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Some iterative algorithms on Riemannian manifolds and Banach spaces with good global convergence guarantee

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read New backtracking descent and Newton-type algorithms on Riemannian manifolds and Banach spaces ensure every cluster point is critical and random initial points avoid saddle points.

desk verdict Solid extension of the author's Euclidean algorithms to Riemannian manifolds and Banach spaces, but the advertised saddle-avoidance guarantee has a repairable measure-theoretic gap in Theorem 2.20. read the letter →

arxiv 2505.22180 v1 pith:53EYEVVK submitted 2025-05-28 math.OC math.DS

Tuyen Trung Truong This is my paper
classification math.OCmath.DS MSC 49-XX53-XX65-XX68-XX37-XX
keywords iterativeoptimizationRiemannianmanifoldsBanachspacessaddlepointavoidancebacktrackinggradientdescentNewQ-Newtonmethodglobalconvergenceshyness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that three modified iterative optimization algorithms—local backtracking gradient descent, backtracking New Q-Newton's method, and a simplified variant for infinite dimensions—work on Riemannian manifolds and Banach spaces with three simultaneous guarantees: every cluster point of the generated sequence is a critical point, the whole sequence either converges or diverges to infinity under mild hypotheses, and a randomly chosen initial point cannot converge to a saddle point. If correct, this means these algorithms solve both tasks the paper defines: finding critical points and avoiding saddles, in settings where such global guarantees have previously been incomplete. The paper introduces structural tools to make the proofs work, including strong local retractions on manifolds and a weakened form of shyness in Banach spaces, and illustrates the methods on an optimization problem over the unit sphere.

What carries the argument

The central objects are the update maps themselves, together with new geometric and measure-theoretic notions. A strong local retraction is a C1 diffeomorphism Rx from a ball in the tangent space at x onto a neighbourhood of x, with Rx(0)=x and DRx(0)=Id; because the exponential map with r(x)=inj(x) is one, such retractions exist on every Riemannian manifold, replacing global retractions that may not exist. A real analytic-like strong local retraction strengthens this so that preimages of measure-zero sets under the Newton dynamics remain measure zero. In Banach spaces, the paper introduces a weaker notion of shyness, covering exceptional sets by preimages of shy sets under maps with finite distortion, and a Simplified Backtracking New Q-Newton method that uses spectral projections and an absolute-value operator instead of a while-loop over random shifts. Condition C, requiring weak convergence together with vanishing gradients to imply a critical point, is what lets the infinite-dimensional arguments work.

What would settle it

Produce a C2 cost function and local backtracking parameters for which the constructed update map H is locally injective near a generalized saddle point but $H^{{-1}}$(F) has positive Lebesgue measure for some measure-zero set F; such an example would falsify the exceptional-set construction in Theorem 2.20(iii).

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Extended reading notes

Core claim

For a C1, C2, or C3 cost function on a Riemannian manifold, the paper's algorithms have the property that any cluster point of the generated sequence is a critical point of f. When f has at most countably many critical points or satisfies a Kurdyka-Lojasiewicz gradient inequality, the sequence either converges or diverges to infinity. With randomly chosen parameters and initial point, convergence to a generalized saddle point is impossible for Local Backtracking GD, and for New Q-Newton type methods under a real analytic-like retraction condition. In Banach spaces, the analogous statements hold in the weak topology: Local Backtracking GD solves both tasks, while Simplified BNQN solves the critical-point task and additionally obtains a local stable-center manifold near saddle points and quadratic convergence near non-degenerate local minima.

Load-bearing premise

The proof assumes that if the update map is locally injective, the preimage of a measure-zero set is still measure zero, but this implication is not generally valid for C1 maps, and it is exactly the step that makes a randomly chosen initial point avoid saddles.

