{"id":"6c030b43-73e4-4090-9350-08b93b66d1df","arxiv_id":"1908.06783","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Extended RSDFO is a population-based optimizer on Riemannian manifolds that combines local searches into mixtures to escape local regions, with a finite-step global convergence theorem on compact manifolds.","lead":"This paper builds a new family of optimization algorithms that search over curved spaces called Riemannian manifolds by combining several local searches into one global mixture. It claims the method improves its expected score at every step and, on compact curved spaces, eventually explores the whole space and reaches a global optimum in finitely many steps.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 8.1's boundary-exploration hypothesis is not guaranteed by the implemented acceptance-rejection exploration, so the abstract's unconditional finite-step global convergence claim is not established.","rationale":"The reader's weakest_assumption identifies exactly this gap: Theorem 8.1 assumes exact generation of a boundary exploration point, while the implemented exploration can fail to produce any such point. My analysis of the proof confirms that the boundary-exploration condition is the sole mechanism driving finite-step exhaustion of the manifold: without it, the sequence of explored regions can stall, and the finite-step global convergence conclusion collapses. The theorem's proof is internally coherent conditional on that assumption; the concern is not a flaw in the theorem but a mismatch between the theorem's hypothesis and the algorithm's implemented behavior, compounded by the abstract presenting the result unconditionally. This justifies a CONDITIONAL verdict: the core theoretical construction has merit and the conditional theorem is likely correct, but the paper must either amend the algorithm to guarantee boundary exploration or clearly state that the convergence result applies only to the idealized version satisfying the hypothesis. A concrete test can settle whether the implemented exploration actually satisfies the hypothesis in generic and adversarial configurations.","tokens_in":62418,"tokens_out":8189,"duration_ms":88690,"concrete_test":"Run Extended RSDFO on S^2 with the exploration step exactly as in Section 9.2 (sampling 50 points on 5 randomly chosen centroids with epsilon_b = 1). At each iteration, compute whether ∂W_k is nonempty and whether any accepted boundary point was generated. Start from a configuration where W_k is a union of two closed geodesic balls that meet at a single point, so ∂W_k is that single point. If in any run the algorithm fails to generate a boundary point while ∂W_k is nonempty, the theorem's hypothesis is not guaranteed by the implementation. A sharper analytical check: compute the measure of ∂W_k under the uniform distribution on the union of sphere boundaries for this configuration; if it is zero, acceptance-rejection cannot satisfy the assumption with positive probability.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that Extended RSDFO converges globally in finitely many steps on compact connected Riemannian manifolds. The proof of Theorem 8.1 depends critically on the assumption that, whenever the explored region W_k is not all of M, the algorithm generates at least one exploration point on ∂W_k. This is the mechanism that forces the explored region to exhaust M in finitely many steps. However, the practical exploration step described in Section 6.3.4 and implemented in Section 9 does not guarantee this. The exploration distribution is defined as U = unif(∂W_k), but points are generated by sampling from the boundaries of individual geodesic spheres and then rejecting those that lie in the interior of any explored ball (Equation (46)). The paper explicitly states: \"if all the sampled points are rejected, then Extended RSDFO will not sample new boundary points\" (Section 9, setup). For configurations where ∂W_k has measure zero with respect to the sampling distributions (e.g., two geodesic balls meeting at a single point), the acceptance-rejection procedure almost surely produces no boundary point, yet ∂W_k is nonempty. Thus the theorem's hypothesis is not satisfied by the algorithm as implemented. The theorem itself is a valid conditional statement, but the abstract presents the convergence result without this essential caveat, so the unconditional claim for the implemented algorithm is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":62714,"tokens_out":12476,"duration_ms":141699,"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":[{"comment":"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.","section":"Section 8, Theorem 8.1; Sections 6.3.4 and 9"},{"comment":"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.","section":"Section 7, Equations (48), (52)-(55)"},{"comment":"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.","section":"Section 7, Remark 7.2; Section 6.3.2, Equation (40)"},{"comment":"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.","section":"Sections 