ measurable with respect to the stated sigma-algebras, its risk under $P$ is the extended nonnegative integral $R_{P} (g) : = \int_{Y \times Q} ℓ (t, g (y)) P (d y, d t) .$ For a randomized reconstruction rule given by a probability kernel $K$ from $Y$ to $Q$ , define $R_{P} (K) : = \int_{Y \times Q} ⁣ (\int_{Q} ℓ (t, a) K (y, d a)) P (d y, d t) .$ The inner integral is measurable in $(y, t)$ by kernel integration; a deterministic rule is the special case $K_{y} = δ_{g (y)}$ . When $P_{t r a i n}$ and $P_{t e s t}$ are specified joint laws on the same input-target space, the corresponding training-law risk and test-law risk are $R_{P_{t r a i n}} (K)$ and $R_{P_{t e s t}} (K)$ . Either value may be $+ \infty$ . This definition imposes no inequality between the two risks: any such comparison requires an additional assumption relating the joint laws or the rule's behavior under them.

Definition references

Forward map, admissible truths, target map, and target metric space

Data-only point-estimator convention

Learned point rules, posterior samplers, and fitting/test-law distinction

Measurable spaces for data and targets

Product sigma-algebra for joint input-target laws and losses

A joint evaluation law is a probability measure

Measurability of losses and point rules

Randomized reconstruction rules are probability kernels

Kernel integration gives a measurable conditional risk

Statistical risk is expected nonnegative loss under the observation law

Martin, §5.3.1, complete lecture notes

Hoff, §§4.1–4.2, complete lecture notes

Zheng et al., ICLR 2025 InverseBench paper

Depends on

Used by

Zero training loss can fail under an operator shift Counterexample
Benchmark risk, compute budgets, and uncertainty records Definition
A posterior mean can lie outside a nonconvex signal class Example
A train/test prior shift changes reconstruction risk Example
An exact posterior sample need not minimize point risk Example
Prior-matched and out-of-distribution cases Example
Finite-sample generalization of learned inverse maps Open problem
Finite unrolling alone gives no regularization convergence or risk guarantee Proposition
Scope of InverseBench findings Remark

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ryan Martin, Lecture Notes on Advanced Statistical Theory, §5.3.1, printed pp. 104–105 (PDF pp. 105–106) (standard reference, not scraped)
Peter Hoff, Statistical Decision Problems, §§4.1–4.2, PDF pp. 7–8 (standard reference, not scraped)
Hongkai Zheng et al., InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciences, ICLR 2025 (standard reference, not scraped)