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A Framework for Population-Based Stochastic Optimization on Abstract Riemannian Manifolds

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Pith's one-line read A population-based stochastic search can reach the global optimum of any compact Riemannian manifold in finitely many steps, using only local geometry and no ambient embedding.

desk verdict Impressive geometric framework for manifold population-based optimization, but the headline finite-step global convergence claim relies on an exploration assumption the implementation doesn't guarantee. read the letter →

arxiv 1908.06783 v3 pith:OS5ZW7FS submitted 2019-08-19 math.OC

Robert Simon Fong , Peter Tino This is my paper · ORCID
classification math.OC MSC 90C5690C1553B2062B10
keywords ManifoldOptimizationPopulation-basedStochasticDerivative-FreeinformationgeometrymixturedensitiesglobalconvergenceJacob'sladderRiemannian
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

This paper claims that a population-based stochastic search can solve black-box optimization problems on abstract Riemannian manifolds — curved spaces treated on their own terms, without an ambient Euclidean space to embed them in. The central result is that on connected compact manifolds the algorithm's explored region fills the entire manifold in finitely many steps, under the stated condition that each iteration contributes an exploration point on the boundary of the region explored so far; the global optimum is then reached in finite time rather than merely approached as a stationary-point limit. The route runs through information geometry: probability densities are inherited onto the manifold locally through orientation-preserving diffeomorphisms (the exponential map being the special case), then patched into mixture densities whose parameter space forms a product statistical manifold. That product structure yields monotone improvement of expected fitness and a first-principles derivation of the search-coefficient update. If the claims hold, manifold optimization no longer needs completeness, global parametrizations, or logarithms between arbitrary points — and problems on spaces such as Jacob's ladder, a surface of infinite genus with no polynomial-constraint description, become tractable.

What carries the argument

The load-bearing object is the family of mixture densities $\mathcal{L}_V$ over totally bounded subsets $V$ of the manifold. Each component is a locally inherited density, pulled back through an orientation-preserving diffeomorphism from a coordinate patch (the Riemannian exponential map is the special case); these are glued into mixtures $\tilde{p}(x) = \sum_{\alpha} \phi_\alpha \tilde{p}_\alpha(x)$ on a finite cover of $V$. Theorems 5.1–5.3 show $\mathcal{L}_V = S_0 \times \tilde{S}_1 \times \cdots \times \tilde{S}_\Lambda$ as a product statistical manifold, with the mixture divergence decomposing into a coefficient part and a component part — this is what decouples coefficient updates from component updates. On the coefficient simplex the paper installs the modified Fisher metric $G = F^{-1} + \epsilon_0 I$ (Equation (48)), whose natural-gradient fixed point, Equation (54), is proportional to the relative expected fitnesses $E_\alpha$; the practical update (Equation (40)) is the limiting case. Finally, the convergence argument of Theorem 8.1 is carried by the exploration distribution: sampling on geodesic-sphere boundaries and rejecting interior points places each new exploration centroid at least a uniform distance $j_M$ from all previous ones, and compactness then forces the explored region to exhaust $M$ in finitely many steps.

What would settle it

Run Extended RSDFO on the 2-sphere with the Section 9 exploration rule — sample points uniformly on geodesic-sphere boundaries and reject those lying in the interior of the explored region — and count the iterations in which every sampled point is rejected, so that no boundary centroid is added. The paper explicitly allows this fallback ('if all the sampled points are rejected, then Extended RSDFO will not sample new boundary points'), and it violates the hypothesis of Theorem 8.1 that at least one exploration point is generated whenever the boundary is nonempty; the empirical frequency of such stalled rounds at increasing manifold dimension would show whether the finite-step guarantee holds for the implemented algorithm or only for the idealized boundary oracle.

