{"id":"0a558dc1-c942-46f0-8da7-b38bf5de8bfe","arxiv_id":"2412.06342","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A latent-space tracking controller is derived for continuous-time affine nonlinear systems, combining identifiable representation learning with feedback linearization, and extended to uncontrollable environmental latents.","lead":"This paper trains a neural network to recover an unmeasured state from high-dimensional observations, then uses a classic feedback linearization controller to make the state track a desired path. If the learned model is exact, the authors prove the tracking error goes to zero, which could give stability guarantees for robots and spacecraft that perceive through images or neural features.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 is false as stated: for injective g with a singular Jacobian, e.g. g(q)=q^3, zero-loss differentiable h cannot exist, so the identifiability and tracking guarantees have no valid premise.","rationale":"The advertised contribution is a provable tracking guarantee from high-dimensional observations. Theorem 3.1 is the linchpin: Theorem 3.3 inherits \\tau being a diffeomorphism from it, and the multi-environment extension inherits the same identifiability step. The q^3 example shows the zero-loss premise can fail for a system satisfying every stated assumption, even with infinite data and exact optimization. This is a correctness risk internal to the argument, not a disagreement with community consensus. It is closely related to the reader's weakest_assumption (exact zero-loss) but distinct: the problem is not merely missing approximation bounds; the loss may have no zero at all. I therefore mark partial agreement. The defect is localized and fixable by adding an immersion assumption and completing the global-injectivity proof, and the synthetic experiments use a random MLP g that plausibly satisfies it, so I would keep the reader's CONDITIONAL verdict rather than move to REJECT.","tokens_in":17170,"tokens_out":19591,"duration_ms":193277,"concrete_test":"Paper-and-pencil check: instantiate the scalar system k=1, F=0, B=1, g(q)=q^3, and attempt to solve (5) with a differentiable h and \\hat b>0. Writing zero-loss equations shows the u-coefficient identity 3q^2 h'(q^3)=\\hat B(h(q^3)); evaluating at q=0 gives \\hat B(h(0))=0, impossible, so no global minimizer can achieve zero loss. If the authors add the assumption rank J_g=nk everywhere, verify the counterexample is excluded and re-prove Theorem 3.1, including a separate induction showing \\tau is injective from the triangular positive-diagonal Jacobian rather than from nonsingularity alone.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central premise of Theorem 3.1 is that minimizing (5) reaches zero loss and hence gives \\dot{\\hat z}=\\hat F(\\hat z,u) with \\hat B(\\hat z)\\ge \\hat h_b>0. This premise is not implied by the stated assumptions. Consider a valid instance with k=1, n=1, F(q)=0, B(q)=1, and x=g(q)=q^3. g is injective and C^\\infty, as required. If a differentiable h achieved zero loss, then for \\hat q=h(q^3), comparing \\dot{\\hat q}=3q^2 h'(q^3)\\dot q with \\hat F(\\hat q)+\\hat B(\\hat q)u at q=0 forces \\hat B(h(0))=0, contradicting the strict positivity constraint. Thus no zero-loss differentiable model exists, and the 'actual model' invoked in the proof (h=g^{-1}) is not differentiable at 0. The theorem therefore overstates its domain: identifiability needs g to be an injective immersion (full-rank Jacobian everywhere) and a differentiable h to exist. This is a structural gap, not a finite-sample issue. A secondary flaw is the assertion that nonsingular J(z) makes \\tau=h\\circ g a global diffeomorphism; injectivity and global invertibility need a separate argument, e.g., induction on the lower-triangular blocks.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies continuous-time latent dynamic systems of the form (1)-(2), where the true latent state z is observed through an unknown injective mixing function g. The authors propose to train a decoder h and a model (Fhat, Bhat) by minimizing the latent predictive loss (5), then to use the feedback-linearizing controller (19) on the learned coordinates. The main theoretical results are Theorem 3.1 (identifiability of q up to a componentwise transformation and diffeomorphism tau = h composed with g), Theorem 3.3 (asymptotic tracking e -> 0), and Theorem 3.4 (extension to uncontrollable environmental latents using multiple training environments). The claims are tested in simulation on a rigid-body spacecraft attitude model with high-dimensional observations produced by a random multilayer perceptron.","tokens_in":17461,"tokens_out":14556,"duration_ms":145064,"significance":"The