{"id":"385bf69a-e160-4b87-a054-e605b66e52e9","arxiv_id":"2507.06381","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The gradient flow of recurrent networks can be written as -P K P* Err, and the low rank of K explains dynamic collapse to low-dimensional attractors.","lead":"This paper decomposes the gradient descent training of recurrent neural networks into two operators, one that maps errors to parameter updates and one that propagates changes through time. The authors show that the parameter operator is low-rank, which explains why trained recurrent networks collapse to low-dimensional attractors, and they use the decomposition to predict how multi-task representations align.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"K's low rank bounds each individual gradient step, but the paper never shows that the cumulative update span—not task geometry—drives final hidden-state collapse.","rationale":"Proposition 1 is a clean local identity and the KPFlow package is a real contribution; I do not dispute the spectral observation that K is low-rank. The concern is about the inference from that observation to the causal mechanism of collapse. The reader already flags the continuous-flow/ADAM gap, which is adjacent; my check makes the gap concrete by tracking the accumulated updates rather than a single linearized step. The cumulative-span experiment is cheap: it requires storing per-iteration gradients or δz vectors and performing PCA on them. If the test supports the paper, the causal claim would be substantially strengthened; if not, the paper would remain a useful descriptive toolkit but the headline mechanism would need revision. The reader's CONDITIONAL verdict therefore remains appropriate, so no verdict change is recommended.","tokens_in":31212,"tokens_out":7116,"duration_ms":90854,"concrete_test":"Record per-iteration δz = -PKP*Err (or the equivalent parameter gradients) during the 5000-step memory-pro runs; compute the effective dimension of the cumulative subspace span{δz_1,...,δz_s} as s grows, and compare it with the final hidden-activity dimension (Figures 2E-G). If the cumulative update dimension grows well beyond effrank(K) while final dynamics still collapse, the K-bottleneck is not the cause. As a stronger control, replace K by the identity in the update (δf = -P*Err, the parameter-free flow of Appendix B.4) and simulate the same task; if the resulting hidden dynamics collapse to a similar low-dimensional ring, collapse is a property of the task/loss, not of K.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Observation 3.1 reports that K (hence PKP*) has small effective rank at any snapshot, and concludes that 'dynamic collapse appears in such models due to the bottle-necking of dimension by the operator K in PKP*.' The logical gap is that after S GD steps, z_S = z_0 + Σ_s δz_s, and the dimension of the span of all updates is bounded only by Σ_s rank(K_s), which can grow linearly with S even if every K_s has effective rank about 7. Collapse of the final hidden activity (Figures 2E-G) is therefore not a formal consequence of a single-step rank bound. The task itself—two scalar stimuli on a circle—has intrinsic low-dimensional structure, so the same collapse might occur for any reasonable training signal, independent of K. A causal test is missing. The paper defines the parameter-free baseline PP* (Appendix B.4) but never uses it to check whether removing the K bottleneck changes collapse; this is exactly the experiment that would separate network-structure effects from task effects.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes KPFlow, an operator-theoretic decomposition of the gradient flow that trains recurrent dynamical systems. The central identity, Proposition 1, states that the hidden-state perturbation produced by one gradient step is δz = -[P ◦ K ◦ P*](Err), where K is a parameter-space kernel and P is the linearized flow propagator. The authors apply this decomposition to RNNs and GRUs trained on a memory-pro task, reporting that K has much lower effective rank than P and concluding that this low-rank bottleneck causes dynamic collapse. They also study a four-task multitask setting, using sub-blocks of K and P to define interference matrices that anticipate the alignment of learned representations. The paper includes an implementation package, extensive experiments across initial weight scales and model families (RNNs, GRUs, ReLU RNNs, Hodgkin-Huxley networks), and two theoretical theorems about the rank of K and the structure of P.","tokens_in":31405,"tokens_out":14459,"duration_ms":170811,"significance":"If the central claim holds, KPFlow would provide a useful unifying lens on recurrent-network