Distance Measure Based on an Embedding of the Manifold of K-Component Gaussian Mixture Models into the Manifold of Symmetric Positive Definite Matrices

Amit Vishwakarma; KS Subrahamanian Moosath

REVIEW 3 major objections 5 minor 15 references

Distance Measure Based on an Embedding of the Manifold of K-Component Gaussian Mixture Models into the Manifold of Symmetric Positive Definite Matrices

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read The paper embeds the manifold of K-component Gaussian mixture models into the manifold of symmetric positive definite matrices, proving that the affine-invariant metric on the matrix manifold is always a lower bound for the Fisher-Rao…

desk verdict The block embedding is clean but the central lower-bound theorem is unproved and the proposed distance is not invariant under component relabeling, so it is not a distance between GMM densities. read the letter →

arxiv 2501.07429 v1 pith:ZSRXHZFT submitted 2025-01-13 math.DG cs.LG

Amit Vishwakarma , KS Subrahamanian Moosath This is my paper · ORCID

classification math.DGcs.LG MSC 53B1253B2062B1062H30

keywords GaussianmixturemodelssymmetricpositivedefinitematricesFisher-Raometricaffine-invariantinformationgeometrymanifoldembeddingtextureclassification

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

Comparing two Gaussian mixture models is a standing problem in machine learning because the natural geodesic distance (Fisher-Rao) and the KL divergence both lack closed forms for mixtures. This paper tries to make GMM comparison tractable by embedding the whole K-component mixture into a single symmetric positive definite matrix, then using the affine-invariant metric on the matrix manifold as the distance between GMMs. The core claims are that this embedding makes the GMM manifold a submanifold of the SPD manifold, that the GMM manifold with the pullback metric is isometric to the embedded submanifold, and that the resulting affine-invariant distance is always a lower bound for the Fisher-Rao distance, becoming exact when means are fixed and mixing coefficients are uniform. The authors use this distance as a similarity measure in texture classification, reporting accuracies near 99% on KTH-TIPS, 92% on UIUC, and 93% on UMD, above the KL-based baselines they compare against.

What carries the argument

The load-bearing object is the explicit block embedding of Theorem 4: for θ = (π_k, μ_k, Σ_k), S = [[A, X],[X^T, B]] with A = diag(Σ_k + π_k μ_k μ_k^T), X = diag(π_k μ_k), and B = diag(π_k). The proof machinery is the Schur complement criterion for positive definiteness (Gallier's theorem), Sherman-Morrison inversion for (Σ_k + π_k μ_k μ_k^T)^{-1}, and the Lu-Shiou block inverse formula, which together yield the pullback metric ds² = Σ_k [ ½(dπ_k/π_k)² + π_k dμ_k^T $Σ_k^{{-1}}$ dμ_k + ½ tr(($Σ_k^{{-1}}$ dΣ_k)²) ]. That metric, coupled with the comparison between geodesic distance inside the embedded submanifold and geodesic distance in the ambient SPD manifold, is what produces the lower bound of Theorem 7.

What would settle it

Take a two-component GMM with equal weights, distinct means, and the same covariance; swap the two components to get a second parameter tuple that defines the same density. If the affine-invariant distance between the two embedded SPD matrices is nonzero, the proposed measure is not a distance between probability distributions, because identical distributions would be assigned positive separation.

Watch

Extended reading notes

Core claim

The paper's central discovery, stated in its own terms, is that the map f(θ) = S with S = [[A, X],[X^T, B]] is an embedding of the statistical manifold M of K-component GMMs into SP D_{K(n+1)}(R), and that (M, f^*ρ_{f(M)}) is isometric to (f(M), ρ_{f(M)}). Theorem 7 then asserts that the Fisher-Rao metric g_M dominates the induced affine-invariant metric: g_M ≥ ρ_{f(M)}, so the geodesic distance in the SPD manifold is a genuine lower bound for the Rao distance between GMMs, with equality when the means are fixed and the weights are uniform. The paper identifies this lower bound as a distance measure on the GMM manifold and employs it as the similarity measure for GMMs, demonstrating the approach on texture recognition benchmarks.

Load-bearing premise

The identification of the GMM manifold with the ordered parameter space Θ in Section 3.3 — since GMM densities are invariant under reordering components, while both the embedded matrix S and the distance change when components are permuted, the proposed distance is a distance on parameter tuples, not on probability distributions, unless that identification is justified differently.