contribution is potentially valuable: it connects identifiable representation learning with a stability-certified feedback controller, offering an alternative to RL for latent-space control. The proof strategy is transparent, and the simulations include observation noise and control deviation studies. However, the central theoretical premise, namely the existence of a zero-loss differentiable model, is not a consequence of the stated assumptions, and the proof of the global diffeomorphism property is incomplete. The significance is therefore conditional on strengthening the assumptions and closing the proof gaps; under those strengthened assumptions the framework would be a useful step for safety-relevant latent-space control.","major_comments":[{"comment":"The assertion that solving (5) gives \\dot{\\hat z} = \\hat F(\\hat z,u) assumes a zero-loss model exists, but this is not guaranteed by the stated assumptions. The proof cites 'the actual model' as h = g^{-1}; however, g is only assumed injective and C^k, so g^{-1} need not be differentiable. Concrete counterexample: k=1, n=1, F=0, B=1, x = g(q) = q^3. For any differentiable h, equality in (5) forces \\hat B(h(q^3)) u = 3q^2 h'(q^3) u for all q,u; at q=0 this gives \\hat B(h(0)) = 0, contradicting \\hat B \\ge \\hat b_h > 0. The theorem needs an explicit realizability assumption, e.g., that g is an injective immersion with a C^k left inverse on its image, or that a C^k h attaining zero loss exists. This is load-bearing because Theorem 3.3 transfers its conclusion through Theorem 3.1.","section":"§3.1.1, proof of Theorem 3.1"},{"comment":"Nonsingularity of J(z) for all z only establishes that tau = h composed with g is a local diffeomorphism by the inverse function theorem; it does not establish that tau is a global diffeomorphism on R^{nk}. Global injectivity and surjectivity need separate arguments, for example properness or an induction exploiting the triangular structure. Since Theorem 3.3 uses invertibility of tau to convert convergence of \\hat e into convergence of the real tracking error e, this gap should be closed, or the statement weakened to a version where injectivity plus continuity is shown to suffice for the tracking error transfer.","section":"§3.1.1, final paragraph of the proof of Theorem 3.1"},{"comment":"The statement 'Theorem 3.1 ensures ... \\forall i = 0,...,k-2, \\dot{\\hat q}_{d,i} = \\hat q_{d,i+1}' is asserted without proof. For arbitrary desired trajectories z_d(t), this identity is not automatic; it requires the triangular structure of tau to imply that the components \\hat q_{d,i} form a prolonged jet, i.e., that \\hat q_{d,i+1} equals the total time derivative of \\hat q_{d,i} along any smooth curve. Please add the missing argument or a lemma establishing this derivative identity, because equation (21) for \\hat e depends on it.","section":"§3.1.2, proof of Theorem 3.3"},{"comment":"Even under a realizability assumption, the paper does not analyze the effect of the nonzero training loss that any finite-capacity or finite-sample learner will have. Theorems 3.1 and 3.3 require the exact equality \\dot{\\hat z} = \\hat F(\\hat z,u) for all z,u; no approximation-error, generalization, or perturbation bound is given. As a result, the asymptotic tracking theorem does not apply to the trained models in the experiments, which have positive loss (Figure 2). Please either add a robustness result for the closed loop under a bounded model error \\|\\dot{\\hat z} - \\hat F(\\hat z,u)\\| \\le \\varepsilon, or explicitly restrict the claims to the idealized zero-loss setting and revise the experimental discussion accordingly.","section":"§3.1.1, objective (5)"}],"minor_comments":[{"comment":"The block index convention is easy to misread; with the natural convention J^{i,j} = \\partial \\hat q_{i-1}/\\partial q^{(j-1)}, the recursion after Eq. (15) is consistent. I recommend defining the block index explicitly to remove ambiguity.","section":"§3.1.1, Eqs. (12)-(15)"},{"comment":"In the sentence containing Eq. (28), 'G_{k+1}(s)' should read 'G_{l+1}(s)'.","section":"§3.2, proof of Theorem 3.4"},{"comment":"The text says '\\hat z (and hence \\hat q_i) is inferred from \\hat x', but \\hat x is not defined; it should say 'inferred from x'.","section":"§3.1.2, controller (19)"},{"comment":"The displayed chain in Eq. (16) is garbled; it should read \\hat q_{k-1} = \\Delta B q^{(k-1)} rather than the repeated equalities shown.","section":"§3.1.1, Eq. (16)"},{"comment":"The subplot labels in Figure 1 are inconsistent: panels (a), (c), (e) should label the angles as \\theta_x, \\theta_y, \\theta_z rather than 'x(rad)', 'y(rad)', 'z(rad)', and panels (b), (d), (f) should label angular rates as \\omega_x, \\omega_y, \\omega_z.","section":"§4.2.1, Figure 1"},{"comment":"Reference [51] is an unpublished self-citation with no preprint identifier; please provide an arXiv number or publication status, or remove the reliance on it if the relevant results are proved in this manuscript.