training, connecting neural tangent kernel ideas with Lyapunov/optimal-control theory and offering practical SVD-based diagnostics for representation collapse. The manuscript has notable strengths: Proposition 1 is an exact, proven gradient-flow identity, not a heuristic fit; the experimental scope is substantial (hundreds of RNN/GRU training runs, multitask sweeps, and several architecture extensions); and the authors provide a PyTorch package with notebooks intended to reproduce the figures. The proposed PP* parameter-free baseline in Appendix B.4 is conceptually valuable even though it is not deployed. However, the paper's headline causal claim—that the low rank of K drives dynamic collapse 'over and above the task'—is not established, because the reported evidence concerns single-step operators rather than cumulative training dynamics and lacks a control experiment. The theoretical results also contain statements that are either unproven or false in the claimed generality. The paper is therefore best viewed as a promising but overclaimed contribution that requires substantial revision.","major_comments":[{"comment":"The conclusion that dynamic collapse is caused by the low rank of K does not follow from the single-step decomposition. At each training step, rank(δz_s) ≤ rank(K_s), but after S steps the hidden state is z_0 + Σ_s δz_s, whose effective dimension is not bounded by the rank of any single K_s; it can grow linearly with S even if every K_s has effective rank about 7. The collapse measured in Figures 2E-G is a property of the final hidden activity, not of a one-step update, so the paper needs an argument or experiment that connects the per-step rank bound to the cumulative state. Furthermore, the low rank of K is partly inherited from the task: for the RNN, K's kernel is the Gram matrix of x(t) and σ(z(t|x)) (Eqs. (9)-(10)), and the memory-pro task uses only two scalar stimuli on a circle. The PP* parameter-free baseline introduced in Appendix B.4 is the natural control, but it is never computed or used. Concretely, the authors should compare hidden-state collapse under PKP* training versus PP* training (or training with K replaced by a high-rank approximation). Without such a control, the claim in the abstract that collapse is a consequence of network structure 'over and above the nature of the underlying task' is not supported.","section":"Section 3.1, Observation 3.1 and Eq. (5)"},{"comment":"The proof in Appendix D.4 establishes the algebraic factorization P = U V U^{-1} via the semigroup property of the state-transition matrix, but it does not prove the subsequent claim that the full singular values of U are σ_j(t|x) = exp(λ_j(x)t), with λ_j the j-th Lyapunov exponent. For a time-varying or non-normal product of Jacobians, the finite-time singular values of Φ(t,0|x) are not equal to exponentials of the asymptotic Lyapunov exponents; at best one can write σ_j(t) = exp(λ_j^{FT}(t) t) for time-dependent finite-time exponents. If λ_j is intended to be the finite-time exponent, the statement is definitional; if it is intended to be the asymptotic exponent, it is false in general for the RNN/GRU models considered. Since this part of Theorem 2 is used to support the claim that P is high rank and to connect P to Lyapunov theory, it should either be corrected with explicit assumptions (e.g., normal, time-invariant Jacobians) or removed from the theorem statement.","section":"Section 4, Theorem 2 and Appendix D.4"},{"comment":"Property (3) of Theorem 1 does not follow from the proof and is false in the stated generality. In the notation of Appendix D.3, the kernel of K_j is a(t,x,t0,x0) · D_q f_outer(t|x) D_q f_outer(t0|x0)^T, where a is the Gram kernel of f_inner. The rank of this operator is controlled by the joint variation of D_q f_outer and f_inner, not by the effective dimension of f_inner alone. For example, take f_inner_j to be a constant scalar (effective dimension 1) and D_q f_outer(t|x) to be a time-varying full-rank n×n matrix; then K_j can have rank n, not 1. The proof's statement that the rank is 'bounded above by the effective dimension of f_inner' needs either a strong restriction (such as D_q f_outer being the identity, which covers the RNN and Hodgkin-Huxley examples) or a corrected bound involving both factors. Because Section 4 cites Theorem 1 as the theoretical motivation for the low rank of K, this is a load-bearing issue rather than a cosmetic one.","section":"Section 4, Theorem 1 and Appendix D.3"},{"comment":"The experiments train with ADAM, adaptive learning rates, momentum, and gradient clipping, while Proposition 1 is an identity for continuous gradient flow with an infinitesimally small learning