{"id":"1801d31b-144e-43dd-8809-d813d4bf4e25","arxiv_id":"2607.22892","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under simple growth conditions on the potential and mirror map, Mirror Langevin diffusions satisfy Poincaré or log-Sobolev inequalities, and a new exactly-stationary Gibbs sampler converges weakly to the diffusion and mixes in chi-square at rate (1 - c0*epsilon)^k.","lead":"Mirror Langevin diffusions sample from a target density by running a Langevin process on a curved space defined by a convex function. This paper gives new tail conditions that guarantee fast exponential convergence, and introduces a two-step Gibbs sampler that keeps the target exactly stationary while approximating the diffusion with explicit mixing bounds.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5's chi-square rate rests on an unproven Poincaré inequality for the push-forward ν; Assumption 1 does not imply it, and without it the comparison gives no spectral gap.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing point found here: Theorem 5 requires a Poincaré inequality for the push-forward measure ν, and nothing in Assumption 1 or in the Lyapunov results of Theorems 1–2 supplies it. My stress test confirms this by an explicit family (V(x)=(d+1)log(1+‖x‖²), u(x)=c‖x‖²/2) satisfying all regularity and ellipticity assumptions while ν has polynomial tails, so no finite Poincaré constant exists. In the proof, Step 4's SDPI bound is only useful when ν has a finite Poincaré constant; otherwise the comparison gives no positive spectral gap and the claimed chi-square contraction rate of order ε does not follow. This is a genuine soft spot in the central claim about the Markov chain, though the theorem as formally stated is conditional and therefore not logically false. The minor issues noted by the reader — the Example 1 KL rate, the sign typo in the contraction display, and the lower-bound justification — are secondary and do not invalidate the conditional result. My recommendation does not move the reader's CONDITIONAL verdict; it sharpens what the revision must address: either prove that ν satisfies a Poincaré inequality under checkable conditions on (V,u), or state explicitly that the theorem's mixing guarantee is only as strong as the user's ability to certify that external condition. I also note a small algebraic display issue in the proof of Theorem 5 Step 2, where det(∂x*/∂x) should appear with exponent −1/2 for the displayed equality to e^{-F(y)}; this is consistent with Lemma 2 and is fixable.","tokens_in":30017,"tokens_out":13107,"duration_ms":114864,"concrete_test":"Construct the explicit instance in dimension d≥2: u(x)=c‖x‖²/2 with c>0 and V(x)=(d+1)log(1+‖x‖²), normalized so that e^{-V} is a probability density. Verify that Assumption 1 and Theorem 5(ii)–(iv) hold, and that ν has density proportional to (1+c^{-2}‖y‖²)^{-(d+1)}, which fails the Poincaré inequality. Then compute the HGR maximal correlation S of (X, X+√s Z) for X∼ν and s=σ0²ε: by using test functions f_R(x)=x_1 1_{‖x‖≤R} and letting R→∞, show that S=1 for every ε>0. This establishes that the Gaussian-smoothed chain from ν has contraction coefficient 1, so the two-step chain cannot have the Θ(ε) spectral gap asserted in Theorem 5 unless assumption (i) is separately certified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 5 is a chi-square contraction rate of order (1 - c0*epsilon) for the two-step Gibbs chain. The proof transfers a Poincaré inequality from ν = (∇u)_#μ to the discrete Dirichlet form of the chain, and the only input supplying that PI is Theorem 5(i), which assumes e^{-F} satisfies a Poincaré inequality. This is not a consequence of Assumption 1. Assumption 1 imposes global uniform ellipticity and bounded derivatives on u and bounded second derivatives on V, but condition (18) of Theorem 1 — the paper's own sufficient condition for a PI — requires x·∇V - αx^T∇²u x → ∞. Bounded second derivatives of V do not force this: with V(x)=(d+1)log(1+‖x‖²) and u(x)=c‖x‖²/2, Assumption 1 holds but condition (18) fails, and ν has polynomially decaying tails, hence no Poincaré inequality. In Step 4 of the proof, the Klartag–Ordentlich bound S² ≤ (1+σ0²ε/c_F)^{-1} becomes vacuous when c_F = ∞, and the comparison argument yields no positive spectral gap. Thus the O(1/ε) mixing time is not guaranteed for natural designs satisfying all the other hypotheses; the theorem's applicability depends entirely on an external, unverified condition on ν.