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REVIEW 3 major objections 4 minor 53 references

Fitting, Comparison, and Alignment of Trajectories on Positive Semi-Definite Matrices with Application to Action Recognition

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that a skeleton-only geometric pipeline—Gram-matrix trajectories on a fixed-rank PSD manifold, denoised by curve fitting and aligned with a Global Alignment Kernel—matches or beats deep learning baselines on three…

desk verdict The closed-form distance in Eq. (4) is wrong as stated (it ignores reflections), and the paper never says whether the headline numbers used the correct M1 or the incorrect M2. read the letter →

arxiv 1908.00646 v3 pith:B3HS3SZQ submitted 2019-08-01 cs.CV

Benjamin Szczapa , Mohamed Daoudi , Stefano Berretti , Alberto Del Bimbo , Pietro Pala , Estelle Massart This is my paper · ORCID
classification cs.CV MSC 68T4553B20
keywords actionrecognitionskeleton-basedGrammatricespositivesemidefinitemanifoldRiemanniangeometryglobalalignmentkernelcurvefittingSVM
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 claims that human actions can be recognized from body skeletons alone by representing each video frame as a Gram matrix of joint coordinates and treating the whole action as a trajectory on the manifold of fixed-rank positive-semidefinite matrices. To compare and classify such trajectories, it replaces the Dynamic Time Warping used in prior work with a Global Alignment Kernel, built on a quotient-manifold distance that has a closed form when the skeleton is two-dimensional. The fitted, aligned trajectories are fed to a standard SVM through a kernel the paper asserts is positive semidefinite. If the claim holds, the approach provides a parameter-light, interpretable alternative to deep models: it reports 97.99% on UTKinect, 96.16% on KTH, and 92.44% on UAV-Gesture, the last improving on the provided deep baseline while using only skeletons.

What carries the argument

The central object is the manifold $S_+(d,n)$ of $n\times n$ positive-semidefinite matrices of fixed rank $d$, realized as the quotient $\mathbb{R}^{n\times d}_*/O_d$: each Gram matrix $G=ZZ^T$ corresponds to an equivalence class of landmark matrices under right multiplication by orthogonal matrices. The distance between two such classes is the minimal Frobenius distance $\min_{Q\in O_d}\|Z_jQ-Z_i\|_F$, which for $d=2$ admits the closed form involving the entries of $Z_j^T Z_i$. Around this geometry, two mechanisms do the work: curve fitting by composite cubic Bézier curves (with a parameter $\lambda$ trading data fidelity against acceleration) denoises the trajectories, and the Global Alignment Kernel—a positive-semidefinite kernel that sums over all possible temporal alignments—is applied to the resulting distance matrix to produce the PSD kernel $k(i,j) = \frac{1}{2}e^{-D(i,j)/\sigma^2}/(1-\frac{1}{2}e^{-D(i,j)/\sigma^2})$ that is fed to an SVM.

What would settle it

Take any dataset used in the paper, compute the full similarity matrix $K$ from the GAK scores, and test its smallest eigenvalue; if $K$ has a negative eigenvalue, the kernel is not PSD for this manifold distance and the standard SVM argument fails, even if classification accuracy stays high. A sharper test is to sample pairs of Gram matrices in $S_+(d,n)$, compute the local kernel matrix $k(i,j) = \frac{1}{2}e^{-D(i,j)/\sigma^2}/(1-\frac{1}{2}e^{-D(i,j)/\sigma^2})$, and check whether it is positive semidefinite for many random configurations.

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Extended reading notes

Core claim

The central claim is that the combination of three ingredients—the quotient geometry of $S_+(d,n)$ identified with $\mathbb{R}^{n\times d}_*/O_d$, composite cubic Bézier curve fitting, and the Global Alignment Kernel—turns Gram-matrix trajectories into a competitive action classifier with a theoretically valid PSD kernel. The paper's key identity is the closed-form Riemannian distance for $d=2$: $d(G_i,G_j)^2 = \operatorname{tr}(G_i) - 2\sqrt{(a+d)^2+(c-b)^2} + \operatorname{tr}(G_j)$, where $Z_j^T Z_i = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, which arises from minimizing the Frobenius distance over the orthogonal group. The paper claims this PSD kernel supports a standard SVM, rather than the proximity SVM required by DTW, and reports accuracy gains on all three datasets, with the largest margin on UAV-Gesture.

Load-bearing premise

The pipeline's theoretical justification rests on the assumption that the Global Alignment Kernel built from the quotient-manifold distance is positive semidefinite; the paper cites this property from the Euclidean time-series setting [8] without proving it for this manifold distance.