rate. Appendix E.1 acknowledges that ADAM's solutions may differ from true gradient flow, but the paper does not verify that the actual finite-step updates along the training trajectory are well approximated by -P K P* Err at each snapshot. The linearization check in Appendix C compares two ways of computing P for small perturbations, which is not the same as validating the composed operator against the real optimizer update. The authors should report, for representative runs, the cosine similarity or relative error between the empirical Δz_s and the operator-predicted δz_s over training. This would directly address whether the K-bottleneck mechanism describes the trajectories that produce Figures 2 and 4.","section":"Appendix E.1 and Section 3.1"}],"minor_comments":[{"comment":"The text says 'since K is positive definite (see Theorem 1 below)', but Theorem 1 states that each K_j is positive semi-definite, and K is typically rank-deficient. Please correct this inconsistency.","section":"Main text after Eq. (7)"},{"comment":"Corollary 1 is stated for a fixed linear readout Wout, yet the paragraph immediately before says that in this work the output weights are always trained with GD. The proof later notes that training Wout adds a coupled update. Please clarify the status of the corollary and explicitly separate the fixed-readout and trained-readout cases.","section":"Corollary 1 and Appendix D.2"},{"comment":"There are several typos: the y-axis in Figure 3 reads '# of Principle Functions' instead of 'Principal Functions'; Appendix B.1 contains 'general detials'; and Section 3.1 says 'drawn as (x, y) coordinates on a a circle' with a doubled article. These should be corrected.","section":"Figure 3 and throughout"},{"comment":"The symbol U is used both for the fundamental operator in Theorem 2 and for the left singular-vector operator in the SVD notation of Appendix B.4. This notation collision is confusing in a paper that relies heavily on SVDs; please disambiguate, for example by calling the fundamental operator Φ or F.","section":"Theorem 2 and Appendix B.4"},{"comment":"The loss in Eq. (14) is written with an unsquared norm, while the proof of Corollary 1 assumes a squared-error loss. Please make the definition consistent or explain the normalization.","section":"Appendix B.1, Eq. (14)"},{"comment":"The checklist states that details of the computer setup are provided in the Appendix, but the appendix text included in this version does not contain such details. Please either add the compute information or revise the checklist answer.","section":"Checklist item 8"}],"recommendation":"major_revision","confidential_remarks":"This is a borderline case: the empirical package is strong and the operator decomposition is a genuinely useful idea, but the paper's central causal claim is not yet proven and two of the three theoretical statements (Theorem 1 and Theorem 2) need correction. The missing PP* control experiment is, in my view, essential for the abstract's 'over and above the task' claim; without it, the paper reads as a descriptive tool paper rather than a mechanistic explanation. I would encourage the editor to send the revision back to the same referees if the authors add the control and fix the theorems, since the remaining claims are then likely to be sound. The scope fits a machine-learning venue well, given the practical package and the focus on training dynamics."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper gives a clean, exact decomposition of the gradient flow in recurrent networks—δz = −PKP*Err—plus a working PyTorch package and a lot of experiments. The empirical observation that K is dramatically lower rank than P across RNNs and GRUs is real and worth knowing. But the paper's central causal claim, that K's low rank is what drives dynamic collapse, is asserted, not shown. The missing experiment is the one that would test it.\n\nThe decomposition itself is a repackaging of the standard adjoint/backprop equations, but that's fine: the value is in studying the two operators' spectra separately. The multi-task block decomposition and the interference matrix are a nice diagnostic. Proposition 1 is a correct identity, proven in the appendix. The software is a genuine asset, and the experiments are extensive: 500 RNNs, GRU sweeps, ReLU, even a Hodgkin-Huxley derivation.