{"id":"6e999f60-c877-492e-aa46-1eee8e8cc3cf","arxiv_id":"2501.04913","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A geodesic Hamiltonian Monte Carlo sampler for Kronecker-structured separable covariance matrices, built on the affine-invariant metric and a regularized or orthogonalized pullback metric.","lead":"A new Hamiltonian Monte Carlo sampler, SGLMC, targets Bayesian inference for matrix normal models whose covariance is the Kronecker product of two smaller matrices. It builds a curved geometry on the covariance components and shows how to move along geodesics, with higher effective sample sizes per iteration than Stan or Gibbs in the reported experiments.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"SGLMC-O and the weighted metric GW constrain |Σ2|=1 but target the posterior slice at that constraint, not the marginal posterior of the identified parameters; the claimed equivalence to normalized Gibbs in Figure 5 is unsupported and likely false.","rationale":"The reader's weakest assumption was the missing proof of reversibility/detailed balance for the generalized leapfrog under GO and the inconsistent printed Algorithm 2; those are real and would themselves justify conditional acceptance. However, the most load-bearing concern is more structural: even with a perfectly reversible integrator, the constrained metrics GO and GW appear to target the slice of the full posterior at |Σ2|=1 rather than the marginal posterior of the identified parameters. This follows from the definition of the Lagrangian target in Equation (31) together with the constraint used in Proposition 2.3/Algorithm 2, and it does not depend on pseudocode typos. The paper's own discussion in Section 4.2 and Figure 5 acknowledges a 'bias' but asserts it is equivalent to normalizing Gibbs samples; that equivalence is not shown and is contradicted by the non-factorizing scale dependence of the IW posterior. I still recommend a conditional verdict because the regularized metric and the P-VL gradient machinery are plausible contributions, and the flaw may be fixable by adding the missing Jacobian term or by redefining the target as the conditional slice and restricting the claims accordingly. The proposed computational check would settle whether the concern lands. Agreement with the reader is partial: we share the validity worry, but the target-distribution issue is deeper than the leapfrog reversibility point and should be addressed explicitly in revision.","tokens_in":26376,"tokens_out":28091,"duration_ms":264311,"concrete_test":"Re-run the Section 5 synthetic experiment (d1=15, d2=6, same IW priors) with a correct reversible implementation of SGLMC-O and with regularized SGLMC (G_α, α=0.95). For every regularized-SGLMC sample (Σ1,Σ2), transform to the identified gauge (A,U)=(|Σ2|^{1/d2}Σ1, Σ2/|Σ2|^{1/d2}); SGLMC-O already outputs |Σ2|=1, so take (A,U)=(Σ1,Σ2). Compare the joint posterior of (A,U), using summaries of A⊗U such as log|A|, κ(A), and tr(A^{-1} Σ_k A_k tr(U^{-1}B_k)), between the two samplers. If the distributions differ beyond Monte Carlo error, SGLMC-O targets the slice rather than the posterior marginal, and the orthogonal and weighted metrics require a Jacobian correction or must be removed from the claims.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.2/Algorithm 2 (GO) and Proposition 2.3's 'additional constraint |Σ2|=1' (also for GW) run geodesic HMC on the submanifold {|Σ2|=1}. In the Lagrangian formulation (31), the position marginal of exp(-E) is π(q): the (1/2) log|G(q)| term in E cancels the √|G(q)| normalization of the velocity distribution. Hence the stationary distribution is the full posterior density π(q) evaluated on the slice |Σ2|=1, not the posterior of the identified parameters (A,U)=(|Σ2|^{1/d2}Σ1, Σ2/|Σ2|^{1/d2}), which requires integrating out the scale. These densities differ whenever the prior is not scale-invariant. For the IW priors (22), the scale 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); the integral over s is not proportional to the integrand at s=1. The Section 4.2/Figure 5 claim that the bias is 'equivalent to normalizing Gibbs samples of Σ2' is not proven, and normalizing only Σ2 changes the product Σ1⊗Σ2 instead of mapping to the same identified gauge. If this lands, the high ESS/it values for GO and GW in Table 1 describe a distribution that is not the posterior, undermining the efficiency claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":26798,"tokens_out":22293,"duration_ms":204181,"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":[{"comment":"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.","section":"Sec. 4.2, Fig. 5, Table 1"},{"comment":"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.","section":"Sec. 4.2, Algorithm 