Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition

Zhenhua Lin

REVIEW 2 major objections 6 minor 41 references

Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Log-Cholesky metric gives SPD matrices a flat, bi-invariant geometry with closed-form averages and parallel transport.

desk verdict Log-Cholesky is a real, useful addition to SPD geometry; the central derivations hold up, and the gaps are small. read the letter →

arxiv 1908.09326 v1 pith:VBBGOZMC submitted 2019-08-25 math.DG math.STstat.TH

Zhenhua Lin This is my paper · ORCID

classification math.DGmath.STstat.TH MSC 47A6426E6053C3522E9932F4553C2215A22

keywords symmetricpositivedefinitematricesCholeskydecompositionLog-Choleskymetricbi-invariantFrechetmeanswellingeffectparalleltransportLiegroup

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

Symmetric positive definite matrices encode covariance or diffusion information, but standard averaging can inflate determinants. This paper argues that pushing a carefully chosen metric from Cholesky space to SPD space yields a Riemannian metric with unusually concrete properties. If the construction is right, practitioners get a geometry where averages are easy to compute, geodesics and parallel transport have closed forms, and the determinant of any average stays between the determinants of the inputs. The paper also shows the SPD manifold becomes an abelian Lie group with a bi-invariant metric under this operation.

What carries the argument

The central object is the Cholesky decomposition $S(L)=LL^\top$, used as an isometry between Cholesky space $L^+$ and the SPD manifold. On $L^+$ the paper defines the metric $\tilde g_L(X,Y)=\sum_{i>j}X_{ij}Y_{ij}+\sum_j X_{jj}Y_{jj}L_{jj}^{-2}$. In the coordinate chart that logs the diagonal, this metric has constant coefficients, so all Christoffel symbols vanish; geodesics, exponentials, and the group operation are then computed by simple formulas and pushed forward to SPD matrices.

What would settle it

Take $P_1=\begin{pmatrix}\epsilon^2&0\\0&1\end{pmatrix}$ and $P_2=\begin{pmatrix}1&0\\0&\epsilon^2\end{pmatrix}$ and compute the Log-Cholesky geodesic midpoint $\gamma(1/2)$. The claim predicts $\det\gamma(1/2)=\sqrt{\det P_1\det P_2}=\epsilon^2$; any computed determinant larger than $\max\{\epsilon^2,\epsilon^2\}=\epsilon^2$ would refute the swelling-free claim.

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Extended reading notes

Core claim

On the space of lower-triangular matrices with positive diagonal, the paper puts a metric on each tangent space by combining the Euclidean inner product on the strictly lower-triangular part with the squared relative change on the diagonal. Through the Cholesky map $P=LL^\top$, this metric becomes the Log-Cholesky metric on SPD matrices. The paper proves that this metric is bi-invariant for a commutative group operation on SPD matrices, has identically zero sectional curvature, and admits closed-form geodesics, exponential and logarithmic maps, parallel transport, and Fréchet averages. It also proves that $\log\det$ of the average equals the average of $\log\det$'s, so the determinant of the Log-Cholesky average lies between the minimum and maximum determinant of the averaged matrices.

Load-bearing premise

Existence and uniqueness of the Log-Cholesky average rests on a theorem that requires the manifold to be complete, connected, and nonpositively curved; the paper proves flatness and simple connectedness but leaves the completeness check implicit.

Editorial extensions

If this is right

The Log-Cholesky average of $P_1,\dots,P_n$ has a closed form involving only Cholesky factors and matrix logarithms of diagonal blocks, so no numerical optimization is needed.
The determinant identity $\det E_n = \left(\prod_i \det P_i\right)^{1/n}$ means the average never inflates dispersion, for any number of matrices.
Parallel transport along geodesics has a closed formula that uses only Cholesky factors and inverses of triangular matrices, making it substantially cheaper than the Log-Euclidean approach.
The SPD manifold becomes flat and abelian, so geodesic interpolation between two SPD matrices keeps determinants between the endpoint determinants.
Because the Log-Cholesky average shares its determinant with the Log-Euclidean and affine-invariant averages, it offers a fast route to averages with the same determinant behavior.

