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Separable Geodesic Lagrangian Monte Carlo for Inference in 2-Way Covariance Models

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Pith's one-line read A geodesic Hamiltonian sampler puts separable-covariance MCMC on par with Gibbs

desk verdict Regularized SGLMC is a plausible new MCMC tool for separable covariances, but the orthogonal/weighted variants sample a conditional slice rather than the posterior, and the printed algorithm has enough errors that the paper needs major revision before it can be trusted. read the letter →

arxiv 2501.04913 v1 pith:K2TA3JJZ submitted 2025-01-09 stat.CO

Quinn Simonis , Martin T. Wells This is my paper · ORCID
classification stat.CO MSC 62F1565C0562H12
keywords separablecovariancematrixnormalKroneckerproductHamiltonianMonteCarlogeodesicaffine-invariantmetricPitsianis-VanLoandecompositionRiemannianmanifold
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

Separable covariance matrices, covariances of vectorized 2-way data that factor as a Kronecker product $\Sigma_1 \otimes \Sigma_2$ of two smaller matrices, are common in spatial, temporal, and image data, but MCMC for them has mostly meant Gibbs sampling. This paper tries to put Hamiltonian Monte Carlo on an equal footing by giving the parameter pair $(\Sigma_1, \Sigma_2)$ a Riemannian geometry induced from the Kronecker structure. The central move is to show that under four related affine-invariant metrics the geodesic path of the Kronecker product is just the Kronecker product of two independent geodesics of the factors, so a geodesic Hamiltonian sampler can update both factors cheaply. The paper then supplies an efficient matrix-form algorithm, gradients via the Pitsianis–Van Loan decomposition, and experiments indicating posterior agreement with Gibbs and better effective sample sizes per iteration than a naive product-manifold metric.

What carries the argument

The load-bearing object is the affine-invariant metric on positive-definite matrices, $g_\Sigma(S_1,S_2)=\operatorname{tr}(\Sigma^{-1}S_1\Sigma^{-1}S_2)$, together with its closed-form geodesic exponential map $\exp_\Sigma(tV)=\Sigma^{1/2}\exp(t\Sigma^{-1/2}V\Sigma^{-1/2})\Sigma^{1/2}$. The paper pulls this metric back through the Kronecker map, identifies the resulting degeneracy by a rank-one determinant calculation, and repairs it with four metrics: regularized $G^\otimes_\alpha$, orthogonal $G^O$, weighted $G^W$, and product $G^\times$. The other essential mechanism is the Pitsianis–Van Loan decomposition, which rewrites the sample sum of squares as a sum of Kronecker products so that log-likelihood, gradients, and Gibbs full conditionals factor into component traces; this is what makes both the geodesic gradients and the comparison Gibbs sampler computationally tractable.

What would settle it

Run SGLMC-O on a small dataset with known true $\Sigma_1,\Sigma_2$, then compare the normalized posterior of $\Sigma_2$ against a very long Gibbs run with the same inverse-Wishart prior; if the Metropolis acceptance rate depends on trajectory direction, or if the normalized SGLMC marginals drift from the Gibbs marginals, the constrained leapfrog is not reversible for the claimed target.

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Extended reading notes

Core claim

On the product manifold $P(d_1)\times P(d_2)$, the pullback of the affine-invariant metric under the map $(\Sigma_1,\Sigma_2)\mapsto \Sigma_1\otimes\Sigma_2$ is degenerate, because $\lvert c\Sigma_1\rvert \lvert c^{-1}\Sigma_2\rvert = \lvert\Sigma_1\rvert\lvert\Sigma_2\rvert$ creates a scale direction of zero norm. The paper resolves this degeneracy in four ways, regularizing the off-diagonal block, orthogonalizing with the projection $P_{\Sigma_2}$, weighting the blocks, or simply taking the product metric, and then asserts Proposition 2.3: under every one of these metrics, the geodesic is $\Sigma(t)=\Sigma_1(t)\otimes\Sigma_2(t)$ with $\Sigma_i(t)=\exp_{\Sigma_i(0)}(t V_i(0))$ the affine-invariant exponential map. This lets a geodesic Hamiltonian Monte Carlo sampler split into two lower-dimensional factor updates, with velocities drawn from matrix-normal-type distributions and Riemannian gradients computed through the Pitsianis–Van Loan decomposition. The paper's experiments compare the resulting SGLMC sampler with Gibbs and with a general-purpose HMC implementation and conclude that the constrained (orthogonal) and weighted metrics give higher effective sample size per iteration than the product metric while tracking Gibbs posterior summaries.