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies on the authors' unpublished reference [51] for the learning approach; if that work contains essential lemmas, the authors should make it available. The novelty relative to [51] appears to be the continuous-time formulation and the feedback linearization controller, which is a reasonable contribution for the journal. The main concern is the mismatch between the strength of the assumptions and the strength of the claims; the results may become correct after adding suitable realizability and regularity assumptions and a perturbation analysis, so I view this as fixable within the scope of a major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the idea is good and the paper is worth engaging, but the central theorem overreaches. The continuous-time setting, the tracking controller without RL, and the multi-environment extension for uncontrollable latents are genuinely new relative to the discrete-time and non-control prior work. The derivation of the chain structure from zero-loss identification is clever, and the authors honestly flag that the validation is synthetic. A serious referee should not desk-reject this.\n\nThe soft spots are real, though. The reader's indexing concern after (15) does not actually land—block column k-1 is q^{(k-2)}, so the independence conclusion is fine—but the stress-test example does: an injective g with a singular Jacobian, like g(q)=q^3, makes a zero-loss differentiable h impossible. The theorem assumes the global minimizer of (5) has exactly zero loss because \"the actual model\" attains it, but the actual left-inverse is not differentiable at the rank-deficient point. So as stated, Theorem 3.1 is false; it needs an injective-immersion assumption or an explicit condition guaranteeing a differentiable left-inverse. That is load-bearing, because the feedback linearization guarantee in Theorem 3.3 inherits the diffeomorphism conclusion.\n\nTwo more issues are worth naming. First, nonsingular Jacobian gives only local diffeomorphism; the proof does not supply the global injectivity argument for tau=h∘g, so the global conclusion is not established. Second, Theorem 3.4 calls tau_zs=h∘g_zs a diffeomorphism from R^{nk+l} to R^{nk}; the dimensions do not match, so that statement is ill-posed. The multi-environment proof itself is plausible once the notation is corrected. The exact-zero-loss premise and the absence of finite-sample or approximation-error bounds are also acknowledged gaps, though the paper is upfront about being a theoretical idealization.\n\nWho is this for? Researchers working on identifiable representation learning for control. With the identifiability theorem repaired, the paper would be a useful building block; as it stands, the main guarantee is conditional on assumptions that the stated hypotheses do not imply. The experiments are illustrative rather than convincing evidence, and no code or data are provided. I would send it to peer review, but the revision needs a corrected Theorem 3.1 and a clean statement of the multi-environment result.","headline":"A promising combination of identifiable representation learning and feedback linearization, but the main identifiability theorem is stated too broadly and needs a fix before the tracking guarantees hold.","tokens_in":17975,"tokens_out":10050,"would_cite":false,"duration_ms":96731,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B18","93C10","93D15","93B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a continuous-time latent dynamic system can be identified up to an invertible componentwise transformation and tracked asymptotically by feedback linearization built on the learned model, including when…","keywords":["latent dynamic systems","identifiable representation learning","feedback linearization","asymptotic tracking","spacecraft attitude control","affine nonlinear systems","uncontrollable environmental latents","continuous-time systems"],"falsifier":"Train the model on a finite dataset from a concrete system in the assumed class and record both the achieved value of (5) and the closed-loop tracking error $e(t)$ under (19). If the empirical loss stays strictly positive while $e(t)$ still converges to zero, the zero-loss premise is stronger than needed; if the error fails to decay whenever the loss is positive, the premise is doing the load-bearing work. A cleaner check is to compute, for a smooth injective $g$, whether the infimum of (5) over the