{"id":"38127ba8-b69f-47cc-9372-46afa0111c1e","arxiv_id":"2505.09485","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A single-loop Riemannian stochastic smoothing method with recursive momentum attains O(epsilon^-3) iteration complexity for Lipschitz nonsmooth terms, and O~(epsilon^{-max{theta+2,2theta}}) under an error-bound condition for constraints.","lead":"Two new algorithms minimize nonsmooth objectives on curved geometries by smoothing the nonsmooth part and using momentum to cut gradient noise. The authors prove one algorithm reaches an epsilon-accurate solution in O(epsilon^-3) steps, matching the best rates known for ordinary flat-space problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.4's momentum-error bound requires isometric vector transport, which Definition 2.3 and Assumption 1 do not provide; the O(epsilon^{-3}) proof is unsupported as written.","rationale":"The reader's weakest assumption correctly identifies the isometric vector-transport gap in Lemma 4.4. My independent reading confirms that Definition 2.3 provides only linearity and the identity property, and Assumption 1 does not state isometry. The proof's first equality in Eq. (4.9) needs ||T_{x_k}^{x_{k-1}}(v)|| = ||v|| to conclude that the transported momentum term retains norm ||epsilon_{k-1}||. Without this, the momentum recursion acquires an additional constant, and the delicate summation in Lemma 4.5 no longer gives the clean K^{2/9} bound that feeds into Theorem 3.1. I also noticed a second gap in the same lemma: Assumption 1.D bounds E||grad f_tilde(x,xi) - T_y^x grad f_tilde(y,xi)|| by a first moment, while the proof uses a squared version E||.||^2 <= \\tilde{L}^2||zeta||^2. This is repairable either by strengthening Assumption 1.D to mean-squared smoothness or by exploiting boundedness of the gradients on the compact manifold, so it is secondary to the isometry issue. Both gaps are structural but not obviously fatal: they can likely be fixed by adding standard assumptions and adjusting constants. The verdict CONDITIONAL is therefore appropriate, and my stress-test does not move it. The concrete test I propose settles whether the missing isometry is actually needed by checking a specific vector transport on the sphere, a manifold covered by Assumption 1.A.","tokens_in":23610,"tokens_out":8211,"duration_ms":82430,"concrete_test":"Re-derive Lemma 4.4 using only the stated Definition 2.3 and Assumption 1, tracking the coefficient of E||epsilon_{k-1}||^2. If it is C_T^2 (1 - a_k)^2 with C_T^2 = sup ||T_y^x(v)||^2 / ||v||^2, compute C_T for a standard projection vector transport on the unit sphere: take x = (1,0,0), y = (0,1,0), and v = (0,1,0) in T_x S^2. Projection of v onto T_y S^2 gives zero, so C_T >= 0 and the transport is not isometric. If any such pair gives C_T != 1, the first equality in Eq. (4.9) is false without an explicit isometry or bounded-transport assumption, and Lemma 4.5 must be reworked before the O(epsilon^{-3}) claim is accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Lemma 4.4, specifically the first equality in Eq. (4.9), writes E[||epsilon_k||^2 | F_k] = (1 - a_k)^2 ||epsilon_{k-1}||^2 + E[||a_k(...) + (1 - a_k)Z_k||^2 | F_k], and the text justifies this by the 'isometry property of vector transport'. However, Definition 2.3 postulates only linearity and T_{0_x} xi = xi; it does not require ||T_{x_k}^{x_{k-1}}(v)|| = ||v||, and Assumption 1 contains no isometry or uniform norm-ratio bound. Linearity is sufficient to make the conditional cross-terms vanish, but it does not make the first term's squared norm equal to ||epsilon_{k-1}||^2. Without isometry the recursion in Lemma 4.4 carries an extra multiplicative constant C_T^2 in front of (1 - a_k)^2 E||epsilon_{k-1}||^2, and the telescoping step in Lemma 4.5, Eq. (4.12), no longer yields the quoted 18\\tilde{L}^2 K^{2/9}(E sum ||G_k||^2)^{1/3} bound. The two-case argument in Theorem 3.1 then loses its O(epsilon^{-3}) conclusion. Separately, the same proof uses E||grad f_tilde(x_k, xi_k) - T grad f_tilde(x_{k-1}, xi_k)||^2 <= \\tilde{L}^2 tau_{k-1}^2 ||G_{k-1}||^2, whereas Assumption 1.D only bounds the first moment E||.