\n\nThe soft spots, in proportion: First, Theorem 2 claims the singular values of the fundamental operator U are exp(λ_j t). That's not proven, and at finite time it's generally false for non-normal fundamental matrices. It should be weakened to an asymptotic statement, or labeled empirical. Second, and more important, the causal step. Observation 3.1 shows each individual GD step is filtered through a low-rank K. But after S steps, the cumulative update span is bounded by the sum of the ranks, which can easily grow to full dimension. The tasks used are intrinsically low-dimensional, so collapse might be task geometry, not K. The paper defines the PP* baseline in Appendix B.4 but never runs it. That is exactly the control that would separate network structure from task structure. Third, the abstract says collapse is 'a result of network structure, over and above the nature of the underlying task,' but the collapse experiments use one task. That's an overclaim. Fourth, the theory is continuous-time gradient flow, while training uses ADAM with clipping. The appendix acknowledges this, so it's an honest limitation, not a hidden one.\n\nOverall, the descriptive machinery is solid and worth having. The explanatory story needs a good deal more work. The paper deserves a serious referee; with a fixed Theorem 2, a PP* control, and softened claims, it could be a nice contribution. I'd bring it to a reading group to chew on the causal gap, and I'd cite the operator decomposition if I worked on RNN training dynamics.","headline":"Useful operator toolbox with a real empirical observation, but the collapse mechanism is asserted, not shown.","tokens_in":31896,"tokens_out":3958,"would_cite":true,"duration_ms":43002,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that dynamic collapse in recurrent networks is forced by the low rank of a single operator, $K$, that filters every learning update: $\\delta z = -[P \\circ K \\circ P^*](\\mathrm{Err})$.","keywords":["KPFlow","gradient-flow decomposition","dynamic collapse","neural tangent kernel","recurrent neural networks","multi-task interference","Lyapunov exponents","Hilbert–Schmidt operators"],"falsifier":"Run the memory-pro sweep again while recording, at every gradient-descent iteration, the empirically observed hidden-state change $\\delta z$ alongside the decomposition's prediction $-PKP^*(\\mathrm{Err})$ for the same error signal, and compare their effective ranks and cosine alignment across the full range of initial weight scales $g$, including the chaotic regime. If the real updates stay high-dimensional while $K$ remains low rank, or if the predicted and actual updates diverge sharply for large $g$, the claim that $K$'s rank drives collapse would not describe the actual training trajectory.","tokens_in":31043,"feed_emoji":"🧠","tokens_out":11105,"duration_ms":101955,"temperature":0.7,"pith_summary":"This paper claims that the low-dimensional attractors recurrent networks converge to under gradient descent—the phenomenon of dynamic collapse—are largely forced by the training dynamics themselves, not by the structure of the task. The authors decompose each step of learning for any recurrent model into two operators acting on per-trial trajectories: $P$, the linearized flow propagator, and $K$, a parameter operator analogous to the neural tangent kernel. The central identity is $\\delta z = -[P \\circ K \\circ P^*](\\mathrm{Err})$: an error signal is propagated backward, filtered through the parameters, and propagated forward again into hidden-state changes. Because $K$ is dramatically lower rank than $P$, it acts as a bottleneck that collapses the updates into a low-dimensional subspace, and the same operator machinery quantitatively measures how objectives interfere during multi-task training. If the paper is right, dynamic collapse is a generic prediction of the architecture, and the speed of training plus the sharing of attractors across tasks can be read off from the spectrum of these two operators.","feed_headline":"A single low-rank operator explains RNN dynamic collapse","feed_subtitle":"The parameter operator K is so low rank it forces trained networks into low-dimensional attractors.","key_machinery":"The central object is the KPFlow decomposition itself: the factorization of the hidden-state gradient flow as $\\delta z = -[P \\circ K \\circ P^*](\\mathrm{Err})$ acting on the space $\\mathbb{T}$ of per-trial hidden trajectories—3-tensors over trials, times, and hidden units. $P$ is the linearized flow (propagation) operator, which solves the variational dynamics forward in time; its adjoint $P^*$ is the costate/adjoint backpropagation step, and $P$ is block-diagonal over trials. $K$ is the parameter operator, an integral operator that averages over trials and integrates over time, mapping adjoint signals to vector-field perturbations through the parameter Jacobians; it is the only operator that mixes trials. The load-bearing property is the rank asymmetry: by Theorem 1, each weight contributes a positive semi-definite Hilbert–Schmidt kernel operator