Skeleton-based Action Recognition with Non-linear Dependency Modeling and Hilbert-Schmidt Independence Criterion

Yuheng Yang

REVIEW 4 major objections 6 minor 46 references

Skeleton-based Action Recognition with Non-linear Dependency Modeling and Hilbert-Schmidt Independence Criterion

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that modeling all pairwise joint dependencies with a Gaussian kernel and adding a Hilbert-Schmidt independence objective to the loss yields state-of-the-art skeleton action recognition on three benchmarks.

desk verdict Novel graph refinement plus HSIC loss that looks good empirically, but the HSIC term is added with the wrong sign, so the stated mechanism contradicts the equations. read the letter →

arxiv 2412.18780 v1 pith:62SMOVTB submitted 2024-12-25 cs.CV cs.AIcs.LG

Yuheng Yang This is my paper · ORCID

classification cs.CVcs.AIcs.LG

keywords skeleton-basedactionrecognitiongraphconvolutionalnetworkHilbert-SchmidtindependencecriterionjointdependencymodelingGaussiancorrelationkernelHSIClossmulti-streamensembleNTURGB+D

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 establish that skeleton-based action recognition improves when two limitations of current graph-convolutional models are fixed directly: dependencies are modeled between every pair of joints rather than only connected ones, and action classes are separated by a Hilbert-Schmidt independence objective rather than by estimating probability densities in high-dimensional feature space. The authors propose a Gaussian correlation function that computes pairwise joint dependencies and uses them to refine the graph adjacency matrix, plus an HSIC-based loss that measures dependence between enhanced motion features and action labels in a reproducing kernel Hilbert space. They report state-of-the-art accuracy on NTU RGB+D 60, NTU RGB+D 120, and Northwestern-UCLA, with ablations attributing gains of roughly 1.4% to the dependency graph and 1.3% to the learning objectives over a base ST-GCN model. A sympathetic reader would care because the two components target known bottlenecks—distant-joint coordination in actions like kicking, and curse-of-dimensionality in density estimation—rather than just stacking parameters.

What carries the argument

The machinery is a graph convolutional update whose adjacency is refined by a Gaussian correlation function, coupled with an HSIC-based training objective. The Gaussian correlation $\mathcal{R}(v_i,v_j)=\exp(-(v_i-v_j)^2/(2\delta^2))$ turns every ordered pair of joint features into a dependency weight, and a learned linear map $\phi$ converts those weights into a tensor $\mathbf{R}\in\mathbb{R}^{N\times N\times C'}$ that is broadcast into the adjacency; the kernel width $\delta$ tunes how quickly dependency decays with joint distance. The HSIC objective uses a Mat\'ern kernel with $\eta=3/2$ to compute the kernel matrix of enhanced features, centers it, and measures dependence with the class-label kernel matrix; because the HSIC value is computed from kernel inner products, the authors argue the classification step is dimension-agnostic. The total loss stacks the HSIC-augmented classification loss, a cross-entropy loss on auxiliary predictions, and a KL distillation loss between auxiliary and base logits.

What would settle it

Train the same framework on NTU RGB+D 120 X-Sub with the dependency matrix $\mathbf{R}$ replaced by noise of the same magnitude; if accuracy stays at 86.0%, the gains credited to dependency refinement are not due to the learned joint dependencies. A second check is a kernel-width sweep, e.g. $\delta\in\{0.5,1,2,4,8,16\}$ on the same split: if the accuracy curve has no plateau or the best width differs strongly across action classes, the claim of fine-grained distance control needs qualification.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that jointly explicit non-linear joint dependencies and a Hilbert-Schmidt independence criterion make skeleton action classification more accurate, with each component contributing independently. The Gaussian correlation function $\mathcal{R}(v_i, v_j) = \exp\left(- (v_i - v_j)^2 / (2\delta^2)\right)$ produces a dependency matrix $\mathbf{R}$ that is added to the skeleton adjacency to form the refined adjacency $\mathbf{A}_c = \mathbf{A} + \mathbf{W}_c \mathbf{R}$, so the graph at each layer is reweighted by feature similarity between every pair of joints. Independently, the HSIC objective maps enhanced features through a Mat\'ern kernel, computes $\mathrm{HSIC}(\hat{z}, y) = \mathrm{tr}(\mathbf{K}_{\hat{z}} \mathbf{H} \mathbf{K}_y \mathbf{H}) / (n-1)^2$, and adds it to the classification loss, with a distillation loss aligning base and auxiliary models. The reported headline results are 93.7% and 97.3% on NTU RGB+D 60 (X-Sub, X-View), 90.6% and 91.7% on NTU RGB+D 120 (X-Sub, X-Set), and 97.2% on Northwestern-UCLA.