Reading between the lines

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

Beyond the paper: because the Log-Cholesky coordinates are globally Euclidean, standard Euclidean statistical tools could be applied to the log-Cholesky representation and mapped back, with an exact geodesic interpretation rather than an approximation.
Beyond the paper: the abelian Lie group structure suggests that Fourier-style analysis or convolution-type operations on SPD-valued data may be meaningful, though the paper does not develop this direction.
Beyond the paper: the closed-form Log-Cholesky average could serve as a fast initialization or surrogate for affine-invariant or Log-Euclidean means in large-scale pipelines, especially when computational cost is the bottleneck.

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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

The Log-Cholesky metric is a genuine new construction, not a repackaging. The Cholesky-space Lie group with the componentwise operation and the diagonal log scaling is natural, and pushing it forward to SPD matrices gives a bi-invariant metric with closed-form geodesics, parallel transport, and Fréchet mean. I checked the main derivations: the geodesic formula, the pushforward metric, the Lie group structure, and the mean formulas are all correct. The no-swelling property follows from the determinant identity, not from circular reasoning.

The novelty claim holds against the cited work. Dryden et al. use Euclidean distance on Cholesky factors, which swells; Grubisic and Pietersz are on correlation matrices with a different structure. The log-Cholesky operation on the diagonal is not in those papers.

The soft spots are minor. Proposition 9 invokes Bhattacharya-Patrangenaru for the Fréchet mean, which requires completeness. The paper proves flatness and simple connectedness but not completeness. As the stress test notes, completeness is immediate from the global isometry ψ(L)=⌊L⌋+log D(L), so the theorem applies. It should have been stated, but it is not a load-bearing flaw. Similarly, surjectivity of the group exponential in Lemma 6 is implicit but holds because the group is a vector group in those coordinates.

The bigger weakness is the numerical section. The timing numbers (9.3ms vs 0.85ms vs 0.2ms) are presented without code, without specifying how many terms of the Log-Euclidean series were used, and without a protocol. The stability claims about 10^10 and 10^15 condition numbers are also anecdotal. These claims are peripheral to the geometry, but they need to be reproducible if the paper is going to make computational promises.

There is a small typo in Table 2: the metric row repeats W instead of V in the second argument. Easily fixed.

The paper is for anyone doing statistical analysis or imaging with SPD matrices who wants a faster alternative to Log-Euclidean or affine-invariant metrics. The central contribution is solid, and the presentation is clear. It deserves a serious referee; the main referee asks should be to fill the completeness step, clean up the numerical experiments, and add code or a detailed protocol.

Referee Report

2 major / 6 minor

Summary. The paper introduces a new Riemannian metric, the Log-Cholesky metric, on the manifold of symmetric positive definite matrices. The construction first defines a commutative Lie group operation X⋄Y = ⌊X⌋+⌊Y⌋+D(X)D(Y) and a bi-invariant metric on the Cholesky space L+ of lower-triangular matrices with positive diagonal, then pushes both forward to SPD matrices via the Cholesky parametrization S(L)=LL^T. The paper derives closed-form expressions for geodesics, Riemannian exponential and logarithmic maps, geodesic distance, and parallel transport on both L+ and S+_m, and shows that (S+_m,⋄) is an abelian Lie group with bi-invariant Log-Cholesky metric. It then studies Fréchet means, proving existence and uniqueness under a finite second-moment condition and giving closed forms for the Log-Cholesky mean and average. The determinant identity log det(mean) = E log det(Q) is established, implying that the average's determinant lies between the minimum and maximum determinants of the input matrices, so the average avoids the swelling effect. The paper also makes computational-efficiency and numerical-stability claims relative to the affine-invariant and Log-Euclidean metrics.

Significance. The mathematical core is sound and the paper delivers a genuinely useful, parameter-free Riemannian structure on SPD matrices with closed-form geodesics, exponential and logarithmic maps, parallel transport, and Fréchet mean, together with a bi-invariant metric on an abelian Lie group. The determinant identity and the resulting no-swelling property are proved directly from the definitions rather than assumed. This gives the SPD community an additional computational tool, particularly for large-scale or computation-heavy applications where the simplicity of the formulas matters. The novelty—combining Cholesky coordinates with a logarithmic treatment of the diagonal—is modest but real, and the paper is careful to note the lack of congruence invariance relative to the affine-invariant metric. The main mathematical claims are checkable and, apart from the completeness gap noted below, correct; the numerical claims are less well supported but are peripheral to the mathematical contribution.

major comments (2)

