Alphabeta MathHow 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.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Proximal mapping
Definition
Let be proper, closed and convex, and let . For , define the set-valued proximal mapping The existence and uniqueness theorem on this page proves that this set is a singleton for every admissible ; its unique member is also denoted .
Depends on
Used by
- Certified inexact proximal-gradient and FISTA updates Algorithm
- Chambolle–Pock primal-dual hybrid gradient Algorithm
- Diagonal AdaGrad for online convex losses Algorithm
- Randomized proximal coordinate descent with certified inner solves Algorithm
- Proximal-gradient mapping and projected-gradient mapping Definition
- The two-pixel total-variation proximal map Example
- Coordinate proximal subproblems are attained uniquely and descend Lemma
- Diagonal AdaGrad subproblem has a unique minimizer Lemma
- The proximal map of an indicator is Euclidean projection Lemma
- The proximal map of the l1 norm is soft thresholding Lemma
- Three-point inequality for the proximal-gradient model Lemma
- Vanishing aggregate-model error controls the proximal candidate Lemma
- Existence, resolvent characterization, and firm nonexpansiveness of the proximal map Theorem
- Proximal bundle accumulation under vanishing aggregate-model error Theorem
Dependency tree · two levels
18 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
- Neal Parikh and Stephen Boyd, Proximal Algorithms, §1.1 (standard reference, not scraped)
- Amir Beck, First-Order Methods in Optimization, SIAM, 2017, Chapter 6 (standard reference, not scraped)