{"id":"1188226e-362b-42b1-9752-08c993901796","arxiv_id":"2411.14321","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"An incremental Koopman algorithm that grows both dataset and latent dimension enables linear-MPC walking on five simulated legged robots, but its monotonic convergence theorem assumes the learned embedding is already the true eigenfunction basis.","lead":"Each training round adds new data and a wider latent space to a learned Koopman linear model, letting a simple linear MPC controller walk simulated humanoids and quadrupeds. The result offers a possible route to replacing nonlinear MPC or reinforcement learning with a scalable linear model, but the promised convergence proof assumes the hardest part of the learning problem.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed O(1/sqrt(n)) convergence rate depends on Assumption 7, which assumes the learned embedding equals the first n Koopman eigenfunctions; the training loss in Eq. 6 does not enforce this, so the theorem's main rate is conditional on the very property the algorithm is supposed to discover.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing condition: Assumption 7 in Appendix D.3. My stress-test confirms that this is the point where the central theoretical claim is least secure. The theorem's projection error rate is derived by assuming the learned embedding is the first n true Koopman eigenfunctions; under that assumption the proof becomes a spectral truncation bound. The actual algorithm's loss in Eq. (6) contains no orthogonality penalty, no eigenfunction constraint, and no term that would force the learned operator to act as the true Koopman operator on the dominant subspace. Thus the theorem does not establish convergence of the method as implemented; it establishes a rate for an idealized EDMD model whose dictionary is already the correct one. This is a substantive correctness risk for the advertised theoretical contribution, not merely a stylistic issue. It does not, however, undermine the empirical claims: the tracking and prediction tables, the ablations, and the computational cost comparison provide evidence that the iterative data- and dimension-expansion recipe works well in the tested IsaacLab environments. The absence of error bars and the omission of the RL bootstrap policy as a baseline are real but secondary issues; the primary reason for a conditional verdict is the mismatch between the theory and the algorithm. The on-policy data collection also conflicts with the i.i.d. Assumption 1, as the reader notes, but that issue can in principle be repaired with mixing or ergodic assumptions; Assumption 7 is more fundamental because it assumes away exactly the learning problem. My recommendation is therefore unchanged from the reader's CONDITIONAL verdict: the empirical stack is promising, but the theoretical claim must be restated to cover the actual training objective, or the monotonic-convergence claim should be removed or weakened. The concrete test proposed above would settle whether the learned embeddings in practice satisfy the eigenfunction structure assumed by the theorem.","tokens_in":23292,"tokens_out":5775,"duration_ms":58940,"concrete_test":"Train the actual Incremental Koopman pipeline on a benchmark system whose Koopman eigenfunctions are analytically known (e.g., a controlled Duffing oscillator or pendulum), then evaluate the learned embedding on a held-out test set: (1) compute the Gram matrix G = E[Phi(s)Phi(s)^T] and check whether it is close to the identity; (2) compute the eigenfunction residual r_i = ||K Phi_i - lambda_i Phi_i||_{L2} for the learned K and Phi; (3) measure the empirical projection error onto the learned span as n grows. If the Gram matrix is far from I, the residuals are not small, or the projection error does not decay as O(1/sqrt(n)), Assumption 7 is violated and Theorem 1(b) does not apply to the trained model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical claim is Theorem 1(b): the linear approximation error decreases as O(sqrt(ln(n)/m)) + O(1/sqrt(n)). The projection-error part O(1/sqrt(n)) is derived in Appendix D.3 under Assumption 7: the embedding functions Phi = [phi_1, ..., phi_n] correspond to the first n eigenfunctions of the true Koopman operator K associated with the largest eigenvalues. Inspection of Eqs. (44)-(48) shows that, once Assumption 7 is granted, the projection error reduces to the spectral tail sum sum_{i=n+1}^infinity c_i lambda*_i phi*_i. That is a truncation bound for an exact eigenfunction basis, not a learned-model error bound. The implemented algorithm trains g, A, and B end-to-end by minimizing the multi-step prediction loss L_koopman in Eq. (6); this objective does not constrain the learned coordinates to be orthogonal (Assumption 4) nor does it require K Phi(s) = Phi(f(s)) on the dominant eigenspace. Nothing in the algorithm verifies that the learned operator's eigenvectors correspond to the true Koopman eigenfunctions. Therefore the O(1/sqrt(n)) rate holds only if the network has already solved the representation problem that the method is intended to learn. The paper's statement in Section 3.4 that the assumptions are 'generally mild' is not defensible for Assumption 7, which is a realizability assumption about the learned dictionary. The empirical results may still stand, but the advertised monotonic-convergence guarantee is not established for the actual trained model; it is a conditional statement about an idealized EDMD model with a perfect eigenfunction dictionary.