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Extended reading notes

Core claim

The central discovery is Theorem 8.1: on a compact connected Riemannian manifold $(M,g)$, if Extended RSDFO generates at least one exploration point from the boundary of the explored region whenever that boundary is nonempty, then the global convergence condition of Equation (57) holds within finitely many steps. Concretely, there is a finite integer $N$ such that $\sup_{\alpha \in \Lambda_N} E^\alpha_N = E^*$, the best attainable expected fitness, so the algorithm does not merely converge to a stationary point in the limit — the global optimum is attained in the explored region after finitely many iterations. The key structural claim supporting this is that the family of mixture densities $\mathcal{L}_V$ over a totally bounded subset $V$ of the manifold is a product statistical manifold of the mixture-coefficient simplex and the locally inherited component families, which lets the algorithm evolve mixture coefficients and component parameters independently while comparing solution quality across disjoint tangent spaces.

Load-bearing premise

The proof of Theorem 8.1 assumes that whenever the explored region is not yet the whole manifold, the algorithm can always create a new search point exactly on the boundary of that region, with every search ball at least a fixed positive radius — but the implemented exploration samples boundary points randomly and simply adds none if all samples fall inside the explored region, and under continuous sampling the probability of landing exactly on the boundary is zero.

Editorial extensions

If this is right

  • On any connected compact Riemannian manifold, Extended RSDFO's expected fitness is monotone non-decreasing across iterations (Proposition 6.1), and with boundary exploration it reaches the global optimum in finitely many steps rather than converging only to a stationary point (Theorem 8.1).
  • Because mixture coefficients and component parameters live on independent factors of a product statistical manifold, solution quality can be compared across disjoint tangent spaces — something the paper argues is impossible for single-centroid Riemannian SDFO methods such as Riemannian CMA-ES.
  • Extended RSDFO's computations are strictly local geodesic-ball computations, so the manifold need not be complete and no Riemannian logarithm map between arbitrary points is required; the paper shows the latter assumption is what cripples Riemannian PSO on large manifolds.
  • The experiments on the sphere, Grassmann manifolds, and Jacob's ladder indicate the method combines the global-exploration behaviour of Riemannian PSO with the local-optimum accuracy of Riemannian CMA-ES, at the price of extra function evaluations and boundary-point evaluations.
  • Jacob's ladder — a countably infinite connected sum of tori without a global ambient representation — constitutes a problem class that constraint-based Euclidean optimization cannot formulate, so any success there is evidence for intrinsic manifold optimization as such.

Reading between the lines

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

  • The finiteness argument is essentially topological: boundary centroids pinned at a uniform distance $j_M$ plus compactness forces exhaustion. It should transfer to any exploration rule that guarantees a point at least a fixed distance beyond the explored region — for example deterministic, low-discrepancy boundary sampling — which would also remove the measure-zero difficulty the paper's own rando
  • Reading the algorithm through the product structure $\mathcal{L}_V = S_0 \times \prod_\alpha \tilde{S}_\alpha$ suggests a design principle for population-based manifold optimizers: evolve coefficients by natural gradient on the simplex under $G = F^{-1} + \epsilon_0 I$ and evolve components separately. Deriving component-parameter updates from the same fixed-point condition would extend the paper'
  • A testable extension is to separate the two sources of difficulty the paper's Grassmann experiments conflate: the cost of estimating local component models in higher dimension, and the density of centroid coverage needed for exploration. Budgeting boundary samples rather than centroid count per iteration would isolate which one drives the sharp drop in success rate from $\mathrm{Gr}(2,4)$ to $\mat
  • The Jacob's ladder construction is reusable as a stress test for any intrinsic manifold optimizer: a countably infinite connected sum of tori with a glued exponential map is a manifold on which no ambient constraint formulation exists, so comparisons there measure genuine manifold behaviour rather than embedding artefacts.
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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

4 major / 4 minor

Summary. The manuscript develops an information-geometric framework for population-based stochastic optimization on Riemannian manifolds and introduces Extended RSDFO, an algorithm that combines locally inherited probability densities into mixture densities over totally bounded subsets of the search space. The paper claims that the expected fitness of Extended RSDFO improves monotonically (Proposition 6.1) and that the algorithm converges globally in finitely many steps on connected compact Riemannian manifolds (Theorem 8.1). It also reports experiments on the sphere, Grassmannian manifolds, and a novel Jacob's-ladder test problem, comparing Extended RSDFO with Riemannian trust-region, Riemannian CMA-ES, and Riemannian PSO.