chosen function class is zero; a positive infimum leaves Theorem 3.1 with no case to apply.","tokens_in":16951,"feed_emoji":"🛰️","tokens_out":13097,"duration_ms":120802,"temperature":0.7,"pith_summary":"This paper tries to establish a full pipeline from high-dimensional observations to asymptotic tracking for a class of latent dynamic systems, where an unknown smooth injective map $x=g(z)$ hides a low-dimensional state whose dynamics are affine in an available control $u$. The authors show that minimizing a latent-velocity prediction error recovers the true controllable latent $q$ up to an invertible componentwise transformation, making the composition $\\tau=h\\circ g$ a diffeomorphism. On the identified model they build a feedback linearization controller and prove that the real tracking error $e=z-z_d$ converges to zero as $t\\to\\infty$ whenever the chosen gains make $A$ Hurwitz. The same guarantee extends to settings with uncontrollable environmental latents by training in several environments whose uncontrollable dynamics differ. If correct, the result offers a training procedure that does not depend on goals or reference trajectories, in contrast to reinforcement learning approaches for controlling spacecraft or robots from images or neural-network outputs.","feed_headline":"Tracking error converges to zero from high-dimensional observations","feed_subtitle":"A model trained once on raw observations plus feedback linearization drives states to the reference trajectory, no RL needed.","key_machinery":"The load-bearing object is the pair of learned modules $(h,\\hat F,\\hat B)$ that minimizes the velocity-matching objective (5), together with the Jacobian of the composition $\\tau=h\\circ g$. The proof's central step is differentiating the matched dynamics with respect to $u$: this forces the upper-right $n\\times n$ blocks of the Jacobian to vanish and the diagonal blocks to equal $\\hat B(\\hat z)B^{-1}(z)$, giving a lower triangular Jacobian. Recursion over the derivative levels then shows $\\hat q_0$ depends only on $q$ and the Jacobian is nonsingular, so $\\tau$ is a diffeomorphism and identification is componentwise invertible. The controller is a feedback linearization law, i.e., a control that uses the estimated model to cancel the estimated nonlinearity and impose the linear error dynamics $\\dot{\\hat e}=A\\hat e$; choosing the gain matrices $K_i$ so that $A$ is Hurwitz turns asymptotic tracking for the estimated coordinates into asymptotic tracking for the real coordinates.","core_discovery":"The central claim is Theorem 3.1 plus Theorem 3.3: for a continuous-time affine nonlinear latent dynamic system with injective mixing $g$, solving the learning objective (5) gives an exact fitted model $\\dot{\\hat z}=\\hat F(\\hat z,u)$, identifies the true $q$ up to an invertible componentwise transformation $\\hat q_0=\\tau_q(q)$, and makes $\\tau=h\\circ g$ a diffeomorphism. With that model, the feedback linearizing controller (19) makes the estimated tracking error obey $\\dot{\\hat e}=A\\hat e$, so if all eigenvalues of $A$ have negative real parts, the real tracking error $e=z-z_d$ goes to zero as $t\\to\\infty$. For linear constant-$B$ systems, Corollary 3.2 sharpens identification to scaling and translation, and to exact coefficients when the dynamics are linear. Theorem 3.4 extends the same guarantees to the presence of uncontrollable environmental latents: training in $l+1$ environments with nonsingular differences among the latent-environment dynamics makes $\\tau_{zs}$ depend only on $z$, so tracking still holds in an unseen stable test environment.","pith_inferences":["The zero-loss premise is stronger than any finite neural network can meet on a continuum; a perturbation analysis that tracks a positive residual in (5) into the closed-loop error would turn the guarantee into a practical bound, and the paper does not supply it.","The identifiability argument differentiates the matched dynamics with respect to $u$, so the affine-in-control structure is doing real work: a system nonlinear in $u$ would break the step that isolates $\\hat B$, and the approach would need a different identification route.","The multi-environment result suggests a concrete recipe for image-based control: collect data under several different environmental dynamics (for instance, different disturbance or lighting regimes), train on the same objective (5), and the controllable latents should be decorrelated automatically; this can be tested in simulation before spacecraft deployment.","The forward-difference approximation (6) used to estimate $\\dot{\\hat z}$ introduces a bias the theory does not model; replacing it with a differentiable ODE solver or a trajectory-level loss would reduce