|| <= \\tilde{L}||zeta||; a squared mean-smoothness condition or an additional boundedness argument is needed. Both are absent from the stated assumptions, so the central rate is currently conditional on unstated strengthenings.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two single-loop stochastic smoothing algorithms for the composite nonsmooth Riemannian problem (1.1). For Lipschitz h, Algorithm 3.1 uses a STORM+-style recursive momentum estimator for grad f and the Moreau envelope for h, with an adaptive step size τ_k; the paper claims an optimal O(ε^{-3}) iteration complexity at an ε-stationary point in the sense of Definition 2.9. For h equal to the indicator of a convex set, Algorithm 3.2 uses a truncated momentum estimator and, under Assumption 4 with parameter θ≥1, claims a complexity of O~(ε^{-max{θ+2,2θ}}). The analysis relies on retraction smoothness of the smoothed problem (Lemma 4.3) and on variance bounds for the momentum estimator (Lemmas 4.4–4.5 and 4.8), together with an error-bound induction for the constraint-violation measure in the constrained case.","tokens_in":24012,"tokens_out":7601,"duration_ms":68542,"significance":"If the central proofs are made rigorous, the results would be a meaningful advance: an optimal, fully single-loop, O(1)-samples-per-iteration rate for nonsmooth stochastic Riemannian optimization, extending Euclidean STORM+ results to compact submanifolds and improving on the O(ε^{-5}) and O(ε^{-3.5}) bounds of [33] and [14]. The unified treatment of two nonsmooth regimes, the explicit stationary-point definitions, and the comparison with previous oracle bounds are useful contributions. However, as written the paper is not self-contained: three load-bearing gaps (isometry of the vector transport, a missing second-moment smoothness condition, and undefined initialization of the momentum estimator) currently prevent verification of the stated rates.","major_comments":[{"comment":"The first equality in (4.9) is justified by 'the isometry property of vector transport', but Definition 2.3 and Assumption 1 only postulate linearity and the zero-transport identity, not norm preservation. Without an isometry condition (or a uniform bound on ||T_{x_k}^{x_{k-1}}(v)||/||v||), the conditional variance recursion carries an extra multiplicative constant in front of (1-a_k)^2 E||ε_{k-1}||^2; Lemma 4.5's telescoping in (4.12) then fails, and the two-case argument in Theorem 3.1 no longer yields the O(ε^{-3}) bound. The proof requires either an explicit isometric vector transport assumption or an additional uniform norm-bound assumption on T.","section":"Lemma 4.4 / Eq. (4.9)"},{"comment":"The last inequality in (4.9) uses E||grad \\tilde f(x_k, ξ_k) - T_{x_k}^{x_{k-1}} grad \\tilde f(x_{k-1}, ξ_k)||^2 ≤ \\tilde L^2 τ_{k-1}^2 ||G_{k-1}||^2, but Assumption 1.D only bounds the first moment E||·|| ≤ \\tilde L||ζ||. A first-moment bound does not imply the corresponding second-moment bound without additional assumptions, such as a squared mean-smoothness condition or a uniform bound on the stochastic gradients. Since this step is used immediately in the base variance bound (4.7), the rate statement in Theorem 3.1 depends on an unstated strengthening of Assumption 1.D.","section":"Assumption 1.D / Eq. (4.9)"},{"comment":"Both algorithms are formally incomplete at k=1: the momentum estimator δ_1 is never computed or initialized, yet step 2 of Algorithm 3.1 uses G_1 = δ_1 + P(∇c^T ∇h_{μ1}) before any rule producing δ_1; the same holds for δ_1 in Algorithm 3.2. The convergence proofs further use ε_0 = δ_0 - grad f(x_0) via Lemma 4.4, but δ_0 is undefined. The statement needs an explicit initialization (for example, sample ξ_1 and set δ_1 = grad \\tilde f(x_1, ξ_1), or set δ_1 = grad \\tilde f(x_0, ξ_1)) and a corresponding base-case treatment of the error recursion.","section":"Algorithm 3.1 / Algorithm 3.2"},{"comment":"The theorem's displayed Lyapunov function Φ_k = E[F_{μ_k}(x_k)] + ||δ_k - grad f(x_k)||/(16τ_0 \\tilde L^2) is inconsistent with the proof, which uses Φ_k = E[F_{μ_k}(x_k)] + C E||δ_k - grad f(x_k)||^2/τ_{k-1} with C = 1/(16\\tilde L^2). The theorem statement also references τ_0, which is never defined, and omits the square on the norm. Since the telescoping argument in (4.42)–(4.46) uses