[§4.1, proof of Proposition 9] The proof invokes Theorem 2.1 of [8], which requires a complete, simply connected, nonpositively curved manifold. The proof establishes simple connectedness via the diffeomorphism ψ and zero sectional curvature via Proposition 8, but it never states or proves completeness of (L+,~g). This is load-bearing for the existence and uniqueness conclusion. The gap is easily closed: ψ(L)=⌊L⌋+log D(L) is a global isometry from (L+,~g) to Euclidean (L,‖·‖_F), since dψ_L(X)=⌊X⌋+D(L)^{-1}D(X), so (L+,~g) is complete; completeness of (S+_m,g) then follows from the isometry S. Please add this argument before invoking [8].
[§3.4, Lemma 6] The lemma as stated applies to an arbitrary abelian Lie group with a bi-invariant metric, but the proof assumes the existence of Y with exp(Y)=p^{-1}q. This requires the group exponential to be surjective, which holds for connected abelian Lie groups but should be stated explicitly; for a disconnected abelian Lie group, p^{-1}q need not lie in the identity component. The later application to L+ is safe because L+ is isomorphic to the vector group (L,+) via ψ, but the lemma should be restated with a connectedness hypothesis, or with the exponential-image condition made explicit, so that the proof is valid as written.

minor comments (6)

[Table 2] In the row labeled 'Riemannian metric', the second argument of ~g is written as (D_P L)(W), but it should be (D_P L)(V); the displayed formula should read g_P(W,V) = ~g_{L(P)}((D_P L)(W), (D_P L)(V)).
[§3.4, Proposition 7] In the expression for τ_{P,Q}(W), the second bracket uses D(K)D(L^{-1})D(X) while the first uses D(K)D(L)^{-1}D(X). Since D(L^{-1})=D(L)^{-1} for L∈L+, the two are equal, but the notation should be made consistent.
[§3.1, proof of Proposition 3] The map x is defined as 'x : L+ → R', but the intended codomain is R^{m(m+1)/2}; please correct the typo.
[§4.1, proof of Proposition 10] The phrase 'minimizes F2j(ex)' should read 'minimizes F2j(e^x)' or 'minimizes F2j(exp(x))' for clarity.
[§2.1] The text says 'log(P) is a symmetric metric for an SPD matrix P'; this should be 'symmetric matrix' rather than 'symmetric metric'.
[§3.4 and §4.2] The numerical timing and stability claims (9.3ms, 0.85ms, 0.2ms; expected relative difference 3.3×10^-2; largest-eigenvalue ratios 10^10 and 10^15) are not accompanied by a full experimental protocol or code. Since the abstract advertises computational efficiency and numerical stability, please supply a reproducible procedure or clearly mark these statements as indicative and soften the corresponding claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all central formulas are derived from explicit definitions and standard external theorems, with no fitted parameters, no self-citations, and no prediction that reduces to an input by construction.

full rationale

The paper's derivation chain is self-contained. The Log-Cholesky metric is defined explicitly by equation (3.1) as the pushforward of the metric g~ on Cholesky space, and all subsequent claims are consequences of that definition plus standard Riemannian geometry. The geodesic formula, exponential/logarithmic maps, and distance function in Section 3.1 are obtained by direct computation in a global chart in which the metric coefficients are constant, as shown in the proof of Proposition 3. The group operation and bi-invariance are verified by explicit calculation of left and right translations in Section 3.3. Parallel transport in Proposition 7 is derived from the stated bi-invariant metric and an abelian Lie group argument in Lemma 6, which is not a renaming of the conclusion. The no-swelling property is not assumed: Proposition 11 and Corollary 13 derive det(En(P1,...,Pn)) = exp((1/n) sum log det Pi) from the explicit closed form of the Log-Cholesky mean, so the determinant bound (4.8) is a proved consequence rather than an input. There are no fitted constants, no benchmark-tuned parameters, and no self-citations in the load-bearing argument. The only implicit point noted by a careful reader is completeness in Proposition 9: Theorem 2.1 of [8] requires a complete, simply connected, nonpositively curved manifold, and the paper explicitly establishes simple connectedness and zero sectional curvature but leaves completeness implicit. This is a minor expository gap, not a circular step, because completeness follows immediately from the global coordinate isometry psi(L) = floor(L) + log D(L) with constant identity metric. Accordingly, the paper receives a circularity score of 0.

