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REVIEW 3 major objections 5 minor 161 references

LieBN: Batch Normalization over Lie Groups

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A single normalization recipe now claims provable mean and variance control on every Lie group.

desk verdict Useful extension of the authors' ICLR LieBN with real new metrics and careful experiments, but the advertised variance-control guarantee does not hold as stated: scaling can push the Fréchet mean off the identity. read the letter →

arxiv 2607.08783 v3 pith:HXPLPXRE submitted 2026-06-13 cs.LG cs.AI

Ziheng Chen , Yue Song , Rui Wang , Xiao-Jun Wu , Nicu Sebe This is my paper
classification cs.LGcs.AI
keywords LiegroupsRiemannianbatchnormalizationinvariantmetricsSPDmatricesrotationcorrelationFréchetmeanvariancecontrol
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 build batch normalization for neural network layers whose activations live on Lie groups—matrix manifolds that are also groups, such as symmetric positive definite matrices, rotation matrices, and correlation matrices. It claims that centering by left or right translation, scaling radially through the tangent space at the identity, and biasing back give provable control of both the Riemannian (Fréchet) mean and the sample variance under any left-, right-, or bi-invariant metric. Prior Riemannian normalizations were tied to specific geometries and typically controlled the mean without a matching variance guarantee. The paper instantiates the scheme on nine geometries, including a newly proposed right-invariant SPD metric, and reports accuracy gains when its layers are plugged into standard SPD and rotation networks.

What carries the argument

Algorithm 1 is the load-bearing mechanism: centering by left translation (or right translation under right-invariant metrics), radial scaling through the identity, and biasing by translation. The critical identity is Equation (16), Σᵢ wᵢ d²(φ_s(Pᵢ), E) = s² Σᵢ wᵢ d²(Pᵢ, E), which converts the batch dispersion into a controllable scalar multiple. For pullback metrics from Euclidean spaces—LEM, LCM, and the four correlation geometries—the entire algorithm is computed in the codomain and collapses to standard Euclidean batch normalization. The paper also contributes a new right-invariant SPD metric, CRIM, built from the Cholesky-product group operation, giving a non-left-invariant geometry on t

What would settle it

Center a batch of rotations on SO(3) with known Fréchet mean, apply φ_s(P) = Exp_E[s Log_E(P)] with s = 2, then recompute the batch mean and variance of the scaled samples: if the mean departs measurably from the identity, or the variance is not s² times the original, the claimed sample-variance control fails outside an infinitesimal neighborhood.

Watch

Extended reading notes

Core claim

The paper's central claim is that batch normalization on a Lie group can be decomposed into three operations: sending every sample to the group identity by translating with the inverse of the Fréchet mean; scaling the dispersion with the radial map φ_s(P) = Exp_E[s Log_E(P)]; and biasing toward a learnable parameter by a group translation. Proposition 7 supplies the two identities that carry the argument: translations move the sample mean homogeneously, and distances to the identity scale as d(φ_s(P), E) = s d(P, E), so the weighted sum-of-squares dispersion around the identity contracts or expands by s². The paper therefore claims a general framework—LieBN—that normalizes both first- and se

Load-bearing premise

After centering a batch at the identity, the proof that radial scaling controls the variance assumes the scaled batch's Fréchet mean remains at the identity (Section 4.1, following Proposition 7); because radial scaling and group composition do not commute on non-abelian geometries like (θ,α,β)-AIM, CRIM, and SO(n), this is asserted but not proved and is generally false there.

Editorial extensions

If this is right

  • Every Lie group carries at least one invariant metric, so LieBN applies wherever a network layer outputs group-valued activations, not just to SPD matrices.
  • Under log-Euclidean, log-Cholesky, and correlation metrics—all pullbacks of Euclidean spaces—the normalization reduces to ordinary Euclidean batch normalization, inheriting its computational profile.
  • The new right-invariant SPD metric (CRIM) and the power-deformed variants expand the menu of SPD geometries; the paper reports that the best geometry varies by data set.
  • On rotation groups, centering can use either left or right translation; both normalize the statistics, and the reported experiments show faster convergence for the rotation network with either choice.
  • Because LieBN controls both the mean and the variance, it directly addresses the covariate-shift problem for manifold-valued activations that earlier SPD-specific methods only partially handled.

Reading between the lines

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

  • If the hidden assumption that radial scaling keeps the Fréchet mean at the identity fails away from the small-dispersion regime, the s² variance law is only approximately true; a two-pass version that re-centers after scaling would restore the guarantee and is a direct test of the paper's boundary.
  • A quantitative stress test suggests itself: on SO(3) or CRIM, measure how the scaled batch mean drifts from the identity as s moves away from 1; the drift curve would delimit where LieBN's guarantee is exact and where it becomes approximate.
  • The same translation-centering pattern could be exported to layer-, instance-, or group-normalization variants for manifold-valued activations, but each port must independently re-examine whether scaled samples keep their mean at the identity.
  • The theory and the nine instantiations together suggest that the main practical obstacle is not the choice of normalization recipe but the choice of geometry: geodesic distance, mean, and variance all change with the metric, so the same batch normalizes differently under each.
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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 / 5 minor

Summary. The paper proposes LieBN, a Riemannian batch-normalization layer for Lie-group-valued activations. The framework centers a batch by left/right translation to the identity, scales the centered points radially in the tangent space at the identity, and then applies a learned biasing translation. Centering and biasing are justified by the fact that left/right translations are isometries under invariant metrics; scaling is justified by Proposition 7, which gives an identity for the weighted dispersion around the identity. The authors instantiate the framework on SPD manifolds under four metrics (including a new right-invariant metric CRIM), on SO(n), and on full-rank correlation matrices under four metrics, and they report experiments on radar, action recognition, EEG, and skeleton-based tasks. The paper claims a theoretical guarantee for controlling both the Riemannian sample mean and variance on general Lie groups.