the proof's version, the statement of Theorem 3.4 as written is not well defined.","section":"Theorem 3.4 / proof of Theorem 3.4"}],"minor_comments":[{"comment":"The phrases 'an stochastic algorithm' and 'for exist algorithms' are typographical errors; the latter should presumably read 'for existing algorithms'.","section":"Abstract and Section 1"},{"comment":"The line 'We denote l(t) = O(l(t))' appears to be a typo; the intended statement is presumably l(t) = O(\\bar l(t)) or a similar asymptotic comparison between two different functions.","section":"Section 1.2, Notations"},{"comment":"The constant 863√3 in (3.10) conflicts with the 864√3 in (3.6) and with the doubling of (4.24); the displayed constants should be reconciled.","section":"Corollary 3.2, Eq. (3.10)"},{"comment":"The indexing of the momentum parameters is not stated consistently: Theorem 3.1 sets a_1=1 and a_{k+1}=k^{-2/3} for k≥1, while Lemma 4.5's summations use a_k and a_{k-1} without defining a_0; the base-case index should be spelled out.","section":"Theorem 3.1 and Lemma 4.5"},{"comment":"Calling Algorithm 3.1 'parameter-free' is confusing in view of the explicit schedules for a_k and μ_k; the intended meaning (no manually tuned step-size sequence) should be clarified in the text.","section":"Remark 2"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely salvageable: the identified gaps correspond to standard assumptions in the Riemannian stochastic optimization literature, and the central STORM+-style argument is recognizable. However, a thorough revision is needed to state the vector-transport isometry, the second-moment smoothness condition, and the initialization of δ_1 explicitly, and to reconcile the indexed quantities in Theorem 3.4 with the proof. The comparison table and the claimed rates are interesting, but they cannot be accepted without these repairs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a serious attempt at the O(epsilon^-3) single-loop rate for nonsmooth Riemannian optimization, and the combination of dynamic Moreau smoothing with a STORM+ momentum estimator on compact submanifolds is genuinely new, as is the truncated-momentum variant for indicator constraints. Second, the central proof of that rate has a load-bearing gap: Lemma 4.4 uses the isometry property of vector transport, but Definition 2.3 only postulates linearity and the identity property, and Assumption 1 does not state isometry or any uniform norm-ratio bound. Without ||T(v)|| = ||v||, the recursion carries an extra constant and the telescoping in Lemma 4.5 does not give the quoted bound.\n\nWhat the paper does well: the algorithmic design is clean, the per-iteration cost is genuinely O(1) samples, and the claimed rates would indeed improve on the cited O(epsilon^-5) and O(epsilon^-3.5) results. The handling of the indicator case with the error-bound condition is thoughtful and the high-level strategy—decaying smoothing parameter, adaptive step sizes, momentum variance reduction—is coherent. The paper does not reduce to an earlier result by its own equations. The self-citations are used as baselines, not as inputs to the main theorem.\n\nWhere the soft spots are, in order of severity. (1) Lemma 4.4's first equality in Eq. (4.9) requires isometric vector transport; this is absent from Definition 2.3 and Assumption 1. The stress-test note is correct. (2) The same lemma's last step needs E||grad tilde f(x_k,xi)-T grad tilde f(x_{k-1},xi)||^2 <= tilde L^2 tau^2, while Assumption 1.D only bounds the first moment. A squared version or a separate boundedness argument is needed; as written, the inequality does not follow. (3) There are indexing inconsistencies: tau_0 and mu_0 are never defined, and Theorem 3.4's schedule uses mu_k = k^{-omega}, tau_k = c_tau (k+1)^{-omega}, a_k = c_a k^{-2 omega}, but Algorithm 3.2's input line lists {mu_k}_k, {a_k}_k, {tau_k}_k without initial values, and the proof sometimes writes tau_{k-1} with k starting at 1. These are typos, but they matter for a theory paper. (4) The problem statement in the introduction allows h to be weakly convex, but Assumption 2 and Lemma 2.6 require convexity; the theorems only cover the convex case, so the abstract's 'weakly-convex' framing is misleading.