Pith. sign in

REVIEW 3 major objections 4 minor 26 references

The Convergence of Dynamic Routing between Capsules

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Dynamic routing between capsules has a hidden concave objective.

desk verdict The objective identification for dynamic routing is real and checkable, but the advertised convergence to a local optimum is asserted, not proved, and is false in a simple interior case. read the letter →

arxiv 2501.06240 v1 pith:KPPCERRY submitted 2025-01-08 cs.LG math.OC

Daoyuan Ye , Juntao Li , Yiting Shen This is my paper · ORCID
classification cs.LGmath.OC MSC 90C2690C3068T07
keywords capsulenetworksdynamicroutingconvergenceanalysisconcaveobjectivefunctionnonlineargradientdescentconvexoptimizationpolarization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to settle a question that has lingered around capsule networks: does the iterative routing procedure actually optimize something? It answers yes. The dynamic routing algorithm is cast as a nonlinear gradient method minimizing a specific concave energy over matrices whose rows are probability distributions, and the paper argues this yields a rigorous convergence proof. If the argument is right, the routing's well-known tendency to polarize link strengths is not a side effect but a mathematical necessity, and the practical problem becomes redesigning the objective's constraints rather than tuning iteration counts.

What carries the argument

The engine of the paper is a descent inequality for a concave energy built from $\psi(z)=z-\arctan z$. The routing couplings $C$ are row-stochastic, and the energy $\Psi(C)=-\sum_{j=1}^N \psi(\|\hat{U}_j C(:,j)\|)$ is shown to be concave by composing this increasing convex $\psi$ with the norm. The update is rewritten as a nonlinear gradient step on the logit matrix $B$: $B(r+1)=B(r)-\nabla_C\Psi(C(r))$, then $C(r+1)=\nabla_B\Phi(B(r+1))$, with $\Phi(B)=\sum_i \phi(B(i,:))$ the sum of log-sum-exp functions. The proof uses a decomposition $\Psi=F-G$ with $F=\Phi^*$ and $G=\Phi^* - \Psi$; Lemma 2.4 converts convexity of $F-\frac12\|C\|_F^2$ and $G-\frac12\|C\|_F^2$ into $\Psi(C(r))-\Psi(C(r+1))\ge\|C(r)-C(r+1)\|_F^2$. This inequality is what carries the monotonicity and the claimed convergence.

What would settle it

Run Algorithm 2 on a random fixed set of prediction vectors $\hat{u}_{j|i}$ and record $C(r)$ and $\Psi(C(r))$ for many iterations; if $C(r)$ ever enters a non-convergent cycle or drifts forever while $\Psi(C(r))-\Psi(C(r+1)) \ge \|C(r)-C(r+1)\|_F^2$ holds in floating point, the convergence-to-a-local-optimum claim fails even though the descent inequality holds.

Watch

Extended reading notes

Core claim

The central claim is that the dynamic routing algorithm from the original capsule architecture is the realization of a nonlinear gradient method on a constrained optimization problem with an explicit closed-form objective. The objective is $E(C) = -\sum_{j=1}^{N} \psi(\|\hat{U}_j C(:,j)\|)$ with $\psi(z) = z - \arctan z$, minimized over coupling matrices $C$ whose rows satisfy $C(i,:)e=1$ and $C(i,j)\ge 0$. In matrix form the routing update becomes $B(r+1)=B(r)-\nabla_C \Psi(C(r))$ followed by $C(r+1)=\nabla_B \Phi(B(r+1))$, where $\Phi$ is the sum of log-sum-exp row functions; Theorem 3.1 establishes the energy decrease $\Psi(C(r))-\Psi(C(r+1)) \ge \|C(r)-C(r+1)\|_F^2$, so the energy is strictly monotone and bounded. The paper concludes that the iterates converge to a local optimum of this concave problem, giving dynamic routing the mathematical convergence proof it previously lacked.

Load-bearing premise

The load-bearing premise is that a bounded, monotonically decreasing energy $\Psi$ together with the routing update forces the coupling matrices themselves to converge to a local optimum; the paper states this as a consequence of a basic property of discrete dynamical systems but does not prove the limit exists at the boundary where couplings become one-hot.