Significance. If the variance-control guarantee were valid, this would be a valuable unification: the mean-centering mechanism is elegant, and the pullback-based implementation (Theorem 17) is practical and covers a broad range of geometries. The paper is also strong in breadth, with nine instantiated geometries, a new right-invariant SPD metric, a public PyTorch toolbox, and extensive experiments. However, the advertised second-moment guarantee is the load-bearing theoretical contribution, and it is not established as stated. The mean-centering part is sound; the variance part rests on an unproved and generally false assumption that radial scaling preserves the Fréchet mean at the identity.

major comments (3)
  1. [Section 4.1, Eq. (16)] Equation (16) is correct as a statement about dispersion around the identity: ∑ w_i d²(φ_s(P_i), E) = s² ∑ w_i d²(P_i, E). But this does not imply control of the Fréchet variance after scaling, because the Fréchet variance of the scaled batch is min_Q ∑ w_i d²(φ_s(P_i), Q), and equality with s² times the original variance requires the Fréchet mean of the scaled batch to remain E. The text 'As the post-centering mean is E, Equation (16) implies that Equation (13) can control the sample variance' is therefore a non sequitur. A concrete counterexample on SO(3) with the bi-invariant metric: take the batch {exp(θu), exp(-θu)} with θ=π/3 and weights 1/2. The identity I is the Fréchet mean and v²=θ². For s=2, Equation (16) holds, but φ_s maps the batch to rotations by ±2π/3, whose Fréchet mean is the rotation exp(πu), not I, and whose Fréchet variance is (π/3)², not 4(π/3)². The batch was initi
  2. [Abstract and Section 1] The abstract and introduction state that the framework 'provides a theoretical guarantee for normalizing the Riemannian sample mean and variance across general Lie groups.' In view of the gap above, this claim is too strong. The variance guarantee is known to hold for the pullback/commutative geometries treated in Appendix D (LEM, LCM, and the correlation metrics), where the scaling reduces to Euclidean scaling in the codomain, but it is not established for AIM, CRIM, or SO(n), which are central instantiations of the paper. The authors should either restrict the variance-guarantee claim to the geometries where it is true, or add explicit assumptions and revise the abstract, introduction, and conclusion accordingly.
  3. [Algorithm 1 and Section 4.1] Algorithm 1 computes the Fréchet variance v²_b of the centered batch and then applies the scaling s/√(v²_b+ε). Even if Proposition 7 were extended to establish that the scaled batch has mean E, the algorithm would still need a guarantee that v²_b is computed in a region where the Fréchet mean is unique and where the Log/Exp maps at E are valid for all scaled points. The paper's blanket assumption that 'the involved Exp, Log, and WFM are well-defined' does not cover the post-scaling configuration; the SO(3) example satisfies the pre-scaling assumptions but violates the variance normalization. The assumptions need to be stated explicitly and must be sufficient for the variance claim.
minor comments (5)
  1. [Section 4.1] The text refers to 'Theorem 6' and 'Theorem 7' when the statements are Propositions 6 and 7. Please harmonize the cross-references.
  2. [Section 5.1.2] The section title contains a typo: 'SPD Right-Inv ariant Metrics' should read 'SPD Right-Invariant Metrics.'
  3. [Algorithm 1] The running-mean update M_r ← WFM({1-γ,γ},{M_r,M_b}) is only well-defined if M_r and M_b lie in a common strongly convex ball. This assumption is used implicitly and should be stated alongside the other regularity assumptions.
  4. [Section 6.1, Table 9 and Section 6.3, Table 22] The 'Bestθ' columns and the correlation matrix powers are selected per dataset from small candidate sets, but the validation protocol for this selection is not described. As written, the reader cannot tell whether these hyperparameters were chosen on a validation split or on the test folds; this should be clarified to avoid overstating the empirical gains.
  5. [Figure 3] The t-SNE visualization is qualitative. Since the paper's main theoretical claim concerns Fréchet mean and variance control, a quantitative report of the batch mean and variance before and after LieBN would provide more direct evidence for the claimed normalization effect.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation; the main theoretical gap is a non-sequitur, not a reduction to inputs.

full rationale

The derivation chain in Sections 4.1-4.3 is not circular. Mean normalization follows from the isometry of left/right translations (Prop. 7, Eq. 15; Prop. 10), and Eq. (16) is a direct computation from the definition of phi_s and the Riemannian distance/norm at E. The paper's leap from Eq. (16) to 'Equation (13) can control the sample variance' is not a circular step: it would require the additional fact that the Fréchet mean of the scaled samples remains E, which is neither proved nor stated, and the Fréchet variance (Definition 4) is not defined as the sum in Eq. (16). This is a correctness gap, not a case where the conclusion is assumed in the premises. The uses of the authors' previous work (e.g., Chen et al. 2024b,d) are for provenance or for facts also supported by independent external references (Lin 2019; Thanwerdas and Pennec), so no load-bearing self-citation chain forces the result. The reported tuning of theta and matrix powers per data set is standard hyperparameter selection rather than a fitted input being presented as a theoretical prediction. No equation or theorem in the paper reduces its target statement to its own definition.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper's central mathematical framework rests on standard Lie-group/Riemannian facts plus one unstated ad hoc premise: radial scaling preserves the identity as the Fréchet mean. The fitted quantities are the geometry-deformation parameters θ, the matrix-power exponents for correlation LieBN, and the choice K=1 for Fréchet mean iterations. No new particles, forces, or speculative physical entities are introduced.