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Mirror Langevin diffusions (MLD) on Hessian manifolds, i.e., Langevin diffusions intrinsic to the metric g=∇²u with stationary density μ=e^{-V}. The main theoretical results are: (i) Lyapunov-function sufficient conditions for the MLD to satisfy a Poincaré inequality (Theorem 1) or a logarithmic Sobolev inequality (Theorem 2), intended to cover targets that are not strongly log-concave; (ii) a two-step Gibbs-sampler Markov chain with transition density r_ε that is exactly stationary with respect to μ (Theorem 3); (iii) weak convergence of the continuous-time interpolation of this chain to the MLD as ε↓0 (Theorem 4); and (iv) a χ²-divergence contraction rate of the form (1-c₀ε)^k for the discrete chain, under the additional hypothesis that the push-forward measure ν=(∇u)_#μ satisfies a Poincaré inequality and F is L-smooth (Theorem 5). The proofs use Laplace-expansion computations, comparison of discrete Dirichlet energies, and the strong data processing inequality of Klartag and Ordentlich.","tokens_in":30254,"tokens_out":27575,"duration_ms":248847,"significance":"If the results hold, the paper makes a useful contribution. The Lyapunov conditions in Theorems 1–2 are checkable and avoid the often-untractable curvature-dimension verification; the construction of an exactly stationary Markov chain approximation to the MLD is elegant and improves on biased Euler-type discretizations. The Laplace computations in Section 4 are careful, and the observation that the relevant third derivatives vanish at the minimizer is a genuinely useful simplification. The SDPI-based comparison in Theorem 5 gives the correct diffusion-consistent time scale and connects the Gibbs sampler to entropic optimal transport. The main limitation is that Theorem 5's rate is conditional on a Poincaré inequality for the push-forward ν, a condition that is not derived from the paper's other assumptions and can fail in natural examples; this restricts the advertised 'guaranteed convergence rate' more than the abstract and introduction suggest.","major_comments":[{"comment":"The Poincaré inequality for ν=(∇u)_#μ is an external hypothesis that is not implied by Assumption 1 or by L-smoothness of F. For example, take d≥1, V(x)=(d+1)log(1+‖x‖²), and u(x)=c‖x‖²/2. Then u and u* satisfy the uniform Hessian and derivative bounds of Assumption 1, and F is L-smooth, but ν has density proportional to (1+‖y/c‖²)^{-(d+1)}, which has polynomial tails. Since a Poincaré inequality implies exponential concentration of 1-Lipschitz functions, ν has no Poincaré inequality, i.e., c_F=∞. In Step 4 the Klartag–Ordentlich bound S²≤(1+σ₀²ε/c_F)^{-1} then becomes vacuous, and the comparison argument yields no positive spectral gap. The theorem is correctly stated as a conditional result, but the paper should either supply checkable sufficient conditions for Theorem 5(i) — for example, by applying the dual version of Theorem 1 to the dual MLD, which would give a Lyapunov condition of the form y·∇F(y)-αyᵀ∇²u*(y)y→∞ — or explicitly delimit Theorem 5 as a transfer theorem whose applicability must be verified case by case.","section":"Theorem 5(i) and its proof, Step 4"}],"minor_comments":[{"comment":"The classical Langevin diffusion for a 1-strongly convex potential converges in KL at rate 2, not 1, under the Bakry–Émery criterion. The comparison in Example 1 should therefore be 2γ versus 2, rather than 2γ versus 1. The qualitative conclusion that the MLD can be arbitrarily faster