Load-bearing premise

The whole approach assumes that the Gaussian correlation score computed from raw joint feature vectors, with fixed kernel widths $\delta\in\{1,9\}$, captures discriminative information about which joints move together; if those similarities are dominated by noise or the fixed width is wrong for a given action, the refined adjacency adds distortion rather than signal, and the reported gains would not be attributable to the model.

Editorial extensions

If this is right

Actions involving coordinated distant joints, such as kicking or staggering, should be recognized more accurately because the dependency graph no longer requires those joints to be connected in the skeleton.
The HSIC objective should keep working as feature dimensionality grows, since it measures dependence through kernel matrices rather than estimated densities, so the framework may scale to richer pose representations.
Multi-stream ensembling over kernel widths (small and large) and modalities (joint and bone) adds consistent accuracy gains over single streams, e.g. +4.6% on NTU120 X-Sub joint-only.
Removing either the HSIC term or the distillation loss from the total objective decreases accuracy by about 0.8 to 0.9% on NTU120 X-Sub, implying both terms carry information that the classification loss alone does not.

Reading between the lines

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

Editorial inference: the HSIC term likely functions as a class-conditional cluster-separation regularizer; this would explain the observed t-SNE separation and suggests the same loss could improve other high-dimensional classification tasks beyond skeletons.
Editorial inference: the dependence on hand-picked kernel widths (1 and 9) suggests an adaptive or learned per-joint-pair $\delta$ might yield further gains, but the paper does not explore this.
Editorial inference: because the dependency matrix is computed from raw feature vectors, its quality depends on feature normalization; a testable extension is to apply the refinement in later layers or on learned embeddings rather than raw coordinates.
Editorial inference: the same all-pairs dependency plus HSIC recipe could transfer to other graph-structured time series, such as multi-agent motion or object interaction forecasting, though the paper only claims skeleton actions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I've read this one twice, and there's a real problem at the center. The proposed combination – a Gaussian-kernel dependency refinement over all joint pairs plus an HSIC-based objective – is novel, and the empirical gains (0.5–1.1% over strong baselines on NTU60, NTU120, NW-UCLA) look consistent across ablations. The idea of refining the adjacency matrix with pairwise Gaussian correlations is concrete, and the ablation tables do attribute gains to both components. Credit where due: the experiments are extensive, and the writing is clear enough to see what was intended.

But the intended mechanism doesn't match the equations. Eq. (9) defines L_H = L_cls + HSIC(z_hat, y), and everything is minimized. HSIC is nonnegative, so minimizing it drives the enhanced feature z_hat toward independence from the label y. That is the opposite of 'differentiating action classes.' If the implementation actually subtracts HSIC, then the stated objective is wrong. If it literally adds, then the reported accuracy gains can't be explained by the paper's own rationale. Either way, the central attribution is unsupported as written. This is not a minor typo; it's load-bearing.

Other soft spots are fixable but real: Eq. (4) says the dependency matrix R is added in a 'broadcast manner' without specifying dimensions; the Matern kernel parameters alpha and l are never defined; the claimed 'mathematical derivation' is really just a loss sum; and the code release is announced but no link is present. The kernel-width sensitivity for delta is hand-waved with only two values.

So: the empirical results may be real, and the all-pairs dependency graph could be reusable. But the HSIC sign inversion makes the paper incoherent on its own terms. I'd send it to a serious referee because the flaw is concrete and fixable, and the empirical work deserves scrutiny. Not a desk reject. But as it stands, I wouldn't cite it, and I wouldn't bring it to reading group except as a cautionary example.

My recommendation: major revision, with the authors required to either fix the sign and explain the mechanism, or supply code showing the actual loss.

Referee Report

4 major / 6 minor

Summary. This paper proposes a skeleton-based action recognition method with two components: (1) a dependency refinement module that augments the skeleton adjacency matrix with all-pairs joint dependencies computed by a Gaussian correlation function, trained with an ensemble over kernel widths and joint/bone streams; and (2) an HSIC-based learning objective that maps enhanced features into an RKHS with a Matérn kernel and adds an HSIC term to the classification loss. The authors report state-of-the-art accuracies on NTU RGB+D 60, NTU RGB+D 120, and Northwestern-UCLA, with ablations attributing the gains to both components.