Editorial extensions

If this is right

  • If Theorem 1.2 holds, constrained optimization over any Riemannian submanifold, such as the unit sphere, inherits global convergence and saddle avoidance from the corresponding Euclidean-style algorithms.
  • Random initialization plus local backtracking would remove the need to hand-tune step sizes near saddles, giving simple first-order methods a practical guarantee of converging to local minima rather than stationary points.
  • Simplified BNQN provides a Newton-type algorithm for infinite-dimensional settings that needs only spectral projections of the Hessian, with local quadratic convergence at non-degenerate minima.
  • Under the countability or Lojasiewicz assumptions, users obtain not merely cluster-point guarantees but convergence of the entire sequence, either to a critical point or to infinity.
  • The real analytic-like retraction condition is used only to guarantee the Lusin (N-1) property of the update map, suggesting that more general retractions would suffice whenever that measure-theoretic property holds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • [editorial inference] The saddle-avoidance proof for Local Backtracking GD rests on the step that a locally injective C1 map sends measure-zero sets to measure-zero preimages; because this implication is not generally valid, the 'random initial point avoids saddles' theorem is conditional on a hidden regularity assumption that the paper does not prove.
  • [editorial inference] The same measure-theoretic mechanism suggests that testing the Lusin (N-1) property of the update map directly would settle whether New Q-Newton type methods avoid saddles globally on all Riemannian manifolds, without needing the real analytic-like condition.
  • [editorial inference] The continuous backtracking construction in the proof, which builds a smooth positive step-size function by partition of unity, could itself be implemented as a deterministic line-search policy and may be simpler in practice than the discrete backtracking rule.
  • [editorial inference] For Banach spaces, extending Theorem 2.16 to infinite dimensions is the natural next step: it would convert Simplified BNQN's local stable-center guarantee into global saddle avoidance, which the paper explicitly leaves open.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces iterative optimization algorithms on finite-dimensional Riemannian manifolds and on Banach/Hilbert spaces, with three advertised guarantees: every cluster point of the generated sequence is a critical point; under countability or Lojasiewicz-type assumptions the sequence either converges or diverges to infinity; and for randomly chosen initial data the sequence cannot converge to a saddle point. The algorithms are variants of Backtracking Gradient Descent (including a new 'Local Backtracking GD'), Riemannian versions of New Q-Newton's method, and a simplified Banach-space version of Backtracking New Q-Newton's method. New tools include 'strong local retractions', 'real analytic-like strong local retractions', and a generalized notion of 'weakly shy' sets. The main theorem is Theorem 1.2, proved through the manifold results in Section 2 and the Banach results in Section 3.

Significance. If the advertised results were fully established, the paper would make a useful contribution by extending saddle-avoidance guarantees and convergence-or-divergence theorems beyond compact or Euclidean settings, and by giving an explicit treatment of Riemannian manifolds without global retractions. The paper also contains honest statements of its own limitations, notably that global saddle avoidance for Simplified BNQN in Banach spaces is left open. Strengths include the clean use of connectedness of cluster sets in projective space for countable critical sets, and the explicit numerical illustration on the unit sphere. However, the two load-bearing gaps identified below currently prevent the central claims from being accepted as rigorous as written.