Editorial extensions

If this is right

  • On UTKinect-Action3D, the best configuration reaches 97.99% accuracy, above the 96.48% of the direct Gram-matrix predecessor [18].
  • On KTH-Action, using GAK instead of DTW improves accuracy from 94.49% to 96.16% in the same pipeline, and curve fitting adds about one point on this noisy 2D data.
  • On UAV-Gesture, the approach reaches 92.44%, outperforming the deep-learning P-CNN baseline of 91.9% while using only skeletons.
  • The closed-form distance for two-dimensional skeletons halves the alignment computation time relative to the general quotient distance, making the geometric pipeline practical on modest hardware.
  • Because the kernel is claimed positive semidefinite, the same pipeline can use a standard SVM, avoiding the pairwise-proximity SVM that DTW-based methods require.

Reading between the lines

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

  • If the PSD claim transfers, this recipe of quotient metric plus global alignment should apply to any landmark-trajectory task, such as facial expression or sign-language recognition, without retraining a metric.
  • The paper's own results hint that curve fitting helps on noisy 2D data and can slightly hurt on clean 3D data; an adaptive choice of $\lambda$ per dataset might push accuracy higher.
  • A natural testable extension is to swap the halved Gaussian local kernel for other positive-semidefinite local kernels and check whether the Global Alignment Kernel remains PSD on the quotient manifold, isolating how much of the gain comes from the kernel choice.
  • For three-dimensional skeletons, an analogous closed-form distance derived from the $3\times 3$ matrix $Z_j^T Z_i$ would likely speed up 3D pipelines the same way the 2D formula does.
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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 manuscript proposes a skeleton-based action recognition pipeline. Each frame's landmarks are mapped to a Gram matrix in the fixed-rank positive semidefinite cone S_+(d,n); a sequence becomes a trajectory on that manifold. The trajectory is denoised with composite Bézier curve fitting, sequences are aligned with a Global Alignment Kernel built from a Riemannian distance, and the resulting kernel matrix is fed to a standard SVM. Experiments on UTKinect-Action3D, KTH-Action, and UAV-Gesture report accuracies of 97.99%, 96.16%, and 92.44% respectively, which the authors describe as competitive with state-of-the-art methods and better than their direct predecessor [18] on UTKinect and better than the P-CNN baseline on UAV-Gesture.

Significance. The paper is a useful engineering contribution if the mathematical components hold: it provides a non-deep, interpretable alternative that achieves competitive accuracy from skeletons only, and the use of GAK rather than DTW is a sensible step toward a valid SVM kernel. However, the closed-form distance in Section 3.2 is incorrect for d=2, and the PSD property of the GAK on this manifold is asserted rather than proved; both issues bear directly on the reported numbers. The paper also includes detailed timing tables and comparison tables, which help reproducibility.

major comments (3)
  1. [§3.2, Eq. (4), and Appendix] Theorem 3.2 is false as stated. The distance in Eq. (3) minimizes over the full orthogonal group O_2, which includes reflections, but the proof in the Appendix parameterizes Q only as a rotation matrix. For Z_i = I_2 and Z_j = [[0,1],[1,0]], both Gram matrices equal I_2, so the true quotient distance is 0, whereas Eq. (4) returns 2. The correct maximization of tr(Z_i^T Z_j Q^T) over O_2 is the nuclear norm of Z_i^T Z_j, which equals sqrt((a+d)^2+(c-b)^2) only when det(Z_j^T Z_i) >= 0; otherwise it is larger. This error directly affects the d=2 experiments on KTH and UAV-Gesture.
  2. [§5.1, Eqs. (6)-(8)] The paper asserts without proof that the resulting global alignment kernel is positive semi-definite on S_+(d,n). The cited PSD result in Cuturi et al. [8] applies to local kernels that are positive definite; no argument or reference is given that exp(-d^2/sigma^2) is positive definite for the quotient/Bures metric on fixed-rank PSD matrices. Since the SVM step relies on the kernel matrix being PSD, this is a load-bearing gap. Please provide a proof or a specific citation, or state explicitly that a non-PSD proximity matrix is used with a suitable variant such as ppfSVM.
  3. [§6, Tables 3 and 5] The accuracy tables for KTH and UAV-Gesture do not state whether Eq. (3) (M1) or Eq. (4) (M2) was used. Because Eq. (4) is incorrect, the reported accuracies are ambiguous; if M2 was used, the numbers may change under the correct O_2 distance. Please specify the metric used for each reported result and rerun the experiments if M2 was used.
minor comments (4)
  1. [Algorithm 1] There is an indexing mismatch: the loops that fill k(i,j) range over i=0..tau1 and j=0..tau2, while the recurrence for M uses indices i=1..tau1+1 and j=1..tau2+1 without a consistent offset; please clarify the indexing convention.
  2. [Appendix, Eq. (10)] In Eq. (10) the orthogonal group is written as O_p; it should be O_2, consistent with Section 3.2.
  3. [§6] The hyperparameters lambda and sigma are tuned per dataset but the selection criterion is not described; please report how these values were chosen (for example, via a validation split) to mitigate overfitting concerns.
  4. [§5.1] The sentence 'This strategy assures us that the kernel is positive semi-definite' overstates the conclusion; at most the construction inherits the property if the local kernel is positive definite, which is exactly the point that needs to be established.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pipeline is an empirical system whose accuracy claims are measured against external benchmarks, and no prediction reduces to a fitted input or to a self-citation chain.

