{"id":"2c02bc78-12e3-4df8-b1ee-a1c03b51d9a0","arxiv_id":"2507.16055","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"An intrinsic Riemannian proximal gradient method is shown to converge sublinearly on geodesically convex problems and linearly on strongly convex problems over Hadamard manifolds.","lead":"The paper presents a new optimization algorithm for nonsmooth convex problems on curved spaces, using a proximal map defined directly on the manifold. It claims convergence rates matching Euclidean proximal gradient methods and tests the algorithm on matrix and hyperbolic-space problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.7's linear-rate proof uses inequality (16), which is derived under g geodesically convex (Assumption 4.3); the theorem is stated only under Assumption 4.1 plus Assumption 4.6, so the linear-rate claim lacks a stated hypothesis.","rationale":"The reader identified exactly the load-bearing weakness: Theorem 4.7's proof cites equation (16), which was derived in Theorem 4.4 under Assumption 4.3 (g geodesically convex), and Theorem 4.7 omits that assumption. The central claims of the paper are the O(1/k) convex rate and the linear strongly-convex rate; the linear rate is a headline result and its proof is not supported by the hypotheses stated for the theorem. The issue is not merely cosmetic: if Assumption 4.6 is used to imply f itself is geodesically convex, a repaired proof is plausible, but the manuscript does not supply that argument and explicitly builds (16) on g's convexity. The secondary concerns about Lemma 3 — the unstated existence of a minimizer of g and the unstated finite diameter of the initial sublevel set — are also genuine: Assumption 4.1(v) guarantees arg min f is nonempty, not arg min g, and diam(L_{p(0)}) appears without a hypothesis ensuring it is finite. These strengthen the case that the proof structure does not match the theorem statements. I agree with REJECT because the paper's central correctness claims are not established under the stated assumptions. I nevertheless credit the paper's independent strengths: the derivation of the hyperbolic l1 proximal map in Theorems 6.1–6.2 is concrete and checkable, the experiments are reproducible in Julia/Manopt.jl, and the convex-case analysis may be salvageable with additional assumptions. Those strengths do not outweigh the missing-hypothesis gap in the main strong-convexity theorem.","tokens_in":23333,"tokens_out":2618,"duration_ms":28517,"concrete_test":"Independently re-derive inequality (16) in the proof of Theorem 4.4 starting from p(n+1) = prox_{lambda h}(z(p(n))) using only f = g + h strongly geodesically convex and h geodesically convex, with no convexity assumption on g. If the key step minimizing over the geodesic between p(n) and p* requires g's convexity (as the paper's proof of (16) does), then Theorem 4.7 is unproven as stated. Optionally, run a one-dimensional Hadamard example M = R with g(x) = -x^2 on a bounded geodesically convex domain and h(x) = C x^2 with C large enough that f is strongly convex, and compare the observed convergence factor with the claimed linear rate; however, the analytical re-derivation is the decisive check.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim includes Theorem 4.7, the linear convergence rate for strongly geodesically convex objectives. Its proof begins by citing equation (16), which is derived inside Theorem 4.4's proof. That derivation invokes geodesic convexity of f to restrict the min over p to the geodesic between p(n) and p*, and in Theorem 4.4 this convexity comes from Assumption 4.3(i), 'g is geodesically convex', together with h convex in Assumption 4.1(iii). Theorem 4.7 is stated under Assumption 4.1 and Assumption 4.6 only; Assumption 4.3 is not restated. Strong convexity of f does not by itself supply the g-convexity used in the displayed derivation: the proof chain in Theorem 4.4 explicitly uses g's convexity to justify the step leading to (16), and Theorem 4.7 does not assume it. One could repair the proof by observing that Assumption 4.6 (f strongly geodesically convex) implies f is geodesically convex and re-deriving (16) for f directly, but the proof as written does not take that route; it literally inherits an inequality whose stated provenance is Assumption 4.3. The