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an incremental (continual) Koopman-operator learning algorithm for legged-robot dynamics, in which the training dataset is repeatedly augmented with on-policy failure data and the latent state dimension is increased. The resulting linear model is used as the dynamics inside a linear MPC controller. The authors claim a theoretical guarantee that the linear approximation error converges monotonically as data size m and latent dimension n grow, with rate O(sqrt(ln(n)/m)) + O(1/sqrt(n)), and support the method with experiments on five simulated legged robots across flat and rough terrain, comparing against several Koopman and neural-dynamics baselines and ablating the data-increment and dimension-increment components.","tokens_in":23509,"tokens_out":3247,"duration_ms":31132,"significance":"If the empirical results hold, the paper makes a useful contribution: it demonstrates that a learned linear Koopman model, refined by an iterative data- and dimension-expansion procedure, can serve as the dynamics model for a whole-body MPC controller on high-dimensional legged robots, an application where Koopman methods have previously been limited to low-dimensional systems. The ablations in Section 4.4-4.5 provide evidence that both the dataset increment and the dimension increment contribute to the reported tracking performance. The paper also provides code at a public repository, which is a practical strength. However, the central theoretical claim — the advertised monotonic convergence guarantee and the O(1/sqrt(n)) rate — is established only under a realizability condition that the learned embedding exactly matches the dominant eigenfunctions of the true Koopman operator, a property the training loss does not enforce and the paper does not verify. The empirical claims are also weakened by the absence of reported variance and by the capped survival metric.","major_comments":[{"comment":"The O(1/sqrt(n)) projection-error rate in Theorem 1(b) is derived under Assumption 7, which states that the learned embedding functions Phi = [phi_1, ..., phi_n] are exactly the first n eigenfunctions of the true Koopman operator K. This is precisely the property that the end-to-end neural-network training with the loss in Eq. (6) is supposed to discover; nothing in the loss enforces orthogonality (Assumption 4) or that K Phi(s) = Phi(f(s)) on the dominant eigenspace, and the paper does not verify this condition after training. Consequently, the rate in Eq. (49) is a truncation bound for an exact eigenfunction basis, not a bound for the actually learned model. The statement in Section 3.4 that these assumptions are 'generally mild' is not defensible for Assumption 7. The theoretical convergence claim for the proposed algorithm is therefore not established as stated.","section":"Appendix D.3, Assumption 7 and Eq. (44)-(49)"},{"comment":"The main theorem (Theorem A.3/A.4) assumes that data samples s_1, ..., s_m are i.i.d. from a fixed distribution (Assumption 1). The algorithm described in Section 3.1, however, collects incremental data on-policy: the MPC controller built from the current dynamics T^(j) generates the next dataset D^(j+1)_incre. This changes the sampling distribution across iterations and creates dependence between the collected samples and the previously learned model. The theorem therefore does not directly apply to the iterative, on-policy data collection procedure that is central to the method. The authors should either prove the convergence result under an adaptive-sampling formulation or explicitly state the theorem as applying only to a simplified off-policy setting, with the on-policy procedure treated as an empirical heuristic.","section":"Section 3.1 and Assumption 1"},{"comment":"The abstract claims that 'the linear approximation error of our method converges monotonically,' but Theorem 1 does not establish monotonic convergence. It provides a limit statement (Eq. 7) and an asymptotic upper bound (Eq. 8) on the error for a given (n, m) pair; it does not show that the error sequence over the algorithm's iterations is monotone decreasing. The theory needs to be stated more precisely, and the wording should be aligned with what is actually proven.","section":"Theorem 1 and abstract"},{"comment":"The main tracking results (Tables 1 and 2) and the per-task tables in Appendix E.5 report single-seed averages without standard deviations or confidence intervals. Since the comparisons include stochastic training and simulation, the reader cannot assess whether the reported improvements are statistically meaningful. In addition, the