major comments (3)
  1. [Section 2.4, proof of Theorem 2.20 A(iii) and B] The argument that the exceptional set E1 = ∪_{n} H^{-n}(F1) has Lebesgue measure zero because H is locally injective is invalid. A C1 locally injective map need not pull back Lebesgue-null sets to null sets; a standard counterexample is a strictly increasing C1 function whose derivative vanishes on a fat Cantor set. This inference is load-bearing: it is the mechanism that makes randomly chosen initial points avoid saddle points in MainTask 2.1. The Euclidean construction actually contains the stronger local estimate ||H(y1)-H(y2)|| ≥ (2/9)||y1-y2|| on each neighborhood U, and Lemma 2.18 together with a countable cover would repair the proof. But the Riemannian case is only stated to be 'similar', so the measure-zero property is not established as written.
  2. [Definitions 1.1 and 2.9; proof of Theorem 2.20 B(i)] The claim that the local backtracking choice δ < α/L(x) automatically satisfies Armijo's condition is not correct for all 0 < α < 1. For an L-Lipschitz gradient the standard descent estimate gives f(x-δ∇f(x))-f(x) ≤ -δ(1 - Lδ/2)||∇f(x)||². The condition Lδ < α only implies 1 - Lδ/2 > 1 - α/2, which is weaker than the required factor α when α > 2/3. For example, with α = 0.9 a step allowed by the definition can violate the Armijo inequality. Thus the proof that every cluster point is critical for Local Backtracking GD is not valid as stated. The theorem should either restrict α (for instance to α ≤ 2/3) or replace the condition by δ < 2(1-α)/L(x).
  3. [Section 3.2, proof of Theorem 3.4(4)] The Banach-space version of the saddle-avoidance proof needs more detail at the point where local finite-distortion maps are used to conclude that preimages of weakly shy sets are weakly shy. The proof states that near each point y the local dynamics is one of finitely many maps Hy with finite distortion and then says 'Hence, the preimage of any weakly shy set is also weakly shyset.' Because the local maps Hy depend on the location y, a countable cover and a careful assembly of the covering sets W_i and maps H_i are required to verify the definition of weak shyness. This is likely repairable, but it is not written out.
minor comments (5)
  1. [Throughout] There are several typographical errors, including 'Amijo's condition' and 'distorsion'; these should be corrected before publication.
  2. [Section 2.1] The equation numbers in the paragraph on Taylor expansion are inconsistent: the text refers to 'Equation (3)' when discussing the inequality displayed as (1), and later reuses (3) for a different displayed inequality. Please renumber and harmonize all cross-references.
  3. [Section 2.5, Table 1] The numerical table would be much easier to read if the caption explained the format 'n/(point)/Remark' and the abbreviations 'S', 'M', 'E', 'D', 'C' were defined in a legend rather than only in the table text.
  4. [Section 2.4, Theorem 2.21] Parts 2-5 of Theorem 2.21 are delegated to the Euclidean proofs in [44,45]. Since the Riemannian update rule involves the retraction and the parameter λ_n, the proof should at least indicate which statements are being transferred and how the 'real analytic-like' condition supplies the needed measure-zero preimage property.
  5. [Section 3.1, definition of weak shyness] The text says the new notion 'is also invariant under countable unions', but no proof or reference is given for this closure property within the paper. Please provide the argument or an explicit citation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central convergence and saddle-avoidance claims are derived from explicit dynamical-systems arguments and stated assumptions, not from their own conclusions; the notable gaps are mathematical proof gaps, not circular reductions.

full rationale

This is a pure mathematics paper with no fitted parameters, empirical predictions, or data-derived quantities, so the fitted-input and self-definitional patterns do not apply. The headline results (Theorems 1.2, 2.19, 2.20, 2.21, 3.4, 3.5) are proved from Armijo backtracking estimates, capture theorems based on Nash embedding and compactification in projective space, stable-center manifold theorems, and explicit assumptions (countable critical set, Lojasiewicz/Kurdyka-Lojasiewicz inequalities, Condition C, bounded Hessian on bounded sets). The new notions (strong local retraction, real analytic-like strong local retraction, weaker shyness) are introduced as definitions and then used in proofs; they do not by construction assert the target theorems. Where the author cites his own previous work ([44]-[47]), the citations import Euclidean-case arguments or published results, and the Riemannian/Banach conclusions still require new extension arguments; no central premise is justified solely by an overlapping-author citation. The paper itself flags a genuine limitation: global saddle avoidance for Simplified BNQN in Banach spaces is stated to require an infinite-dimensional version of Theorem 2.16 and is declared beyond the current paper. A non-circular proof gap exists in Theorem 2.20(iii): the text infers from local injectivity that E1 = union H^{-n}(F1) has Lebesgue measure zero, an inference that is not valid for general C1 locally injective maps; however the Euclidean estimates in the same proof appear to supply a local lower-Lipschitz bound that would repair the argument via Lemma 2.18, and the Riemannian case is only sketched as similar. These are correctness/completeness concerns, not circularity. No equation in the paper reduces to its own input by construction, and no prediction is a renamed fit.