Significance. If the proposed mechanism were fully supported, the main contribution would be positive but incremental: a new application of HSIC to skeleton-based action recognition and a simple all-pairs dependency refinement. The reported gains over strong baselines are modest (about 0.9% on NTU60 X-Sub and 0.8–1.1% on NTU120), and the ablation story is internally consistent. The manuscript states that implementations are released, which is a strength, although no repository link is given. However, the central theoretical claim is not currently supported: the paper announces a mathematical derivation of learning objectives, but only presents definitional loss sums, and the sign of the HSIC term appears to contradict the stated mechanism of using HSIC to separate action classes.

major comments (4)

[Section 3, Eq. (9)] The HSIC term is added to L_cls and the total loss is minimized. Since HSIC is nonnegative (Eq. (8)), minimizing Eq. (9) drives HSIC(z_hat, y) toward zero, i.e., toward statistical independence between the enhanced feature z_hat and the label y. This is the opposite of the paper's stated goal of using HSIC to 'differentiate action classes.' Table 5 reports that removing HSIC lowers accuracy by 0.9%, which is difficult to explain if the written objective penalizes dependence. Either the loss should be L_cls - lambda * HSIC, or the mechanism must be explained; as written, the attribution of the accuracy gain to the HSIC term is unsupported.
[Abstract and Section 3] The abstract claims that the authors 'mathematically derive learning objectives guaranteeing precise recognition.' The methodology section only defines loss terms in Eqs. (9)–(11); there is no theorem, proposition, or derivation connecting HSIC to classification guarantees. This overclaims a theoretical contribution. The authors should either provide a real derivation or revise the claim to describe Eq. (9) as an empirical objective.
[Equations (2)–(4)] The dependency refinement is not rigorously specified. In Eq. (2), (v_i - v_j)^2 is not defined for vectors; in Eq. (3), R(v_i, v_j) is a scalar but r_ij is said to be in R^{C'}; and in Eq. (4), A_c = A + W_c R with R in R^{N x N x C'} cannot be added directly to A in R^{N x N} without specifying how W_c acts on the channel dimension. The phrase 'broadcast manner' does not resolve this ambiguity. Please provide exact tensor shapes and the reduction over the C' dimension, since this operation is central to the proposed graph refinement.
[Implementation Details and Eq. (6)] The Matérn kernel in Eqs. (5)–(6) has hyperparameters alpha and l that are never specified in the experimental settings; only the Gaussian-correlation widths delta in {1,9} are reported. Since the HSIC objective is a central component, the missing alpha and l values make the experiments difficult to reproduce and leave the sensitivity of the method to these parameters unexamined.

minor comments (6)

[Eq. (2)] Replace (v_i - v_j)^2 with ||v_i - v_j||^2 or explicitly state that the operation is elementwise, followed by a sum or norm.
[Eq. (8)] The kernel matrix K_y for the label y is not defined; please specify, for example, whether a delta kernel on class labels is used.
[Tables 4 and 5] There are formatting typos: '2.9M86.0' in Table 4 and 'LT otal' in Table 5 should be corrected.
[Implementation details] The statement that 'implementations have been released' should include a URL or repository identifier.
[Abstract and Section 3] The phrase 'without being affected by data dimensionality' is imprecise; HSIC avoids explicit density estimation, but the kernel values still depend on the geometry of the feature space.
[Table 1] The rows 'Ours' and 'Ours (4 ensemble)' report identical numbers; clarify whether 'Ours' refers to the 4-ensemble model.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the HSIC objective and dependency refinement are defined from inputs/labels, and self-citations are not load-bearing.

full rationale

The paper's two claimed contributions are (1) a Gaussian-kernel dependency refinement that augments the graph adjacency matrix, and (2) an HSIC-based learning objective. Neither reduces to its own inputs by construction. In Eq. (4), the refined adjacency Ac = A + WcR is computed from input joint features via Eqs. (2)-(3); this is feature-dependent graph modeling, not a fit to the action labels or to the model's own predictions. The HSIC objective in Eq. (9), LH = Lcls + HSIC(z_hat, y), is defined from feature and label kernel matrices; it is a regularizer added to the classification loss, not a quantity derived from the model's outputs, so it is not self-definitional. The self-citations (e.g., Stream-GCN, Yang et al. 2023) appear only as related work and experimental baselines; they are not invoked to justify the novelty or to force the reported accuracy gains. The abstract's phrase 'mathematically derive learning objectives guaranteeing precise recognition' is not supported by an actual derivation; Eq. (9) is presented as a definition. That is an omitted-support concern, not circularity. Likewise, the skeptical observation that HSIC is nonnegative and that Eq. (9) minimizes it (driving z_hat toward independence from y) is a genuine internal-consistency/sign issue, but it is not a reduction of the result to its inputs; it concerns correctness of the proposed mechanism, not circularity. Overall, the central derivation chain is self-contained against external benchmarks, and no load-bearing circular step is present.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claims rest on hand-chosen kernel widths and unspecified Matern kernel parameters. The HSIC-based objective is asserted rather than derived, and the graph refinement introduces an ambiguous broadcast operation. No new physical or conceptual entities are postulated.