2"},{"comment":"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.","section":"Prop. 3.2, Eqs. (25)-(26)"},{"comment":"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.","section":"Lemma 2.2, Eq. (11)"},{"comment":"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.","section":"Lemma 2.3, Eq. (31)"}],"minor_comments":[{"comment":"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.","section":"Prop. 2.2"},{"comment":"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.","section":"Sec. 8, proof of Prop. 2.3"},{"comment":"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.","section":"Algorithm 1"},{"comment":"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.","section":"Algorithm 2"},{"comment":"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.","section":"Secs. 5 and 8.5"},{"comment":"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.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper contains a promising geometric idea—Proposition 2.3 and the P-VL gradient machinery could be useful—but the submitted version is not yet publishable. The most serious issue is the target-distribution mismatch for the constrained metrics GO and GW; the authors should verify whether any of the reported posterior agreements survive after correcting the projection and sign errors. If the constrained variants cannot be made to target the posterior, the paper should be refocused on the regularized metric, whose validity is less in doubt. The inconsistencies in Algorithms 1 and 2 are substantial but fixable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The regularized version of SGLMC is a genuine extension of geodesic LMC to Kronecker-structured covariances, and the central geometric observation—that the pullback of the affine-invariant metric is degenerate because of scale indeterminacy—is correct and useful. Proposition 2.3's claim that the regularized metric's geodesics coincide with product-manifold geodesics is plausible from the Euler-Lagrange equations; I don't have a substantive objection. The PVL-based gradient computation is also sensible. So there is real material here.\n\nThe problem is the orthogonalized and weighted metrics. The projection P_Σ2 preserves the determinant of Σ2, but it does not set it to 1; it fixes whatever determinant the initial state has. The sampler therefore targets the full posterior conditioned on that determinant, not the marginal posterior of the identified parameters. I checked the change of variables for the IW priors in (22): the scale does not integrate out, so the slice density is not proportional to the marginal. And normalizing Gibbs samples by only rescaling Σ2 does not map to the same identified gauge—it changes the product. The Figure 5 equivalence claim is unsupported, and the high ESS/it numbers for GO/GW in Table 1 describe a different distribution. That is the load-bearing flaw for those two variants. It is fixable if the authors redefine the target and compare on that basis, but as written it's misleading.\n\nThere are also printed errors that would block a referee: Lemma 2.2's projection formula subtracts a multiple of V2 rather than Σ2, and the d1/d2 weights are swapped in GO; Algorithm 2 applies P_Σ1 to the V2 update, reuses qj in the second half-kick, and has the acceptance assignment garbled. Proposition 3.2 has sign inconsistencies in the gradient. None of these are fatal to the regularized sampler, but they make the unverified claims hard to check.\n\nNo code or data is provided, and the empirical comparison omits one Stan run and reports no error bars. For a computational methods paper that matters.\n\nWho is this for: people doing Bayesian inference for matrix-normal or separable covariance models who want an alternative to Gibbs and are willing to think about the nonidentifiability. The regularized variant deserves a serious referee. The orthogonal/weighted variants need rework before I would trust them. My recommendation: send to peer review, but with major revision expected—fix the target distribution, correct the printed algorithm and equations, and ship the code.","headline":"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.","tokens_in":27262,"tokens_out":11195,"would_cite":false,"duration_ms":103033,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","65C05","62H12"],"pacs":[],"model":"deepseek-v4-flash","headline":"A geodesic Hamiltonian sampler puts separable-covariance MCMC on par with Gibbs","keywords":["separable covariance","matrix normal","Kronecker product","Hamiltonian Monte Carlo","geodesic Monte Carlo","affine-invariant metric","Pitsianis-Van Loan decomposition","Riemannian manifold"],"falsifier":"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.","tokens_in":26177,"feed_emoji":"📐","tokens_out":9611,"duration_ms":93035,"temperature":0.7,"pith_summary":"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.","feed_headline":"Geodesic sampler matches Gibbs for two-way covariance models","feed_subtitle":"Splitting covariance into Kronecker factors lets Hamiltonian Monte Carlo move along cheap product-manifold geodesics.