{"id":"bc06013c-8155-4e2f-84b6-5e2bbf2f89fb","arxiv_id":"2412.20487","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper unifies existing multimodal VAE aggregation schemes as barycenters under KL divergence and introduces WB-VAE, which aggregates unimodal posteriors via the 2-Wasserstein barycenter.","lead":"This paper reframes how multimodal variational autoencoders combine information from different data types, showing that the two standard combination rules are special cases of a more general 'barycenter' framework. It then proposes a new combination rule based on Wasserstein geometry, which averages the per-modality distributions in a way that preserves their shapes, and shows modest gains on three benchmarks.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equal-weight Wasserstein barycenter cannot fuse evidence: its variance is an average of unimodal variances, whereas the true joint posterior should be sharper than every unimodal posterior; the paper's central motivation for WB-VAE is therefore unsupported.","rationale":"I read the paper as making two claims: (i) PoE and MoE are KL barycenters, and (ii) replacing KL by the 2-Wasserstein distance yields a principled aggregation that 'preserves geometry' and works empirically. Claim (i) is correct and a legitimate conceptual contribution. The proposed WB-VAE is a concrete, reproducible algorithm with consistent empirical gains, which counts in its favor. However, claim (ii) rests on an unexamined identification: the W2 barycenter of the unimodal posteriors is treated as a good approximation to the true joint posterior. In the simplest Bayesian setting this identification is false because the W2 barycenter averages standard deviations while the joint posterior should reduce them. The reader's weakest_assumption is the same: equal weights and no error bound. My proposed synthetic check would settle it in a few lines of computation. Because the empirical results are promising and the issue is addressable (e.g., learned weights, including a prior expert, or a controlled comparison), I keep the conditional verdict rather than escalating to rejection. Proposition 1's proof problem is real but secondary; regardless of it, the W2 posterior-fidelity gap remains.","tokens_in":18155,"tokens_out":14828,"duration_ms":157035,"concrete_test":"Use a scalar two-modality Gaussian generative model: z ~ N(0,1), x_m = z + epsilon_m with epsilon_m ~ N(0, tau_m^2) and tau_1 != tau_2. Compute the exact unimodal posteriors q_m(z|x_m), the exact joint posterior p(z|x_1,x_2) (Gaussian, precision 1 + 1/tau_1^2 + 1/tau_2^2), and the equal-weight WB aggregate from Eq. (7), PoE, and MoE. Sweep tau_2/tau_1 from 1 to 100 and report D_KL(p_true || q_agg) for each aggregation. If WB's KL is never lower than PoE's and its variance stays above the true posterior variance, then the equal-weight W2 choice is not a valid joint-posterior approximation and the paper's theoretical motivation for WB-VAE fails; if WB beats PoE in some interpretable regime, the concern is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 is correct: PoE is the reverse-KL barycenter and MoE is the forward-KL barycenter. The load-bearing weakness is the next step, in the section 'Multimodal VAE from Wasserstein Barycenter': the paper assumes that the equal-weight Bures-Wasserstein barycenter of unimodal Gaussian posteriors approximates the true joint posterior. For conditionally independent modalities, the true joint posterior has precision prec_true = prec_prior + sum_m prec_m, so its covariance is smaller than every unimodal covariance. The isotropic WB aggregate (Remark 2, Eq. 7) instead sets the standard deviation to the arithmetic mean, sigma_tilde = (1/M) sum_m sigma_m, which is at least the smallest unimodal sigma. The aggregate can therefore never be sharper than the sharpest expert and dilutes informative modalities when uninformative ones are present. This is a structural property, not a tuning issue. Proposition 1 does not close the gap: its Jensen inequality only bounds the divergence of the mixture, and its ELBO argument is tautological; neither statement relates the barycenter to the true joint posterior. The paper also offers no learned weights and no error bound, so the claimed 'geometry-preserving' advantage is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a