reader's secondary concerns are also valid: Lemma 3 requires q* in arg min g and a finite diam(L_{p(0)}), neither of which follows from Assumption 4.1(v), which only asserts that arg min f is nonempty. These assumptions appear in the proof but are absent from the standing assumptions and from the theorem statements that use R and R_alpha. The mismatch in Theorem 4.7 is the single most load-bearing concern because the linear rate is a headline contribution and the proof is unsupported under the stated hypotheses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an intrinsic Riemannian proximal gradient method (CRPG) on Hadamard manifolds for minimizing f = g + h, where g is L-smooth and h is convex and possibly nonsmooth, using the manifold proximal map for the nonsmooth step. The main theoretical contributions are an O(1/k) sublinear convergence rate in function values for geodesically convex objectives (Theorem 4.4), an O(1/ε) iteration complexity for reaching ε-stationary points (Theorem 4.5), and a linear convergence rate for strongly geodesically convex objectives (Theorem 4.7). The paper also derives Riemannian generalizations of the Euclidean prox-grad inequalities and reports numerical experiments on SPD matrices and hyperbolic spaces, comparing constant and backtracking step sizes against existing algorithms.","tokens_in":23683,"tokens_out":17714,"duration_ms":203371,"significance":"If the main results are correct, the paper offers a genuinely intrinsic Riemannian proximal gradient method and a useful generalization of the fundamental prox-grad inequality, with rates that account for curvature through the quantity ζ1,κl. The numerical experiments are well chosen and support the practical value of the method. However, the theoretical core currently has several load-bearing hypothesis gaps: Theorem 4.7 inherits an inequality proved under an assumption it does not state, and the main rates depend on finiteness and existence conditions that are not part of the standing assumptions. These issues are fixable within the scope of the manuscript, so the paper does not warrant outright rejection, but it needs a substantial revision.","major_comments":[{"comment":"The proof of Theorem 4.7 begins by invoking inequality (16), which is established in the proof of Theorem 4.4 under Assumption 4.3(i), i.e., g geodesically convex. Theorem 4.7 is stated only under Assumptions 4.1 and 4.6, and µf-strong convexity of f does not imply convexity of g. The derivation of (16) uses the convexity of g in an essential way, when g(p(n)) is replaced by g(p) − (grad g(p(n)), log_{p(n)} p); this step has no counterpart under the stated assumptions. The linear rate (25) is therefore unsupported as written. The theorem should either explicitly add the assumption that g is geodesically convex, or the proof should supply a direct derivation of (16) that does not rely on convexity of g.","section":"§4.2, Theorem 4.7"},{"comment":"Lemma 3 requires the existence of q* ∈ arg min_M g and a finite diameter of the sublevel set L_{p(0)}, but neither condition is stated in Assumptions 4.1 or 4.3; Assumption 4.1(v) only guarantees that the minimizer set of f is nonempty. Moreover, the set L_{p(0)} is never defined in the paper. The quantities R := diam(L_{p(0)}) and Rα appear in the central rates (13), (14), (22), and (25), so if L_{p(0)} is unbounded or g has no minimizer, the rates become vacuous or the bound (11) is undefined. These are load-bearing hypotheses and should be stated explicitly, with L_{p(0)} properly defined.","section":"§4, Lemma 3 and Theorem 4.4"},{"comment":"The proof of the ε-stationarity complexity invokes Lemma 4.8 of the authors' prior nonconvex paper (Bergmann, Jasa, John, Pfeffer, 2025) without stating the lemma or its hypotheses. The paper itself notes that the step-size conditions in that lemma differ from those used here and asserts that the result remains valid because sufficient decrease is already guaranteed by Lemma 1; this assertion is not demonstrated. Since this lemma is the bridge from the decrease bound (24) to the subgradient-norm bound, the complexity claim in (22) is not self-contained. The lemma should be stated and its conditions verified under Assumptions 4.1 and 4.3.","section":"§4.1, Theorem 4.5"}],"minor_comments":[{"comment":"A Hadamard