{"id":"b3b62b03-623e-4297-ae8f-bdc8760a92e8","arxiv_id":"1908.04583","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors propose Bregman Itoh–Abe discrete gradient methods, prove that their iterates converge to Clarke stationary points under box constraints, and demonstrate faster convergence on sparse linear systems and denoising.","lead":"This paper discretizes the inverse scale space flow with Itoh–Abe discrete gradients, producing Bregman discrete gradient methods for non-smooth, non-convex optimization. It proves convergence to Clarke stationary points and shows numerically that the methods find sparse solutions faster than standard SOR.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 4.5 assumes without justification that the implicit coordinate updates of (4.3) move in the descent direction; admissible left-moving roots exist, so the stationarity guarantee is not established for all solutions of (4.3).","rationale":"The central claim is that all accumulation points of the amended scheme (4.3) are Clarke stationary. The proof's contradiction argument depends critically on coordinate updates never moving opposite to the assumed descent direction. Because the scheme is implicit and the objective may be nonconvex, multiple scalar roots of the update inclusion can exist; the paper defines no selection rule. Without monotonicity, inequality (4.5) can fail with the wrong sign, so the contradiction does not follow. The scalar example above makes this precise and shows the proof gap is real, not merely cosmetic. On Assumption 3.1(b), I read it as used in the final step from coordinate-wise directional derivative conditions to Clarke stationarity, so it is not superfluous, but it is indeed false in general and therefore restricts the theorem's scope; this reinforces the conditional status of the result. The dissipative structure, well-posedness arguments, and numerical demonstrations remain solid and are not called into question by this concern. The reader's conditional verdict is therefore appropriate, and no change to that verdict is needed.","tokens_in":19324,"tokens_out":24219,"duration_ms":241603,"concrete_test":"Run the scalar test: define V(t)=-t+t^3 for |t|<=2 and smooth C^1 quadratic tails outside so that V is bounded below and locally Lipschitz; let J(t)=t^2/2, C=R, tau=1, x0=0, p0=0. Solve the amended inclusion (4.3) exactly and list all roots of y=1-y^2 (y≈0.6183 and y≈-1.6180), verifying that both are admissible updates. Then simulate the method with a root-finder (e.g., scipy.optimize.brenth) using the left root as the bracket, and record the sequence x^k. If x^1<0, the monotonicity assertion in the proof of Theorem 4.5 is false for an admissible sequence; if the sequence accumulates at a point with V^o(x*;e)<0 and x*<u, the theorem itself fails as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 4.5, after assuming V^o(x*;e_i)<-eta and x*_i<u_i, the proof asserts 'since x^{k+1}_i >= x^k_i' for a block of iterates and uses this to infer that the constraint component q~ is zero and to bound the sum in (4.5). No justification for this monotonicity is given. Lemma 4.1 only guarantees existence of some y>x when the coordinate directional derivative is negative; the implicit inclusion (4.3) can have additional admissible roots with y<x. Concrete scalar example: take V(t)=-t+t^3 on [-2,2], extended by smooth quadratic tails so that V is bounded below and locally Lipschitz, with J(t)=t^2/2, tau=1, x=0, p=0. The update condition y = -(V(y)-V(0))/y becomes y=1-y^2, whose roots are y≈0.618 and y≈-1.618; both satisfy the inclusion, so an admissible sequence may move left. Thus the proof does not cover all sequences solving (4.3). Separately, the final inference from coordinate-wise derivative conditions to Clarke stationarity is exactly Assumption 3.1(b); that assumption is not generally true (e.g., V(x)=max{x1,x2}+(x1+x2)^2 at 0 satisfies (a) but not (b)), so the theorem is conditional on a nontrivial structural property even if the monotonicity gap is repaired.