Editorial extensions

If this is right

The GMM manifold embeds as a submanifold of SP D_{K(n+1)}(R) of dimension K/2 (n+1)(n+2) − 1, with the pullback metric making the embedding an isometry.
For any two GMMs, the affine-invariant distance between their embedded matrices is a computable lower bound for the Fisher-Rao geodesic distance, and the bound is tight when means are fixed and mixing coefficients are uniform.
When means are fixed and weights are uniform, the embedded submanifold is geodesic, so the proposed distance exactly equals the Fisher-Rao distance within that submanifold.
Because the distance is computed by closed-form matrix logarithms of the SPD manifold, no Monte Carlo sampling or component-wise matching is needed for GMM comparisons.

Reading between the lines

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

If the ordered-parameter identification is dropped, the distance is likely to be a distance on quotient objects — GMM densities modulo permutation — only after symmetrization over component orderings; one testable extension is to define d_sym(p,q) = min over permutations σ of d(f(θ_p), f(θ_σ(q))) and check whether the triangle inequality still holds.
The lower bound could serve as a fast surrogate in algorithms that currently approximate Rao distances or KL divergences in estimation, clustering, and gradient-based fitting of GMMs, in the same spirit that the single-Gaussian Calvo-Oller bound supports hypothesis tests.
The equality case (fixed means, uniform weights) suggests a submanifold where several inference problems become exactly solvable in closed form, which might provide a testbed for calibration of approximate methods on the full GMM manifold.

Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

Claim #1: The paper's central discovery, stated in its own terms, is that the map f(θ) = S with S = [[A, X],[X^T, B]] is an embedding of the statistical manifold M of K-component GMMs into SP D_{K(n+1)}(R), and that (M, f^*ρ_{f(M)}) is isometric to (f(M), ρ_{f(M)}). Theorem 7 then asserts that the Fisher-Rao metric g_M dominates the induced affine-invariant metric: g_M ≥ ρ_{f(M)}, so the geodesic distance i

/-- @claim 1 The paper's central discovery, stated in its own terms, is that the map f(θ) = S with S = [[A, X],[X^T, B]] is an embedding of the statistical manifold M of K-component GMMs into SP D_{K(n+1)}(R), and that (M, f^*ρ_{f(M)}) is isometric to (f(M), ρ_{f(M)}). Theorem 7 then asserts that the Fisher-Rao metric g_M dominates the induced affine-invariant metric: g_M ≥ ρ_{f(M)}, so the geodesic distance i -/
def central_claim : Prop :=

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main construction is a block SPD embedding for ordered K-component GMM parameter tuples, extending Calvo-Oller's single-Gaussian embedding and Popović et al.'s component-wise schemes. The block matrix is explicitly defined, positivity is proved via Schur complements, and the pullback metric computation in Appendix B is explicit and checkable. Theorem 6's isometry between the parameter manifold with the pullback metric and its image is correct but tautological.

The problems start with Theorem 7. Appendix D does not relate the Fisher-Rao metric of the GMM to the pullback affine-invariant metric; it only invokes the isometry from Theorem 6, which is about the pullback metric. The inequality g_M >= rho_{f(M)} is asserted, not derived, and the equality case is likewise unsupported: the geodesic submanifold property concerns intrinsic versus extrinsic distance in SPD, not Fisher-Rao.

More fundamentally, the embedding is not an embedding of GMM densities. The density p(x;theta) is unchanged by permuting components, but f(theta) changes to a simultaneously permuted block matrix. The affine-invariant distance is not invariant under relabeling only one argument, so d(f(theta_1), f(theta_2)) can change when theta_2 is relabeled while the density is unchanged. The map is injective only on ordered parameter tuples, as Appendix C shows. Thus the proposed quantity is a distance on ordered parameter space, not between GMM distributions; the comparison with the Fisher-Rao distance is ill-posed because Fisher-Rao is defined on densities. The abstract's 'distance between GMMs' is therefore not what the paper delivers.

The experiments are weak support: five classes per dataset, no error bars or splits, and the text claims to outperform the DPLM baseline that never appears in the results table. The abstract reports 98% for UIUC while the table says 98.82%; minor inconsistencies like that do not help.

Credit where due: the block matrix algebra is sound and the embedding is new, though natural. This could serve as a similarity measure for ordered GMM representations with fixed labels, if Theorem 7 is dropped. As a claim about GMM geometry, the central result fails. I would desk reject this version; the salvageable version is a smaller, honest paper.