Editorial extensions

If this is right

  • Dynamic routing is not an unsupervised heuristic: it is minimizing $E(C)$ under row-stochastic constraints, so its behavior can be analyzed with existing optimization theory.
  • Each routing iteration strictly decreases the energy by at least $\|C(r)-C(r+1)\|_F^2$, so the algorithm cannot cycle while the descent inequality holds.
  • Polarization is intrinsic: minimizing a concave function over the simplex pushes couplings toward the boundary, which explains why extra iterations typically do not improve and can hurt classification.
  • The proof machinery generalizes to a broad class of capsule routing algorithms that share the same energy form, giving them a common convergence framework.
  • Changing the constraints can split the optimization into separable subproblems, potentially reducing routing's computational cost.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An immediate corollary not developed in the paper: because a concave function over a polytope attains its minimum at an extreme point, the observed one-hot polarization of couplings is the expected outcome of running the routing to convergence; any fix must change the objective or constraints, not the iteration count.
  • If the missing convergence step cannot be supplied, the paper's theorem should be read as a monotonicity result rather than a full convergence proof, since a bounded monotone energy alone does not force the iterates themselves to converge.
  • A testable extension: add a small entropy regularizer to the objective or soften the softmax temperature, and measure whether the energy landscape's boundary minima become interior, reducing the polarization the experiments visualize.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the dynamic routing algorithm of capsule networks (Sabour et al., 2017). Its central claims are that the routing procedure minimizes the concave objective Ψ(C) = -Σ_j ψ(||U_j C(:,j)||) with ψ(z) = z - arctan(z) under the constraint that each row of C is a probability vector, that the routing updates constitute a nonlinear gradient method, and that the algorithm converges, with a rigorous proof given in Theorem 3.1. The paper also reports numerical experiments visualizing the monotone behavior of Ψ and the polarization of coupling coefficients.

Significance. If correct, the paper would give the first closed-form objective for dynamic routing, resolve the tension between routing's empirical success and the absence of a known objective, and explain polarization as a consequence of minimizing a concave function over the simplex. The identification of Ψ and the derivation of the monotonicity inequality in Section III.B are genuine contributions: the algebraic equivalence between Algorithm 1 and the matrix-form update is sound, and the inequality Ψ(C(r)) - Ψ(C(r+1)) ≥ ||C(r) - C(r+1)||²_F is a useful quantitative statement. However, the headline convergence claim is not established, and the stated conclusion that C(r) converges to a local optimum is false as written. The paper's main theorem therefore does not deliver what the abstract promises.

major comments (3)
  1. [§III.C, Theorem 3.1] The proof's final inference is invalid. The paper shows Ψ(C(r)) - Ψ(C(r+1)) ≥ ||C(r) - C(r+1)||²_F and then asserts: 'Based on the basic property of the convergence of discrete dynamical system, we can get the convergence of C(r).' This is not a theorem and is not proved. Monotone decrease of a bounded scalar sequence implies convergence of Ψ(C(r)), and the square-summability of successive differences gives ||C(r+1) - C(r)|| → 0, but on a compact set this only yields subsequential convergence, not convergence of the full iterate sequence. The missing step is load-bearing because it is the bridge from monotonicity to the abstract's claim of a 'strictly proved' convergence.
  2. [§III.C, Theorem 3.1 / Theorem 3.2] The stated conclusion that C(r) converges to 'the local optimal solution' is false. Since Ψ is concave, interior stationary points are maxima or saddle points of the minimization problem, not minima. A concrete counterexample is M = N = 2, U_1 = [u, -u], U_2 = [v, -v], and B(0) = 0. Then C(0) has all entries 1/2, every s_j = 0, every v_j = 0, and B(r+1) = B(r) for all r. The algorithm is fixed at C(0), where Ψ = 0. Because ψ(z) ≥ 0 for all z, Ψ(C) ≤ 0 for every feasible C, so C(0) is a global maximum of the minimization objective, not a local minimum. The boundary or one-hot regime emphasized by the experiments is likewise not analyzed; the fixed-point argument used in the interior does not extend to it.
  3. [§III.C, Corollary 3.2 and Lemma 2.4] The convexity assertions in Corollary 3.2 are stated without a domain. From Corollary 2.2, φ*(y) - (1/2)y^T y is convex only on the probability simplex {y ≥ 0, Σ y_i = 1}, not on all of R^N. The proof of Theorem 3.1 applies Lemma 2.4 with F(C) = Φ*(C), and the omission of the domain leaves the application unjustified as written. The inequality can be recovered by restricting to the feasible set of C, but the manuscript must state this domain explicitly and verify that the updates remain in it.
minor comments (4)
  1. [§IV.B, Figure 1] The text says 'the line above in blue color represents the value of C in each iteration,' but Equation (16), which the experiment reports, is the energy function Ψ(C); this should be clarified to avoid confusing the matrix C with the scalar objective.
  2. [§III.C, Theorem 3.1] The phrase 'strictly monotone decreasing' is stronger than what the inequality proves; the inequality Ψ(C(r)) - Ψ(C(r+1)) ≥ ||C(r) - C(r+1)||²_F gives non-strict monotonicity unless consecutive iterates are distinct.
  3. [§II.A, Theorem 2.1] The statement reads 'ψ′(z) > 0, ψ′(z) ≥ 0'; the second condition should be ψ''(z) ≥ 0, as used in the proof.
  4. [§III.C, Corollary 3.2 proof] Equation (21) contains a typo: 'ϕ(B(i,ˆi))' should be 'φ(B(i,:))'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the energy function is reverse-engineered from the routing updates, not assumed as an input to derive them.