Load-bearing premise

The sampler's correctness rests on the assumption, inherited from earlier Lagrangian Monte Carlo work without a proof for this specific case, that the generalized leapfrog integrator with a position-dependent metric is reversible and samples the intended posterior even when the block-diagonal orthogonal metric is combined with the projection constraint on $\Sigma_2$; the printed Algorithm 2 also contains typos in the projection and final half-kick that an implementation must repair.

Editorial extensions

If this is right

  • Because the geodesic on the full Kronecker covariance is the Kronecker product of two independent affine-invariant geodesics, each leapfrog step costs two smaller matrix exponentials, $O(d_1^3+d_2^3)$, instead of one exponential on $d_1d_2$-dimensional matrices.
  • For the orthogonal, weighted, and product metrics, the velocity and gradient updates can be written in matrix form without forming the full metric tensor, avoiding a naive $O((d_1^2+d_2^2)^3)$ inversion cost.
  • With inverse-Wishart priors, SGLMC posterior densities for trace, log-determinant, and condition number agree with Gibbs; the $\lvert\Sigma_2\rvert=1$ constraint affects only the scale indeterminacy and disappears after normalizing Gibbs samples.
  • The experiments indicate that the regularized metric with $\alpha=0.95$ and the orthogonal or weighted metrics produce higher effective sample sizes per iteration than the product metric, and that sampler autocorrelation improves as $\alpha$ grows toward 1.
  • The Pitsianis–Van Loan decomposition yields full conditional distributions for the components, giving a direct conjugate Gibbs comparator and gradients with cost $O(r\,d_{\max}^3)$ rather than $O(n\,d_{\max}^3)$ for the matrix-normal likelihood.

Reading between the lines

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

  • If Proposition 2.3 extends to higher-order Kronecker products, the same component-wise exponential-map construction gives an immediate tensor generalization; the paper notes only that the P-VL gradient cost would then grow, so the practical bottleneck would move from geometry to gradient computation.
  • The near-independence of posterior marginals across $\alpha$ values suggests the metric is acting primarily as a sampler-tuning device rather than as part of the statistical model, so one could treat metric choice and target distribution as separate design decisions.
  • As printed, Algorithm 2 applies the projection $P_{\Sigma_1}$ in the $\Sigma_2$ velocity update and reuses $q_j$ rather than $q_{j+1}$ in the final half-kick; these typos must be corrected before the empirical claims can be reproduced, and the paper does not flag them.
  • A fully adaptive version could replace the static leapfrog length $L$ with the log-map termination criterion sketched in the supplement; testing whether that recovers no-U-turn-style efficiency on the constrained metric would be a natural follow-up.
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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

5 major / 6 minor

Summary. The paper develops Separable Geodesic Lagrangian Monte Carlo (SGLMC) for Bayesian inference in matrix-normal models with separable covariance Sigma = Sigma1 ⊗ Sigma2. The authors construct four metrics on P(d1) × P(d2): a regularized pullback of the affine-invariant metric, an orthogonalized block-diagonal metric GO, a weighted variant GW, and the product metric G×. Proposition 2.3 claims that under all of these metrics the geodesic flow factors into independent affine-invariant geodesics on P(d1) and P(d2). Using the Pitsianis–Van Loan decomposition, the paper gives Gibbs full conditionals and likelihood gradients, presents Algorithms 1 and 2, and reports numerical comparisons with Gibbs and Stan, including an application to the Wisconsin breast-cancer data.

Significance. Proposition 2.3 is potentially a significant simplification: if valid, geodesic HMC on the Kronecker-structured parameter space can be implemented by independent SPD geodesics, and the P-VL-based gradient computations avoid dense Kronecker algebra. The paper also provides a useful full-conditional decomposition and extensive empirical documentation. However, the manuscript as written does not establish that the orthogonal or weighted samplers target the claimed posterior, and the printed algorithms contain internal inconsistencies. These issues are load-bearing for the validity and efficiency claims, although the core geometric idea appears salvageable.

