Multiple Riemannian Manifold-valued Descriptors based Image Set Classification with Multi-Kernel Metric Learning

Josef Kittler; Rui Wang; Xiaojun Wu

REVIEW 3 major objections 4 minor 56 references

Multiple Riemannian Manifold-valued Descriptors based Image Set Classification with Multi-Kernel Metric Learning

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

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Pith's one-line read The paper argues that modeling image sets with covariance, subspace, and Gaussian descriptors at once, fused through Riemannian kernels and multi-kernel metric learning, improves classification over single-geometry and equal-weight…

desk verdict The multi-kernel fusion idea is a real but modest extension; the paper's blanket superiority claim is refuted by its own YTC table, so it needs major revision before I'd trust the empirical story. read the letter →

arxiv 1908.01950 v1 pith:GXNKM2QL submitted 2019-08-06 cs.CV

Rui Wang , XiaoJun Wu , Josef Kittler This is my paper · ORCID

classification cs.CV

keywords imagesetclassificationRiemannianmanifoldsmulti-kernelmetriclearningcovariancedescriptorsGrassmannmanifoldGaussiandistributionembeddingtraceratiooptimizationvideo-basedrecognition

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

The paper sets out to show that an image set is best represented not by one geometric model but by three complementary ones at once: a covariance matrix on the SPD manifold, a linear subspace on the Grassmann manifold, and a Gaussian distribution embedded in another SPD manifold. The proposed algorithm maps each descriptor into a Hilbert space with its own Riemannian kernel, then learns a shared low-dimensional metric through a trace-ratio objective with adaptive per-descriptor gating weights. The reported experiments on face recognition, object categorization, emotion recognition, and dynamic scene classification are offered as evidence for the intended conclusion that multi-geometry, adaptively weighted fusion beats single-descriptor and equal-weight alternatives.

What carries the argument

The load-bearing object is the multi-kernel metric learning objective with adaptive gating. Equation (18) maximizes the trace ratio $R_b(W)/R_w(W)$ under a learned projection $W$, with three Riemannian kernels supplying Gram matrices $K^q$; the gating model of Eq. (28) is a softmax over per-descriptor scores, making each local kernel region's weight data-dependent. The optimization alternates an iterative trace-ratio solver for the projection $E$ with gradient ascent for the gating parameters $\delta_q$ and $\rho_q$.

What would settle it

Compute the combined kernel matrix $K = \sum_{q=1}^{3} \xi_q K^q$ on held-out pairs and test positive semidefiniteness; negative eigenvalues would show the three RKHS embeddings cannot act as one common feature space, undercutting the projection step.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a fusion recipe: encode each image set with three Riemannian manifold-valued descriptors, map each into a reproducing kernel Hilbert space using the Log-Euclidean kernel for covariance matrices and Gaussian-embedded SPD matrices and the Projection Metric kernel for linear subspaces, and then learn a projection into a common lower-dimensional subspace by maximizing the trace ratio of between-class to within-class scatter while a softmax gating model assigns per-descriptor weights. The paper claims this leads to improved image set classification; in the reported tables the proposed method has the highest accuracy on ETH-80, AFEW, and MDSD, while on YTC the earlier MMML method is listed at 76.70 percent against the proposed 74.82 percent.

Load-bearing premise

The derivation assumes the three kernel feature maps all live in a single common Hilbert space, so that one projection matrix can act on them together; the paper does not state or prove that common-space condition.

Editorial extensions

If this is right

On the reported evidence, fusing the three descriptors beats each single descriptor on all four datasets, which is the paper's main argument for their complementarity.
The adaptive per-descriptor weights are claimed to give an advantage over equal-weight fusion methods such as HERML and MMML, with the ETH-80, AFEW, and MDSD results consistent with that advantage.
Because the same framework is applied to four very different tasks, the fusion recipe is presented as task-independent rather than tuned specifically to face recognition.
The iterative trace-ratio and gradient-ascent procedure is reported to stabilize after a small number of iterations, so the method is usable without deep networks or manifold backpropagation.

Reading between the lines

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

A testable extension is to examine whether the learned gating weights concentrate on one descriptor per dataset; the ablation results suggest Grassmann features matter most for faces and Gaussian features for scenes, implying a dataset-adaptive attention pattern.
The paper does not exploit temporal order; injecting frame order into the set descriptors is a natural extension that could strengthen video-based tasks.
The common-space assumption behind the derivation could be tested by checking positive semidefiniteness of the weighted sum of the three kernel matrices, since an indefinite sum would mean the projection step has no clean RKHS interpretation.