6.3 and 7.2"}],"minor_comments":[{"comment":"The manuscript contains many typographical artifacts and misspellings, including '/f_irst', 'de/f_ined', 'neighhbourhood', 'experiement', and 'Jaccob's ladder'; please proofread carefully.","section":"Global"},{"comment":"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.","section":"Section 9, Tables 4-6"},{"comment":"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.","section":"Section 6.3.5"},{"comment":"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.","section":"Section 9.2"}],"recommendation":"major_revision","confidential_remarks":"The main editorial risk is that the advertised finite-step global convergence claim is presented as a property of the implemented algorithm, while the proof relies on a boundary-exploration assumption that the implementation does not guarantee. The paper's geometric framework and experiments are valuable, but the convergence theorem should either be stated as conditional or the exploration mechanism should be strengthened to satisfy the hypothesis. The metric inconsistency in Section 7 should also be resolved before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper builds something genuinely useful: a rigorous information-geometric framework for mixture densities on Riemannian manifolds, using orientation-preserving bundle morphisms to inherit statistical structure locally and then glue it into a product manifold over totally bounded subsets. That construction is novel and likely to outlive the algorithm itself. The monotonic expected-fitness result (Proposition 6.1) is sound under the stated elitist selection, and the multi-centroid idea with boundary exploration is a sensible way to escape local traps. The Jacob's ladder experiment is a thoughtful piece of motivation, even if the manifold is only represented locally.\n\nThe soft spot is exactly where the stress-test lands. Theorem 8.1 is a valid conditional statement: if the algorithm generates a boundary point whenever the explored region is not all of M, then it exhausts a compact connected manifold in finitely many steps. But the implemented exploration samples from geodesic spheres and rejects points in the interior of any explored ball. When the boundary of the explored region is lower-dimensional (e.g., two tangent balls), the acceptance-rejection procedure almost surely misses it. The paper's own text says that if all sampled points are rejected, no new boundary point is added. So the hypothesis of Theorem 8.1 is not guaranteed by the implementation, and the abstract's unconditional claim of finite-step global convergence is not established. This is fixable: either prove a positive-measure condition on the boundary, or explicitly add boundary points when they exist, or weaken the theorem and abstract.\n\nThe reverse-engineered metric in Section 7 is a minor circularity. Defining G = F^{-1} + epsilon_0 I and then showing that natural gradient has the fitness-proportional fixed point is fine as a consistency check, but calling it a derivation from first principles overstates the case. The experiments also lack error bars and code, so the comparative claims are suggestive rather than conclusive.\n\nNone of this invalidates the core construction. The convergence theorem is true under its assumption; the problem is the disconnect between that assumption and the practical sampling. This paper deserves a serious referee, but the authors should be asked to reconcile the theorem with the implementation and to tone down the abstract. I would bring it to a reading group if people care about manifold optimization or information geometry, and I would not cite it in its current form.\n\nRecommendation: send to peer review, with major revision expected.","headline":"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.","tokens_in":63217,"tokens_out":2667,"would_cite":false,"duration_ms":29766,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C56","90C15","53B20","62B10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Manifold Optimization","Population-based Stochastic Optimization","Derivative-Free Optimization","information geometry","mixture densities","global convergence","Jacob's ladder","Riemannian geometry"],"falsifier":"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.","tokens_in":62179,"feed_emoji":"🎯","tokens_out":19656,"duration_ms":166896,"temperature":0.7,"pith_summary":"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.","feed_headline":"Reaches the global optimum on compact curved spaces in finite steps","feed_subtitle":"The search expands from local patches until a compact curved space is fully covered, then locks onto the best region.