DefinitionDefinition: Literature-sourcedProof: 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.

Bregman distances for convex regularizers

Definition

Let $X$ be a real Banach space, let $R : X \to (- \infty, + \infty]$ be proper and convex, choose $x^{†} \in dom R$ , and choose $ξ^{†} \in \partial R (x^{†})$ . For $x \in dom R$ , the Bregman distance generated by $ξ^{†}$ from the reference point $x^{†}$ to $x$ is

$D_{R}^{ξ^{†}} (x, x^{†}) : = R (x) - R (x^{†}) - ⟨ ξ^{†}, x - x^{†} ⟩_{X^{*}, X} .$

The subgradient inequality makes this quantity nonnegative, and $D_{R}^{ξ^{†}} (x^{†}, x^{†}) = 0$ . The definition fixes the orientation: the first argument is the comparison point and the second is the reference point carrying the chosen subgradient.

Definition references

Banach subgradients and dual pairings

Fenchel–Young equality and inequality

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, equation (1.25)

Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, Chapter 2

Depends on

Used by

Dependency tree · two levels

6 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

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, equation (1.25) (standard reference, not scraped)
Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, Chapter 2 (standard reference, not scraped)