Significance. If the convergence result held for the implemented algorithm, it would be a notable contribution: finite-step global convergence on general compact Riemannian manifolds is a strong guarantee for a population-based derivative-free method, and the product statistical-manifold construction (Theorem 5.1, Remark 5.4) is a useful geometric framework. The paper also introduces a genuinely non-embeddable test manifold, Jacob's ladder, which helps motivate intrinsically manifold-native optimization. The experimental study is extensive and the paper gives credit where due by providing explicit geometric derivations and a relatively reproducible setup. However, the advertised global-convergence claim is not established for the algorithm as implemented, and the geometric derivation of the mixture-coefficient update contains a metric inconsistency; the significance is therefore real but conditional on resolving these issues.

major comments (4)
  1. [Section 8, Theorem 8.1; Sections 6.3.4 and 9] Theorem 8.1 is proved under the hypothesis that, whenever the explored region W_k is not all of M, the algorithm generates at least one exploration point on the boundary of W_k, and the proof additionally asserts a uniform positive lower bound j_M on all geodesic-ball radii. Neither condition is guaranteed by Algorithm 3 as described. The exploration procedure in Section 6.3.4 samples from boundaries of individual geodesic spheres and rejects points lying in interiors (Equation (46)); Section 9 explicitly states that if all sampled points are rejected, no new boundary point is added. Thus the theorem is a conditional statement about an idealized boundary-exploration oracle, not about the implemented algorithm. The abstract's unconditional phrasing, 'converges globally eventually in finitely many steps on connected compact Riemannian manifolds', is therefore unsupported for the implemented method. The proof's assertion j_M > 0 also needs an explicit assumption, since j_x ≤ inj(x) and the algorithm could in principle choose radii with no uniform positive lower bound.
  2. [Section 7, Equations (48), (52)-(55)] The metric used in the natural-gradient computation is not consistently defined. Equation (48) defines G := F^{-1} + ε0 I, but the Riemannian metric immediately below is gξ(Y,Z) := Σ yα zα (φ + ε0), which corresponds to the diagonal matrix diag(φ + ε0). For the simplex Fisher information matrix F, the matrix F^{-1} is not diagonal, so its inverse is not diag(1/(φ + ε0)). The inverse metric used in Equation (52) is therefore not the inverse of the matrix defined in Equation (48). The fixed-point derivation in Equations (53)-(55) and the claimed recovery of Equation (40) in Remark 7.2 rely on this ambiguity, so the 'first principles' derivation of the mixture-coefficient update needs to be corrected.
  3. [Section 7, Remark 7.2; Section 6.3.2, Equation (40)] The claim that Equation (40) is 'rigorously derived from first principles' in Remark 7.2 is circular in the present form. Equation (40) is introduced before Section 7 as the definition of the update, and the natural-gradient fixed point derived in Section 7 is, by construction, proportional to the expected fitnesses Eα. The metric in Equation (48) is explicitly selected so that its natural-gradient fixed point favors the interior point with coordinates proportional to relative fitness (see the bullet list in Section 7). The derivation therefore shows consistency between the chosen metric and the pre-existing update rule, but it does not independently derive Equation (40). This is a framing and justification issue that should be corrected.
  4. [Sections 6.3 and 7.2] The paper states 'without loss of generality' that f can be made strictly positive by translation, but the mixture-coefficient updates in Equations (40), (54), and (55) are not invariant under adding a constant to f. If f is replaced by f + C, then every Eα changes to Eα + C, and the normalized coefficients change unless C = 0. The same translation dependence appears in the minimization counterpart in Equation (56). This invalidates the 'without loss of generality' claim and further weakens the natural-gradient justification of the update rule.
minor comments (4)
  1. [Global] The manuscript contains many typographical artifacts and misspellings, including '/f_irst', 'de/f_ined', 'neighhbourhood', 'experiement', and 'Jaccob's ladder'; please proofread carefully.
  2. [Section 9, Tables 4-6] The relaxed exploration parameter ϵ_b is introduced in the experiments but is not analyzed in the theoretical sections; please clarify how sampling from boundaries of smaller geodesic spheres (ϵ_b < 1) interacts with the convergence assumptions in Theorem 8.1.
  3. [Section 6.3.5] The theoretical termination criterion (boundary of the explored region empty) differs from the practical termination criterion used in the experiments (all local RSDFO streams terminate); the paper should state explicitly that the convergence analysis applies only to the former.
  4. [Section 9.2] In the experimental setup, the local RSDFO core is described with budgets of 'parents' and 'offsprings', but these terms are not defined in the RSDFO framework of Section 3.1, which makes the setup harder to reproduce.