whose effective rank is bounded by the effective dimension of the quantity it acts on, so for an RNN, $K$'s rank tracks the effective dimensions of the inputs $x(t)$ and activities $\\sigma(z(t|x))$ and stays small; by Theorem 2, $P = U V U^{-1}$ has singular values from Lyapunov exponents $\\exp(\\lambda_j t)$ and Volterra factors $O(1/j)$, hence high rank. Because $PKP^*$ inherits $K$'s low rank, the paper argues, learning is squeezed into a low-dimensional subspace, which is the mechanism for dynamic collapse; in the multi-task case the block decomposition of $K$ and $P$ yields the interference matrix $M$ measuring cosine alignment between direct and interfering corrections.","core_discovery":"On the paper's own terms, the discovery is that the full gradient flow of the hidden state during gradient-descent training factors exactly as $\\delta z = -[P \\circ K \\circ P^*](\\mathrm{Err})$, where $P$ maps perturbations of the network's vector field into perturbations of the trajectory through the linearized flow and $K$ maps adjoint (error) signals into vector-field perturbations by projecting through the parameter Jacobians. The structural fact that carries the argument is a rank asymmetry: $K$ is a sum of positive semi-definite Hilbert–Schmidt integral operators whose effective rank is bounded by the effective dimension of the dynamical quantities each weight multiplies, making it very low rank, while $P$ is consistently high rank, with singular values governed by Lyapunov exponents and the Volterra operator. The paper concludes that dynamic collapse appears in trained RNNs and GRUs because the operator $K$ bottlenecks the dimension of the composite update $PKP^*$, and that this collapse is a property of the network structure that shows up over and above the task. In the multi-task setting, $P$ is block-diagonal over trials and the off-diagonal blocks $P_i K_{ij} P_j^*$ quantify how each sub-task's error interferes with each other sub-task's update; the resulting interference measure anticipates the groups of tasks that later share aligned activity subspaces.","pith_inferences":["Editorial inference: the paper's mechanism suggests a design principle the authors only gesture at—architectures that raise the effective rank of $K$ (richer input encoding, higher-dimensional activity, or parameterizations that keep the $J_\\theta$ Gram matrices full-rank) should train faster and collapse less; a direct test would compare collapse depth across architectures matched for task but di","Editorial inference: the block structure of $PKP^*$ could serve as an online diagnostic for continual and transfer learning—computing the interference matrix at each step would reveal which stored tasks are being overwritten before the loss shows it, enabling task-order or reweighting decisions during training.","Editorial inference: since $K$'s end-of-training rank stabilizes to a similar low value across initializations, the surviving eigenmodes of $K$ may bound the number of distinct attractors or prototypes a trained network can separate, connecting this operator bottleneck to neural collapse in classification networks as a special case."],"forward_implications":["Dynamic collapse is a structural prediction: for any recurrent architecture whose parameter operator $K$ is low rank, gradient descent should converge to low-dimensional attractors on essentially any task, not just the memory and delay tasks studied here.","Training speed tracks the effective rank of $K$: larger initial weight scale $g$ yields higher-rank $K$, less filtering of the adjoint signal, and faster convergence, so the spectrum of $K$ at initialization predicts the convergence time of a run.","Sub-task sharing can be predicted early: the off-diagonal blocks $P_i K_{ij} P_j^*$ and the interference matrix $M$ indicate, from the first few iterations, which groups of tasks will end up with aligned activity subspaces, before the alignment is visible in the hidden states themselves.","The decomposition carries over to any weight-based dynamical model—GRUs, LSTMs, neural ODEs, and biophysical networks such as Hodgkin–Huxley models—so the rank bound on $K$ gives a general formula for when gradient training of a recurrent system will collapse its dynamics."],"supporting_citations":[{"why":"Supplies the adjoint-method equations and the neural-ODE framing from which Proposition 1's gradient-flow factorization is algebraically derived.","marker":"[1]"},{"why":"Documents the low-dimensional attractor dynamics in trained RNNs, the phenomenon the paper claims to explain mechanistically.","marker":"[3]"},{"why":"Provides the memory-pro and multi-task paradigms and the shared-dynamical-motif findings that the