survival metric TSur is capped at 200 steps, and many entries — including the authors' own — are close to that cap, so the metric has limited sensitivity to differences in long-horizon robustness. Finally, the PPO policy used to generate the initial dataset D^(0) is not included as a control baseline; without it, the reader cannot separate the contribution of the Koopman-MPC controller from the quality of the initial data collection.","section":"Section 4, Tables 1-2 and Appendix E.5"}],"minor_comments":[{"comment":"The MPC cost uses P z_{t+k} as the predicted state, but the state vector x_t is defined only in Section 4.1. It would be clearer to explicitly define the relation between x_t, z_t, and P before the cost is introduced.","section":"Section 2.2, Eq. (5)"},{"comment":"The pseudocode initializes the optimizer and scheduler inside the TrainKoopman function, but the text in Section 3.3 does not describe how the latent-space dimension n^(j) is mapped to the network architecture, especially when the dimension is increased by Delta_n. A sentence clarifying this would help reproducibility.","section":"Section 3.3, Algorithm 1"},{"comment":"The expression for E_pre(k) in the metrics paragraph uses x_t for the predicted state and x*_t for the reference, but the same notation is used later for the system state and reference trajectory. Please make the distinction explicit in the equation or in the text.","section":"Section 4.1, metric definitions"},{"comment":"In the definition of Theorem A.1, the statement 'for any g in H_N' uses the subscript N inconsistently with the earlier notation H_n; please use a consistent index.","section":"Appendix D.1"},{"comment":"The statement in Section 5 that the method 'encounters an explosion in latent dimensionality' is not backed by any quantitative analysis of how n grows across iterations; the computational-cost comparison in Appendix C.1 only reports wall-clock times, not the dimensional growth. A brief analysis or a bound on the number of iterations would be useful.","section":"Section 5 and Appendix C.1"},{"comment":"There are a number of typographical errors and inconsistencies, e.g., 'catagorized' in Section 1, 'reconsrtuction' in Section 3.2, 'incre' for the incremental dataset in Section 3.1, and the inconsistent use of 'Tsur' and 'TSur'. A careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's empirical contribution is potentially valuable, and the idea of combining iterative data collection with latent-dimension growth is interesting. However, the central theoretical claim, which is a prominent part of the abstract and Section 3.4, rests on an assumption that is essentially the representation problem the method is supposed to solve. I would recommend that the authors either (a) prove a convergence rate under weaker, checkable conditions, or (b) recast the theory as an analysis of the EDMD estimator under ideal conditions and explicitly label the on-policy continual procedure as a practical heuristic. The experimental sections would also benefit from error bars and a more sensitive survival metric. Given that the empirical core is plausible and the ablations are informative, major revision seems appropriate rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the empirical recipe is real: alternating failed-tracking data collection with latent-dimension growth lets a learned linear Koopman model drive QP-based MPC for whole-body locomotion on five simulated legged robots, and the ablations show both increments matter. Second, the advertised convergence guarantee does not cover what the algorithm actually learns. The O(1/sqrt(n)) projection rate follows only if the learned embedding already equals the first n eigenfunctions of the true Koopman operator (Assumption 7); the end-to-end loss in Eq. 6 never enforces that, and the paper itself shows the rate is just spectral tail truncation once you grant the premise. So the theory is an idealized EDMD statement, not a justification for the trained model.\n\nWhat's genuinely new here is the continual-lifting procedure and the whole-body Koopman/MPC demonstration. To my knowledge no one has used Koopman-linearized whole-body dynamics with linear MPC on high-dimensional legged robots at this scale. The code is public, the baselines (DKUC, DKAC, NNDM with NMPC, DKRL) are sensible, and the per-task tables in the appendix are consistent with the aggregate table. Credit where due: the ablations in Sections 4.4 and 4.5 support the claim that both dataset expansion and dimension expansion contribute, and the method beats the baselines by wide margins on most suites.\n\nThe soft spots, in order of seriousness. (1) Assumption 7 is a realizability assumption; without it the theorem reduces to a known result from Korda and Mezic. The paper calls the assumptions 'generally mild,' but this one is doing all the work. (2) Assumption 1 (i.i.d. samples) is inconsistent with the on-policy data collection in Algorithm 1, where the current MPC model generates the next dataset. This is fixable by framing as an approximation or by relaxing to ergodic trajectory data, but as stated it is a mismatch. (3) The main tracking tables report single-seed averages without variance; the survival metric caps at 200 steps, so the headline TSur numbers are close to ceiling. (4) The RL policy used to bootstrap D(0) is not included as a baseline, which weakens the comparison for practitioners.