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arxiv: 2604.18202 · v1 · submitted 2026-04-20 · 🧮 math.DS · cs.LG

Centre manifold theorem for maps along manifolds of fixed points

Lachlan Ewen MacDonald This is my paper

Pith reviewed 2026-05-10 04:00 UTC · model grok-4.3

classification 🧮 math.DS cs.LG
keywords centre manifold theoremmanifolds of fixed pointsmanifold with boundarydiscrete dynamical systemsgradient descentmatrix factorization
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The pith

The centre manifold theorem extends to maps along a manifold-with-boundary of fixed points.

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

The paper proves a centre manifold theorem for a discrete map when the fixed points form a manifold with boundary rather than an isolated point. This matters for systems in which equilibria form a continuum with edges, because the theorem reduces the local dynamics to a lower-dimensional map on the centre manifold. The result is then applied to gradient descent with large step sizes on two-layer matrix factorization problems.

Core claim

We prove that under suitable smoothness and spectral conditions, a map with a manifold-with-boundary of fixed points admits a locally invariant centre manifold tangent to the centre bundle along the manifold. This manifold is used to reduce the dynamics near the fixed-point set. The theorem is applied to study the behavior of large-step-size gradient descent iterates in two-layer matrix factorization.

What carries the argument

The centre manifold tangent to the centre directions along the manifold-with-boundary of fixed points, which remains invariant under the map and captures the non-hyperbolic dynamics.

Load-bearing premise

The map is sufficiently smooth and the linearization along the fixed-point manifold has a spectral gap that cleanly separates centre, stable, and unstable directions.

What would settle it

A concrete C^1 map possessing a manifold-with-boundary of fixed points for which no locally invariant centre manifold exists near a boundary point.

read the original abstract

We prove a centre manifold theorem for a map along a manifold-with-boundary of fixed points, and provide an application to the study of gradient descent with large step size on two-layer matrix factorisation problems.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a centre manifold theorem for a map along a manifold-with-boundary of fixed points and applies the result to analyse gradient descent with large step sizes on two-layer matrix factorisation problems.

Significance. If the theorem holds under the stated conditions, the result supplies a reduction tool for non-hyperbolic dynamics near continua of equilibria that include boundaries; the matrix-factorisation application could then yield concrete statements about the long-term behaviour of large-step GD near rank-deficient loci.

major comments (2)
  1. [§3] §3 (Statement of the main theorem): the required spectral-gap and invariance hypotheses are stated only locally in the interior of the manifold-with-boundary; no uniform control near the boundary is proved or assumed, yet the application in §5 sends trajectories arbitrarily close to that boundary.
  2. [§5] §5 (GD application): the linearisation of the large-step GD map at points of the fixed-point manifold is never computed explicitly, so it is impossible to verify that the centre spectrum remains on the unit circle while the stable/unstable parts satisfy a uniform gap as rank deficiency is approached.
minor comments (2)
  1. [§2] The definition of the centre bundle in §2 is given only in local coordinates; a coordinate-free formulation would clarify the statement for readers.
  2. [References] Several standard references on centre-manifold theorems for maps (e.g., Vanderbauwhede 1989, Carr 1981) are omitted from the bibliography.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below and outline the revisions we plan to make to strengthen the paper.

read point-by-point responses

  1. Referee: [§3] §3 (Statement of the main theorem): the required spectral-gap and invariance hypotheses are stated only locally in the interior of the manifold-with-boundary; no uniform control near the boundary is proved or assumed, yet the application in §5 sends trajectories arbitrarily close to that boundary.

    Authors: We acknowledge that the spectral gap and invariance conditions in Theorem 3.1 are formulated in a pointwise manner for points in the interior of the manifold-with-boundary. The theorem itself is local in nature, constructing the center manifold in a neighborhood of each point. However, to ensure the application in §5 is rigorous, where trajectories can approach the boundary, we will add a uniform spectral gap assumption near the boundary and prove that it holds for the specific gradient descent map under consideration. This will be incorporated as an additional hypothesis in the theorem statement and verified in the application section. revision: yes

  2. Referee: [§5] §5 (GD application): the linearisation of the large-step GD map at points of the fixed-point manifold is never computed explicitly, so it is impossible to verify that the centre spectrum remains on the unit circle while the stable/unstable parts satisfy a uniform gap as rank deficiency is approached.

    Authors: In §5, the analysis of the linearization relies on the algebraic structure of the two-layer matrix factorization and the form of the gradient descent update, allowing us to determine the spectrum without a full explicit matrix representation at every point. Nevertheless, to facilitate verification, we will include an explicit computation of the Jacobian in a new subsection or appendix, demonstrating that the center eigenvalues lie on the unit circle and that the spectral gap condition holds uniformly as the rank deficiency parameter varies and approaches the boundary. This explicit calculation will confirm the applicability of the center manifold theorem. revision: yes

Circularity Check

0 steps flagged

No circularity: standard proof of extended centre manifold theorem from stated assumptions

full rationale

The paper claims to prove a centre manifold theorem for maps along a manifold-with-boundary of fixed points, plus an application to gradient descent on matrix factorisation. No equations, fitted parameters, or 'predictions' appear in the abstract or reader's summary. The derivation is a mathematical proof deriving invariance and tangency properties from hyperbolicity, spectral gap, and smoothness hypotheses on the fixed-point manifold. These are external assumptions, not self-defined or fitted to the target result. No self-citation load-bearing steps, ansatz smuggling, or renaming of known results are indicated. The application is presented as an illustration rather than a fitted prediction. The result is therefore self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all technical content is deferred to the unavailable full text.

pith-pipeline@v0.9.0 · 5307 in / 1013 out tokens · 27944 ms · 2026-05-10T04:00:56.086058+00:00 · methodology

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Reference graph

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