barycentric reformulation of aggregation functions in multimodal VAEs. It argues that product-of-experts (PoE) and mixture-of-experts (MoE) aggregation are specific instances of barycenters obtained by minimizing reverse and forward KL divergences, respectively (Theorem 1). It then introduces WB-VAE, which aggregates unimodal Gaussian posteriors via the 2-Wasserstein barycenter, with a closed-form isotropic-Gaussian solution (Remark 2), and a mixture variant MWB-VAE (Remark 3). The method is evaluated on PolyMNIST, MNIST-SVHN-TEXT, and CelebA, reporting competitive classification accuracy, generation coherence, and log-likelihood.","tokens_in":18393,"tokens_out":4960,"duration_ms":47987,"significance":"The barycentric interpretation of PoE and MoE in Theorem 1 is a genuine and clearly stated theoretical contribution: the derivation in Appendix A.2 correctly identifies PoE as the weighted reverse-KL barycenter and MoE as the weighted forward-KL barycenter, and it provides an information-theoretic vocabulary (zero-forcing versus mass-covering) that is useful for reasoning about existing methods. However, the central methodological claim of the paper, namely that the equal-weight Wasserstein barycenter of unimodal posteriors is a sound approximation to the true joint posterior, is not supported and is in fact contradicted by the basic statistical property that a joint posterior under conditional independence should be sharper than every unimodal posterior. The proposed WB-VAE's aggregation rule, Eq. (7), averages standard deviations and therefore cannot produce a distribution sharper than the sharpest expert. This is a structural flaw in the proposed method, not a tuning issue. The empirical results, while sometimes favorable on classification and coherence metrics, do not test the sharpening property and cannot compensate for the lack of a theoretical justification.","major_comments":[{"comment":"The equal-weight isotropic Wasserstein barycenter cannot fuse evidence. In Remark 2, Eq. (7) gives \\tilde{\\sigma} = (1/M)\\sum_m \\sigma_m, so the aggregate standard deviation is an arithmetic mean of the unimodal standard deviations. In contrast, for conditionally independent modalities, the true joint posterior has precision 1/\\sigma_{\\text{true}}^2 = 1/\\sigma_{\\text{prior}}^2 + \\sum_m 1/\\sigma_m^2, implying \\sigma_{\\text{true}} < \\min_m \\sigma_m. Thus the proposed WB-VAE posterior is never sharper than the sharpest unimodal posterior, and it dilutes informative modalities when uninformative ones are present. This is a structural property of the chosen divergence and weights, not a tuning issue. The paper's claim that the Wasserstein barycenter 'strikes a balance' or 'preserves geometry' does not address this failure to meet the defining requirement of a joint posterior in a multimodal VAE.","section":"Multimodal VAE from Wasserstein Barycenter, Remark 2 and Eq. (7)"},{"comment":"The claim that 'for any divergence measure d(q_{\\phi_m}, \\cdot) that is convex on q_{\\phi_m}, the resultant barycenter guarantees a valid ELBO' is either trivial or unproved. The ELBO inequality log p(X_{1:M}) \\geq L(\\theta,\\phi; X_{1:M}) holds for any approximate posterior q(z|X_{1:M}) whatsoever, regardless of how q is constructed, because it is equivalent to the nonnegativity of D_{KL}(q||p_{\\theta}(z|X_{1:M})). The proof in Appendix A.1 indeed only notes that D_{KL}(f(\\mathcal{M}(...))||p) \\geq 0, which is always true and does not use the barycenter minimization or Jensen's inequality in any substantive way. The additional statement that the left-hand side of Eq. (5) 'defines a scalable inference' while the right-hand side requires 2^M inference networks is also unsupported: the mixture \\sum_m \\lambda_m q_{\\phi_m} is not generally the minimizer of the right-hand side except in the special forward-KL case. Proposition 1 therefore provides no connection between the barycenter construction and the quality of the approximation to the true joint posterior.","section":"Proposition 1 and Appendix A.1"},{"comment":"The paper chooses the barycenter weights \\lambda_m = 1/M without any theoretical or empirical justification, stating only that 'it is typically safe' to do so. No learned-weight scheme, no error bound, and no condition are given under which the equal-weight Wasserstein barycenter is close to the