\n\nWho this is for: researchers working on Riemannian nonsmooth optimization, especially those tracking the best known iteration complexity for stochastic first-order methods. A serious referee can sort out the gaps, and the paper is worth referee time. My recommendation: send it out, but flag Lemma 4.4 and Assumption 1.D in the review request. The O(epsilon^-3) rate is plausible and likely repairable, but it is not established as written.","headline":"Plausible and well-structured attempt at an optimal single-loop rate for nonsmooth Riemannian problems, but the central Lemma 4.4 currently rests on an unstated vector-transport isometry and a squared-smoothness condition that is not part of the assumptions.","tokens_in":24591,"tokens_out":1840,"would_cite":false,"duration_ms":20045,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65K05","65K10","90C05","90C26","90C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single-loop stochastic smoothing algorithm reaches an $\\varepsilon$-stationary point of a nonsmooth Riemannian problem in $O(\\varepsilon^{-3})$ iterations for Lipschitz nonsmooth terms, and in…","keywords":["nonsmooth Riemannian optimization","stochastic smoothing","Moreau envelope","recursive momentum","variance reduction","single-loop algorithm","iteration complexity","error bound condition"],"falsifier":"Give a compact submanifold and a retraction/vector-transport pair satisfying Definition 2.3 but not preserving tangent-vector norms, and compute the left and right sides of inequality (4.7) on a one-step example; if the inequality fails, the estimator-error analysis needs an additional isometry assumption. Alternatively, implement Algorithm 3.1 on a sphere or Stiefel manifold with a factory vector transport, measure $\\|T_x^y v\\|/\\|v\\|$, and test whether the empirical error recursions obey the claimed bounds.","tokens_in":23369,"feed_emoji":"🎯","tokens_out":9800,"duration_ms":85217,"temperature":0.7,"pith_summary":"The paper establishes worst-case iteration guarantees for solving nonsmooth stochastic optimization problems on compact Riemannian manifolds, where the objective is an expected smooth term plus a nonsmooth term composed with a nonlinear map. Its first algorithm targets Lipschitz nonsmooth terms and proves the optimal $O(\\epsilon^{-3})$ iteration complexity to reach an $\\epsilon$-stationary point, improving on earlier stochastic Riemannian rates of $O(\\epsilon^{-5})$ and $\\tilde{O}(\\epsilon^{-3.5})$. Its second algorithm handles constraints encoded as an indicator function of a convex set and, under an error-bound condition with parameter $\\theta\\ge 1$, reaches $\\tilde{O}(\\epsilon^{-\\max\\{\\theta+2,2\\theta\\}})$, matching the best-known Euclidean results. Both methods are single-loop, use only $O(1)$ samples per iteration, and dynamically shrink the smoothing parameter, so iteration complexity and sample complexity coincide. A sympathetic reader would care because these rates are the first to match deterministic benchmarks in the stochastic Riemannian nonsmooth setting without nested loops or large batches.","feed_headline":"A single loop reaches optimal O(epsilon^-3) on manifolds","feed_subtitle":"Dynamic smoothing plus recursive momentum matches the best-known complexity for nonsmooth Riemannian optimization.","key_machinery":"The load-bearing mechanism is the interaction of three ingredients. First, the Moreau envelope $h_\\mu$ with proximal map $\\mathrm{prox}_{\\mu h}$ turns the nonsmooth term into a smooth surrogate whose gradient is $(x-\\mathrm{prox}_{\\mu h}(x))/\\mu$, with the smoothing parameter $\\mu_k$ decreased over time. Second, a recursive momentum estimator $\\delta_k=\\mathrm{grad}\\,\\tilde f(x_k,\\xi_k)+(1-a_k)T_{x_k}^{x_{k-1}}(\\delta_{k-1}-\\mathrm{grad}\\,\\tilde f(x_{k-1},\\xi_k))$ transports previous gradient information across tangent spaces to reduce variance without inner loops or large batches. Third, the adaptive step $\\tau_k=1/(\\sum_{i=1}^k\\|G_i\\|^2/a_{k+1})^{1/3}$ in Algorithm 3.1 couples step size to accumulated gradient norms; Algorithm 3.2 instead uses a truncated momentum estimator projected onto a ball of radius $L_f$ plus polynomially decaying step and smoothing