Circularity Check

1 steps flagged · score 4.0 of 10

The main circularity is in Section 7: the modified metric is chosen so that its natural-gradient fixed point equals the already-adopted proportional-fitness update (Eq. 40), which is then called a first-principles derivation. Theorem 8.1 is conditional and independent.

  1. self definitional [Section 7, Equation (48) through Remark 7.2; compare Equation (40) in Section 6.3.2]
    "Natural gradient ascent on the closure S0 under the modified metric, as we will discuss in the subsequent subsection, favours the interior point of the simplex with coordinates proportional to the relative weights of the vertices. ... if ... ϵ0 ... sufficiently small ... we retrieve Equation (40). This fixed point therefore directly reflects the relative expected fitness ..."

    Equation (40) was introduced directly as the coefficient update: 'it is natural to assign to the individual mixture coefficients φα a value proportional to Eα. In particular, in line 7 ... φ^{k+1}_α = E_α / Σ E_α.' The metric G = F^{-1} + ϵ0 I was then introduced with the stated purpose that natural gradient ascent on the simplex 'favours the interior point ... with coordinates proportional to the relative weights of the vertices.' Under this metric the stationarity condition (Eq. 53) is E_α/(φ_α + ϵ0) = c, which rearranges to φ_α ∝ E_α and, in the small-ϵ0 limit, is exactly Equation (40). Thus the 'derived' fixed point is not an independent consequence of information geometry; it is a restatement of the heuristic update that motivated the choice of metric.

full rationale

The paper's central information-geometric constructions (Sections 4 and 5) are largely self-contained: the pulled-back dualistic structures and the product statistical manifold result for mixture densities are derived from stated definitions, and no load-bearing self-citation chain appears. Theorem 8.1 is a conditional statement: if the algorithm generates a boundary exploration point whenever the explored region is a proper subset, compactness forces exhaustion of M in finitely many steps. That argument is independent, though the abstract omits the boundary-generation hypothesis. The implementation in Section 9 explicitly admits that the acceptance-rejection exploration can fail to produce boundary points ('if all the sampled points are rejected, then Extended RSDFO will not sample new boundary points'), so the theorem's hypothesis is not guaranteed by the implemented algorithm; this is a correctness gap rather than a circularity. The significant circular step is in Section 7: the modified metric is chosen precisely so that its natural-gradient fixed point equals the proportional-fitness update already adopted in Equation (40), and Remark 7.2 then labels this retrospective equivalence a first-principles derivation. Because that derivation is forced by the metric choice rather than by independent principles, the score is 4; the remaining convergence theorem and experimental comparisons still carry independent content.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard Riemannian and information-geometric background, on domain assumptions about the objective and manifold, and on an idealized boundary-exploration condition that is not guaranteed by the implemented algorithm. Several hyperparameters are hand-chosen in experiments. No new physical entities are introduced.