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a discrete-gradient discretisation of the inverse scale space flow, called the Bregman Itoh–Abe method, for constrained optimisation problems min_{x in C} V(x) with possibly non-convex, non-smooth V. The scheme (3.7)/(4.3) is shown to be unconditionally dissipative with respect to a symmetric Bregman distance (3.6); existence of updates is claimed in Lemma 4.1, uniqueness is proved under convexity or small-step weak convexity in Proposition 4.2, and Theorem 4.5 asserts that all accumulation points of the amended scheme are Clarke stationary under Assumption 3.1. The paper also derives closed-form sparse SOR variants, notes equivalences to SOR/coordinate descent for quadratic problems, and provides numerical experiments for sparse linear systems and a student-t regularised denoising problem.","tokens_in":19663,"tokens_out":26155,"duration_ms":262544,"significance":"The dissipative identity (3.6) is a clean and valuable contribution: it gives unconditional energy decay for arbitrary time steps and a natural Bregman generalisation of the Itoh–Abe discrete gradient. The closed-form BSOR updates and the numerical comparisons are useful and clearly presented. If Theorem 4.5 were fully established, the method would provide a derivative-free, dissipative scheme with Clarke-stationarity guarantees for a broad class of non-smooth, non-convex objectives. The paper also honestly includes a counterexample (Appendix A) showing why the unmodified scheme needs the amendment in (4.3). However, the proof of the main theorem has a load-bearing gap, and the final stationarity inference relies on a strong assumption whose role is not made explicit.","major_comments":[{"comment":"The step 'since x^{k+1}_i >= x^k_i' is asserted without proof and is not a consequence of the scheme. Lemma 4.1 guarantees only existence of some solution of the implicit inclusion; Proposition 4.2 gives uniqueness only under convexity or small-step weak convexity, and neither is assumed in Theorem 4.5. In general (4.1) admits multiple admissible roots, including roots with y<x even when V^o(x;e_i)<0. For example, take V(t)=-t+t^3 extended by smooth quadratic tails so that V is bounded below and locally Lipschitz, J(t)=t^2/2, tau=1, x=0, p=0; the scalar inclusion p-(V(y)-V(0))/y in partial J(y) becomes y=1-y^2, whose roots are y about 0.618 and y about -1.618, both admissible. Thus an admissible sequence may move left, and the bound (4.5) does not follow for all sequences solving (4.3). The subsequent inference that the constraint component tilde q is zero is also not justified when an iterate is stationary at a lower bound, where partial chi_{[l_i,u_i]} is non-zero. The authors need either to impose a root-selection rule that guarantees the monotone root, or to prove the stationarity claim for every admissible root by a different argument.","section":"Section 4.3, proof of Theorem 4.5 (around (4.4)–(4.5))"},{"comment":"The proof establishes only the coordinate-wise inequalities V^o(x*;±e_i)>=0 (with the appropriate box sign), and then concludes Clarke stationarity. This conclusion is exactly Assumption 3.1(b), but that assumption is never cited or used in the proof. Assumption 3.1(b) is not a general property of locally Lipschitz functions: for V(x)=max{x_1,x_2}+(x_1+x_2)^2 at x=0, one has V^o(0;e_i)=1 and V^o(0;-e_i)=0 for i=1,2, while V^o(0;(-1,-1))=-1<0, so 0 is not Clarke stationary on C=R^2 although the coordinate-wise conditions hold. The theorem is therefore conditional on a nontrivial structural property, and the proof must either invoke (b) explicitly or replace it with a direct verification of stationarity on the tangent cone.","section":"Section 4.3, final inference of Theorem 4.5"}],"minor_comments":[{"comment":"'all accumulation points of the scheme (4.3) are nonstationary' should read 'stationary'; as printed it states the opposite of the theorem.","section":"Section 4.3, opening sentence"},{"comment":"'we leave include this in the appendix' is ungrammatical; it should be 'we leave this to the appendix'.","section":"Section 5.2, end of the paragraph after (5.5)"},{"comment":"The proof uses inequalities between sets, e.g. 'partial j(y) >= partial j(x)+mu(y-x)' and 'p-tau v^o(x;1) > partial j(y)'; since partial j(y) is an interval, these should be written for the relevant endpoints or in terms of support functions to be unambiguous.","section":"Lemma 4.1 proof"},{"comment":"Assumption 3.1(b) deserves a remark: it is a genuine restriction (see Major Comment 2), and the paper should indicate where it is satisfied by the examples in Section 7.","section":"Assumption 3.1(b)"},{"comment":"The notation 'tau = 2/diag(A)' is a vector relation; the componentwise meaning should be stated explicitly, although (5.4) makes the intended interpretation clear.","section":"Section 7.1"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the gap in the proof of Theorem 4.5; this is serious but potentially fixable. If the monotonicity of the coordinate updates cannot be justified for all admissible roots, the theorem should be weakened or a selection rule added. The manuscript is otherwise within scope, and the numerical and algorithmic parts are solid."