Referee Report

3 major / 5 minor

Summary. The paper proposes an embedding of K-component Gaussian Mixture Models (GMMs) into the manifold of symmetric positive definite (SPD) matrices, proves several geometric properties of the embedded submanifold, and claims that the affine-invariant distance on the embedded matrices is a lower bound for the Fisher-Rao distance on the GMM manifold. The authors also report texture-classification experiments using this distance. The central mathematical claims are Theorem 6 (embedding and isometry) and Theorem 7 (lower bound g_M ≥ ρ_{f(M)}).

Significance. If the main theorems were correct, the paper would provide a computable, closed-form similarity measure for GMMs and a lower bound for the Fisher-Rao distance, which would be a useful contribution to information geometry and machine learning. The direct computation of the pullback of the affine-invariant metric (Theorem 5) is a useful exercise, and the positive-definiteness proof in Appendix A is checkable. However, the two load-bearing claims—that the construction gives a distance between GMM densities and that it lower-bounds the Fisher-Rao distance—are not established. The proposed distance is not invariant under component relabeling, and the proof of Theorem 7 conflates the pullback metric with the Fisher-Rao metric. These are foundational issues, not presentation issues.

major comments (3)

[Section 3.3, Theorem 4, Appendix C] The map f is not well-defined on the statistical model S = {p(x;θ)} because GMM densities are invariant under permutations of the K components, while f uses the ordered tuple θ = (π_k, μ_k, Σ_k). For any permutation σ, p(x; σ·θ) = p(x; θ) but f(σ·θ) = P_σ S P_σ^T for a block permutation matrix P_σ, and generally P_σ S P_σ^T ≠ S. The affine-invariant distance on SPD is invariant under simultaneous conjugation of both arguments but not under permutation of a single argument, so d(f(θ_1), f(θ_2)) changes when θ_2 is relabeled even though the two densities are identical. Appendix C proves injectivity only for ordered parameter tuples, not for the quotient by the symmetric group, so Theorem 6 does not establish an embedding of the manifold of GMM densities into SPD. Consequently, the abstract's claim of a 'distance between GMMs' is not a distance between probability distributions.
[Appendix D, Theorem 7] The proof of the lower bound g_M ≥ ρ_{f(M)} is not valid. Theorem 6(2) proves an isometry between (M, f^*ρ_{f(M)}) and (f(M), ρ_{f(M)}), i.e., it identifies the pullback of the induced affine-invariant metric with the induced metric on the image. This says nothing about the Fisher-Rao metric g_M. The proof then asserts that 'g_M corresponds to the Riemannian distance between points in f(M) induced by the affine-invariant metric ρ,' which is precisely the unproved claim. The metric computed in Theorem 5 is the pullback of the affine-invariant metric, not the Fisher-Rao metric of a GMM; the Fisher-Rao metric for a mixture model involves integrals over products of component densities and does not reduce to a sum of per-component terms as in Eq. (24). Thus the central inequality is unsupported.
[Section 3.1 vs. Section 3.3] The definition of a statistical manifold in Section 3.1 explicitly requires that the parametrization θ ↦ p_θ be injective. The GMM parameterization in Section 3.3 is not injective because of the permutation symmetry of the components. The paper does not address this and identifies the 'manifold of GMMs' with the ordered parameter space Θ of Eq. (13). This is not a minor technicality: it breaks the embedding claim (since f is not injective on densities) and it undermines the interpretation of the Fisher-Rao distance, which is defined on densities, not on arbitrary labelings of the same density.

minor comments (5)

[Abstract and Section 5.6] The reported accuracy for KTH-TIPS is inconsistent: the abstract and Section 5.6 state 98% and 98.82%, respectively, while Table 1 lists 98.82. Please harmonize these numbers.
[Theorem 5] In the displayed formula for ds^2, the first term inside the sum is written as (dπ_K/π_K)^2, but the sum index is k; this should be (dπ_k/π_k)^2.
[Section 5.5] The symbol K is used both for the number of GMM components and for the number of nearest neighbors in the K-nearest-neighbors classifier (K=5). This overloading is confusing and should be avoided.
[Section 5.6] The experimental section does not report the train/test splits, the number of images per class, the number of EM restarts, or any variance measures; it also compares only against KL-WA and KL-MB, despite the text claiming superiority over DPLM. Additional experimental detail is needed for the results to be reproducible and interpretable.
[Throughout] There are numerous typographical errors, including 'Intordutcion', 'similairty', 'satistfies', 'comapre', 'Furure', and 'simialrity'. The manuscript also contains the template placeholder 'Editor: My editor', which should be removed.