free parameters (3)
  • Deformation factor θ for SPD metrics = Per-dataset selections from {±0.5, ±1, ±1.5}; e.g., AIM θ=1.5 on HDM05 and FPHA, LCM θ=0.5 on HDM05, CRIM θ=-0.5 on FPHA
    θ interpolates metric families and is chosen per dataset to maximize accuracy; the paper's best-result tables depend on this tuning.
  • Matrix power exponent in LieBN-Cor = Table 22: HDM05 ECM=0.75, LECM=0.5, OLM=0.5, LSM=-0.5; FPHA ECM=-0.5, LECM=-0.25, OLM=-0.25, LSM=-0.25.
    These exponents are applied before correlation normalization and are selected per dataset and metric; they are fitted hyperparameters, not derived quantities.
  • Number of Fréchet mean iterations K = 1
    The ablation in Section 6.4.1 shows K=1 is the default and gives best accuracy in several settings; accuracy is non-monotonic in K (e.g., AIM on HDM05 drops from 68.16% at K=1 to 52.88% at K=20), so K=1 is a tuned approximation rather than a converged statistic.
assumptions (5)
  • standard math The weighted Fréchet mean exists and is unique inside a geodesic ball of radius r_cvx(M).
    Invoked after Definition 4 in Section 2.1; relies on Afsari (2011) and is assumed throughout the paper.
  • standard math Every Lie group admits left- and right-invariant metrics.
    Used in Section 2.1 (footnote 1) to justify the existence of the metrics on which LieBN is built.
  • domain assumption The intrinsic Riemannian Gaussian density N(M, σ²) in Equation (11) is the correct manifold analogue of the Euclidean Gaussian.
    Needed for the population-level claims in Propositions 6 and 10; this is a modeling choice, not a theorem of Riemannian geometry.
  • ad hoc to paper After centering to E and radial scaling φ_s, the sample Fréchet mean remains E.
    Unstated premise in Section 4.1 after Equation (16). The paper equates dispersion around E with the Fréchet variance of the scaled batch, which requires the scaled mean to stay at E; this is not proven and is false in general for non-abelian or non-pullback geometries.
  • ad hoc to paper One Karcher-flow iteration is sufficient to estimate the Fréchet batch mean in practice.
    Algorithm 1 and the implementation use K=1 iteration; the ablation shows this is not converged (accuracy degrades with more iterations on several datasets), so the batch statistics used during training are approximate.

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Cite this review

Pith. "Pith review of LieBN: Batch Normalization over Lie Groups." pith.science (2026). https://pith.science/paper/HXPLPXRE

@misc{pith2026260708783,
  author       = {Pith},
  title        = {Pith review of: LieBN: Batch Normalization over Lie Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HXPLPXRE}},
  note         = {Machine review of arXiv:2607.08783}
}
read the original abstract

Manifold-valued measurements are prevalent in various machine learning tasks. Recent advances have extended Deep Neural Networks (DNNs) to operate on manifolds. These extensions have been accompanied by normalization techniques tailored to different geometries, collectively referred to as Riemannian normalization. However, most existing Riemannian normalization methods are either designed for specific manifolds or fail to effectively normalize manifold-valued sample distributions. To address these limitations, we propose LieBN, a framework for Riemannian Batch Normalization (RBN) over Lie groups. Our approach leverages the theoretically convenient left- and right-invariant metrics, which naturally exist in every Lie group, and provides theoretical guarantees for controlling the Riemannian mean and variance. We instantiate LieBN across nine distinct geometries: four on the Symmetric Positive Definite (SPD) manifold, one on the group of rotation matrices, and four on the manifold of full-rank correlation matrices. Notably, among the SPD metrics, we introduce a novel right-invariant metric and extend three existing Lie group structures via matrix power deformation. Experiments on different manifolds validate the effectiveness of our framework. The code is available at https://github.com/GitZH-Chen/LieBN.git.

Figures

Figures reproduced from arXiv: 2607.08783 by the authors.

Figure 1
Figure 1. Illustration of LieBN on the SPD, rotation, and correlation Lie groups. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Minimal examples of applying LieBN [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 2
Figure 2. Minimal examples of applying LieBN. • A novel right-invariant metric on the SPD manifold, which is the first non-trivial right-invariant SPD metric; • Concrete instantiations of our LieBN framework on different geometries: four on SPD manifolds, one on the rotation manifold, and four on the correlation manifold; • Validation of the effectiveness of our LieBN framework by extensive experiments on different geometries… view at source ↗
Figures from the paper (6 more)
Figure 3
Figure 3. Figure 3: Visualization of input and output 30 × 30 SPD matrices in LieBN using 2 × 2 Riemannian t-SNE embeddings. The first row shows the input and output under different metrics. Due to the significant difference in magnitude between the t-SNE embeddings of LieBN’s input and o…
Figure 3
Figure 3. Figure 3: Visualization of input and output 30 × 30 SPD matrices in LieBN using 2 × 2 Riemannian t-SNE embeddings. The first row shows the input and output under different metrics. Due to the significant difference in magnitude between the t-SNE embeddings of LieBN’s input and o…
Figure 4
Figure 4. Figure 4: Test accuracy curves corresponding to Tab. 9. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 4
Figure 4. Figure 4: Test accuracy curves corresponding to Table 12. [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: Comparisons of LieBN and LieAlgebraBN on the LieNet backbone. [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]
Figure 5
Figure 5. Figure 5: Comparisons of LieBN and LieAlgebraBN on the LieNet backbone. [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]