manifold is a complete, simply connected manifold with nonpositive sectional curvature, not nonnegative as written. This appears to be a typo but should be corrected because the subsequent analysis and the experiments rely on nonpositive curvature.","section":"Assumption 4.1(i)"},{"comment":"Please define L_{p(0)} explicitly as the sublevel set (e.g., {p ∈ M : f(p) ≤ f(p(0))}) and state the boundedness condition needed for the diameter R = diam(L_{p(0)}) to be finite.","section":"§4.1, before Lemma 3"},{"comment":"The statement uses a bound g+ ≥ ∥grad g(p)∥ for all p ∈ L_{p(0)}; this should be listed as an explicit assumption, since L_{p(0)} is not known to be compact under the standing assumptions.","section":"Theorem 4.5"}],"recommendation":"major_revision","confidential_remarks":"This is a borderline paper: the core method and the geometric inequalities are valuable, but the proof of the linear-rate theorem is missing a stated hypothesis, and the main rates depend on hidden finiteness/existence assumptions. These issues appear fixable in revision; I would not recommend rejection unless the authors cannot resolve the Theorem 4.7 hypothesis gap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real contribution that is not ready as stated. The intrinsic proximal gradient update — prox of h composed with a gradient step on the manifold — is a genuine novelty relative to Huang–Wei and Chen et al., and the prox-grad inequalities in Section 5 are a credible generalization of Beck's Euclidean toolkit, with an honest discussion of why they do not telescope on Hadamard manifolds. The derivation of the ℓ1 proximal map on hyperbolic space is a practical spin-off worth keeping. Citations to prior work are accurate; the distinction from tangent-space and embedding formulations is real.\n\nThe problems are concentrated in the two headline theorems. Theorem 4.4 depends on Lemma 3, which requires q* in arg min g and finite diam(L_{p(0)}). Neither is anywhere in Assumption 4.1, which only asserts arg min f is nonempty. A convex L-smooth g can lack a minimizer, and sublevel sets can be unbounded even when minimizers exist. The rate expressions contain R and R_alpha; without finiteness they are vacuous. This is a missing-hypothesis error, not a subtle one.\n\nTheorem 4.7 is worse. Its proof opens by citing inequality (16), which was derived inside Theorem 4.4 under Assumption 4.3 (g geodesically convex). Theorem 4.7 is stated only under Assumption 4.6 (f strongly convex). Strong convexity of f does not imply g is convex, so the proof has no license to use (16). You could fix the statement by adding Assumption 4.3, but that changes the theorem. As written, the linear-rate claim is unsupported.\n\nThe stress-test note is right on both counts. The paper's central ideas are not wrong — the algorithm is plausible and the proofs probably go through with the additional assumptions spelled out — but the theorems as stated do not follow from their hypotheses. That is a load-bearing flaw in the presentation, not a cosmetic one.\n\nWho gets value: researchers in Riemannian optimization, especially those working on splitting methods or proximal maps on Hadamard manifolds. It deserves a serious referee because the method is novel and the gaps are the sort a careful revision can close. I would send it out, but the report should insist on corrected theorem statements and explicit boundedness/convexity assumptions. I would not cite it in its current form.","headline":"Genuinely new intrinsic proximal-gradient method, but both headline rate theorems are stated without hypotheses their proofs actually need; a serious revision could fix it.","tokens_in":24224,"tokens_out":5996,"would_cite":false,"duration_ms":62528,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","49Q99","49M30","65K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"An intrinsic proximal gradient step on curved spaces converges at O(1/k), and linearly for strongly convex costs.","keywords":["proximal gradient method","Riemannian optimization","Hadamard manifolds","geodesic convexity","strong convexity","proximal map","splitting methods","convergence rate"],"falsifier":"Check the derivation of inequality (16) on a nonconvex perturbation: on $\\mathbb{H}^2$ take $h$ as the indicator of a geodesic ball and $g(p)=-c\\,d(p,q)^2$ so that $f$ is strongly convex for small $c$ while $g$ is not geodesically convex; if the claimed linear bound (25) still holds for Algorithm 1, the missing Assumption 4.3 is a repairable gap, and if the rate deteriorates or fails, Theorem 4.7 as stated is not supported.","tokens_in":23105,"feed_emoji":"📉","tokens_out":11018,"duration_ms":102900,"temperature":0.7,"pith_summary":"The paper introduces a proximal gradient method that works directly on a Riemannian manifold, for problems whose objective splits into a smooth part $g$ and a possibly nonsmooth part $h$. On Hadamard manifolds (complete, simply connected spaces of nonpositive curvature), it claims a sublinear $O(1/k)$ rate in function values for geodesically convex objectives (Theorem 4.4), an $O(1/\\varepsilon)$ iteration bound for reaching $\\varepsilon$-stationary points (Theorem 4.5), and a linear rate when $f=g+h$ is strongly geodesically convex (Theorem 4.7). The construction's point is that the nonsmooth proximal step is taken by the manifold's own proximal map rather than by a Euclidean proxy in a tangent space or an embedding. If the rates hold, an intrinsic splitting method matches the guarantees of the Euclidean proximal gradient method.","feed_headline":"Curved-space proximal gradient converges at O(1/k)","feed_subtitle":"Stays on the manifold, avoids tangent-space subproblems, and proves linear rates for strongly convex costs.","key_machinery":"The central object is the intrinsic proximal map $\\mathrm{prox}_{\\lambda h}(\\cdot) = \\arg\\min_{q\\in\\mathcal{M}} h(q) + \\frac{1}{2\\lambda}d^2(\\cdot,q)$, applied after a gradient step $z(p)=\\exp_p(-\\lambda\\,\\mathrm{grad}\\,g(p))$. The iteration $p^{k+1}=\\mathrm{prox}_{\\lambda(k)h}(z(p(k)))$ keeps both steps on the manifold, avoiding tangent-space or embedding formulations. The convergence analysis is carried by Riemannian cosine inequalities with curvature-dependent constant $\\zeta_{1,\\kappa_l}$, which replace the Euclidean law of cosines, together with the sufficient decrease condition coming from $L_g$-smoothness of $g$ and the proximal characterization $\\frac{1}{\\lambda}\\log_{T_\\lambda(q)} z(q)\\in\\partial h(T_\\lambda(q))$.","core_discovery":"The central claim is that Algorithm 1, whose update is $p^{k+1} = \\mathrm{prox}_{\\lambda(k) h}(\\exp_{p(k)}(-\\lambda(k)\\,\\mathrm{grad}\\, g(p(k))))$, converges in function values at rate $O(1/k)$ when $g$ is geodesically convex, and linearly when $f$ is $\\mu_f$-strongly geodesically convex. The proof separates into two regimes governed by the ratio $\\lambda(k)\\Delta(k)/(\\zeta_{1,\\kappa_l}(D(k))\\,d^2(p(k),p^*))$: when the ratio is at least one the function error halves each step, and otherwise a $1/k$ shrinkage follows. The same machinery yields an $O(1/\\varepsilon)$ complexity for $\\varepsilon$-stationarity. The paper also derives Riemannian analogues of the fundamental prox-grad inequality of the Euclidean theory and shows that, unlike in Euclidean space, these inequalities alone do not yield Fejér monotonicity or a convergence rate for the composite method.","pith_inferences":["The two-regime character of Theorem 4.4 suggests an accelerated variant might be built by alternating halving steps with slow shrinkage steps; the paper does not propose such an acceleration, but its proof structure is compatible with an epoch-based analysis.","The curvature constant $\\zeta_{1,\\kappa_l}(R_\\alpha)$ grows with the diameter of the initial sublevel set, so a testable prediction is that the observed linear-rate constant degrades as data spread increases on a fixed hyperbolic space; plotting iteration counts against $R_\\alpha$ would check this.","The fixed-point formula for the $\\ell^1$-prox on $\\mathbb{H}^n$ (Theorems 6.1 and 6.2) is self-contained; it can be plugged into other manifold first-order methods whose only manifold-specific ingredient is a prox.","An interesting open direction is whether the $O(1/\\varepsilon)$ stationarity complexity remains valid under the weaker assumption that only $f$, not $g$, is geodesically convex, since the proofs currently use convexity of $g$ in inequality (16)."],"forward_implications":["For convex $f$, reaching $\\varepsilon$-accuracy in function value costs at most $O(\\max\\{\\log(\\Delta(0)/\\varepsilon), 