true joint posterior. This is a load-bearing omission because the weights are the only free parameter of the proposed aggregation function: with equal weights, the method reduces to a simple average of means and standard deviations, which is not a principled approximation of the joint posterior. The claim that different weights 'may lead to a joint posterior that maintains diverse shapes and structures' is speculative and not tested. A concrete test would be to measure the entropy or precision of \\tilde{q} as the number of modalities increases and compare it to the true conditional-independent posterior; such a test is absent.","section":"Multimodal VAE from Wasserstein Barycenter, paragraph after Eq. (7)"},{"comment":"The empirical evaluation does not resolve the structural concern above. The reported log-likelihoods (Table 3) show WB-VAE is on par with or worse than PoE-VAE, and the classification/coherence gains of WB-VAE and MWB-VAE can be explained by the smoothing effect of averaging rather than by successful joint-posterior approximation. The claim that WB-VAE 'strikes a balance' between zero-forcing and mass-covering is a restatement of the averaging behavior, not evidence that the aggregate posterior sharpens with additional modalities. The paper needs a direct evaluation of the sharpening property, for instance by computing the precision or differential entropy of \\tilde{q}(z|x_{1:M}) as a function of M, or by comparing against the analytical joint posterior under conditional independence. Without such evidence, the central motivation for WB-VAE remains unsupported.","section":"Experiments, Tables 1-4 and Figure 3"}],"minor_comments":[{"comment":"The phrase 'to for multimodal representation learning' contains a typo; it should read 'for multimodal representation learning'.","section":"Abstract"},{"comment":"The reconstruction term E_{q}[\\log p_{\\theta}(X_{1:M}|z)] is written with a single joint likelihood; for conditionally independent modalities it should factorize as \\sum_m E_{q}[\\log p_{\\theta_m}(x_m|z)]. As written, the connection between the ELBO and the unimodal decoders is ambiguous.","section":"Method, Eq. (3)"},{"comment":"The statement 'convex on q_{\\phi_m}' is imprecise. The Jensen inequality in Eq. (5) requires d(\\cdot, q) to be convex in its first argument for fixed q, not merely 'convex on' the unimodal distributions. Please clarify the convexity assumption and state the argument ordering consistently.","section":"Proposition 1 and Eq. (5)"},{"comment":"The proof of Remark 2 correctly derives \\tilde{\\mu} = \\sum_m \\lambda_m \\mu_m and \\tilde{\\sigma} = \\sum_m \\lambda_m \\sigma_m by differentiating the 1D squared 2-Wasserstein distance. However, the sentence 'the same results can also be derived by leveraging Proposition 1' is misleading, because Proposition 1's Jensen inequality does not identify the minimizer; remove or rephrase it.","section":"Appendix A.3"},{"comment":"There are several typos: 'Folliwing' should be 'Following' in the MNIST-SVHN-TEXT dataset description; 'pf 32' should be 'of 32' in Section B.3; and 'V AE' appears with inconsistent spacing throughout. The caption of Figure 2 also contains an empty placeholder '( )' after 'Wasserstein barycenter'.","section":"Experiments and Supplementary Material"},{"comment":"The paper cites Sutter, Daunhawer, and Vogt (2021) for the generalized multimodal ELBO but does not discuss how the proposed barycentric formulation relates to the sub-sampled ELBO objective or to the mixture-of-product-of-experts construction beyond the brief remark in Appendix A.4. A more detailed comparison would help position the contribution.","section":"Related Work"}],"recommendation":"reject","confidential_remarks":"The theoretical result in Theorem 1 (PoE and MoE as reverse- and forward-KL barycenters) is sound and could be the basis of a useful short paper on the information-theoretic interpretation of existing aggregation functions. However, the current manuscript's main proposed method, WB-VAE, rests on the equal-weight Wasserstein barycenter, which cannot sharpen the posterior relative to its unimodal experts and therefore cannot approximate a joint posterior under conditional independence. This is a load-bearing structural error rather than a missing experiment or a tuning issue, and it cannot be fixed within the current scope without abandoning the proposed aggregation rule. The empirical results favoring WB-VAE on classification and coherence are plausibly explained by averaging effects and do not test the sharpening property. I therefore recommend rejection, while noting that the PoE/MoE barycenter interpretation is a publishable contribution if separated from the Wasserstein-barycenter application."