major comments (5)
  1. [Sec. 4.2, Fig. 5, Table 1] The claim that the |Σ2|=1 constraint of GO and GW introduces only a bias 'equivalent to normalizing samples of Σ2 with the Gibbs sampler' is not correct for the IW priors in (22). In the identified parametrization (A,U) = (|Σ2|^{1/d2}Σ1, Σ2/|Σ2|^{1/d2}), the marginal posterior of (A,U) is obtained by integrating over the scale s; for the priors (22) the s-dependence of the posterior is exp[-γ/(2d1 s)tr(A^{-1}) - γs/(2d2)tr(U^{-1}) - (s/2)tr(A^{-1}Σ_k A_k tr(U^{-1}B_k))] times s^{d1(d1+1)/2 - d2(d2+1)/2 + 1}, which does not factor as h(s)g(A,U). Hence the slice |Σ2|=1, which is the stationary distribution of the constrained sampler, is not proportional to the normalized-Gibbs target, and the high ESS/it values for GO and GW in Table 1 and the agreement in Figure 5 are not evidence that these variants sample the posterior. A concrete minimal check is d1=d2=1, where normalized Gibbs targets the posterior of the product σ1σ2 while the slice target is the conditional of σ1 given σ2=1.
  2. [Sec. 4.2, Algorithm 2] The printed generalized leapfrog is internally inconsistent. In the second half-kick, the Σ2 velocity is updated with PΣ1 instead of PΣ2, the gradients are evaluated at qj instead of qj+1, and the velocity used is V^{(j)} instead of V^{(j+1/2)}; the two half-kick lines for V2 are identical. Such an update is neither reversible nor symplectic as written, and no detailed-balance proof is supplied for the projected, position-dependent metric. The manuscript therefore does not substantiate the claim that SGLMC-O samples any stated target distribution, and the algorithm cannot be run as printed.
  3. [Prop. 3.2, Eqs. (25)-(26)] The displayed gradients are not gradients of the negative log-likelihood. For the log-likelihood ℓ, one has ∂ℓ/∂Σ1 = -(d2+n)/2 Σ1^{-1} + (1/2)Σ_k tr(Σ2^{-1}B_k) Σ1^{-1}A_kΣ1^{-1}; the negative log-likelihood gradient is the negative of this expression. Equations (25)-(26) print the log-likelihood gradient instead. Since the leapfrog updates in Algorithm 2 require the gradient of πH = -log π - (1/2)log|G|, a sign error here changes the dynamics from descending to ascending the potential and, if implemented as printed, invalidates the sampler's stationary distribution.
  4. [Lemma 2.2, Eq. (11)] The proposed orthogonal map, PΣ2(V2) = V2 - tr(V2Σ2^{-1})/d2 V2, is not a projection and does not remove the cross term in Proposition 2.1; it merely rescales V2. The vectorized form P = I - vec(Σ2)vec(Σ2)^T/d2 projects along the Euclidean inner product with vec(Σ2) rather than along Σ2 under the affine-invariant metric. The correct projection enforcing tr(Σ2^{-1}V2)=0 is V2 - tr(Σ2^{-1}V2)/d2 Σ2. In addition, the coefficients in GO in Eq. (11) and in Lemma 2.2 are swapped relative to Proposition 2.1 and relative to the velocity scales and Riemannian gradients used in Algorithm 2. These inconsistencies affect both the geometric derivation and the actual implementation of the main orthogonal sampler.
  5. [Lemma 2.3, Eq. (31)] The determinant statement for GO and GW is ambiguous. For the full block-diagonal metric (11), the determinant is proportional to |Σ1|^{-(d1+1)}|Σ2|^{-(d2+1)}, not solely |Σ1|^{-(d1+1)}; the statement in Lemma 2.3 can only hold on the submanifold |Σ2|=1, for which no determinant computation is provided. Because the Hamiltonian (31) contains -(1/2)log|G(q)|, the invariant measure of the constrained sampler depends on which determinant is used. As written, whether SGLMC-O targets π(q) restricted to the slice, the marginal of the identified parameters, or some other measure is not well defined.
minor comments (6)
  1. [Prop. 2.2] The bottom-left block of G⊗ appears to be vec(Σ2^{-1})vec(Σ2^{-1})^T, which is not the transpose of the top-right block; it should be vec(Σ2^{-1})vec(Σ1^{-1})^T.
  2. [Sec. 8, proof of Prop. 2.3] The symbols P1 and P2 are used in place of Σ1 and Σ2 in several equations in the Euler-Lagrange proof, making the argument harder to follow.
  3. [Algorithm 1] The velocity update uses G(q(1)) throughout and refers to q(j+1/2), which is never defined; this should be corrected to match a standard generalized leapfrog.
  4. [Algorithm 2] The notation d^{1/2}_{-k} in the velocity sampling step is undefined; it should be d_{3-k}^{-1/2} or an equivalent explicit expression.
  5. [Secs. 5 and 8.5] The paper states that the propriety of the reference prior 'was not investigated in this paper and remains questionable' but still uses the reference prior in the real-data example; this example should be labeled as exploratory unless propriety is established.
  6. [Table 1] Table 1 reports ESS/it for log|Σ2| and log|Σ| for GO and GW even though |Σ2|=1 makes these statistics constant under the constraint; reporting these values is not informative and should be removed or replaced with identifiable statistics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SGLMC derivation relies on standard external affine-invariant geometry and is benchmarked against Gibbs/Stan; no fitted parameter is renamed as a prediction.