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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

Read this if you care about Riemannian image-set classification. What is genuinely new here is the fusion of three geometric descriptors—covariance, linear subspace, and Gaussian—with learned per-kernel softmax gating and a jointly optimized projection via trace-ratio. That specific combination does not appear in the cited prior work, including the authors' own MMML and HERML, both of which use equal weights or fewer descriptor types. The paper is organized well enough, and it does some things right: per-descriptor ablation, parameter sensitivity for the subspace dimension, convergence curves, and an honest admission that a theoretical convergence proof is missing. The self-comparison to MMML is not circular, and the citation pattern is unremarkable.

The soft spots are substantial. First, the empirical claim. The abstract says extensive results justify superiority over the state of the art, and Section V-B explicitly says the proposed algorithm shows better performance than the compared methods on YTC. Table I gives the proposed method 74.82% and MMML 76.70% on that same dataset. That is a direct internal contradiction. It is not a missing error bar or a weak baseline; the paper's own evidence refutes its central claim as written. A reader can still salvage a weaker claim—competitive on YTC, best on the other three sets—but the authors did not make that claim.

Second, the kernelized derivation in Section IV-B is shaky. Equations (23)-(26) require all three feature maps to live in a single common Hilbert space so that one projection matrix W (or its kernelized counterpart E) can act on all of them. The paper never states or proves that assumption. If each Riemannian kernel induces its own RKHS with no shared codomain, then the projection is not well-defined and the scatter matrices in (25)-(26) lack a clear basis. This is a correctness risk, not a cosmetic issue.

Third, reproducibility is weak: no code is actually available despite the promise, and no error bars are given, so the numbers cannot be checked. The convergence issue is honestly flagged, but the optimization alternates between a trace-ratio problem and gradient updates without any guarantee; the experiments show stability on two datasets, which is evidence but not proof.

Bottom line: this is not ready to be accepted as is, but it deserves a serious referee rather than a desk reject. The flaws are concrete and identifiable, and the fusion formulation is worth engaging with. I would not cite the reported numbers yet; I might cite the method idea if the math and claims are cleaned up.

Referee Report

3 major / 4 minor

Summary. The paper proposes an image-set classification framework that represents each image set simultaneously by a covariance matrix, a linear subspace, and a Gaussian distribution, which live on three different Riemannian manifolds. These heterogeneous descriptors are mapped into reproducing kernel Hilbert spaces via three Riemannian kernels, and a multi-kernel metric learning objective with adaptive gating weights is used to learn a common low-dimensional subspace for classification. The method is evaluated on four tasks: video-based face recognition (YTC), set-based object categorization (ETH-80), video-based emotion recognition (AFEW), and dynamic scene classification (MDSD). The Abstract, Section V, and the Conclusion claim that the reported results justify superiority over state-of-the-art methods.

Significance. If the empirical claim and the derivation were both sound, the work would be a useful contribution to image-set classification: it combines three complementary geometric descriptors, provides a kernel-based fusion mechanism, and includes ablations and a complexity analysis. The paper is best understood as an extension of the authors' earlier MMML framework, and it is commendable that the comparison includes that earlier method. However, the central headline claim is directly contradicted by the paper's own Table I, and the kernelized derivation has a load-bearing gap concerning the common Hilbert space assumption. These issues are central rather than cosmetic, so the paper in its current form does not support its main conclusion.

major comments (3)