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful core of this paper is the reframing: PoE and MoE are exactly reverse- and forward-KL barycenters, and Theorem 1 plus Appendix A.2 makes that precise. That is a nice unification and gives a vocabulary for discussing what those aggregation functions do. The Wasserstein extension is a plausible heuristic, and the experiments are encouraging: MWB-VAE beats or matches MoPoE-VAE on three benchmarks, with means and stds reported for the main tables. I believe the empirical work is honest enough to take seriously.\n\nBut the paper oversells the theoretical motivation. Proposition 1 is basically tautological: any aggregation that produces a valid distribution gives a valid ELBO, because the KL to the true posterior is nonnegative. It does not say why a Wasserstein barycenter should be closer to the true joint posterior than, say, PoE or MoE. More concretely, in the isotropic Gaussian case the equal-weight Bures-Wasserstein barycenter (Remark 2, Eq. 7) takes the arithmetic mean of the standard deviations. That means the aggregate is never sharper than the sharpest expert. Under conditional independence the true joint posterior has precision equal to the sum of the unimodal precisions, so it is sharper than every expert. This mismatch is structural, not a tuning issue. The paper's claim that the Wasserstein distance 'preserves geometry' is qualitative and never quantified. A simple experiment comparing the barycenter to the true posterior in a controlled setting, or learning the weights, would help.\n\nTwo more soft spots. MWB-VAE's objective (Remark 3) is underspecified: how subsets are sampled, how the bilevel optimization is implemented, and how the mixture is used in training are not explained, so the results are hard to reproduce from the text. And some PolyMNIST plots in Fig. 3 show trends without error bars. Both are fixable.\n\nI don't think this is a fatal case. The barycentric viewpoint is genuinely useful, and the empirical gains, even if modest, are consistent. The paper should be sent to peer review, but the authors should be asked to either provide theoretical support for the Wasserstein posterior or soften the claims, and to spell out the MWB-VAE training procedure.","headline":"A clean barycentric reframing of PoE/MoE, plus a Wasserstein aggregation that works empirically but lacks the theoretical grounding the paper claims.","tokens_in":18988,"tokens_out":2575,"would_cite":true,"duration_ms":25605,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that product-of-experts and mixture-of-experts aggregation in multimodal VAEs are both instances of the same barycenter problem—minimizing a weighted KL divergence in opposite directions—and introduces the Wasserstein…","keywords":["multimodal VAE","barycenter","Wasserstein barycenter","product of experts","mixture of experts","KL divergence","missing modalities","latent representation learning"],"falsifier":"In a two-modality synthetic example where the true posterior is computable—say a shared latent $z$ observed through Gaussian likelihoods with very different variances—compare the equal-weight Wasserstein barycenter of the two unimodal posteriors with the true posterior in squared 2-Wasserstein distance. If the distance stays small across a wide range of variance imbalances, the equal-weight prescription is validated; if it grows sharply when one modality is much more informative than the other, the fixed weights are the weak point and a learned weighting would be needed.","tokens_in":17931,"feed_emoji":"📐","tokens_out":8448,"duration_ms":73035,"temperature":0.7,"pith_summary":"Multimodal VAEs must combine a separate approximate posterior per modality into one joint posterior. This paper argues that the standard combination rules are not heuristic expert mixtures but are exactly the barycenters of the unimodal posteriors: product-of-experts minimizes a weighted reverse KL divergence and