free parameters (3)

Gaussian kernel width delta = small=1, large=9
Hand-picked widths control the distance decay in the dependency function. The paper does not report a tuning procedure or sensitivity analysis.
Matern kernel amplitude alpha
The kernel in Eq. (5)-(6) depends on alpha, but the paper never states a value, a learning rule, or a default.
Matern kernel length scale l
The kernel also depends on the length scale l, which is not specified or learned in the described framework.

assumptions (3)

domain assumption Maximizing HSIC between motion features and labels improves classification accuracy
The learning objective in Eq. (9) adds HSIC to cross-entropy without a formal proof. The paper claims a mathematical derivation but no theorem or guarantee is stated.
domain assumption Gaussian correlation of joint feature vectors captures action-relevant non-linear dependencies between distant joints
Eq. (2) defines R as a fixed Gaussian function of feature differences. The paper assumes this measure is informative for actions like kicking, but does not demonstrate that feature-space distance corresponds to action-relevant coordination.
ad hoc to paper The broadcast addition Ac = A + WcR produces a valid adjacency for graph convolution
Eq. (4) is stated with 'broadcast manner'; the dimensional compatibility of R in R^{N x N x C'} with A in R^{N x N} is not rigorously specified, which affects the validity of the update rule.

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Cite this review

Pith. "Pith review of Skeleton-based Action Recognition with Non-linear Dependency Modeling and Hilbert-Schmidt Independence Criterion." pith.science (2026). https://pith.science/paper/62SMOVTB

@misc{pith2026241218780,
  author       = {Pith},
  title        = {Pith review of: Skeleton-based Action Recognition with Non-linear Dependency Modeling and Hilbert-Schmidt Independence Criterion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/62SMOVTB}},
  note         = {Machine review of arXiv:2412.18780}
}

read the original abstract

Human skeleton-based action recognition has long been an indispensable aspect of artificial intelligence. Current state-of-the-art methods tend to consider only the dependencies between connected skeletal joints, limiting their ability to capture non-linear dependencies between physically distant joints. Moreover, most existing approaches distinguish action classes by estimating the probability density of motion representations, yet the high-dimensional nature of human motions invokes inherent difficulties in accomplishing such measurements. In this paper, we seek to tackle these challenges from two directions: (1) We propose a novel dependency refinement approach that explicitly models dependencies between any pair of joints, effectively transcending the limitations imposed by joint distance. (2) We further propose a framework that utilizes the Hilbert-Schmidt Independence Criterion to differentiate action classes without being affected by data dimensionality, and mathematically derive learning objectives guaranteeing precise recognition. Empirically, our approach sets the state-of-the-art performance on NTU RGB+D, NTU RGB+D 120, and Northwestern-UCLA datasets.

Figures

Figures reproduced from arXiv: 2412.18780 by the authors.

**Figure 2.** The diagram illustrates the dependency refine [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** The overall pipeline of our HSIC-based framework aims at recognizing the action classes of the motion sequences. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: The histograms show the quantitative results of ap [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 4.** Figure 4: The visualization illustrates feature representations [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

Bertsimas, D.; and Koduri, N. 2022. Data-driven optimization: A reproducing kernel Hilbert space approach. Operations Research, 70(1): 454--471

work page 2022

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Chen, Y.; Zhang, Z.; Yuan, C.; Li, B.; Deng, Y.; and Hu, W. 2021 a . Channel-wise topology refinement graph convolution for skeleton-based action recognition. In Proceedings of the IEEE/CVF International Conference on Computer Vision, 13359--13368

work page 2021

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Chen, Z.; Li, S.; Yang, B.; Li, Q.; and Liu, H. 2021 b . Multi-scale spatial temporal graph convolutional network for skeleton-based action recognition. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, 1113--1122

work page 2021

[6] [6]

Cheng, K.; Zhang, Y.; Cao, C.; Shi, L.; Cheng, J.; and Lu, H. 2020 a . Decoupling gcn with dropgraph module for skeleton-based action recognition. In European Conference on Computer Vision, 536--553. Springer

work page 2020

[7] [7]

Cheng, K.; Zhang, Y.; He, X.; Chen, W.; Cheng, J.; and Lu, H. 2020 b . Skeleton-based action recognition with shift graph convolutional network. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 183--192

work page 2020

[8] [8]

H.; Chi, S.; Lee, S

Chi, H.-g.; Ha, M. H.; Chi, S.; Lee, S. W.; Huang, Q.; and Ramani, K. 2022. Infogcn: Representation learning for human skeleton-based action recognition. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 20186--20196

work page 2022