free parameters (5)
  • epsilon_0 in modified simplex metric G = F^{-1} + epsilon_0 I = not specified, arbitrarily small positive
    Introduced in Equation (48) to define a metric whose natural-gradient fixed point Equation (54) equals the fitness-proportional update Equation (40) in the small-epsilon_0 limit.
  • N_random and N_cull = 2 and 2 on S2/Grassmann; 6 and 3 on Jacob's ladder
    Chosen by hand; they control the number of new and retained centroids and therefore the exploration-exploitation balance.
  • exploration rate tau(k) = printed as 0.6 e^{0.015 k}, contradicting stated non-increasing requirement
    Controls the mixture between elitist sampling and exploration; the formula in Section 9 is inconsistent with its stated purpose.
  • exploration radius factor epsilon_b = 1.0, 0.4, or 0.5 in different experiments
    Scales the geodesic balls used for boundary exploration and changes performance, e.g., global-minimum rate on S2 rises from 128/200 to 154/200.
  • local RSDFO budgets and Monte Carlo sample sizes = parents/offsprings and MC samples listed per experiment
    Set per benchmark and affect the stochastic estimates of expected fitness used by the algorithm.
assumptions (5)
  • domain assumption M is a connected and orientable Riemannian manifold, and compact for the convergence theorem.
    Assumed in Sections 4 and 8; orientability is needed for volume-form constructions and compactness is used in Theorem 8.1.
  • domain assumption The objective f is bounded above, strictly positive by translation, and attains a global optimum.
    Used in Section 8 and Remark 7.3; positivity is required for the fitness-proportional mixture coefficient update.
  • ad hoc to paper Exploration generates a centroid exactly on the boundary of the explored region whenever the boundary is nonempty, with all geodesic ball radii uniformly bounded below by a positive constant.
    The proof of Theorem 8.1 relies on this. The acceptance-rejection procedure in Section 6.3.4 and the simplified implementation in Section 9 can reject all sampled points and add no boundary centroid.
  • domain assumption Local component families satisfy Conditions 1 and 2: distinct proper support and no functional dependency between components.
    Stated in Section 5.3.1; these conditions make the mixture parametrization injective and an immersion. Condition 1 can be met by choosing the open cover, and Condition 2 is automatic for unconstrained mixtures.
  • standard math Standard statistical-manifold background is taken as given: Fisher metric, dual connections, Bregman divergence, and the existence of a divergence generating a dualistic structure.
    Section 4.1 and Theorem 5.1 rely on results from Amari and Nagaoka, Eguchi, and Matumoto without reproving them.

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Pith. "Pith review of A Framework for Population-Based Stochastic Optimization on Abstract Riemannian Manifolds." pith.science (2026). https://pith.science/paper/OS5ZW7FS

@misc{pith2026190806783,
  author       = {Pith},
  title        = {Pith review of: A Framework for Population-Based Stochastic Optimization on Abstract Riemannian Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OS5ZW7FS}},
  note         = {Machine review of arXiv:1908.06783}
}
read the original abstract

We present Extended Riemannian Stochastic Derivative-Free Optimization (Extended RSDFO), a novel population-based stochastic optimization algorithm on Riemannian manifolds that addresses the locality and implicit assumptions of manifold optimization in the literature. We begin by investigating the Information Geometrical structure of statistical model over Riemannian manifolds. This establishes a geometrical framework of Extended RSDFO using both the statistical geometry of the decision space and the Riemannian geometry of the search space. We construct locally inherited probability distribution via an orientation-preserving diffeomorphic bundle morphism, and then extend the information geometrical structure to mixture densities over totally bounded subsets of manifolds. The former relates the information geometry of the decision space and the local point estimations on the search space manifold. The latter overcomes the locality of parametric probability distributions on Riemannian manifolds. We then construct Extended RSDFO and study its structure and properties from a geometrical perspective. We show that Extended RSDFO's expected fitness improves monotonically and it's global eventual convergence in finitely many steps on connected compact Riemannian manifolds. Extended RSDFO is compared to state-of-the-art manifold optimization algorithms on multi-modal optimization problems over a variety of manifolds. In particular, we perform a novel synthetic experiment on Jacob's ladder to motivate and necessitate manifold optimization. Jacob's ladder is a non-compact manifold of countably infinite genus, which cannot be expressed as polynomial constraints and does not have a global representation in an ambient Euclidean space. Optimization problems on Jacob's ladder thus cannot be addressed by traditional (constraint) optimization methods on Euclidean spaces.

Figures

Figures reproduced from arXiv: 1908.06783 by the authors.