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. The core combination—discretizing the inverse scale space flow with Itoh–Abe discrete gradients—is genuinely new, and the dissipation structure is as clean as advertised. But the main convergence theorem has a real gap: the proof asserts monotone coordinate updates, and that assertion is not justified and can fail when the implicit update has multiple admissible roots.\n\nWhat the paper does well. The dissipative identity (3.6) is correct and unconditional in the step size, which is the kind of structural guarantee discrete-gradient people care about. The well-posedness lemma is plausible and the amended scheme (4.3), which forgets constraint subgradients, comes with a real counterexample (Appendix A) showing the unmodified scheme can fail. That is honest and useful. The closed-form updates for sparse SOR variants are practical. The citation pattern is sound: the SOR equivalence is credited to Miyatake et al. and the convergence framework extends rather than borrows from Riis et al.\n\nThe soft spots. In the proof of Theorem 4.5, after assuming V^o(x*;ei) < -eta, the paper simply states that x^{k+1}_i >= x^k_i for the next N iterates and uses this to show the constraint subgradient vanishes and to bound the sum (4.5). Neither step is derived. The stress-test example is concrete and correct: with V(t) = -t + t^3 (suitably tail-extended), J = t^2/2, tau = 1, x = 0, p = 0, the equation y = -(V(y)-V(0))/y has roots y ≈ 0.618 and y ≈ -1.618, both satisfying the inclusion. So an admissible sequence can move left. Lemma 4.1 only guarantees existence of some y > x; it does not force the solution to be right-moving. Without a selection rule or a proof that all solutions move in the descent direction, the stationarity claim is not established for the method as written. Separately, Assumption 3.1(b) is not generally true for locally Lipschitz functions, yet the final inference from coordinate-wise conditions to Clarke stationarity on the box effectively relies on it. The theorem is therefore conditional on a structural property that is neither proven nor discussed.\n\nNumbers and data. The experiments are illustrative: no error bars, no code, a few smallish problems. That is minor for this theory-first paper, but worth saying.\n\nBottom line. This is a valuable paper for people working on Bregman methods, geometric integration, and sparse inverse problems. It deserves a serious referee and, with a repaired stationarity proof, likely acceptance. The monotonicity gap is the thing to fix, not the overall direction.","headline":"Genuinely new Bregman discrete-gradient scheme with a clean dissipation structure, but the main stationarity theorem has an unproved monotonicity claim that the stress test exposes as a real gap.","tokens_in":20190,"tokens_out":3838,"would_cite":true,"duration_ms":35907,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49M37","49Q15","65K10","90C26"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes Bregman Itoh–Abe discrete-gradient methods for sparse optimisation and proves that, under its assumptions, all accumulation points of the amended scheme are Clarke stationary.","keywords":["Bregman iteration","inverse scale space flow","discrete gradient methods","Itoh–Abe method","non-smooth optimisation","non-convex optimisation","sparse optimisation","Clarke stationarity"],"falsifier":"Construct a locally Lipschitz, bounded-below objective on a box, together with a Bregman function meeting Assumption 3.1, where the amended scheme (4.3) has $V^o(x^k,e_i)<0$ at some iterate yet chooses $x^{k+1}_i < x^k_i$; or run the amended scheme on the Appendix A construction with the constraint subgradient forgotten and observe a subsequence stuck at $z=0$ while $\\partial_z V = -1$. Either outcome would produce a limit point that is not Clarke stationary and would refute Theorem 4.5.","tokens_in":19113,"feed_emoji":"📉","tokens_out":7446,"duration_ms":71185,"temperature":0.7,"pith_summary":"The