\n\nIf the theory section were rewritten to describe the ideal EDMD case and the monotonic-convergence claim were dropped or qualified, the paper would be stronger. As is, I would send it to review because the empirical direction is valuable and the flaw is in the interpretation, not in the core demonstration. The right referee would ask for variance estimates, a stronger baseline, and an honest statement of what the theorem does and does not prove.","headline":"A genuinely useful incremental Koopman pipeline for legged robots, wrapped in a convergence theorem that assumes away the hard part of learning.","tokens_in":24225,"tokens_out":2717,"would_cite":true,"duration_ms":24276,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Learned Koopman models refined with failure data and higher latent dimensions let linear MPC control five high-dimensional legged robots.","keywords":["Koopman operator","model predictive control","legged robots","continual learning","extended dynamic mode decomposition","linearization","locomotion control","latent space lifting"],"falsifier":"Run the incremental algorithm on a system whose Koopman spectrum decays slowly, or on the robot datasets after checking whether the learned embedding functions are near-orthogonal and eigenfunction-like. If the measured linearization error plateaus as the latent dimension n increases despite scaling samples as m = $\\Omega$(n log n), the O(1/$\\sqrt$(n)) projection-error bound is violated and the central convergence claim fails even if the controller still works.","tokens_in":22919,"feed_emoji":"🤖","tokens_out":7190,"duration_ms":60548,"temperature":0.7,"pith_summary":"This paper tries to establish that a learned linear Koopman model of a high-dimensional legged robot can be refined until it is accurate enough to serve as the dynamics model inside a linear MPC controller. The proposed incremental algorithm alternates between collecting failure-prone tracking data from its own current controller, expanding the training dataset, increasing the latent dimension, and retraining the embedding and linear dynamics end-to-end. The paper proves a convergence result: under stated assumptions on sampling and the embedding, the learned Koopman operator converges to the true one, and the paper claims the linear approximation error converges monotonically, with rate O(sqrt(ln(n)/m)) + O(1/sqrt(n)), so adding data and dimensions jointly shrinks the error. In simulation, linear MPC with these models tracks walking references across flat and rough terrain on five robots: Unitree G1, H1, A1, Go2, and ANYmal D. If the claim holds, it offers a scalable model-based alternative to nonlinear MPC and task-specific reinforcement learning for legged locomotion.","feed_headline":"A learned linear model walks five legged robots","feed_subtitle":"Failed-tracking data and bigger latent spaces shrink Koopman error, so QP-based MPC tracks walks on flat and rough terrain.","key_machinery":"The central object is the finite-dimensional Koopman model z_{t+1} = A z_t + B u_t in a latent state z = [x; g'(x)] that concatenates the original robot state with a neural-network embedding g'. The embedding and matrices are learned end-to-end with a discounted k-step prediction loss plus a small reconstruction term. The mechanism that carries the argument is the iterative loop: each round adds on-policy failed-tracking rollouts to the dataset, raises the latent dimension by $\\Delta$ n, and retrains; theoretically, EDMD is read as an L2 projection of the true Koopman operator onto the span of the embedding functions, so the total error separates into a sampling error bounded by matrix Bernstein concentration and a projection error bounded by spectral decay of the Koopman eigenvalues. A QP-based linear MPC then solves the control problem in the lifted space, sending joint targets to a low-level PD controller.","core_discovery":"The central claim is that the incremental lifting scheme—expanding the dataset with trajectories the current controller fails to track and simultaneously growing the latent dimension—drives a finite-dimensional Koopman model of whole-body legged dynamics toward the true Koopman operator. Theorem 1 formalizes this: with i.i.d. samples, bounded latent states, and orthogonal embedding functions, the learned operator converges to the true operator as the sample size m scales as $\\Omega$(n log n) and n grows; adding assumptions on spectral decay and on the embedding matching the dominant eigenfunctions gives the explicit rate O($\\sqrt$(ln(n)/m)) + O(1/$\\sqrt$(n)). The paper's experimental claim is that linear MPC using this learned model achieves high tracking performance on five high-dimensional legged robots over flat and