full rationale

The paper's chain is an empirical recognition pipeline: skeleton positions are mapped to Gram matrices, trajectories are compared with a quotient-manifold distance, smoothed with composite Bezier fitting, aligned with the Global Alignment Kernel, and classified with SVM. At no point is a quantity defined in terms of the result it is later used to predict. Equation (4) is offered as a closed form of the distance in Eq. (3), not as a fitted outcome, and the positive-semidefiniteness of the GAK kernel is transferred from Cuturi et al. [8], an external source, rather than manufactured from the authors' own fitted values. The hyperparameters lambda and sigma are tuned and reported per dataset, but they are not relabeled as predictions, and the reported accuracies are measured against public benchmarks. Several references involve overlapping authors (e.g., [13], [18], [28]), but these supply algorithmic tools that are also anchored in external publications ([4], [8], [17]) and in the Manopt implementation; none is invoked as a uniqueness theorem forcing the paper's choices. The reviewer-identified mathematical issue with Eq. (4) concerns the correctness of the minimization over O_d, which is a correctness risk, not a circularity: the formula is not equivalent to its input by construction. Therefore no circular step is identified.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new theoretical entities. It relies on established manifold geometry, a published curve fitting method, and a published alignment kernel. The main assumptions are the transfer of the PSD kernel property to this manifold and the choice of hyperparameters per dataset.

free parameters (2)
  • lambda (curve fitting regularization) = 0.5 for UTKinect, 10 for KTH and UAV-Gesture
    Controls the trade-off between data fidelity and curve regularity in the composite cubic blended curve fitting. Values are chosen per dataset to maximize accuracy.
  • sigma (GAK Gaussian kernel bandwidth) = 0.3 or 0.5 for UTKinect, 0.2 for KTH, unreported for UAV-Gesture
    Bandwidth of the halved Gaussian kernel used to define the local alignment kernel. Tuned per dataset; for UAV-Gesture the value is not listed in Table 5.
assumptions (4)
  • standard math The manifold of fixed-rank PSD matrices S+(d,n) is identified with the quotient Rn*d / Od and endowed with the quotient Euclidean metric, yielding the distance in Eq. (2).
    This is a known mathematical result cited to [17,28,30]. The paper uses it as the foundation for all subsequent computations.
  • domain assumption The composite cubic blended curve fitting algorithm from [13] can be applied on S+(d,n) using the Riemannian exp and log maps, and the fitted curves improve classification on noisy data.
    The paper assumes the algorithm transfers to this manifold and that smoothing helps, as shown empirically on KTH but not on UTKinect. No theoretical guarantee is given.
  • domain assumption The local kernel k(i,j) = ktilde / (1 - ktilde), with ktilde = 1/2 exp(-d^2/sigma^2), is positive definite for the quotient manifold distance d.
    The paper cites Cuturi et al. [8] for PSD-ness, but [8] gives conditions on the local kernel that are not verified for the Bures-type distance on S+(d,n). This is an unflagged assumption in the classification step.
  • domain assumption The Gram matrices of skeleton configurations have rank exactly d (the spatial dimension) in the datasets considered.
    Stated in Section 3.1: 'always equal to d in the datasets considered.' If a skeleton is degenerate or has missing joints, the rank could drop, invalidating the manifold geometry.

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Cite this review

Pith. "Pith review of Fitting, Comparison, and Alignment of Trajectories on Positive Semi-Definite Matrices with Application to Action Recognition." pith.science (2026). https://pith.science/paper/B3HS3SZQ

@misc{pith2026190800646,
  author       = {Pith},
  title        = {Pith review of: Fitting, Comparison, and Alignment of Trajectories on Positive Semi-Definite Matrices with Application to Action Recognition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B3HS3SZQ}},
  note         = {Machine review of arXiv:1908.00646}
}
read the original abstract

In this paper, we tackle the problem of action recognition using body skeletons extracted from video sequences. Our approach lies in the continuity of recent works representing video frames by Gramian matrices that describe a trajectory on the Riemannian manifold of positive-semidefinite matrices of fixed rank. In comparison with previous works, the manifold of fixed-rank positive-semidefinite matrices is here endowed with a different metric, and we resort to different algorithms for the curve fitting and temporal alignment steps. We evaluated our approach on three publicly available datasets (UTKinect-Action3D, KTH-Action and UAV-Gesture). The results of the proposed approach are competitive with respect to state-of-the-art methods, while only involving body skeletons.

Figures

Figures reproduced from arXiv: 1908.00646 by the authors.

Figure 1
Figure 1. Overview of the proposed approach - After automatic body skeletons detection for each frame of a sequence, the Gram matrices [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Comparison of two sequences that are confused in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Skeleton with the COCO format [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Comparison of two sequences that are confused in UAV [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.