paper proposes optimisation methods for sparse variational-regularisation problems by discretising the inverse scale space flow with the Itoh–Abe discrete gradient. The resulting Bregman discrete-gradient schemes are unconditionally dissipative: the objective decreases for every choice of time step, with the decrease measured by a symmetric Bregman distance. The main theoretical claim is that, under Assumption 3.1, every accumulation point of the amended scheme is a Clarke stationary point of the objective restricted to a box constraint, even when the objective is non-smooth and non-convex. This matters because it provides convergence guarantees for a derivative-free, structure-preserving family of algorithms, and the numerical examples show sparse reconstructions obtained substantially faster than with plain SOR or Itoh–Abe methods.","feed_headline":"Bregman Itoh–Abe methods converge on non-smooth sparse problems","feed_subtitle":"Coordinate-wise Bregman steps preserve dissipation and still reach stationary points for non-convex sparse objectives.","key_machinery":"The central mechanism is the Bregman Itoh–Abe update, obtained by replacing the gradient in the coordinate-increment discrete gradient with the inverse-scale-space relation $p^{k+1}=p^k-\\tau_k \\bar\\nabla V(x^k,x^{k+1})$. The Itoh–Abe discrete gradient is the coordinate-wise difference quotient (3.4); its mean-value property turns the decrease in $V$ into a symmetric Bregman distance, which by $\\mu$-convexity of $J$ dominates $\\|x^{k+1}-x^k\\|^2$. The amendment (4.3) discards the part of the subgradient coming from the box constraint $\\chi_{[l_i,u_i]}$, keeping the dual variables bounded when iterates sit on a bound.","core_discovery":"The central claim is Theorem 4.5: for an objective $V$ and a Bregman function $J$ satisfying Assumption 3.1, all accumulation points of the amended Bregman Itoh–Abe scheme (4.3) are Clarke stationary points of $V$ restricted to the box $C$. The argument rests on the dissipation identity $V(x^k)-V(x^{k+1}) = \\frac{1}{\\tau_k} D^{\\mathrm{symm}}_J(x^k,x^{k+1}) \\ge \\frac{\\mu}{\\tau_k}\\|x^k-x^{k+1}\\|^2$, which holds for arbitrary positive time steps. In a downhill coordinate direction the proof forces a long block of iterates to keep moving in that direction, so the corresponding dual variable $p_i$ would have to decrease without bound, contradicting boundedness of the coordinate subgradient of $J$ on a compact region. The amended scheme is necessary because the un-amended version can accumulate unbounded constraint subgradients and fail to converge to stationary points, as the appendix demonstrates.","pith_inferences":["Beyond the paper, the dissipation identity suggests a unified Lyapunov analysis: any discrete gradient satisfying the mean-value property, not only Itoh–Abe, could be inserted into the inverse-scale-space flow to obtain an unconditionally dissipative method.","The SOR and coordinate-descent equivalence indicates that Bregman discrete gradients could accelerate randomised coordinate descent for sparse least squares, a variant the paper does not test.","If the missing monotonicity of the implicit coordinate update is established, the same proof would likely transfer to local regularity classes such as prox-regular or lower-$C^2$ functions, since the uniqueness proposition already relies on local convexity bounds.","The imaging experiments exploit sparsity of the residual rather than the solution; a natural extension is transform-domain sparsity, where the Bregman function would be non-separable."],"forward_implications":["For any time-step sequence bounded between positive constants, the method is unconditionally dissipative: $V(x^{k+1}) \\le V(x^k)$ and $\\|x^{k+1}-x^k\\| \\to 0$, so the iterates settle rather than oscillate.","For non-smooth, non-convex objectives satisfying the assumptions, every accumulation point is Clarke stationary restricted to the box, extending the derivative-free Itoh–Abe guarantees to Bregman and inverse-scale-space discretisations.","For quadratic objectives the method reduces to SOR and coordinate descent when the Bregman function is the squared norm, and to a sparse SOR method when the Bregman function includes an $\\ell^1$ term.","In the numerical examples, using a Bregman function that matches the sparsity of the problem substantially accelerates convergence in relative objective and support recovery