full rationale

The paper's central derivation in Section III.B takes the pre-existing dynamic routing Algorithm 1 and rewrites it in matrix form. The squash function is identified as the gradient of ψ(z)=z−arctan z, and the softmax coupling is identified as the gradient of the log-sum-exp function φ. Equation (16) then defines Ψ as the negative sum of the ψ terms, so by construction the routing update becomes B(r+1)=B(r)−∇_C Ψ(C(r)) and C(r+1)=∇_B Φ(B(r+1)). This is a variational reformulation of a given algorithm, not a fitted parameter renamed as a prediction, and not a result that assumes the objective it claims to recover. The monotone descent inequality in Theorem 3.1 is then derived from convex-conjugate properties (Lemma 2.4, Corollary 3.2), which are external convex-analysis facts, not consequences of assuming convergence. No parameters are fitted to data, no load-bearing uniqueness theorem from the authors' prior work is invoked, and no claim is justified solely by a self-citation. The paper's final assertion that C(r) converges to a local optimum relies on an unsupported 'basic property of the convergence of discrete dynamical system' and is open to correctness objections, but that is a proof-completeness gap, not circularity: the stated energy decrease does not reduce by construction to the routing update in a way that makes the conclusion equivalent to the input. Under the circularity taxonomy, the derivation is self-contained and non-circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central derivation is self-contained: one differentiation identity (∇_s ψ(||s||) equals the Sabour squash) plus standard convex analysis assemble the routing loop into a nonlinear gradient descent with explicit objective. The ledger entries that actually carry risk are the unstated domain restriction in Corollary 3.2 (convexity of Φ*(C) - (1/2)tr(C^T C) holds on the row-stochastic simplex, not on all of R^{M×N}) and the unproven convergence-to-local-optimum step at the end of Theorem 3.1. There are no fitted parameters and no invented entities; ψ(z) = z - arctan z is forced by the squash, not chosen to fit data.

assumptions (4)
  • standard math Standard convex analysis background: convexity definitions and criteria, conjugate functions, log-sum-exp conjugacy φ*(y) = Σ y_i ln y_i on the probability simplex, as given in Boyd and Vandenberghe [26].
    Invoked throughout Section II and in Corollaries 3.1 and 3.2; the paper explicitly defers proofs to [26] and [27].
  • domain assumption The routing loop of Algorithm 1 is exactly B(r+1) = B(r) - ∇Ψ(C(r)) with C(r) = ∇Φ(B(r)), which rests on the identity ∇_s ψ(||s||) = squash(s) for ψ(z) = z - arctan z.
    Section III.B, Equation (15) and surrounding text. The identity is verified by direct differentiation and is correct, but it is the hinge that maps the algorithm to the objective.
  • ad hoc to paper Convexity of Φ*(C) - (1/2)tr(C^T C) is asserted without stating the domain; it holds on the row-stochastic simplex where entries are at most 1, but not on all of R^{M×N}.
    Corollary 3.2(2) and the proof of Theorem 3.1. Off the simplex, the Hessian diag(1/y_i - 1) is indefinite when some y_i > 1; the paper never states the restriction to the simplex.
  • ad hoc to paper Monotone decrease of a bounded objective implies convergence of the iterate sequence to a local optimum of the constrained problem (the unnamed 'basic property of the convergence of discrete dynamical system').
    End of Theorem 3.1 proof. This premise carries the headline convergence claim; it is neither proven nor cited to a specific theorem, and the boundary case of diverging logits is unhandled.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Convergence of Dynamic Routing between Capsules." pith.science (2026). https://pith.science/paper/KPPCERRY

@misc{pith2026250106240,
  author       = {Pith},
  title        = {Pith review of: The Convergence of Dynamic Routing between Capsules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPPCERRY}},
  note         = {Machine review of arXiv:2501.06240}
}
read the original abstract

