The proximal point method and its two variants for monotone vector fields in Hadamard spaces

Parin Chaipunya , Fumiaki Kohsaka

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keywords spacesfieldsmonotonevectorhadamardmethodpointproximal

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The pith

The proximal point method and its variants converge for monotone vector fields in Hadamard spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Hadamard spaces are complete metric spaces with non-positive curvature, generalizing Euclidean space in a way that allows for unique geodesics. Monotone vector fields behave like gradients that do not point in opposing directions along lines. The proximal point method is an iterative procedure that finds points where such a field equals zero by solving a regularized subproblem at each step. The paper first studies basic tools like tangent spaces and resolvents in these spaces, then shows that the standard method and two modified versions produce sequences that exist and converge to a zero of the field.

Core claim

We prove existence and convergence of sequences generated by the proximal point method and its two variants for monotone vector fields in Hadamard spaces.

Load-bearing premise

The underlying space is Hadamard (complete CAT(0)) and the vector field is monotone, allowing resolvents to be single-valued and the iteration to be well-defined.

read the original abstract

We prove existence and convergence of sequences generated by the proximal point method and its two variants for monotone vector fields in Hadamard spaces. Before obtaining our results, we investigate some fundamental properties of tangent spaces, resolvents, and monotone vector fields in such spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard properties of Hadamard spaces and monotonicity; no free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption Hadamard spaces are complete CAT(0) metric spaces with unique geodesics
Invoked to define tangent spaces and resolvents
domain assumption Monotone vector fields admit well-defined resolvents in these spaces
Central to the proximal iteration

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation; Foundation.LogicAsFunctionalEquation washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

ρ((1−α)x⊕αy, z)^2 ≤ (1−α)ρ(x,z)^2 + αρ(y,z)^2 − α(1−α)ρ(x,y)^2 (CAT(0) inequality 2.4); quasilinearization ⟨xy, zw⟩ = ½(ρ(x,w)^2 + ρ(y,z)^2 − ρ(x,z)^2 − ρ(y,w)^2).

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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