compared with plain SOR and Itoh–Abe methods.","The un-amended scheme can fail, as the appendix shows, so forgetting the constraint subgradient is a necessary part of the convergence guarantee."],"supporting_citations":[{"why":"Supplies the derivative-free non-smooth, non-convex Itoh–Abe convergence framework that this paper extends to Bregman inverse-scale-space discretisations.","marker":"[45]"},{"why":"Provides the discrete-gradient optimisation foundations, well-posedness results and convergence-rate theory generalised here.","marker":"[21]"},{"why":"Defines the coordinate-increment Itoh–Abe discrete gradient used as the integrator.","marker":"[30]"},{"why":"Establishes the equivalence between SOR-type methods and discrete gradients that underpins the least-squares connections.","marker":"[38]"},{"why":"Provides the Clarke subdifferential and directional-derivative calculus used in the stationarity theorem.","marker":"[16]"},{"why":"Introduces Bregman iterations, the regularisation scheme whose continuous limit is the inverse scale space flow.","marker":"[41]"},{"why":"Introduces the inverse scale space flow that the proposed discrete-gradient methods discretise.","marker":"[50]"}],"fun_headline_variants":["Bregman Itoh–Abe methods converge on sparse non-smooth objectives","Bregman Itoh–Abe: dissipative convergence for sparse non-smooth optima","Coordinate-wise Bregman steps ensure Clarke stationarity on boxes","Dissipative Bregman discrete gradients converge on sparse non-convex","Bregman Itoh–Abe methods hit Clarke stationary points in sparse sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on a monotonicity property it does not establish: whenever a coordinate direction is locally downhill, the implicit update is assumed to move that coordinate forward and never backward, and the box constraint is assumed to contribute nothing to the subgradient during that move.","fun_headline_variants_meta":{"raw":{"variants":["Bregman Itoh–Abe methods converge on sparse non-smooth objectives","Bregman Itoh–Abe: dissipative convergence for sparse non-smooth optima","Coordinate-wise Bregman steps ensure Clarke stationarity on boxes","Dissipative Bregman discrete gradients converge on sparse non-convex","Bregman Itoh–Abe methods hit Clarke stationary points in sparse sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000742,"raw_usage":{"total_tokens":3297,"prompt_tokens":920,"completion_tokens":2377,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":2274}},"tokens_in":536,"tokens_out":2377,"duration_ms":17103,"temperature":1.0,"reasoning_tokens":2274,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:39:02.330586+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a locally Lipschitz, bounded-below objective on a box, together with a Bregman function meeting Assumption 3.1, where the amended scheme (4.3) has $V^o(x^k,e_i)<0$ at some iterate yet chooses $x^{k+1}_i < x^k_i$; or run the amended scheme on the Appendix A construction with the constraint subgradient forgotten and observe a subsequence stuck at $z=0$ while $\\partial_z V = -1$. Either outcome would produce a limit point that is not Clarke stationary and would refute Theorem 4.5.","supporting_citations":[{"cited_title":"A geometric integration approach to nonsmooth, nonconvex optimisation","cited_arxiv_id":"1807.07554","evidence_quote":"Supplies the derivative-free non-smooth, non-convex Itoh–Abe convergence framework that this paper extends to Bregman inverse-scale-space discretisations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the coordinate-increment Itoh–Abe discrete gradient used as the integrator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence between SOR-type methods and discrete gradients that underpins the least-squares connections."},{"cited_title":"Classics in Applied Mathematics","cited_arxiv_id":null,"evidence_quote":"Provides the Clarke subdifferential and directional-derivative calculus used in the stationarity theorem."},{"cited_title":"Multiscale Model","cited_arxiv_id":null,"evidence_quote":"Introduces Bregman iterations, the regularisation scheme whose continuous limit is the inverse scale space flow."},{"cited_title":"In: International Conference on Scale-Space Theories in Computer Vision, pp","cited_arxiv_id":null,"evidence_quote":"Introduces the inverse scale space flow that the proposed discrete-gradient methods discretise."}],"review_version":1}