sequences. Retraction smoothness, established in Lemma 4.3 for both cases, is what lets the algorithms take a single retraction step per iteration and still control function descent.","core_discovery":"On its own terms, the paper's central discovery is that dynamic Moreau smoothing can be combined with a recursive-momentum variance-reduction estimator to make stochastic Riemannian smoothing algorithms match the best deterministic complexity in the Lipschitz case and the best Euclidean complexity in the constrained case. The stationarity measure is a KKT-type condition: a point $x$ is $\\epsilon$-stationary if there is $y$ and $z\\in\\partial h(y)$ with $\\mathbb{E}\\|P_{T_xM}(\\nabla f(x)+\\nabla c(x)^\\top z)\\|\\le\\epsilon$ and $\\|c(x)-y\\|\\le\\epsilon$. For Lipschitz $h$, Algorithm 3.1 with smoothing parameter $\\mu_k=k^{-1/3}$, momentum $a_{k+1}=k^{-2/3}$, and adaptive step $\\tau_k$ achieves the $O(\\epsilon^{-3})$ rate. For $h=\\delta_C$, Algorithm 3.2 with a truncated momentum estimator achieves $\\tilde{O}(\\epsilon^{-\\max\\{\\theta+2,2\\theta\\}})$ under the geometric error-bound assumption $\\|\\mathrm{grad}\\,g(x)\\|\\ge\\zeta\\,\\mathrm{dist}^\\theta(c(x),C)$. A direct corollary is that in both cases the sample complexity equals the iteration complexity because each iteration draws one fresh sample and performs a constant number of gradient evaluations.","pith_inferences":["Editorial inference: the proof's use of the isometry property of vector transport at Eq. (4.9) is not entailed by Definition 2.3, so Algorithm 3.1's advertised rate rests on an implicit assumption that transported vectors keep their norms; a counterexample transport with bounded distortion would test whether the recursion still holds.","Editorial inference: the parameter-free adaptive step and the Moreau smoothing schedule suggest a template for other nonsmooth composite structures, such as weakly convex $h$ with H\\\"older-type growth, where the achievable rate would be governed by the growth exponent in place of $1/2$.","Editorial inference: the truncated momentum estimator in Algorithm 3.2, which projects onto a ball of radius $L_f$, indicates that boundedness of the estimator is the key replacement for convexity of the constraint, so a similar truncation could extend the rate to nonconvex constraint sets satisfying the same geometric error bound.","Editorial note: the paper reports no numerical experiments and defers them to future work, so the practical behavior of the adaptive step and truncation on, say, sparse PCA or robust matrix completion remains untested; a small-scale experiment comparing Algorithm 3.1 with the earlier $\\tilde{O}(\\epsilon^{-3.5})$ method would be the natural next check."],"forward_implications":["If the central claim is right, stochastic nonsmooth Riemannian optimization with Lipschitz composite terms needs no more than $O(\\epsilon^{-3})$ iterations, matching the deterministic Riemannian bound and closing the gap left by the $O(\\epsilon^{-5})$ and $\\tilde{O}(\\epsilon^{-3.5})$ stochastic methods.","The single-loop, one-sample design means the method is practical for online settings: sample complexity and iteration complexity coincide, with no nested subroutines and no periodic restart schedule.","For constraints written as $c(x)\\in C$, the $\\tilde{O}(\\epsilon^{-\\max\\{\\theta+2,2\\theta\\}})$ rate under the error-bound condition transfers the Euclidean variance-reduced constrained-optimization rate to compact submanifolds.","Setting $h\\equiv 0$ recovers the smooth stochastic Riemannian setting, and the analysis removes a logarithmic factor relative to earlier Riemannian recursive-momentum methods.","The stationarity definition requires an exact subgradient $z\\in\\partial h(y)$ and feasibility gap $\\|c(x)-y\\|\\le\\epsilon$, so an $\\epsilon$-stationary point in this paper is a stronger output guarantee than relaxed KKT notions used by prior augmented Lagrangian analyses."],"supporting_citations":[{"why":"Supplies the Moreau-envelope calculus used to define the smoothing: the gradient formula, its Lipschitz bound, and the comparison inequality between two smoothing parameters.","marker":"[6]"},{"why":"Provides the retraction-smoothness