Figure 1
Figure 1. Illustration of parallel transport from x to y within normal neighbourhood Nx of x, described in Section 2.1. The orthonormal basis at x is parallel transported to an orthonormal basis at y. Finally, when restricting our attention to a metric ball within the pre-imagie of the normal neighbourhood, we retrieve a metric ball in M under normal coordinates. Denition 2.1. Given a point x ∈ M, the injectivity radius at x… view at source ↗
Figure 2
Figure 2. Illustration of Riemannian adaptation of optimization algorithms. For each iteration, Riemannian [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Illustration of an iteration of RSDFO. For each iteration, the algorithm initiates with [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: A summary of the induced statistical dualistic geometry of locally inherited parametrized densities [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: 19It is worth noting that for this illustration ( [PITH_FULL_IMAGE:figures/full_fig_p037_5.png]
Figure 5
Figure 5. Figure 5: Illustration (from left to right) of the “full" evolutionary step of Extended RSDFO. At each iteration, [PITH_FULL_IMAGE:figures/full_fig_p038_5.png]
Figure 6
Figure 6. Figure 6: Simplicial illustration of the simplices in evolutionary step of Extended RSDFO (Algorithm 3). [PITH_FULL_IMAGE:figures/full_fig_p043_6.png]
Figure 7
Figure 7. Figure 7: An simplied illustration of the exploration distribution sampling using acceptance-rejection [PITH_FULL_IMAGE:figures/full_fig_p046_7.png]
Figure 8
Figure 8. Figure 8: A simplied illustrated of the mixture distribution in the special case if the search region is su - [PITH_FULL_IMAGE:figures/full_fig_p048_8.png]
Figure 9
Figure 9. Figure 9: Simplicial illustration of the evolutionary step of Algorithm 3. The number on the arrows cor [PITH_FULL_IMAGE:figures/full_fig_p051_9.png]
Figure 10
Figure 10. Figure 10: A demonstration of natural gradient ascent under the metric [PITH_FULL_IMAGE:figures/full_fig_p054_10.png]
Figure 11
Figure 11. Figure 11: Heat map of f on S 2 Injectivity radius on S 2 is inj(x) = π for all x ∈ S 2 [67], and the Riemannian exponential map of S 2 is given by [5]: expx (v) = γv(t)| t=1 = cos (||v|| · t) x + sin (||v|| · t) v ||v|| [PITH_FULL_IMAGE:figures/full_fig_p063_11.png]
Figure 12
Figure 12. Figure 12: Experimental results of one of the experiments on [PITH_FULL_IMAGE:figures/full_fig_p065_12.png]
Figure 13
Figure 13. Figure 13: Illustration of objective function f on GrR(p, n), notice the x-axis is given by the Riemannian distance dGr of x relative to In×p. The injectivity of Grassmannian manifolds GrR(p, n) is π 2 for all x ∈ GrR(p, n) [58, 67], and the Riemannian exponential map of is give…
Figure 14
Figure 14. Figure 14: Experimental results of one of the experiments on [PITH_FULL_IMAGE:figures/full_fig_p068_14.png]
Figure 15
Figure 15. Figure 15: Illustration of Jacob’s ladder surface as an innite connected sum of tori. [PITH_FULL_IMAGE:figures/full_fig_p069_15.png]
Figure 16
Figure 16. Figure 16: Illustration of minor and major rotational angles in the Jacob’s ladder. The vertical smaller [PITH_FULL_IMAGE:figures/full_fig_p070_16.png]
Figure 17
Figure 17. Figure 17: fG: the “global" part of the objective function f on Jacob’s ladder. 0 2 20 40 2 60 Objective value of fL 80 \varphi 100 120 0 0 [PITH_FULL_IMAGE:figures/full_fig_p071_17.png]
Figure 19
Figure 19. Figure 19: Experimental results of one of the experiments on Jacob’s ladder. [PITH_FULL_IMAGE:figures/full_fig_p073_19.png]
Figure 20
Figure 20. Figure 20: Illustration of the “trap” of fL. The local optimal angles (θ ∗, ϕ∗) are marked by the points labelled with ×. The objective value increases along dotted lines on T2 (as well as T1 and the other tori), as the “objective land￾scape", described by [PITH_FULL_IMAGE:figu…

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