[V-B, Table I] The paper's central claim, stated in the Abstract and repeated in the Conclusion, is that extensive experimental results justify superiority over the state of the art. Table I reports 74.82% for the proposed method on YTC, while MMML [49], listed as a compared state-of-the-art method and the authors' own earlier work, achieves 76.70%. The text in Section V-B also states that 'the proposed algorithm shows better classification performance than them on this data.' This is a direct internal contradiction of the blanket-superiority claim on one of the four evaluation datasets. The issue is not a minor wording problem, because the claimed universal superiority is the headline result of the paper; the reported evidence refutes the claim as stated, and the authors would need to revise the claim and explain why the proposed extension underperforms its predecessor on YTC.
[IV-B, Eqs. (23)-(26)] The kernelization step is not rigorously justified. Equations (16)-(17) use a single projection matrix W acting on feature maps φ_i^q, which requires all three descriptor-dependent feature maps to lie in a single common Hilbert space F. However, each Riemannian kernel function induces its own RKHS, and the paper does not state or prove a common-codomain assumption. In Eq. (23), the basis vector w_h is expanded as a linear combination of training samples φ_i^q with a q-dependent index, and Eq. (24) sums over q as if all φ_i^q belong to the same space in which the inner product defining K^q is valid. Without a precise statement of the shared Hilbert space, or a block-diagonal construction of U over the direct sum of the three RKHSs, the projection learned in Eqs. (25)-(27) has no well-defined action across the heterogeneous feature spaces. This is a load-bearing gap because the entire method depends on this derivation.
[IV-C and V-G] The joint optimization of the projection E and the gating parameters (δ_q, ρ_q) is not shown to converge. Section IV-C alternates between solving a trace-ratio problem for E and gradient-ascent updates for the gating parameters, but the combined objective is nonconvex and no convergence guarantee is provided. Section V-G explicitly admits that 'it is hard for us to provide a systematic theoretical proof of convergence behavior,' and only reports objective values on AFEW and YTC. Since the optimization is central to the method, the paper should either provide a convergence argument or demonstrate stability across all four datasets and multiple random initializations; otherwise the reported results may depend on the initialization choices.

minor comments (4)

[V-B] The numerical discrepancy between the proposed method and MMML on YTC should also be discussed in the text; currently MMML is not mentioned in the paragraph comparing the proposed method with LMKML and HERML, even though MMML is the strongest competitor on that dataset.
[V-B through V-E] All experimental tables report only average rates without standard deviations or significance tests. Given that several comparisons are separated by less than one percentage point, error bars are needed to support claims of improvement.
[IV-A, Eq. (13)] The regularization parameter α is fixed to 10^3 in all experiments, but no sensitivity analysis is provided; since this parameter affects the SPD regularization for both the covariance and Gaussian descriptors, its influence on the results should be reported.
[Abstract and Introduction] The phrase 'monotonically increasing' in the Abstract is awkward, and the term 'well-equipped' is used repeatedly to describe Riemannian kernels; the wording should be revised for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the method is a standard supervised multi-kernel metric-learning pipeline evaluated on external benchmark data.

full rationale

The paper's claimed contribution is an empirical image-set classification method that fuses three Riemannian descriptors with learned kernel weights and a projection. The derivation chain in Section IV-B uses standard RKHS kernel constructions: the kernels in Eqs. (5), (8), and (11) are cited from prior external works, and the representer-style expansion in Eq. (23) is the usual kernel trick, although the common-Hilbert-space assumption it relies on is a correctness concern rather than a circular one. The projection E and gating parameters (delta_q, rho_q) are optimized on training data via Eqs. (27)-(42), and classification is then evaluated on held-out test splits (YTC, ETH-80, AFEW, MDSD) against independent baselines; this is standard supervised learning, not a fitted quantity renamed as a prediction. The authors' own prior method MMML [49] appears as a baseline and in Section IV-F as an explicit comparison of technical differences; it is not invoked to justify the method's validity or to forbid alternatives. No self-definitional, fitted-input-as-prediction, or self-citation-load-bearing step is present. The internal Table I inconsistency (Proposed 74.82 vs MMML 76.70 on YTC) undermines the blanket superiority claim in the Abstract, but that is a factual/correctness issue and is outside the scope of circularity analysis.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The method rests on standard manifold kernels and on one unproven common-Hilbert-space assumption, plus several dataset-dependent hyperparameters. The gating parameters are learned from the training data, which is normal but means the adaptive weighting is fit rather than derived.

free parameters (5)

alpha (regularization constant) = 10^3
Added to diagonal of covariance and Gaussian covariance to guarantee positive definiteness (Eq. 13); set to 10^3 in all experiments without sensitivity analysis.
d_w (target subspace dimensionality) = YTC: 70, MDSD: 25, AFEW: 70, ETH-80: 8
Selected per dataset by grid search (Section V-H); results are sensitive to this choice (Fig. 8-9).
gamma (learning rate) = 10^-4
Learning rate for gradient ascent on gating parameters (Eq. 41-42).
B (iteration count) and epsilon (convergence error) = Not specified numerically
Algorithm 1 stops when changes drop below epsilon; the values are never given, so exact training behavior is not reproducible.
delta_q, rho_q (gating model parameters) = Learned from training data (not reported)
Softmax gating weights in Eq. (28)-(30) learned via gradient ascent; they determine the adaptive contribution of each descriptor and are central to the fusion.