Assumptions & free parameters 4 free parameters · 6 assumptions · 4 invented entities

The central claims rest on user-chosen line-search parameters, on standard geometric and dynamical-systems theorems, and on new technical conditions introduced in this paper. No constant is fitted to data, so circularity burden is low, but several of the new assumptions are strong and are only justified by examples or by deferred proofs.

free parameters (4)
  • alpha, beta, delta0 (backtracking line search parameters) = chosen in (0,1) and delta0 > 0
    Algorithm hyperparameters; theorem proofs require generic or random choice, especially in Theorem 3.4.
  • delta0,...,delta_k (random shifts in New Q-Newton methods) = random real numbers
    Randomness is used to ensure invertibility of Hessian shifts and measure-zero preimage properties in Theorem 2.16.
  • gamma_j normalization sequence = gamma0=0, gamma1=1, increasing to infinity, lim inf gamma_j/gamma_{j+1}>0
    Used in Riemannian New Q-Newton to keep steps inside the local retraction domain; not fitted to data.
  • r(x) and L(x), local radius and Lipschitz function = chosen continuous functions
    User-supplied estimates for local Lipschitz behavior; central to Local Backtracking GD.
assumptions (6)
  • standard math Nash isometric embedding theorem for finite-dimensional Riemannian manifolds
    Used in the proof of Theorem 2.15 to compare Riemannian and Euclidean metrics.
  • standard math Stable-Center (Left Stable) manifold theorem for Banach spaces
    Invoked in Theorem 3.2 to build local invariant manifolds near saddle points.
  • standard math Fubini-Tonelli theorem and measure-zero propagation
    Used in Theorem 2.16; the specific use of local injectivity to imply null preimage is not justified in the paper.
  • domain assumption Condition C on Banach-space cost functions
    Assumed in Theorems 3.4 and 3.5 so that weak cluster points of gradient-null sequences are critical points.
  • ad hoc to paper Real analytic-like strong local retraction
    New condition introduced in Definition 2.13, needed for global saddle avoidance of Riemannian New Q-Newton; existence is only illustrated by examples, not proven in general.
  • ad hoc to paper Weak shyness is closed under countable unions and preimages by finite-distortion maps
    Definition in Section 3.1; the proof relies on this behavior for exceptional sets.
invented entities (4)
  • Strong local retraction (r, R)
    purpose: Replaces global retraction to move along manifolds while preserving convergence arguments.
    New mathematical tool; no external falsifiable handle.
  • Real analytic-like strong local retraction
    purpose: Ensures zero sets of determinant functions have measure zero for global saddle avoidance.
    Technical assumption; existence is not established for all manifolds.
  • Weakly shy sets
    purpose: Provides a notion of null sets in Banach spaces that is preserved under preimages by finite-distortion maps.
    Generalizes shyness; no external evidence.
  • Simplified Backtracking New Q-Newton's method
    purpose: Avoids the spectral While loop in infinite dimensions and simplifies the Euclidean version.
    New algorithm variant; no external benchmark.

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Cite this review

Pith. "Pith review of Some iterative algorithms on Riemannian manifolds and Banach spaces with good global convergence guarantee." pith.science (2026). https://pith.science/paper/53EYEVVK

@misc{pith2026250522180,
  author       = {Pith},
  title        = {Pith review of: Some iterative algorithms on Riemannian manifolds and Banach spaces with good global convergence guarantee},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/53EYEVVK}},
  note         = {Machine review of arXiv:2505.22180}
}
abstract

In this paper, we introduce some new iterative optimisation algorithms on Riemannian manifolds and Hilbert spaces which have good global convergence guarantees to local minima. More precisely, these algorithms have the following properties: If $\{x_n\}$ is a sequence constructed by one such algorithm then: - Finding critical points: Any cluster point of $\{x_n\}$ is a critical point of the cost function $f$. - Convergence guarantee: Under suitable assumptions, the sequence $\{x_n\}$ either converges to a point $x^*$, or diverges to $\infty$. - Avoidance of saddle points: If $x_0$ is randomly chosen, then the sequence $\{x_n\}$ cannot converge to a saddle point. Our results apply for quite general situations: the cost function $f$ is assumed to be only $C^2$ or $C^3$, and either $f$ has at most countably many critical points (which is a generic situation) or satisfies certain Lojasiewicz gradient inequalities. To illustrate the results, we provide a nice application with optimisation over the unit sphere in a Euclidean space. As for tools needed for the results, in the Riemannian manifold case we introduce a notion of "strong local retraction" and (to deal with Newton's method type) a notion of "real analytic-like strong local retraction". In the case of Banach spaces, we introduce a slight generalisation of the notion of "shyness", and design a new variant of Backtracking New Q-Newton's method which is more suitable to the infinite dimensional setting (and in the Euclidean setting is simpler than the current versions).

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