rough terrain, with average survival close to the 200-step cap and substantially lower joint and root tracking errors than the compared baselines. The paper further claims this is the first application of linearized whole-body Koopman dynamics to locomotion control of high-dimensional legged robots.","pith_inferences":["A testable extension is to treat the dimension increment Delta n as an adaptive quantity chosen from the measured approximation error, since the paper itself notes the latent dimension can explode if growth is unchecked.","The on-policy data collection creates a distribution shift that the i.i.d. assumption in the theorem ignores; re-weighting incremental data or annealing the dataset mixture would be a natural way to close that gap.","If the convergence proof transfers to other contact-rich, high-dimensional systems, the same failure-data-plus-lifting recipe could give a general route to linear whole-body models for manipulation or soft robots, though the paper only demonstrates locomotion.","The method's reliance on full-body reference trajectories means it would need tele-operation or retargeting data to move to real hardware, which the paper lists as future work."],"forward_implications":["Linear MPC with an incrementally learned Koopman model can track whole-body walking references on five high-dimensional legged robots, on flat and rough terrain, with survival times near the 200-step cap.","Adding failure data and raising latent dimension are both necessary: ablations without them raise joint tracking error and cut survival steps sharply, with roughly seven times higher joint error when the data increment is removed.","The learned model keeps k-step prediction error low over horizons up to 15 steps, where the neural-network dynamics baselines show explosive error growth.","The theoretical rate gives a practical scaling rule: when the latent dimension is increased, sample count should grow at least as n log n to keep the approximation error decreasing."],"supporting_citations":[{"why":"Supplies the L2-projection interpretation of EDMD and the convergence of the finite-dimensional Koopman operator to the true one, which Theorem 1 builds on.","marker":"Korda and Mezić (2018a)"},{"why":"Defines Extended Dynamic Mode Decomposition, the data-driven estimation scheme the paper's closed-form Koopman matrix is based on.","marker":"Williams et al. (2015)"},{"why":"Provides the matrix Bernstein inequality used to bound the sampling error term O(sqrt(ln(n)/m)).","marker":"Tropp (2012)"},{"why":"Is the deep Koopman-with-control baseline (DKUC/DKAC) that the paper extends and compares against on prediction and tracking.","marker":"Shi and Meng (2022)"},{"why":"Establishes the use of linear Koopman predictors in MPC, the control paradigm the paper adopts.","marker":"Korda and Mezić (2018b)"},{"why":"Is the PPO policy used to generate the initial feasible-trajectory dataset and the initial reference repository.","marker":"Schulman et al. (2017b)"}],"fun_headline_variants":["Continual lifting Koopman model enables linear MPC for legged robots","Growing latent spaces refine Koopman dynamics for legged robot control","First linearized Koopman control of high-dimensional legged robots","Iterative Koopman learning shrinks error for linear legged control","From failed tracking to Koopman linearity for legged robot locomotion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rate and monotone-convergence conclusion assume the learned neural embedding functions are the first n eigenfunctions of the true Koopman operator and that the data are i.i.d. from a fixed distribution, while the method's own controller generates the incremental data; if the embedding misses those eigenfunctions, the claimed error bound does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Continual lifting Koopman model enables linear MPC for legged robots","Growing latent spaces refine Koopman dynamics for legged robot control","First linearized Koopman control of high-dimensional legged robots","Iterative Koopman learning shrinks error for linear legged control","From failed tracking to Koopman linearity for legged robot locomotion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000325,"raw_usage":{"total_tokens":1850,"prompt_tokens":1004,"completion_tokens":846,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":752}},"tokens_in":620,"tokens_out":846,"duration_ms":8088,"temperature":1.0,"reasoning_tokens":752,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:20:36.481464+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the incremental algorithm on a system whose Koopman spectrum decays slowly, or on the robot datasets after checking whether the learned embedding functions are near-orthogonal and eigenfunction-like. If the measured linearization error plateaus as the latent dimension n increases despite scaling samples as m = $\\Omega$(n log n), the O(1/$\\sqrt$(n)) projection-error bound is violated and the central convergence claim fails even if the controller still works.","supporting_citations":[],"review_version":1}