Circularity Check

1 steps flagged · score 8.0 of 10

The claimed general lower bound for the Fisher-Rao metric is obtained by identifying the Fisher-Rao metric with the pullback of the affine-invariant metric, making the bound definitional rather than derived.

self definitional [Theorem 7 and Appendix D]
"By theorem 6, f (M) is isometric to M. Therefore, gM corresponds to the Riemannian distance between points inf (M) induced by the affine-invariant metricρ. However, since f (M) is a non-geodesic submanifold of SP DK(n+1)(R), the geodesic distance restricted to f (M) is greater than or equal to the geodesic distance in SP DK(n+1)(R)."

Theorem 6 only establishes an isometry between (M, f^*ρ_{f(M)}) and (f(M), ρ_{f(M)}), i.e. with the pullback of the affine-invariant metric, not between the Fisher-Rao manifold (M, g_M) and f(M). Appendix D replaces g_M, defined in Theorem 7 as the Fisher-Rao metric, with this pullback metric without computing the Fisher information of the GMM density p(x;θ). The conclusion g_M ≥ ρ_{f(M)} therefore holds only if 'Fisher-Rao metric' is redefined to be the pullback metric; the claimed lower bound is the pullback metric itself by construction, not a bound derived from the actual Fisher-Rao metric.

full rationale

The experimental portion is not circular: it uses external texture datasets, standard EM fitting, and no fitted constants are presented as predictions. The paper also does not rely on load-bearing self-citations; the cited embedding of Calvo and Oller is external and the GMM-related citations are background. The circularity is concentrated in the central theoretical claim. The only metric actually computed in the paper is the pullback f^*ρ of the affine-invariant SPD metric (Theorem 5), and Theorem 6.2 merely restates the tautology that a pullback metric makes any embedding an isometry. The abstract and Theorem 7 then advertise this pullback metric as a 'general lower bound for the Fisher-Rao metric.' Appendix D obtains the inequality g_M ≥ ρ_{f(M)} by the sentence 'g_M corresponds to the Riemannian distance... induced by the affine-invariant metric,' i.e. by substituting the pullback metric for the Fisher-Rao metric. No Fisher information matrix for the mixture p(x;θ) = Σ π_k N(x; μ_k, Σ_k) is computed anywhere in the paper, so the lower-bound theorem reduces by definition to an identity between the pullback metric and itself. That is a load-bearing definitional substitution rather than a derivation, which warrants the high circularity score, even though the embedding computations themselves are direct and the experiments are independent.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The theoretical derivation has no fitted constants, but it rests on an ordered parameterization of GMMs, an unproved inequality, and standard SPD geometry. Experimental hyperparameters are user-chosen.

free parameters (3)

number of GMM components K = 5
Chosen for the experiments; the theoretical results hold for general K but the classification accuracy depends on this choice.
number of nearest neighbors k = 5
Chosen for KNN voting in Section 5.5; no sensitivity analysis is provided.
regularization epsilon for SPD matrix inversion = not specified
Added for numerical stability in Section 5.5 but its value is not given.

assumptions (4)

domain assumption The ordered parameter space Θ is a faithful coordinate system for the manifold of K-component GMMs.
Section 3.3 defines Θ and M; however, permutations of (π_k, μ_k, Σ_k) give identical densities, so the coordinates are not injective on the space of distributions. The embedding in Theorem 4 inherits this labeling dependence.
ad hoc to paper The Fisher-Rao metric g_M of the GMM manifold satisfies g_M ≥ ρ_{f(M)} pointwise as quadratic forms.
This is the content of Theorem 7. The appendix proof does not compute the Fisher information matrix and only invokes the isometry from Theorem 6, so the inequality is an unproved assertion assumed for the paper's main distance claim.
standard math The affine-invariant metric on SPD is a Riemannian metric with geodesics, and constant S^{-1}dS characterizes geodesic submanifolds.
Used in Appendix C to analyze total geodesicity of f(M). The argument that non-constant S^{-1}dS implies non-geodesic is incomplete and not a standard criterion.
domain assumption All mixture weights are strictly positive and all covariances are positive definite, so the block matrix S is positive definite.
The embedding proof uses π_k>0 and Σ_k positive definite; boundary cases with zero weights or degenerate covariances are excluded from the manifold.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Distance Measure Based on an Embedding of the Manifold of K-Component Gaussian Mixture Models into the Manifold of Symmetric Positive Definite Matrices." pith.science (2026). https://pith.science/paper/ZSRXHZFT