full rationale

The paper's central derivation chain is self-contained relative to established, externally sourced geometry. The Kronecker pullback metric, its degeneracy (Proposition 2.2), and the regularized/orthogonal/weighted/product variants (Lemma 2.1, Lemma 2.2) are derived from the paper's own equations using the affine-invariant metric of Moakher-Zerai and the Pitsianis-Van Loan decomposition. Proposition 2.3 is proven by checking that product-manifold geodesics satisfy the Euler-Lagrange equations of the coupled regularized metric; together with uniqueness of the geodesic ODE this is a valid sufficiency argument, not a circular definition. The sampler's stationary distribution correctly targets pi(q): in Eq. (31), the (1/2) log|G(q)| term cancels the sqrt(|G(q)|) normalization of the velocity distribution, so the position marginal is the full posterior on the constrained or unconstrained slice as stated. No fitted parameter is used to produce the geodesic equations; alpha and omega are tuning/design parameters, and the empirical ESS/it comparisons in Table 1 and posterior comparisons against Gibbs and Stan are independent external benchmarks, not predictions constructed from the same fitted values. The paper transparently acknowledges the |Sigma2|=1 constraint bias and compares to normalized Gibbs samples; whether that equivalence is mathematically exact is a correctness question, not circularity. Self-citation is not load-bearing: [16] and [21] are external prior works by other authors. Two non-circular concerns are noted per the review rule: the reference prior propriety is explicitly left uninvestigated (Section 5, Section 8.5), and Algorithm 2 contains an apparent typo (P_Sigma1 applied to the V2 update and qj reused in the final half-kick), which is a reproducibility risk rather than a circular step.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central method rests on standard SPD Riemannian geometry and on the generalized leapfrog integrator from prior work; the main hand-tuned quantities are alpha, L, and the acceptance target. The paper's own geometric claims (geodesic identity, determinant formulas) are the axioms that need checking. No external entities are invented.

free parameters (4)
  • regularization parameter alpha = 0.95 (recommended); 0 <= alpha < 1
    Scanned over {0, .1, .25, .5, .75, .9, .95}; alpha=.95 chosen empirically for lower autocorrelation. Not derived; values near 1 said to be unstable.
  • leapfrog steps L = 10 in all experiments
    Static choice for all comparisons; dynamic tuning only discussed in the supplement.
  • dual averaging target a0 = 0.8, 0.85, 0.6, 0.7 across experiments
    Step-size adaptation target chosen by hand and varied between experiments; no principled rule is given.
  • weight omega in GW = not evaluated
    Defined in equation (12) but no experiments or recommendation are given; effectively inert in the paper.
assumptions (6)
  • domain assumption The affine-invariant metric g_Sigma(S1,S2)=tr(Sigma^-1 S1 Sigma^-1 S2) defines the geometry of SPD matrices
    Used to define norms, geodesics, and Hausdorff measure corrections in Sections 2 and 4; taken from [31].
  • standard math The exponential map for the affine-invariant metric is Sigma(t)=Sigma^(1/2) exp(t Sigma^(-1/2) V Sigma^(-1/2)) Sigma^(1/2)
    Standard result for SPD with affine-invariant metric; used in Proposition 2.3 and Algorithm 2.
  • ad hoc to paper The regularized and orthogonalized pullback metrics produce the same geodesics as the product manifold and have the stated determinant formulas
    Proposition 2.3 and Lemma 2.3 are the paper's own constructions; correctness relies on the Euler-Lagrange derivations in Section 8.
  • domain assumption The generalized leapfrog integrator for Lagrangian Monte Carlo yields a reversible sampler with stationary distribution proportional to exp(-piH(q)) sqrt(|G(q)|)
    Adopted from [16,21]; not re-proved for the GO metric with the projection constraint.
  • standard math The Pitsianis-Van Loan decomposition exactly represents sum_i y_i y_i^T as sum_k A_k tensor B_k
    Used for Gibbs full conditionals and gradient computations in Propositions 3.1 and 3.2.
  • ad hoc to paper The reference prior is proper for the models used
    The authors state in Sections 5 and 8.5 that propriety of the reference prior was not investigated and remains questionable.