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Works this paper leans on

161 extracted references · 4 linked inside Pith

  1. [1]

    ICLR , year=

    Gyrogroup Batch Normalization , author=. ICLR , year=

  2. [2]

    Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of

    Ziheng Chen and Yue Song and Xiaojun Wu and Gaowen Liu and Nicu Sebe , booktitle=. Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of

  3. [3]

    Fast and Stable

    Ziheng Chen and Yue Song and Xiaojun Wu and Nicu Sebe , booktitle=. Fast and Stable

  4. [4]

    ICLR , year=

    Riemannian High-Order Pooling for Brain Foundation Models , author=. ICLR , year=

  5. [5]

    Chen, Ziheng and Song, Yue and Wang, Rui and Wu, Xiao-Jun and Sebe, Nicu , booktitle=

  6. [6]

    Ziheng Chen and Yue Song and Yunmei Liu and Nicu Sebe , booktitle=. A

  7. [7]

    Riemannian Multiclass Logistics Regression for

    Chen, Ziheng and Song, Yue and Liu, Gaowen and Kompella, Ramana Rao and Wu, Xiaojun and Sebe, Nicu , booktitle=. Riemannian Multiclass Logistics Regression for

  8. [8]

    Adaptive

    Chen, Ziheng and Song, Yue and Xu, Tianyang and Huang, Zhiwu and Wu, Xiao-Jun and Sebe, Nicu , journal=. Adaptive

Show all 161 references
  1. [9]

    Riemannian Local Mechanism for

    Chen, Ziheng and Xu, Tianyang and Wu, Xiao-Jun and Wang, Rui and Huang, Zhiwu and Kittler, Josef , booktitle=. Riemannian Local Mechanism for

  2. [10]

    Chen, Ziheng and Xu, Tianyang and Wu, Xiao-Jun and Wang, Rui and Kittler, Josef , journal=. Hybrid. 2023 , doi=

    2023

  3. [11]

    Wang, Rui and Wu, Xiao-Jun and Chen, Ziheng and Xu, Tianyang and Kittler, Josef , booktitle=

  4. [12]

    Learning a discriminative

    Wang, Rui and Wu, Xiao-Jun and Chen, Ziheng and Xu, Tianyang and Kittler, Josef , journal=. Learning a discriminative. 2022 , publisher=

    2022

  5. [13]

    Wang, Rui and Wu, Xiao-Jun and Chen, Ziheng and Hu, Cong and Kittler, Josef , journal=

  6. [14]

    Wang, Rui and Hu, Chen and Chen, Ziheng and Wu, Xiao-Jun and Song, Xiaoning , booktitle =. A

  7. [15]

    ICML , year=

    Lorentz group equivariant neural network for particle physics , author=. ICML , year=

  8. [16]

    2017 , publisher=

    A mathematical introduction to robotic manipulation , author=. 2017 , publisher=

    2017

  9. [17]

    Yim, Jason and Trippe, Brian L and De Bortoli, Valentin and Mathieu, Emile and Doucet, Arnaud and Barzilay, Regina and Jaakkola, Tommi , booktitle=

  10. [18]

    NeurIPS , year=

    Riemannian score-based generative modelling , author=. NeurIPS , year=

  11. [19]

    A globally convergent numerical algorithm for computing the centre of mass on compact

    Manton, Jonathan H , booktitle=. A globally convergent numerical algorithm for computing the centre of mass on compact. 2004 , organization=

    2004

  12. [20]

    2019 20th International Radar Symposium (IRS) , pages=

    Complex-valued neural networks for fully-temporal micro-Doppler classification , author=. 2019 20th International Radar Symposium (IRS) , pages=. 2019 , organization=

    2019

  13. [21]

    Geometric Science of Information: 4th International Conference, GSI 2019, Toulouse, France, August 27--29, 2019, Proceedings 4 , pages=

    Second-order networks in PyTorch , author=. Geometric Science of Information: 4th International Conference, GSI 2019, Toulouse, France, August 27--29, 2019, Proceedings 4 , pages=. 2019 , organization=

    2019

  14. [22]

    ICASSP , year=

    Exploring complex time-series representations for Riemannian machine learning of radar data , author=. ICASSP , year=

  15. [23]

    2020 IEEE Radar Conference (RadarConf20) , pages=

    Deep learning and information geometry for drone micro-Doppler radar classification , author=. 2020 IEEE Radar Conference (RadarConf20) , pages=. 2020 , organization=

    2020

  16. [24]

    2019 International Radar Conference (RADAR) , pages=

    A Hermitian Positive Definite neural network for micro-Doppler complex covariance processing , author=. 2019 International Radar Conference (RADAR) , pages=. 2019 , organization=

    2019

  17. [25]

    Gaussian distributions on the space of symmetric positive definite matrices from

    Barbaresco, Fr. Gaussian distributions on the space of symmetric positive definite matrices from. Geometric Science of Information: 5th International Conference , year=