multi-task experiment reproduces and predicts.","marker":"[5]"},{"why":"Provides the convergence-threshold convention for the memory-pro sweep and the prior simplicity-bias results the training-speed comparisons build on.","marker":"[7]"},{"why":"Introduces the neural tangent kernel that $K$ generalizes; the conceptual anchor for calling $K$ the parameter operator.","marker":"[13]"},{"why":"The recurrent NTK extension that the KPFlow decomposition generalizes and distinguishes itself from.","marker":"[14]"},{"why":"Supplies the Lyapunov-exponent machinery used to compute the action of $P$ and to interpret its spectrum.","marker":"[17]"}],"fun_headline_variants":["KPFlow splits GD flow into two operators that explain RNN collapse","Low-rank operator bottleneck forces RNN dynamics to collapse","KPFlow: How GD shapes recurrent networks into low-dim attractors","Operator decomposition reveals why RNN training leads to collapse"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that each real training step is a small perturbation accurately described by the continuous, linearized gradient flow, even though the experiments train with ADAM, adaptive learning rates, and gradient clipping—Appendix E.1 concedes that acceleration and learning-rate adjustment can make ADAM's solutions differ from the infinitesimal-learning-rate gradient flow, so if finite-step updates deviate substantially, especially from chaotic initial conditions at large weight scale $g$, the low rank of $K$ may not shape the actual training trajectory.","fun_headline_variants_meta":{"raw":{"variants":["KPFlow splits GD flow into two operators that explain RNN collapse","Low-rank operator bottleneck forces RNN dynamics to collapse","KPFlow: How GD shapes recurrent networks into low-dim attractors","Operator decomposition reveals why RNN training leads to collapse"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000478,"raw_usage":{"total_tokens":2457,"prompt_tokens":1124,"completion_tokens":1333,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":740,"completion_tokens_details":{"reasoning_tokens":1264}},"tokens_in":740,"tokens_out":1333,"duration_ms":11385,"temperature":1.0,"reasoning_tokens":1264,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:06:31.501135+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the memory-pro sweep again while recording, at every gradient-descent iteration, the empirically observed hidden-state change $\\delta z$ alongside the decomposition's prediction $-PKP^*(\\mathrm{Err})$ for the same error signal, and compare their effective ranks and cosine alignment across the full range of initial weight scales $g$, including the chaotic regime. If the real updates stay high-dimensional while $K$ remains low rank, or if the predicted and actual updates diverge sharply for large $g$, the claim that $K$'s rank drives collapse would not describe the actual training trajectory.","supporting_citations":[{"cited_title":"Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud","cited_arxiv_id":null,"evidence_quote":"Supplies the adjoint-method equations and the neural-ODE framing from which Proposition 1's gradient-flow factorization is algebraically derived."},{"cited_title":"Opening the black box: low-dimensional dynamics in high- dimensional recurrent neural networks","cited_arxiv_id":null,"evidence_quote":"Documents the low-dimensional attractor dynamics in trained RNNs, the phenomenon the paper claims to explain mechanistically."},{"cited_title":"Driscoll, Krishna V","cited_arxiv_id":null,"evidence_quote":"Provides the memory-pro and multi-task paradigms and the shared-dynamical-motif findings that the multi-task experiment reproduces and predicts."},{"cited_title":"The interplay between randomness and structure during learning in rnns","cited_arxiv_id":null,"evidence_quote":"Provides the convergence-threshold convention for the memory-pro sweep and the prior simplicity-bias results the training-speed comparisons build on."},{"cited_title":"Neural tangent kernel: Convergence and generalization in neural networks","cited_arxiv_id":null,"evidence_quote":"Introduces the neural tangent kernel that $K$ generalizes; the conceptual anchor for calling $K$ the parameter operator."},{"cited_title":"The recurrent neural tangent kernel","cited_arxiv_id":null,"evidence_quote":"The recurrent NTK extension that the KPFlow decomposition generalizes and distinguishes itself from."},{"cited_title":"On lyapunov exponents for rnns: Understanding information propagation using dynamical systems tools","cited_arxiv_id":null,"evidence_quote":"Supplies the Lyapunov-exponent machinery used to compute the action of $P$ and to interpret its spectrum."}],"review_version":1}