mixture-of-experts minimizes a weighted forward KL divergence. Seeing them this way exposes why PoE is zero-forcing and MoE is mass-covering, and opens the door to other divergences. The paper proposes the Wasserstein barycenter with equal weights, which for Gaussian unimodal posteriors is a Gaussian whose mean is the weighted mean of the means and whose covariance solves a fixed-point equation (and reduces to elementwise formulas for diagonal covariances). Experiments on PolyMNIST, MNIST-SVHN-TEXT, and CelebA indicate that this Wasserstein aggregation matches or improves latent representation quality and conditional generation coherence, particularly as the number of modalities grows.","feed_headline":"PoE and MoE are barycenters, and Wasserstein beats them","feed_subtitle":"One divergence-minimization view explains both old aggregation rules, and the Wasserstein variant wins on three benchmarks.","key_machinery":"The machinery is the barycenter problem: for distributions $\\{P_m\\}$ and weights $\\{\\lambda_m\\}$, a barycenter is $P_B = \\arg\\min_P \\sum_m \\lambda_m\\, d(P_m, P)$ for a chosen divergence $d$. Choosing $d$ as reverse KL reproduces PoE, choosing it as forward KL reproduces MoE, and choosing it as the squared 2-Wasserstein distance produces the Wasserstein barycenter. In the Gaussian case the Wasserstein barycenter is the Bures-Wasserstein barycenter, whose covariance solves the fixed-point equation $\\tilde{\\Sigma} = \\sum_m \\lambda_m (\\tilde{\\Sigma}^{1/2} \\Sigma_m \\tilde{\\Sigma}^{1/2})^{1/2}$, and in the isotropic diagonal case it reduces to the elementwise formulas $\\tilde{\\mu} = \\sum_m \\lambda_m \\mu_m$ and $\\tilde{\\sigma} = \\sum_m \\lambda_m \\sigma_m$. The barycenter computation is decoupled from the ELBO training, which is what makes the divergence choice a design decision rather than a fixed expert rule.","core_discovery":"The paper establishes a unification result: the two dominant aggregation rules in multimodal variational autoencoders are barycenters under the two directions of KL divergence. Minimizing $\\sum_m \\lambda_m D_{\\mathrm{KL}}(q \\,\\|\\, q_{\\phi_m})$ over $q$, with weights $\\lambda_m$ summing to one, yields the product-of-experts posterior (up to normalization), while minimizing $\\sum_m \\lambda_m D_{\\mathrm{KL}}(q_{\\phi_m} \\,\\|\\, q)$ yields the mixture-of-experts posterior. This is the content of Theorem 1, with the normalization subtlety of PoE handled in Appendix A.2. The paper then replaces KL with the squared 2-Wasserstein distance and defines WB-VAE, whose approximate joint posterior is the Wasserstein barycenter of the unimodal Gaussian posteriors. For isotropic Gaussian encoders this barycenter is explicit: the joint mean is $\\tilde{\\mu} = \\sum_m \\lambda_m \\mu_m$ and the joint scale is $\\tilde{\\sigma} = \\sum_m \\lambda_m \\sigma_m$ in each dimension, interpolating between modalities along the Wasserstein geodesic rather than forcing or erasing support.","pith_inferences":["The paper does not prove that the equal-weight Wasserstein barycenter approximates the true joint posterior; a formal bound on $W_2(\\tilde{q}_{\\mathrm{WB}}, p_{\\theta}(z \\mid X_{1:M}))$, or a counterexample with strongly heteroscedastic modalities, would decide how far the prescription generalizes.","Since the barycentric view separates aggregation from training, the fixed weights $\\lambda_m = 1/M$ could be replaced by learned or uncertainty-scaled weights without changing the framework; that is a direct extension the paper leaves implicit.","The same barycentric lens could be applied to other generative latent variable models beyond VAEs, such as multimodal diffusion models, where an aggregation of per-modality reverse processes could be chosen by a metric divergence rather than by a product rule."],"forward_implications":["PoE and MoE are not competing heuristics but two sides of KL asymmetry: PoE minimizes reverse KL and is therefore zero-forcing, while MoE minimizes forward KL and is therefore mass-covering.","Any divergence that is convex in the unimodal distributions yields a valid ELBO and scalable inference through Jensen's inequality, opening the same barycentric framework to $f$-divergences, Gromov-Wasserstein distances, or other metric divergences.","Because the 2-Wasserstein distance is symmetric, the WB-VAE posterior is neither