framework and Lipschitz bounds for retractions on compact submanifolds used in Lemma 4.3.","marker":"[8]"},{"why":"Introduces the recursive momentum estimator that Algorithm 3.1 adapts to the Riemannian setting.","marker":"[12]"},{"why":"Supplies the adaptive step-size rule and the summation inequality that yield the $O(\\epsilon^{-3})$ rate.","marker":"[28]"},{"why":"Establishes deterministic dynamic smoothing on manifolds, the baseline whose optimal complexity the stochastic algorithm matches.","marker":"[4]"},{"why":"Previous stochastic Riemannian smoothing method with $O(\\epsilon^{-5})$ complexity; the framework and stationarity notion the paper improves.","marker":"[33]"},{"why":"Previous augmented Lagrangian momentum method with $\\tilde{O}(\\epsilon^{-3.5})$ complexity; provides the comparison point for the improved rate and the KKT stationarity concept.","marker":"[14]"},{"why":"Euclidean variance-reduced constrained optimization whose $\\tilde{O}(\\epsilon^{-\\max\\{\\theta+2,2\\theta\\}})$ rate Algorithm 3.2 matches under the error-bound condition.","marker":"[32]"}],"fun_headline_variants":["Single-loop smoothing hits optimal complexity on manifolds","Stochastic smoothing matches deterministic rate on Riemannian manifolds","Optimal O(epsilon^-3) achieved by single-loop Riemannian smoothing","Manifold smoothing: single loop, optimal sample complexity","Riemannian smoothing reaches optimal convergence in one loop"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rate proof assumes the vector transport that moves gradients between tangent spaces preserves norms exactly, although the formal definition only requires linearity and the identity property; under a transport that shrinks or stretches lengths, the momentum-error recursion no longer telescopes and the $O(\\epsilon^{-3})$ rate loses its support.","fun_headline_variants_meta":{"raw":{"variants":["Single-loop smoothing hits optimal complexity on manifolds","Stochastic smoothing matches deterministic rate on Riemannian manifolds","Optimal O(epsilon^-3) achieved by single-loop Riemannian smoothing","Manifold smoothing: single loop, optimal sample complexity","Riemannian smoothing reaches optimal convergence in one loop"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000242,"raw_usage":{"total_tokens":1580,"prompt_tokens":1057,"completion_tokens":523,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":673,"completion_tokens_details":{"reasoning_tokens":441}},"tokens_in":673,"tokens_out":523,"duration_ms":5386,"temperature":1.0,"reasoning_tokens":441,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:31:50.797299+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Give a compact submanifold and a retraction/vector-transport pair satisfying Definition 2.3 but not preserving tangent-vector norms, and compute the left and right sides of inequality (4.7) on a one-step example; if the inequality fails, the estimator-error analysis needs an additional isometry assumption. Alternatively, implement Algorithm 3.1 on a sphere or Stiefel manifold with a factory vector transport, measure $\\|T_x^y v\\|/\\|v\\|$, and test whether the empirical error recursions obey the claimed bounds.","supporting_citations":[{"cited_title":"B¨ohm and S","cited_arxiv_id":null,"evidence_quote":"Supplies the Moreau-envelope calculus used to define the smoothing: the gradient formula, its Lipschitz bound, and the comparison inequality between two smoothing parameters."},{"cited_title":"Boumal, P.-A","cited_arxiv_id":null,"evidence_quote":"Provides the retraction-smoothness framework and Lipschitz bounds for retractions on compact submanifolds used in Lemma 4.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the adaptive step-size rule and the summation inequality that yield the $O(\\epsilon^{-3})$ rate."},{"cited_title":"Beck and I","cited_arxiv_id":null,"evidence_quote":"Establishes deterministic dynamic smoothing on manifolds, the baseline whose optimal complexity the stochastic algorithm matches."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous stochastic Riemannian smoothing method with $O(\\epsilon^{-5})$ complexity; the framework and stationarity notion the paper improves."}],"review_version":1}