  18. [26]

    Is affine-invariance well defined on

    Thanwerdas, Yann and Pennec, Xavier , booktitle=. Is affine-invariance well defined on

  19. [27]

    Geometric Science of Information: 4th International Conference, GSI 2019, Toulouse, France, August 27--29, 2019, Proceedings 4 , pages=

    Exploration of balanced metrics on symmetric positive definite matrices , author=. Geometric Science of Information: 4th International Conference, GSI 2019, Toulouse, France, August 27--29, 2019, Proceedings 4 , pages=. 2019 , organization=

    2019

  20. [28]

    Human action recognition by representing

    Vemulapalli, Raviteja and Arrate, Felipe and Chellappa, Rama , booktitle=. Human action recognition by representing

  21. [29]

    Deep learning on

    Huang, Zhiwu and Wan, Chengde and Probst, Thomas and Van Gool, Luc , booktitle=. Deep learning on

  22. [30]

    Nguyen, Xuan Son , booktitle=. A

  23. [31]

    ICIP , year=

    Frame Attention Networks for Facial Expression Recognition in Videos , author=. ICIP , year=

  24. [32]

    Proceedings of the 20th ACM International Conference on Multimodal Interaction , pages=

    Emotiw 2018: Audio-video, Student Engagement and Group-level Affect Prediction , author=. Proceedings of the 20th ACM International Conference on Multimodal Interaction , pages=

    2018

  25. [33]

    First-person Hand Action Benchmark with

    Garcia-Hernando, Guillermo and Yuan, Shanxin and Baek, Seungryul and Kim, Tae-Kyun , booktitle=. First-person Hand Action Benchmark with

  26. [34]

    Deepkspd: Learning kernel-matrix-based

    Engin, Melih and Wang, Lei and Zhou, Luping and Liu, Xinwang , booktitle=. Deepkspd: Learning kernel-matrix-based

  27. [35]

    ICCV , year=

    Matrix Backpropagation for Deep Networks with Structured Layers , author=. ICCV , year=

  28. [36]

    A review of kernels on covariance matrices for

    Yger, Florian , booktitle=. A review of kernels on covariance matrices for. 2013 , organization=

    2013

  29. [37]

    ICML , year=

    Rectified linear units improve restricted boltzmann machines , author=. ICML , year=

  30. [38]

    ICML , year=

    Batch normalization: Accelerating deep network training by reducing internal covariate shift , author=. ICML , year=

  31. [39]

    CVPR , year=

    Deep Residual Learning for Image Recognition , author=. CVPR , year=

  32. [40]

    CVPR , year=

    Covariance Discriminative Learning: A Natural and Efficient Approach to Image Set Classification , author=. CVPR , year=

  33. [41]

    Huang, Zhiwu and Wang, Ruiping and Shan, Shiguang and Li, Xianqiu and Chen, Xilin , booktitle=. Log-

  34. [42]

    Huang, Zhiwu and Van Gool, Luc , booktitle=. A

  35. [43]

    ICLR , year=

    Riemannian adaptive optimization methods , author=. ICLR , year=

  36. [44]

    Bloom, Victoria and Makris, Dimitrios and Argyriou, Vasileios , booktitle=

  37. [45]

    GeomNet: A Neural Network Based on

    Nguyen, Xuan Son , booktitle=. GeomNet: A Neural Network Based on

  38. [46]

    ICCV , year=

    Dilated Convolutional Neural Networks for Sequential Manifold-valued Data , author=. ICCV , year=

  39. [47]

    Why Approximate Matrix Square Root Outperforms Accurate

    Song, Yue and Sebe, Nicu and Wang, Wei , booktitle=. Why Approximate Matrix Square Root Outperforms Accurate

  40. [48]

    Nguyen, Xuan Son , booktitle=. The

  41. [49]

    ICML , year=

    Hyperplane margin classifiers on the multinomial manifold , author=. ICML , year=

  42. [50]

    A neural network based on

    Nguyen, Xuan Son and Brun, Luc and L. A neural network based on. CVPR , year=

  43. [51]

    NeurIPS , year=

    Hyperbolic neural networks , author=. NeurIPS , year=

  44. [52]

    ICCV , year=

    Delving deep into rectifiers: Surpassing human-level performance on imagenet classification , author=. ICCV , year=

  45. [53]

    Building deep networks on

    Huang, Zhiwu and Wu, Jiqing and Van Gool, Luc , booktitle=. Building deep networks on

  46. [54]

    ECCV , year=

    Momentum batch normalization for deep learning with small batch size , author=. ECCV , year=

  47. [55]

    CVPR , year=

    Domain-specific batch normalization for unsupervised domain adaptation , author=. CVPR , year=

  48. [56]

    ECCV , year=

    Group normalization , author=. ECCV , year=

  49. [57]

    ICLR , year=

    Matrix Manifold Neural Networks++ , author=. ICLR , year=

  50. [58]

    ICLR , year=

    Flow Matching on General Geometries , author=. ICLR , year=

  51. [59]

    Fr. Les. Annales de l'Institut Henri Poincar

  52. [60]

    Shahroudy, Amir and Liu, Jun and Ng, Tian-Tsong and Wang, Gang , booktitle=

  53. [61]

    NeurIPS , year=

    A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices , author=. NeurIPS , year=

  54. [62]