Capsule networks(CapsNet) are recently proposed neural network models with new processing layers, specifically for entity representation and discovery of images. It is well known that CapsNet have some advantages over traditional neural networks, especially in generalization capability. At the same time, some studies report negative experimental results. The causes of this contradiction have not been thoroughly analyzed. The preliminary experimental results show that the behavior of routing algorithms does not always produce good results as expected, and in most cases, different routing algorithms do not change the classification results, but simply polarize the link strength, especially when they continue to repeat without stopping. To realize the true potential of the CapsNet, deep mathematical analysis of the routing algorithms is crucial. In this paper, we will give the objective function that is minimized by the dynamic routing algorithm, which is a concave function. The dynamic routing algorithm can be regarded as nonlinear gradient method to solving an optimization algorithm under linear constraints, and its convergence can be strictly proved mathematically. Furthermore, the mathematically rigorous proof of the convergence is given for this class of iterative routing procedures. We analyze the relation between the objective function and the constraints solved by the dynamic routing algorithm in detail, and perform the corresponding routing experiment to analyze the effect of our convergence proof.

Figures

Figures reproduced from arXiv: 2501.06240 by the authors.

Figure 2
Figure 2. The distribution map on the input of l+1 layer. Sparse points represent the prediction values of different capsules. The asterisks represent the final output of dynamic algorithm for each capsule. In [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 1
Figure 1. The values of objective function Ψ (Equation 16) for all capsules and each capsule during routing process In [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 22 canonical work pages

  1. [1]

    Dynamic routing between capsules,

    S. Sabour, N. Frosst, and G. E. Hinton, “Dynamic routing between capsules,” Advances in neural information processing systems , vol. 30, 2017

    2017

  2. [2]

    Matrix capsules with em routing,

    G. E. Hinton, S. Sabour, and N. Frosst, “Matrix capsules with em routing,” in International conference on learning representations , 2018

    work page 2018

  3. [3]

    Fast capsnet for lung cancer screening,

    A. Mobiny and H. Van Nguyen, “Fast capsnet for lung cancer screening,” in International conference on medical image computing and computer- assisted intervention. Springer, 2018, pp. 741–749

    work page 2018

  4. [4]

    An attention- based bi-gru-capsnet model for hypernymy detection between compound entities,

    Q. Wang, C. Xu, Y . Zhou, T. Ruan, D. Gao, and P. He, “An attention- based bi-gru-capsnet model for hypernymy detection between compound entities,” in 2018 IEEE International Conference on Bioinformatics and Biomedicine (BIBM). IEEE, 2018, pp. 1031–1035

    work page 2018

  5. [5]

    Capsulepose: A variational capsnet for real- time end-to-end 3d human pose estimation,

    N. Garau and N. Conci, “Capsulepose: A variational capsnet for real- time end-to-end 3d human pose estimation,” Neurocomputing, vol. 523, pp. 81–91, 2023

    work page 2023

  6. [6]

    Emotion recognition from multiband eeg signals using capsnet,

    H. Chao, L. Dong, Y . Liu, and B. Lu, “Emotion recognition from multiband eeg signals using capsnet,” Sensors, vol. 19, no. 9, p. 2212, 2019

    work page 2019

  7. [7]

    Hyperspectral image classification using capsnet with well-initialized shallow layers,

    J. Yin, S. Li, H. Zhu, and X. Luo, “Hyperspectral image classification using capsnet with well-initialized shallow layers,” IEEE Geoscience and Remote Sensing Letters , vol. 16, no. 7, pp. 1095–1099, 2019

    work page 2019

  8. [8]

    Remote sensing image scene clas- sification using cnn-capsnet,

    W. Zhang, P. Tang, and L. Zhao, “Remote sensing image scene clas- sification using cnn-capsnet,” Remote Sensing , vol. 11, no. 5, p. 494, 2019

    work page 2019

Show all 26 references
  1. [10]

    Capsules for object segmentation,

    R. LaLonde and U. Bagci, “Capsules for object segmentation,” arXiv preprint arXiv:1804.04241, 2018

    2018 arXiv

  2. [11]

    Ss-3dcapsnet: Self-supervised 3d capsule networks for medical segmentation on less labeled data,

    M. Tran, L. Ly, B.-S. Hua, and N. Le, “Ss-3dcapsnet: Self-supervised 3d capsule networks for medical segmentation on less labeled data,” in 2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI). IEEE, 2022, pp. 1–5

    2022

  3. [12]