@misc{pith2026250107429,
  author       = {Pith},
  title        = {Pith review of: Distance Measure Based on an Embedding of the Manifold of K-Component Gaussian Mixture Models into the Manifold of Symmetric Positive Definite Matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSRXHZFT}},
  note         = {Machine review of arXiv:2501.07429}
}

read the original abstract

In this paper, a distance between the Gaussian Mixture Models(GMMs) is obtained based on an embedding of the K-component Gaussian Mixture Model into the manifold of the symmetric positive definite matrices. Proof of embedding of K-component GMMs into the manifold of symmetric positive definite matrices is given and shown that it is a submanifold. Then, proved that the manifold of GMMs with the pullback of induced metric is isometric to the submanifold with the induced metric. Through this embedding we obtain a general lower bound for the Fisher-Rao metric. This lower bound is a distance measure on the manifold of GMMs and we employ it for the similarity measure of GMMs. The effectiveness of this framework is demonstrated through an experiment on standard machine learning benchmarks, achieving accuracy of 98%, 92%, and 93.33% on the UIUC, KTH-TIPS, and UMD texture recognition datasets respectively.

Figures

Figures reproduced from arXiv: 2501.07429 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 8 canonical work pages

[10]

Kittipat Kampa, Erion Hasanbelliu, and Jos´ e Carlos Pr ´ ıncipe

1109/ICASSP.2007.366913. Kittipat Kampa, Erion Hasanbelliu, and Jos´ e Carlos Pr ´ ıncipe. Closed-form cauchy-schwarz pdf divergence for mixture of gaussians. The 2011 International Joint Conference on Neural Networks , pages 2578–2585,

work page arXiv 2007
[11]

Sivalingam, D

R. Sivalingam, D. Boley, V. Morellas, and N. Papanikolopoulos. Tensor sparse coding for region covariances. In K. Daniilidis, P. Maragos, and N. Paragios, editors, Computer Vision – ECCV 2010 , volume 6314 of Lecture Notes in Computer Science , pages 709–

work page 2010
[15]

URL https://doi.org/10.1007/s11263-009-0220-6

1007/s11263-009-0220-6. URL https://doi.org/10.1007/s11263-009-0220-6 . 21

doi:10.1007/s11263-009-0220-6
[52]

Huan Wan, Hui Wang, Bryan Scotney, and Jun Liu

URL https: //doi.org/10.1007/978-3-642-15561-1_52 . Huan Wan, Hui Wang, Bryan Scotney, and Jun Liu. A novel gaussian mixture model for classification. In 2019 IEEE International Conference on Systems, Man and Cybernetics (SMC), pages 3298–3303,

work page doi:10.1007/978-3-642-15561-1_52 2019
[1990]

doi: 10.1016/0047-259X(90)90026-E

ISSN 0047-259X. doi: 10.1016/0047-259X(90)90026-E. URL https: //www.sciencedirect.com/science/article/pii/0047259X9090026E. Jean-Louis Durrieu, Jean-Philippe Thiran, and Finnian Kelly. Lower and upper bounds for approximation of the kullback-leibler divergence between gaussian mixture models. 2012 IEEE International Conference on Acoustics, Speech and Sig...

arXiv
[1996]

Hershey and Peder A

John R. Hershey and Peder A. Olsen. Approximating the kullback leibler divergence between gaussian mixture models. In 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP ’07 , volume 4, pages IV–317–IV–320,

work page 2007
[2002]

doi: https://doi

ISSN 0898-1221. doi: https://doi. org/10.1016/S0898-1221(01)00278-4. URL https://www.sciencedirect.com/science/ article/pii/S0898122101002784. Leandro Pardo. Statistical inference based on divergence measures . Chapman and Hall/CRC,

doi:10.1016/s0898-1221(01)00278-4
[2003]

doi: 10.1109/ICCV.2003.1238387. Gene H. Golub and Charles F. Van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, 3 edition,

arXiv 2003

Show all 15 references

[2004]

Available online: Accessed 2025-01-07

URL https://www.csc.kth.se/cvap/databases/kth-tips/doc/. Available online: Accessed 2025-01-07. J. Gallier. Schur complements and applications. Texts in Applied Mathematics , pages 431–437,

2025
[2005]