zero-forcing nor mass-covering; it sits between them, and on PolyMNIST its metrics improve roughly linearly as modalities are added, unlike the saturating behavior of the KL-based baselines.","The mixture-of-barycenters variant MWB-VAE outperforms MoPoE-VAE on most evaluated tasks in the paper, including the difficult image-to-text and text-to-image generation directions on CelebA."],"supporting_citations":[{"why":"Introduces product-of-experts aggregation (PoE-VAE), which Theorem 1 identifies as a reverse-KL barycenter.","marker":"Wu and Goodman 2018"},{"why":"Introduces mixture-of-experts aggregation (MoE-VAE), which Theorem 1 identifies as a forward-KL barycenter, and provides the MMVAE experimental setup.","marker":"Shi et al. 2019"},{"why":"Defines MoPoE-VAE, the baseline with the best prior numbers, and supplies the benchmark datasets and evaluation protocol.","marker":"Sutter, Daunhawer, and Vogt 2021"},{"why":"Defines the Wasserstein barycenter and proves the fixed-point equation for Gaussian barycenters used in WB-VAE.","marker":"Agueh and Carlier 2011"},{"why":"Gives the analytic squared 2-Wasserstein distance between Gaussian distributions, the base computation for the Bures-Wasserstein barycenter.","marker":"Knott and Smith 1984"},{"why":"Provides the geodesic-space interpretation of the Wasserstein metric that motivates the geometry-preservation claim.","marker":"Ambrosio, Gigli, and Savaré 2008"},{"why":"Documents the zero-forcing and mass-covering behavior of reverse and forward KL, used in Remark 1 to explain PoE versus MoE.","marker":"Minka et al. 2005"}],"fun_headline_variants":["Barycentric view unifies PoE and MoE, then Wasserstein beats them","PoE and MoE are just KL barycenters - Wasserstein is better","From KL to Wasserstein: new barycenter wins multimodal VAE","One formula explains PoE and MoE; Wasserstein barycenter wins","Wasserstein barycenter trumps KL experts in multimodal VAEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the true joint posterior is well approximated by the equal-weight Wasserstein barycenter of the per-modality Gaussian posteriors, which the paper motivates by geometric intuition about geodesic interpolation but does not support with a closeness guarantee or learned weights.","fun_headline_variants_meta":{"raw":{"variants":["Barycentric view unifies PoE and MoE, then Wasserstein beats them","PoE and MoE are just KL barycenters - Wasserstein is better","From KL to Wasserstein: new barycenter wins multimodal VAE","One formula explains PoE and MoE; Wasserstein barycenter wins","Wasserstein barycenter trumps KL experts in multimodal VAEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000836,"raw_usage":{"total_tokens":3681,"prompt_tokens":1017,"completion_tokens":2664,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":633,"completion_tokens_details":{"reasoning_tokens":2565}},"tokens_in":633,"tokens_out":2664,"duration_ms":17523,"temperature":1.0,"reasoning_tokens":2565,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:20:11.464466+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In a two-modality synthetic example where the true posterior is computable—say a shared latent $z$ observed through Gaussian likelihoods with very different variances—compare the equal-weight Wasserstein barycenter of the two unimodal posteriors with the true posterior in squared 2-Wasserstein distance. If the distance stays small across a wide range of variance imbalances, the equal-weight prescription is validated; if it grows sharply when one modality is much more informative than the other, the fixed weights are the weak point and a learned weighting would be needed.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Wasserstein barycenter and proves the fixed-point equation for Gaussian barycenters used in WB-VAE."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the analytic squared 2-Wasserstein distance between Gaussian distributions, the base computation for the Bures-Wasserstein barycenter."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the geodesic-space interpretation of the Wasserstein metric that motivates the geometry-preservation claim."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the zero-forcing and mass-covering behavior of reverse and forward KL, used in Remark 1 to explain PoE versus MoE."}],"review_version":1}