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

No free parameters are fitted; the metric has a fixed unit weighting between off-diagonal and diagonal components. No new physical entities are introduced. The proof relies on standard tools from matrix analysis, Riemannian geometry and Lie group theory.

assumptions (4)

standard math Cholesky decomposition of an SPD matrix with positive diagonal is unique (Theorem 4.2.5 of Golub and Van Loan).
Establishes the bijection between L+ and S+_m used throughout.
standard math The map S(L)=LL^T is a diffeomorphism between L+ and S+_m.
Allows pushing forward the metric from Cholesky space to SPD space.
standard math On a Lie group with bi-invariant metric, the group exponential map coincides with the Riemannian exponential at the identity, left translations are isometries, and for left-invariant fields Y,Z one has nabla_Y Z = [Y,Z]/2 (Milnor, Morse Theory).
Used in Lemma 6 to derive the parallel transport formula.
standard math Theorem 2.1 of Bhattacharya and Patrangenaru: on a complete simply connected nonpositively curved manifold, a probability measure with finite second moment has a unique Frechet mean.
Used in Proposition 9 to establish existence and uniqueness of Log-Cholesky means.

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Pith. "Pith review of Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition." pith.science (2026). https://pith.science/paper/VBBGOZMC

@misc{pith2026190809326,
  author       = {Pith},
  title        = {Pith review of: Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBBGOZMC}},
  note         = {Machine review of arXiv:1908.09326}
}

read the original abstract

We present a new Riemannian metric, termed Log-Cholesky metric, on the manifold of symmetric positive definite (SPD) matrices via Cholesky decomposition. We first construct a Lie group structure and a bi-invariant metric on Cholesky space, the collection of lower triangular matrices whose diagonal elements are all positive. Such group structure and metric are then pushed forward to the space of SPD matrices via the inverse of Cholesky decomposition that is a bijective map between Cholesky space and SPD matrix space. This new Riemannian metric and Lie group structure fully circumvent swelling effect, in the sense that the determinant of the Fr\'echet average of a set of SPD matrices under the presented metric, called Log-Cholesky average, is between the minimum and the maximum of the determinants of the original SPD matrices. Comparing to existing metrics such as the affine-invariant metric and Log-Euclidean metric, the presented metric is simpler, more computationally efficient and numerically stabler. In particular, parallel transport along geodesics under Log-Cholesky metric is given in a closed and easy-to-compute form.

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Pith tools

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

[1] [8]

Bhattacharya and V

R. Bhattacharya and V. Patrangenaru , Large sample theory of intrinsic and extrinsic sample means on manifolds. I , The Annals of Statistics, 31 (2003), pp. 1–29. 18 Z. LIN

work page 2003

[2] [1]

Aftab and R

K. Aftab and R. Hartley , Lq averaging for symmetric positive-deﬁnite matrices , in 2013 International Conference on Digital Image Computing: Techniques and Applications (DICTA), 2013. LOG-CHOLESKY METRIC 17 Fig. 1. Interpolation of SPD matrices. Top: Euclidean linear interpolation. The associated determinants are 5.40, 17.92, 27.68, 34.69, 38.93, 40.41, ...

work page 2013

[3] [2]

Ando, C.-K

T. Ando, C.-K. Li, and R. Mathias , Geometric means, Linear Algebra and its Applications, 385 (2004), pp. 305–334

work page 2004

[4] [3]

Arnaudon, F

M. Arnaudon, F. Barbaresco, and L. Yang, Riemannian medians and means with applica- tions to radar signal processing , IEEE Journal of Selected Topics in Signal Processing, 7 (2013), pp. 595–604

work page 2013

[5] [4]

Arsigny, P

V. Arsigny, P. Fillard, X. Pennec, and N. Ayache , Geometric means in a novel vector space structure on symmetric positive-deﬁnite matrices , SIAM Journal of Matrix Analysis and Applications, 29 (2007), pp. 328–347

work page 2007

[6] [5]

Barachant, S

A. Barachant, S. Bonnet, M. Congedo, and C. Jutten , Multiclass brain–computer inter- face classiﬁcation by Riemannian geometry, IEEE Transactions on Biomedical Engineering, 59 (2012), pp. 920–928

work page 2012

[7] [6]

Bhatia, Positive Deﬁnite Matrices , Princeton University Press, 2007

R. Bhatia, Positive Deﬁnite Matrices , Princeton University Press, 2007

work page 2007

[8] [7]

Bhatia, T

R. Bhatia, T. Jain, and Y. Lim, On the Bures–Wasserstein distance between positive deﬁnite matrices, Expositiones Mathematicae, 37 (2018), pp. 165–191

work page 2018