that bias and may be necessary when the sampling period is large."],"forward_implications":["The learned model can track any reference trajectory $x_d=g(z_d)$ without retraining or online planning, provided the chosen gains make $A$ Hurwitz.","Controller design reduces to selecting the coefficient matrices $K_0,\\dots,K_{k-1}$; no reward shaping or policy search is needed.","When $B$ is constant and the dynamics are linear, the latent state is identified up to scaling and translation and the system matrix is identified exactly, making the learned model interpretable for model-based design.","With uncontrollable environmental latents, training in $l+1$ environments whose latent-environment dynamics differ by a nonsingular matrix isolates the controllable latents, so tracking succeeds in a stable unseen test environment.","In the spacecraft attitude experiments, the closed-loop tracking errors reach the $10^{-3}$ rad level within a few seconds in the noise-free case, and limited control deviations degrade but do not destabilize the response."],"supporting_citations":[{"why":"Supplies the mechanism-based proof scheme that solving (5) forces $\\dot{\\hat z}=\\hat F(\\hat z,u)$ and yields componentwise invertibility of $\\tau_q$.","marker":"[47]"},{"why":"The discrete-time identifiable representation and model learning result this work extends to continuous time; its identifiability strategy is the direct predecessor of Theorem 3.1.","marker":"[51]"},{"why":"Standard reference for feedback linearization, the method on which controller (19) is built.","marker":"[59]"},{"why":"Source of the rigid-body spacecraft attitude dynamic model (29) used in the experiments.","marker":"[67]"},{"why":"Procedure for using a randomly initialized MLP as the unknown mixing function in the simulations.","marker":"[68]"}],"fun_headline_variants":["No RL: latent model + feedback linearization tracks to zero","High-dim observations yield asymptotic tracking control","Learned latent dynamics enable exact trajectory tracking","Feedback linearization on identified latents drives error to zero"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proofs assume the learning objective (5) reaches a global minimum of exactly zero, so the learned model satisfies $\\dot{\\hat z}=\\hat F(\\hat z,u)$ perfectly on the training states and inputs; no finite-sample, capacity, or approximation-error bound is provided, and the tracking guarantee rests on that exact fit.","fun_headline_variants_meta":{"raw":{"variants":["No RL: latent model + feedback linearization tracks to zero","High-dim observations yield asymptotic tracking control","Learned latent dynamics enable exact trajectory tracking","Feedback linearization on identified latents drives error to zero"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000435,"raw_usage":{"total_tokens":2243,"prompt_tokens":1004,"completion_tokens":1239,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":1190}},"tokens_in":620,"tokens_out":1239,"duration_ms":9924,"temperature":1.0,"reasoning_tokens":1190,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T19:47:41.861392+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the model on a finite dataset from a concrete system in the assumed class and record both the achieved value of (5) and the closed-loop tracking error $e(t)$ under (19). If the empirical loss stays strictly positive while $e(t)$ still converges to zero, the zero-loss premise is stronger than needed; if the error fails to decay whenever the loss is positive, the premise is doing the load-bearing work. A cleaner check is to compute, for a smooth injective $g$, whether the infimum of (5) over the chosen function class is zero; a positive infimum leaves Theorem 3.1 with no case to apply.","supporting_citations":[{"cited_title":"Weakly supervised representation learning with sparse perturbations,","cited_arxiv_id":null,"evidence_quote":"Supplies the mechanism-based proof scheme that solving (5) forces $\\dot{\\hat z}=\\hat F(\\hat z,u)$ and yields componentwise invertibility of $\\tau_q$."},{"cited_title":"Identifiable representation and model learning for latent dynamic sys- tems","cited_arxiv_id":null,"evidence_quote":"The discrete-time identifiable representation and model learning result this work extends to continuous time; its identifiability strategy is the direct predecessor of Theorem 3.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the rigid-body spacecraft attitude dynamic model (29) used in the experiments."},{"cited_title":"Contrastive learning inverts the data generating process,","cited_arxiv_id":null,"evidence_quote":"Procedure for using a randomly initialized MLP as the unknown mixing function in the simulations."}],"review_version":1}