    Kobler, Reinmar and Hirayama, Jun-ichiro and Zhao, Qibin and Kawanabe, Motoaki , booktitle=

  55. [63]

    Shanglin Li and Motoaki Kawanabe and Reinmar J Kobler , booktitle=

  56. [64]

    Exploring the Enigma of Neural Dynamics Through A Scattering-Transform Mixer Landscape for

    Dan, Tingting and Wei, Ziquan and Kim, Won Hwa and Wu, Guorong , booktitle=. Exploring the Enigma of Neural Dynamics Through A Scattering-Transform Mixer Landscape for

  57. [65]

    Rolling rotations for recognizing human actions from

    Vemulapalli, Raviteja and Chellappa, Rama , booktitle=. Rolling rotations for recognizing human actions from

  58. [66]

    A geodesic-based

    Wang, Jing and Sun, Huafei and Li, Didong , journal=. A geodesic-based. 2017 , publisher=

    2017

  59. [67]

    Econometrica , volume=

    A new parametrization of correlation matrices , author=. Econometrica , volume=. 2021 , publisher=

    2021

  60. [68]

    David, Paul and Gu, Weiqing , journal=. A

  61. [69]

    Permutation-invariant

    Thanwerdas, Yann , journal=. Permutation-invariant

  62. [70]

    Automatic differentiation in

    Paszke, Adam and Gross, Sam and Chintala, Soumith and Chanan, Gregory and Yang, Edward and DeVito, Zachary and Lin, Zeming and Desmaison, Alban and Antiga, Luca and Lerer, Adam , booktitle=. Automatic differentiation in

  63. [71]

    IJCV , year=

    Rotation averaging , author=. IJCV , year=

  64. [72]

    Left-invariant

    Zacur, Ernesto and Bossa, Matias and Olmos, Salvador , journal=. Left-invariant

  65. [73]

    Afsari, Bijan , journal=

  66. [74]

    NeurIPS , year=

    Riemannian diffusion models , author=. NeurIPS , year=

  67. [75]

    IFAC Proceedings Volumes , volume=

    A discrete regression method on manifolds and its application to data on SO (n) , author=. IFAC Proceedings Volumes , volume=. 2011 , publisher=

    2011

  68. [76]

    IEEE TPAMI , year=

    Towards a Deeper Understanding of Global Covariance Pooling in Deep Learning: An Optimization Perspective , author=. IEEE TPAMI , year=

  69. [77]

    arXiv preprint arXiv:1607.08022 , year=

    Instance normalization: the missing ingredient for fast stylization , author=. arXiv preprint arXiv:1607.08022 , year=

    arXiv

  70. [78]

    arXiv preprint arXiv:1607.06450 , year=

    Layer normalization , author=. arXiv preprint arXiv:1607.06450 , year=

    arXiv

  71. [79]

    IEEE TPAMI , year=

    On the eigenvalues of global covariance pooling for fine-grained visual recognition , author=. IEEE TPAMI , year=

  72. [80]

    SIMAX , year=

    Theoretically and computationally convenient geometries on full-rank correlation matrices , author=. SIMAX , year=

  73. [81]

    Linear Algebra and its Applications , volume=

    Minh, H. Linear Algebra and its Applications , volume=. 2022 , publisher=

    2022

  74. [82]

    The geometry of mixed-

    Thanwerdas, Yann and Pennec, Xavier , journal=. The geometry of mixed-. 2022 , publisher=

    2022

  75. [83]

    Building Neural Networks on Matrix Manifolds: A

    Nguyen, Xuan Son and Yang, Shuo , journal=. Building Neural Networks on Matrix Manifolds: A

  76. [84]

    Wang, Rui and Wu, Xiao-Jun and Kittler, Josef , journal=

  77. [85]

    arXiv preprint arXiv:2006.08210 , year=

    Hyperbolic neural networks++ , author=. arXiv preprint arXiv:2006.08210 , year=

    2006 arXiv

  78. [86]

    arXiv preprint arXiv:1509.07838 , year=

    Training deep networks with structured layers by matrix backpropagation , author=. arXiv preprint arXiv:1509.07838 , year=

    arXiv

  79. [87]

    Parallel transport on the cone manifold of

    Yair, Or and Ben-Chen, Mirela and Talmon, Ronen , journal=. Parallel transport on the cone manifold of

  80. [88]

    Chakraborty, Rudrasis and Bouza, Jose and Manton, Jonathan H and Vemuri, Baba C , journal=

  81. [89]

    Clinical neurophysiology , volume=

    Brain--computer interfaces for communication and control , author=. Clinical neurophysiology , volume=. 2002 , publisher=

    2002

  82. [90]

    Transfer learning: A

    Zanini, Paolo and Congedo, Marco and Jutten, Christian and Said, Salem and Berthoumieu, Yannick , journal=. Transfer learning: A. 2018 , doi=

    2018

  83. [91]

    2019 , doi=

    Rodrigues, Pedro Luiz Coelho and Jutten, Christian and Congedo, Marco , journal=. 2019 , doi=

    2019

  84. [92]

    Vector-valued distance and

    L. Vector-valued distance and. NeurIPS , year=

  85. [93]

    Controlling the

    Kobler, Reinmar J and Hirayama, Jun-ichiro and Kawanabe, Motoaki , booktitle=. Controlling the

  86. [94]

    NeurIPS , year=

    Attention Is All You Need , author=. NeurIPS , year=

  87. [95]

    NeurIPS , year=

    Imagenet Classification with Deep Convolutional Neural Networks , author=. NeurIPS , year=

  88. [96]

    Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and Kopf, Andreas and Yang, Edward and DeVito, Zachary and Raison, Martin ...