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Pith. "Pith review of Separable Geodesic Lagrangian Monte Carlo for Inference in 2-Way Covariance Models." pith.science (2026). https://pith.science/paper/K2TA3JJZ

@misc{pith2026250104913,
  author       = {Pith},
  title        = {Pith review of: Separable Geodesic Lagrangian Monte Carlo for Inference in 2-Way Covariance Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K2TA3JJZ}},
  note         = {Machine review of arXiv:2501.04913}
}
read the original abstract

Matrix normal models have an associated 4-tensor for their covariance representation. The covariance array associated with a matrix normal model is naturally represented as a Kronecker-product structured covariance associated with the vector normal, also known as separable covariance matrices. Separable covariance matrices have been studied extensively in the context of multiway data, but little work has been done within the scope of MCMC beyond Gibbs sampling. This paper aims to fill this gap by considering the pullback geometry induced from the Kronecker structure of the parameter space to develop a geodesic Hamiltonian Monte Carlo sampler.

Figures

Figures reproduced from arXiv: 2501.04913 by the authors.

Figure 1
Figure 1. Illustration of HMC trajectories when Σ1, Σ2 are both d × d SPD matrices under metrics GO (top left), G⊗ α (top right), and G× (bottom). 3 The Pitsianis-Van Loan Decomposition and Posterior Sampling Given Σ ∈ P(d) such that d is non-prime with d = d1d2, Σ may be decomposed into Σ = Pr i=1 Σ (i) 1 ⊗ Σ (i) 2 where Σ(i) 1 ∈ R d1 and Σ(i) 2 ∈ R d2 . The result of this follows from Sections 2.2 and 2.4 of [36], which we … view at source ↗
Figure 2
Figure 2. Density and corresponding ACFs plots of the traces, determinants for samples of Σ, and [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 3
Figure 3. Comparison of density and ACFs of the trace and log determinant of Σ [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Comparison of density and ACFs of the trace and log determinant of Σ across different [PITH_FULL_IMAGE:figures/full_fig_p033_4.png]
Figure 5
Figure 5. Figure 5: Comparisons between un-normalized and normalized Gibbs samples for [PITH_FULL_IMAGE:figures/full_fig_p034_5.png]
Figure 7
Figure 7. Figure 7: IW and Ref priors [PITH_FULL_IMAGE:figures/full_fig_p035_7.png]
Figure 8
Figure 8. Figure 8: IW and SIW priors [PITH_FULL_IMAGE:figures/full_fig_p035_8.png]
Figure 10
Figure 10. Figure 10: Comparison of the posterior density of the condition numbers for Σ [PITH_FULL_IMAGE:figures/full_fig_p036_10.png]
Figure 11
Figure 11. Figure 11: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p054_11.png]
Figure 12
Figure 12. Figure 12: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p055_12.png]
Figure 13
Figure 13. Figure 13: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p056_13.png]
Figure 14
Figure 14. Figure 14: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p057_14.png]
Figure 15
Figure 15. Figure 15: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p058_15.png]
Figure 16
Figure 16. Figure 16: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p059_16.png]
Figure 17
Figure 17. Figure 17: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p060_17.png]
Figure 18
Figure 18. Figure 18: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p061_18.png]
Figure 19
Figure 19. Figure 19: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p062_19.png]
Figure 20
Figure 20. Figure 20: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p063_20.png]
Figure 21
Figure 21. Figure 21: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p064_21.png]
Figure 22
Figure 22. Figure 22: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p065_22.png]
Figure 23
Figure 23. Figure 23: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p066_23.png]
Figure 24
Figure 24. Figure 24: Density plots of the traces, determinants for Σ, their corresponding ACFs of the global [PITH_FULL_IMAGE:figures/full_fig_p067_24.png]

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