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the geodesic Lagrangian Monte Carlo framework and generalized leapfrog integrator that SGLMC extends to Kronecker-structured covariances.","marker":"[16]"},{"why":"Introduces the Lagrangian dynamics formulation and the position-dependent mass matrix used in Algorithm 1.","marker":"[21]"},{"why":"Provides the affine-invariant metric, its Koszul-style geodesic exponential map, and the duplication-matrix determinant factors that appear in the target density.","marker":"[31]"},{"why":"Defines the Pitsianis–Van Loan approximation of a matrix as a sum of Kronecker products, which the paper uses to split likelihood terms into component traces.","marker":"[39]"},{"why":"Cited as the source of the Pitsianis–Van Loan decomposition's existence and structural properties used in Section 3.","marker":"[36]"},{"why":"Provides the geodesic Monte Carlo on embedded manifolds and the projection-style velocity handling that motivates the orthogonal metric.","marker":"[4]"},{"why":"Earlier work on geodesic convexity of multiway covariances under a product-manifold affine-invariant metric, motivating the product metric choice.","marker":"[40]"},{"why":"Establishes the Gibbs sampling baseline for separable covariance arrays that SGLMC is compared against in the experiments.","marker":"[15]"}],"fun_headline_variants":["Separable geodesic Monte Carlo for 2-way covariance inference","Product-manifold geodesics make covariance sampling split","Hamiltonian Monte Carlo on product manifolds for matrix normal","Geodesic sampler for separable covariances rivals Gibbs","Kronecker-structured covariance sampling via geodesic HMC"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Separable geodesic Monte Carlo for 2-way covariance inference","Product-manifold geodesics make covariance sampling split","Hamiltonian Monte Carlo on product manifolds for matrix normal","Geodesic sampler for separable covariances rivals Gibbs","Kronecker-structured covariance sampling via geodesic HMC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000685,"raw_usage":{"total_tokens":3098,"prompt_tokens":928,"completion_tokens":2170,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":2089}},"tokens_in":544,"tokens_out":2170,"duration_ms":16094,"temperature":1.0,"reasoning_tokens":2089,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:24:04.278835+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Geodesic Lagrangian Monte Carlo over the space of positive definite matrices: with application to Bayesian spectral density estimation","cited_arxiv_id":null,"evidence_quote":"Supplies the geodesic Lagrangian Monte Carlo framework and generalized leapfrog integrator that SGLMC extends to Kronecker-structured covariances."},{"cited_title":"Markov chain Monte Carlo from lagrangian dynamics","cited_arxiv_id":null,"evidence_quote":"Introduces the Lagrangian dynamics formulation and the position-dependent mass matrix used in Algorithm 1."},{"cited_title":"The Riemannian geometry of the space of positive-definite matrices and its application to the regularization of positive-definite matrix-valued data","cited_arxiv_id":null,"evidence_quote":"Provides the affine-invariant metric, its Koszul-style geodesic exponential map, and the duplication-matrix determinant factors that appear in the target density."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Pitsianis–Van Loan approximation of a matrix as a sum of Kronecker products, which the paper uses to split likelihood terms into component traces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cited as the source of the Pitsianis–Van Loan decomposition's existence and structural properties used in Section 3."},{"cited_title":"Geodesic Monte Carlo on embedded manifolds","cited_arxiv_id":null,"evidence_quote":"Provides the geodesic Monte Carlo on embedded manifolds and the projection-style velocity handling that motivates the orthogonal metric."},{"cited_title":"Geodesic convexity and covariance estimation","cited_arxiv_id":null,"evidence_quote":"Earlier work on geodesic convexity of multiway covariances under a product-manifold affine-invariant metric, motivating the product metric choice."},{"cited_title":"Separable covariance arrays via the tucker product, with applications to multi- variate relational data","cited_arxiv_id":null,"evidence_quote":"Establishes the Gibbs sampling baseline for separable covariance arrays that SGLMC is compared against in the experiments."}],"review_version":1}