assumptions (4)

domain assumption The three Riemannian kernel functions (LEM, projection kernel, Gaussian-embedded LEM) are positive definite and preserve the relevant manifold structure.
Invoked in Section III and IV; kernels are taken from [12,14,21,32,33] without proof in this paper.
standard math A d-dimensional Gaussian distribution can be represented uniquely as a point in Sym+_{d+1} via the information-geometric embedding of Eq. (10).
From [31,32]; used in Section III-C and IV-A.
ad hoc to paper There exists a single common Hilbert space F shared by the feature maps of all three kernels, so a single projection matrix W (or E) can be applied across the heterogeneous descriptor spaces.
This is the premise behind Eq. (23)-(26); it is not stated or proven, and the three kernels induce different RKHSs in general.
domain assumption The trace ratio optimizer [48] converges to the global optimum, and this convergence survives interleaving with gradient ascent on the gating parameters.
Global optimality of the trace ratio step is cited from [48]; joint convergence is explicitly not proven in Section V-G.

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Cite this review

Pith. "Pith review of Multiple Riemannian Manifold-valued Descriptors based Image Set Classification with Multi-Kernel Metric Learning." pith.science (2026). https://pith.science/paper/GXNKM2QL

@misc{pith2026190801950,
  author       = {Pith},
  title        = {Pith review of: Multiple Riemannian Manifold-valued Descriptors based Image Set Classification with Multi-Kernel Metric Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GXNKM2QL}},
  note         = {Machine review of arXiv:1908.01950}
}

read the original abstract

The importance of wild video based image set recognition is becoming monotonically increasing. However, the contents of these collected videos are often complicated, and how to efficiently perform set modeling and feature extraction is a big challenge for set-based classification algorithms. In recent years, some proposed image set classification methods have made a considerable advance by modeling the original image set with covariance matrix, linear subspace, or Gaussian distribution. As a matter of fact, most of them just adopt a single geometric model to describe each given image set, which may lose some other useful information for classification. To tackle this problem, we propose a novel algorithm to model each image set from a multi-geometric perspective. Specifically, the covariance matrix, linear subspace, and Gaussian distribution are applied for set representation simultaneously. In order to fuse these multiple heterogeneous Riemannian manifoldvalued features, the well-equipped Riemannian kernel functions are first utilized to map them into high dimensional Hilbert spaces. Then, a multi-kernel metric learning framework is devised to embed the learned hybrid kernels into a lower dimensional common subspace for classification. We conduct experiments on four widely used datasets corresponding to four different classification tasks: video-based face recognition, set-based object categorization, video-based emotion recognition, and dynamic scene classification, to evaluate the classification performance of the proposed algorithm. Extensive experimental results justify its superiority over the state-of-the-art.

Figures

Figures reproduced from arXiv: 1908.01950 by the authors.

**Figure 1.** Schematic diagram of the proposed image set classification framework. The first column in this figure represents the given image sets. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Face frames on YTC dataset TABLE I: Average recognition rates (%) of different methods on YTC dataset Method Years YTC DCC [54] 2007 66.81 MMD [55] 2008 65.30 GDA [12] 2008 65.78 GEDA [53] 2011 66.37 CDL [14] 2012 68.76 RSR [24] 2012 72.77 LMKML [39] 2013 70.31 HERML [33] 2015 71.28 MMML [49] 2018 76.70 PML [23] 2015 67.62 LEML [15] 2015 69.04 SPDML-AIM [16] 2018 64.66 SPDML-Stein [16] 2018 61.57 MMDML [38] 2015 69.… view at source ↗

**Figure 3.** Examples on ETH-80 dataset [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Emotion images on AFEW dataset TABLE III: Emotion recognition results (%) of different methods on AFEW dataset Method Years AFEW DCC [54] 2007 25.78 GDA [12] 2008 29.11 GEDA [53] 2011 29.45 CDL [14] 2012 31.81 RSR [24] 2012 27.49 HERML [33] 2015 32.14 MMML [49] 2018 31…

**Figure 5.** Figure 5: Dynamic scene images on MDSD dataset TABLE IV: Dynamic scene classification results (%) of different methods on MDSD dataset Method Years MDSD DCC [54] 2007 25.92 GDA [12] 2008 30.51 GEDA [53] 2011 30.37 CDL [14] 2012 30.51 DARG [10] 2015 31.62 HERML [33] 2015 32.37 MM…

**Figure 6.** Figure 6: Convergence behavior of the proposed algorithm on AFEW [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Convergence behavior of the proposed algorithm on YTC [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

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