  89. [97]

    Riemannian Batch Normalization for

    Brooks, Daniel and Schwander, Olivier and Barbaresco, Fr. Riemannian Batch Normalization for. NeurIPS , year=

  90. [98]

    Neural architecture search of

    Sukthanker, Rhea and Huang, Zhiwu and Kumar, Suryansh and Endsjo Goron, Erik and Wu, Yan and Van Gool, Luc , booktitle=. Neural architecture search of

  91. [99]

    Statistical computing on manifolds: from

    Pennec, Xavier , booktitle=. Statistical computing on manifolds: from. 2009 , organization=

    2009

  92. [100]

    Differentiating through the

    Lou, Aaron and Katsman, Isay and Jiang, Qingxuan and Belongie, Serge and Lim, Ser-Nam and De Sa, Christopher , booktitle=. Differentiating through the

  93. [101]

    Improving Expressivity of

    Chen, Kaixuan and Liu, Shunyu and Zhu, Tongtian and Qiao, Ji and Su, Yun and Tian, Yingjie and Zheng, Tongya and Zhang, Haofei and Feng, Zunlei and Ye, Jingwen and Song, Mingli , booktitle=. Improving Expressivity of

  94. [102]

    arXiv preprint arXiv:2502.01512 , year=

    Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices , author=. arXiv preprint arXiv:2502.01512 , year=

  95. [103]

    TMLR , year=

    Geometry-Aware visualization of high dimensional Symmetric Positive Definite matrices , author=. TMLR , year=

  96. [104]

    Advances in Applied Mathematics , volume=

    Newton's method, zeroes of vector fields, and the Riemannian center of mass , author=. Advances in Applied Mathematics , volume=. 2004 , publisher=

    2004

  97. [105]

    Proceedings of the London Mathematical Society , volume=

    Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence , author=. Proceedings of the London Mathematical Society , volume=. 1990 , publisher=

    1990

  98. [106]

    IEEE TKDE , year=

    Distribution Knowledge Embedding for Graph Pooling , author=. IEEE TKDE , year=

  99. [107]

    Covariance descriptors on a

    Chen, Kai-Xuan and Ren, Jie-Yi and Wu, Xiao-Jun and Kittler, Josef , journal=. Covariance descriptors on a. 2020 , publisher=

    2020

  100. [108]

    Journal of Mathematical Imaging and Vision , volume=

    Uniform distribution, distance and expectation problems for geometric features processing , author=. Journal of Mathematical Imaging and Vision , volume=. 1998 , publisher=

    1998

  101. [109]

    Statistics on the

    Chakraborty, Rudrasis and Vemuri, Baba C , journal=. Statistics on the

  102. [110]

    IEEE Transactions on Robotics , volume=

    Error propagation on the Euclidean group with applications to manipulator kinematics , author=. IEEE Transactions on Robotics , volume=. 2006 , publisher=

    2006

  103. [111]

    NeurIPS , year=

    Trivializations for gradient-based optimization on manifolds , author=. NeurIPS , year=

  104. [112]

    Riemannian

    Said, Salem and Bombrun, Lionel and Berthoumieu, Yannick and Manton, Jonathan H , journal=. Riemannian

  105. [113]

    Frontiers in Neuroscience , author =

  106. [114]

    2018 , pages =

    Journal of Neural Engineering , author =. 2018 , pages =

    2018

  107. [115]

    Bhatia, Rajendra and Jain, Tanvi and Lim, Yongdo , journal=. On the. 2019 , publisher=

    2019

  108. [116]

    Linear algebra and its applications , volume=

    Geometric means , author=. Linear algebra and its applications , volume=. 2004 , publisher=

    2004

  109. [117]

    SIMAX , year=

    Conic geometric optimization on the manifold of positive definite matrices , author=. SIMAX , year=

  110. [118]

    IEEE TMM , year=

    Deep Manifold-to-Manifold Transforming Network for Skeleton-Based Action Recognition , author=. IEEE TMM , year=

  111. [119]

    Dimensionality Reduction on

    Harandi, Mehrtash and Salzmann, Mathieu and Hartley, Richard , journal=. Dimensionality Reduction on

  112. [120]

    International Journal for Numerical Methods in Engineering , volume=

    Generalized Stochastic Approach for Constitutive Equation in Linear Elasticity: a Random Matrix Model , author=. International Journal for Numerical Methods in Engineering , volume=. 2012 , publisher=

    2012

  113. [121]

    Journal of Elasticity , volume=

    On the Averaging of Symmetric Positive-Definite Tensors , author=. Journal of Elasticity , volume=. 2006 , publisher=

    2006

  114. [122]

    IEEE Journal of Selected Topics in Signal Processing , volume=

    Riemannian Medians and Means with Applications to Radar Signal Processing , author=. IEEE Journal of Selected Topics in Signal Processing , volume=. 2013 , publisher=

    2013

  115. [123]

    Hua, Xiaoqiang and Cheng, Yongqiang and Wang, Hongqiang and Qin, Yuliang and Li, Yubo and Zhang, Wenpeng , journal=. Matrix. 2017 , publisher=

    2017

  116. [124]

    Pennec, Xavier and Fillard, Pierre and Ayache, Nicholas , journal=. A

  117. [125]