1/(\\varepsilon\\delta)\\})$ iterations, where $\\delta$ is the explicit shrinkage constant of Theorem 4.4.","For strongly convex $f$, the function values and iterates converge linearly to the unique minimizer, with complexity $\\frac{4L_g\\zeta_{1,\\kappa_l}(R_\\alpha)}{\\beta\\mu_f}\\log(\\Delta(0)/\\varepsilon)$ up to a factor of two.","The prox-grad inequalities reproduce the known rate for the Riemannian proximal point method when $g=0$ and a comparable Hadamard gradient descent rate when $h=0$.","Because the nonsmooth step uses the manifold proximal map, the method applies to Hadamard manifolds such as $\\mathbb{P}(n)$ and $\\mathbb{H}^n$ without embedding assumptions; the experiments on these spaces show convergence times comparing favorably with CPPA and a projected gradient method.","The new inequalities generalize Beck's fundamental prox-grad inequality to uniquely geodesic manifolds with upper curvature bound, but the paper shows they are insufficient on their own to prove Fejér monotonicity or a rate for Algorithm 1."],"supporting_citations":[{"why":"Supplies the Euclidean proximal gradient framework (rates, prox-grad inequalities, sufficient decrease) that the paper generalizes.","marker":"Beck, 2017"},{"why":"Provides the Riemannian cosine inequalities (Corollary 15, Remark 16) with curvature constants $\\zeta$ used throughout the convergence proofs.","marker":"Martínez-Rubio, Pokutta, 2023"},{"why":"Proposition 22 is the interpolation argument from which the paper derives both the sublinear and linear rate estimates.","marker":"Roux, Martínez-Rubio, Pokutta, 2025"},{"why":"Gives the proximal point characterization on manifolds used to derive the inclusion $\\frac{1}{\\lambda}\\log_{T_\\lambda(q)} z(q)\\in\\partial h(T_\\lambda(q))$.","marker":"Ferreira, Oliveira, 2002"},{"why":"Provides the Hadamard gradient descent rate and the $\\zeta_{1,\\kappa}(D)$-smoothness estimate used to set $L_g$ in the experiments.","marker":"Zhang, Sra, 2016"},{"why":"Prior nonconvex analysis of the same intrinsic method whose lemmas (4.4 and 4.8) the convex proof builds on.","marker":"Bergmann, Jasa, et al., 2025"},{"why":"Lemma 3.1 gives the subdifferential sum rule $\\partial(g+h)=\\mathrm{grad}\\,g+\\partial h$ used to characterize stationarity.","marker":"Bento, Ferreira, Oliveira, 2015"}],"fun_headline_variants":["Riemannian prox-grad without tangent space: O(1/k) rates","Intrinsic proximal gradient converges on curved spaces","New Riemannian prox-grad: linear rates for strongly convex","Curved-space optimization: proximal gradient that stays on manifold","Proximal gradient on Hadamard manifolds: O(1/k) to linear"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the smooth part $g$ is geodesically convex and that the initial sublevel set has finite diameter and contains a minimizer; in the proof of Theorem 4.7 the convexity of $g$ is inherited from inequality (16), which was proved in Theorem 4.4 under Assumption 4.3, but Assumption 4.3 is not restated among the theorem's hypotheses.","fun_headline_variants_meta":{"raw":{"variants":["Riemannian prox-grad without tangent space: O(1/k) rates","Intrinsic proximal gradient converges on curved spaces","New Riemannian prox-grad: linear rates for strongly convex","Curved-space optimization: proximal gradient that stays on manifold","Proximal gradient on Hadamard manifolds: O(1/k) to linear"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000748,"raw_usage":{"total_tokens":3290,"prompt_tokens":858,"completion_tokens":2432,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":474,"completion_tokens_details":{"reasoning_tokens":2341}},"tokens_in":474,"tokens_out":2432,"duration_ms":16927,"temperature":1.0,"reasoning_tokens":2341,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:19:37.994634+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the derivation of inequality (16) on a nonconvex perturbation: on $\\mathbb{H}^2$ take $h$ as the indicator of a geodesic ball and $g(p)=-c\\,d(p,q)^2$ so that $f$ is strongly convex for small $c$ while $g$ is not geodesically convex; if the claimed linear bound (25) still holds for Algorithm 1, the missing Assumption 4.3 is a repairable gap, and if the rate deteriorates or fails, Theorem 4.7 as stated is not supported.","supporting_citations":[],"review_version":1}