Peihua Li, Qilong Wang, and Lei Zhang

doi: 10.1109/TPAMI.2005.151. Peihua Li, Qilong Wang, and Lei Zhang. A novel earth mover’s distance methodology for image matching with gaussian mixture models. In Proceedings of the IEEE International Conference on Computer Vision , pages 1689–1696,

2005 doi
[2010]

doi: 10.1007/978-3-642-15561-1

doi
[2011]

doi: 10.1007/978-1-4419-9961-0

doi
[2019]

doi: 10.1109/SMC.2019.8914215. A. R. Webb. Statistical Pattern Recognition. John Wiley & Sons, Hoboken, NJ, USA,

2019
[2021]

Branislav Popovi´ c, Marko Janev, Lidija Krstanovi´ c, Nikola Simi´ c, and Vlado Deli´ c

doi: 10.3390/math9090957. Branislav Popovi´ c, Marko Janev, Lidija Krstanovi´ c, Nikola Simi´ c, and Vlado Deli´ c. Mea- sure of similarity between GMMs based on geometry-aware dimensionality reduction. Mathematics, 11(1):175,

doi
[2023]

S Rakesh, Nagaratna P Hegde, M Venu Gopalachari, D Jayaram, Bhukya Madhu, Mohd Ab- dul Hameed, Ramdas Vankdothu, and LK Suresh Kumar

doi: 10.3390/math11010175. S Rakesh, Nagaratna P Hegde, M Venu Gopalachari, D Jayaram, Bhukya Madhu, Mohd Ab- dul Hameed, Ramdas Vankdothu, and LK Suresh Kumar. Moving object detection using modified gmm based background subtraction. Measurement: Sensors , 30:100898,

doi

Pith tools

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

[1] [10]

Kittipat Kampa, Erion Hasanbelliu, and Jos´ e Carlos Pr ´ ıncipe

1109/ICASSP.2007.366913. Kittipat Kampa, Erion Hasanbelliu, and Jos´ e Carlos Pr ´ ıncipe. Closed-form cauchy-schwarz pdf divergence for mixture of gaussians. The 2011 International Joint Conference on Neural Networks , pages 2578–2585,

work page arXiv 2007

[2] [11]

Sivalingam, D

R. Sivalingam, D. Boley, V. Morellas, and N. Papanikolopoulos. Tensor sparse coding for region covariances. In K. Daniilidis, P. Maragos, and N. Paragios, editors, Computer Vision – ECCV 2010 , volume 6314 of Lecture Notes in Computer Science , pages 709–

work page 2010

[3] [15]

URL https://doi.org/10.1007/s11263-009-0220-6

1007/s11263-009-0220-6. URL https://doi.org/10.1007/s11263-009-0220-6 . 21

doi:10.1007/s11263-009-0220-6

[4] [52]

Huan Wan, Hui Wang, Bryan Scotney, and Jun Liu

URL https: //doi.org/10.1007/978-3-642-15561-1_52 . Huan Wan, Hui Wang, Bryan Scotney, and Jun Liu. A novel gaussian mixture model for classification. In 2019 IEEE International Conference on Systems, Man and Cybernetics (SMC), pages 3298–3303,

work page doi:10.1007/978-3-642-15561-1_52 2019

[5] [1990]

doi: 10.1016/0047-259X(90)90026-E

ISSN 0047-259X. doi: 10.1016/0047-259X(90)90026-E. URL https: //www.sciencedirect.com/science/article/pii/0047259X9090026E. Jean-Louis Durrieu, Jean-Philippe Thiran, and Finnian Kelly. Lower and upper bounds for approximation of the kullback-leibler divergence between gaussian mixture models. 2012 IEEE International Conference on Acoustics, Speech and Sig...

arXiv

[6] [1996]

Hershey and Peder A

John R. Hershey and Peder A. Olsen. Approximating the kullback leibler divergence between gaussian mixture models. In 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP ’07 , volume 4, pages IV–317–IV–320,

work page 2007

[7] [2002]

doi: https://doi

ISSN 0898-1221. doi: https://doi. org/10.1016/S0898-1221(01)00278-4. URL https://www.sciencedirect.com/science/ article/pii/S0898122101002784. Leandro Pardo. Statistical inference based on divergence measures . Chapman and Hall/CRC,

doi:10.1016/s0898-1221(01)00278-4

[8] [2003]

doi: 10.1109/ICCV.2003.1238387. Gene H. Golub and Charles F. Van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, 3 edition,

arXiv 2003