    NeurIPS , year=

    On Riemannian optimization over positive definite matrices with the Bures-Wasserstein geometry , author=. NeurIPS , year=

  118. [126]

    Dryden, Ian L and Pennec, Xavier and Peyrat, Jean-Marc , journal=. Power

  119. [127]

    Riemannian Geometry of Symmetric Positive Definite Matrices via

    Lin, Zhenhua , journal=. Riemannian Geometry of Symmetric Positive Definite Matrices via

  120. [128]

    Technical report , year=

    Positive definite matrices and the symmetric Stein divergence , author=. Technical report , year=

  121. [129]

    Face recognition on large-scale video in the wild with hybrid

    Huang, Zhiwu and Wang, Ruiping and Shan, Shiguang and Chen, Xilin , journal=. Face recognition on large-scale video in the wild with hybrid. 2015 , publisher=

    2015

  122. [130]

    Neural Computation , volume=

    Long Short-Term Memory , author=. Neural Computation , volume=. 1997 , publisher=

    1997

  123. [131]

    NeurIPS , year=

    Temporal-attentive Covariance Pooling Networks for Video Recognition , author=. NeurIPS , year=

  124. [132]

    Wang, Qilong and Xie, Jiangtao and Zuo, Wangmeng and Zhang, Lei and Li, Peihua , journal=. Deep

  125. [133]

    Differentiation of the

    Murray, Iain , journal=. Differentiation of the

  126. [134]

    IEEE TPAMI , year=

    Fast Differentiable Matrix Square Root and Inverse Square Root , author=. IEEE TPAMI , year=

  127. [135]

    Stochastic gradient descent on

    Bonnabel, Silvere , journal=. Stochastic gradient descent on. 2013 , publisher=

    2013

  128. [136]

    O (n)-invariant

    Thanwerdas, Yann and Pennec, Xavier , journal=. O (n)-invariant. 2023 , publisher=

    2023

  129. [137]

    Geometric deep learning: going beyond

    Bronstein, Michael M and Bruna, Joan and LeCun, Yann and Szlam, Arthur and Vandergheynst, Pierre , journal=. Geometric deep learning: going beyond. 2017 , publisher=

    2017

  130. [138]

    Chakraborty, Rudrasis , journal=

  131. [139]

    Communications on Pure and Applied Mathematics , volume=

    Riemannian center of mass and mollifier smoothing , author=. Communications on Pure and Applied Mathematics , volume=. 1977 , publisher=

    1977

  132. [140]

    Fast and simple computations on tensors with log-

    Arsigny, Vincent and Fillard, Pierre and Pennec, Xavier and Ayache, Nicholas , year=. Fast and simple computations on tensors with log-

  133. [141]

    A panoramic view of

    Berger, Marcel , year=. A panoramic view of

  134. [142]

    1976 , publisher=

    Principles of mathematical analysis , author=. 1976 , publisher=

    1976

  135. [143]

    1992 , publisher=

    Riemannian Geometry , author=. 1992 , publisher=

    1992

  136. [144]

    2013 , publisher=

    Introduction to smooth manifolds , author=. 2013 , publisher=

    2013

  137. [145]

    2016 , publisher=

    Information geometry and its applications , author=. 2016 , publisher=

    2016

  138. [146]

    2011 , publisher=

    An Introduction to Manifolds , author=. 2011 , publisher=

    2011

  139. [147]

    1999 , publisher=

    Lectures on differential geometry , author=. 1999 , publisher=

    1999

  140. [148]

    2019 , publisher=

    Matrix differential calculus with applications in statistics and econometrics , author=. 2019 , publisher=

    2019

  141. [149]

    2008 , publisher=

    Optimization Algorithms on Matrix Manifolds , author=. 2008 , publisher=

    2008

  142. [150]

    2009 , publisher=

    Positive Definite Matrices , author=. 2009 , publisher=

    2009

  143. [151]

    1965 , publisher=

    Thirteen papers on functional analysis and partial differential equations , author=. 1965 , publisher=

    1965

  144. [152]

    Coxeter, Harold Scott Macdonald , year=

  145. [153]

    2016 , publisher=

    Mathematical analysis II , author=. 2016 , publisher=

    2016

  146. [154]

    2013 , publisher=

    Matrix analysis , author=. 2013 , publisher=

    2013

  147. [155]

    Introduction to

    Lee, John M , series=. Introduction to. 2018 , publisher=

    2018

  148. [156]

    , series=

    Hall, Brian C. , series=. 2015 , publisher=

    2015

  149. [157]

    and Darmet, Ludovic and Somon, Bertille and Jahanpour, Emilie and Lotte, Fabien and Ladouce, Simon and Roy, Raphaëlle N

    Hinss, Marcel F. and Darmet, Ludovic and Somon, Bertille and Jahanpour, Emilie and Lotte, Fabien and Ladouce, Simon and Roy, Raphaëlle N. , year =. An

  150. [158]

    Probabilities and statistics on

    Pennec, Xavier , institution=. Probabilities and statistics on

  151. [159]

    2020 , school=

    Deep Learning and Information Geometry for Time-Series Classification , author=. 2020 , school=

    2020

  152. [160]

    2022 , school=

    Riemannian and stratified geometries on covariance and correlation matrices , author=. 2022 , school=

    2022

  153. [161]

    1995 , publisher=

    Riemannian Geometry: A Modern Introduction , author=. 1995 , publisher=

    1995

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