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Riemannian CMA-ES instantiation of the RSDFO core and the main model-based baseline the algorithm must beat on the sphere and Grassmann experiments.","marker":"[24]"},{"why":"Provides the dualistic geometry of statistical manifolds — Fisher metric, conjugate connections, canonical divergence — on which the whole mixture-density framework rests.","marker":"[6]"},{"why":"Describes the intrinsic construction of probability distributions on complete Riemannian manifolds via the exponential map that Section 4.2 extends to general orientation-preserving diffeomorphisms.","marker":"[55]"},{"why":"Defines the Riemannian trust-region baseline used in the experiments.","marker":"[2]"},{"why":"Defines the Riemannian PSO baseline whose arbitrary-pair logarithm-map requirement the paper argues is unnecessary.","marker":"[16]"},{"why":"Supplies the global-convergence condition (Equation (57)) and the same-support argument that the mixture framework adapts to populations spread across tangent spaces.","marker":"[72]"},{"why":"Describes the Riemannian adaptation principle (exponential map or retraction, local tangent-space computation) that the paper identifies as the locality bottleneck it overcomes.","marker":"[5]"},{"why":"Provides the Riemannian-geometry facts — geodesic balls are metric balls, injectivity radius, compactness — used in the finite-step exhaustion proof of Theorem 8.1.","marker":"[44]"},{"why":"Underlies the embedding discussion that motivates optimization on abstract manifolds without ambient Euclidean constraints.","marker":"[68]"},{"why":"Gives the natural-gradient rule whose fixed point on the coefficient simplex yields the mixture-coefficient update.","marker":"[7]"}],"fun_headline_variants":["Extended RSDFO reaches global optimum on compact manifolds in finite steps","Finite-step global optimum on compact manifolds via population-based RSDFO","Global optimum on compact curved spaces in finitely many steps via Extended RSDFO","Compact manifold global optimum in finite steps: Extended RSDFO"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Extended RSDFO reaches global optimum on compact manifolds in finite steps","Finite-step global optimum on compact manifolds via population-based RSDFO","Global optimum on compact curved spaces in finitely many steps via Extended RSDFO","Compact manifold global optimum in finite steps: Extended RSDFO"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001704,"raw_usage":{"total_tokens":6791,"prompt_tokens":1031,"completion_tokens":5760,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":647,"completion_tokens_details":{"reasoning_tokens":5680}},"tokens_in":647,"tokens_out":5760,"duration_ms":39412,"temperature":1.0,"reasoning_tokens":5680,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:35:50.772473+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"The cma-es on rieman- nian manifolds to reconstruct shapes in 3-d voxel images","cited_arxiv_id":null,"evidence_quote":"Supplies the Riemannian CMA-ES instantiation of the RSDFO core and the main model-based baseline the algorithm must beat on the sphere and Grassmann experiments."},{"cited_title":"Methods of Information Geometry","cited_arxiv_id":null,"evidence_quote":"Provides the dualistic geometry of statistical manifolds — Fisher metric, conjugate connections, canonical divergence — on which the whole mixture-density framework rests."},{"cited_title":"Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements","cited_arxiv_id":null,"evidence_quote":"Describes the intrinsic construction of probability distributions on complete Riemannian manifolds via the exponential map that Section 4.2 extends to general orientation-preserving diffeomorphisms."},{"cited_title":"Trust-region methods on Rieman- nian manifolds","cited_arxiv_id":null,"evidence_quote":"Defines the Riemannian trust-region baseline used in the experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Riemannian PSO baseline whose arbitrary-pair logarithm-map requirement the paper argues is unnecessary."},{"cited_title":"On the convergence of a class of estimation of distribution algorithms","cited_arxiv_id":null,"evidence_quote":"Supplies the global-convergence condition (Equation (57)) and the same-support argument that the mixture framework adapts to populations spread across tangent spaces."},{"cited_title":"Optimization algorithms on matrix manifolds","cited_arxiv_id":null,"evidence_quote":"Describes the Riemannian adaptation principle (exponential map or retraction, local tangent-space computation) that the paper identifies as the locality bottleneck it overcomes."},{"cited_title":"Riemannian manifolds: an introduction to curvature","cited_arxiv_id":null,"evidence_quote":"Provides the Riemannian-geometry facts — geodesic balls are metric balls, injectivity radius, compactness — used in the finite-step exhaustion proof of Theorem 8.1."},{"cited_title":"The self-intersections of a smooth n-manifold in 2n-space","cited_arxiv_id":null,"evidence_quote":"Underlies the embedding discussion that motivates optimization on abstract manifolds without ambient Euclidean constraints."},{"cited_title":"Natural gradient works efficiently in learning","cited_arxiv_id":null,"evidence_quote":"Gives the natural-gradient rule whose fixed point on the coefficient simplex yields the mixture-coefficient update."}],"review_version":1}