    Capsnet topology to classify tumours from brain images and comparative evaluation,

    E. Goceri, “Capsnet topology to classify tumours from brain images and comparative evaluation,” IET Image Processing , vol. 14, no. 5, pp. 882–889, 2020

    2020

  4. [13]

    Skin cancer classification using convolutional capsule network (capsnet),

    S. C. Satapathy, M. Cruz, A. Namburu, S. Chakkaravarthy, M. Pitten- dreigh et al. , “Skin cancer classification using convolutional capsule network (capsnet),” Journal of Scientific & Industrial Research , vol. 79, no. 11, pp. 994–1001, 2020

    2020

  5. [14]

    Vehicle-related scene segmentation using capsnets,

    X. Liu, W. Q. Yan, and N. Kasabov, “Vehicle-related scene segmentation using capsnets,” in 2020 35th International Conference on Image and Vision Computing New Zealand (IVCNZ) . IEEE, 2020, pp. 1–6

    2020

  6. [15]

    Comparative study of capsule neural network in various applications,

    T. Vijayakumar, “Comparative study of capsule neural network in various applications,” Journal of Artificial Intelligence , vol. 1, no. 01, pp. 19–27, 2019

    2019

  7. [16]

    Pt-capsnet: A novel prediction-tuning capsule network suitable for deeper architectures,

    C. Pan and S. Velipasalar, “Pt-capsnet: A novel prediction-tuning capsule network suitable for deeper architectures,” in Proceedings of the IEEE/CVF International Conference on Computer Vision , 2021, pp. 11 996–12 005

    2021

  8. [17]

    Capsule networks–a survey,

    M. K. Patrick, A. F. Adekoya, A. A. Mighty, and B. Y . Edward, “Capsule networks–a survey,” Journal of King Saud University-computer and information sciences, vol. 34, no. 1, pp. 1295–1310, 2022

    2022

  9. [18]

    Deep reinforcement learning using capsules in ad- vanced game environments,

    P.-A. Andersen, “Deep reinforcement learning using capsules in ad- vanced game environments,” arXiv preprint arXiv:1801.09597 , 2018

    2018 arXiv

  10. [19]

    Capsule network performance on complex data,

    E. Xi, S. Bing, and Y . Jin, “Capsule network performance on complex data,” arXiv preprint arXiv:1712.03480 , 2017

    2017 arXiv

  11. [20]

    Capsnet comparative performance evaluation for image classification,

    R. Mukhometzianov and J. Carrillo, “Capsnet comparative performance evaluation for image classification,” arXiv preprint arXiv:1805.11195 , 2018

    2018 arXiv

  12. [21]

    Ms-capsnet: A novel multi-scale capsule network,

    C. Xiang, L. Zhang, Y . Tang, W. Zou, and C. Xu, “Ms-capsnet: A novel multi-scale capsule network,” IEEE Signal Processing Letters , vol. 25, no. 12, pp. 1850–1854, 2018

    2018

  13. [22]

    An optimization view on dynamic routing between capsules,

    D. Wang and Q. Liu, “An optimization view on dynamic routing between capsules,” 2018. [Online]. Available: https://openreview.net/ forum?id=HJjtFYJDf

    2018

  14. [23]

    Capsule networks need an improved routing algorithm,

    I. Paik, T. Kwak, and I. Kim, “Capsule networks need an improved routing algorithm,” in Asian Conference on Machine Learning. PMLR, 2019, pp. 489–502

    2019

  15. [24]

    Dynamic capsule attention for visual question answering,

    Y . Zhou, R. Ji, J. Su, X. Sun, and W. Chen, “Dynamic capsule attention for visual question answering,” in Proceedings of the AAAI conference on artificial intelligence , vol. 33, no. 01, 2019, pp. 9324–9331

    2019

  16. [25]

    Group equivariant capsule networks,

    J. E. Lenssen, M. Fey, and P. Libuschewski, “Group equivariant capsule networks,” Advances in neural information processing systems , vol. 31, 2018

    2018

  17. [26]

    Boyd and L

    S. Boyd and L. Vandenberghe, Convex Optimization . Cambridge University Press, 2004

    2004

  18. [27]

    Improved concave-convex procedure and its application to analysis for the stability of hopfield neural network,

    S. Ye and W. Wang, “Improved concave-convex procedure and its application to analysis for the stability of hopfield neural network,” in 2010 3rd International Conference on Advanced Computer Theory and Engineering (ICACTE), vol. 2. IEEE, 2010, pp. V2–173

    2010

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.