is unaffected.","section":"Example 1"},{"comment":"In the sentence 'ηχ²≤(1+σ₀²ε/c_F)^{-2}=1+Θ(ε)', the final equality has the wrong sign: (1+aε)^{-2}=1-2aε+O(ε²), so it should read 1-Θ(ε). As written, the spectral gap 1-ηχ² would be negative. The subsequent conclusion is correct once the sign is fixed.","section":"Theorem 5, Step 4"},{"comment":"The potential u(x)=log∑ᵢ e^{xᵢ} is not strictly convex: its Hessian diag(w)-wwᵀ has the all-ones vector in its kernel. Hence Example 3 is not covered by Theorem 1, which assumes a global diffeomorphism and locally uniformly elliptic Hessian, nor by Assumption 1 later. The example should either be replaced by a uniformly convex approximation or explicitly labelled as formal.","section":"Example 3"},{"comment":"The lower bound e^{F(y)-\\tilde F(y)}≥e^{-Lσ₀²d/2} is valid; the pointwise inequality from L-smoothness has the correct direction, and the Gaussian expectation is in fact (1+Lσ₀²ε)^{-d/2} exp(σ₀²ε‖∇F(y)‖²/(2(1+Lσ₀²ε))), which is bounded below by the displayed constant. No correction is needed here.","section":"Theorem 5, Step 3"}],"recommendation":"major_revision","confidential_remarks":"The central derivations are sound and the paper is likely publishable after revision. My main concern is that Theorem 5, the headline Markov-chain rate result, inherits all of its content from the assumed Poincaré inequality of the push-forward ν; without a checkable condition for that assumption, the result is much narrower than the abstract's 'guaranteed convergence rate' suggests. Adding a dual Lyapunov theorem for ν (or a remark explaining how Theorem 1 applied to the dual MLD supplies such a condition) would substantially strengthen the paper. The remaining issues are local typos and example-level gaps."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the diffusion-side results are genuinely useful: Theorems 1 and 2 give checkable Lyapunov tail criteria, conditions (18) and (23), for Poincaré and log-Sobolev inequalities of mirror Langevin diffusions without strong log-concavity. Second, the headline rate for the Gibbs chain, Theorem 5, is a conditional statement that the paper at times advertises as a guarantee: it assumes a Poincaré inequality for the pushed-forward measure ν=(∇u)_#μ, and nothing else in the paper ensures that. The stress-test example is correct—take V(x)=(d+1)log(1+‖x‖²) and u(x)=c‖x‖²/2; Assumption 1 holds but ν has polynomially decaying tails and no Poincaré inequality, so the SDPI bound is vacuous. The theorem is not false as stated, but its load-bearing hypothesis needs top billing rather than appearing as an item in a list.\n\nWhat is actually new: the tail conditions for functional inequalities on Hessian manifolds; the two-step Gibbs sampler with exact stationarity for every ε, a diffusion limit in Theorem 4, and chi-square contraction when the ν-Poincaré assumption holds; and the observation that the Euclidean specialization recovers the proximal sampler, which is a nice conceptual unification. The proof of Theorem 4, especially the Laplace expansion and the vanishing of third derivatives, is careful and checks out. There are no fitted constants and no circularity; the external theorems are used as black boxes in a standard way.\n\nThe soft spots are the following. The main one is the unverified ν-Poincaré assumption discussed above. There are also fixable glitches: Example 1 quotes the Bakry–Émery KL rate as 1 rather than 2, one displayed contraction-rate sign is off, and the lower-bound comparison inside the proof of Theorem 5 needs a clean rewrite, although the final bound appears repairable. The reverse conditional of the sampler is generally intractable, which limits immediate practical use; the authors acknowledge this, but it undercuts the practical sampling framing.\n\nThis paper is for people working on mirror Langevin diffusions, Hessian geometry, entropy-regularized optimal transport, or MCMC on non-Euclidean geometries. It deserves a serious referee. My recommendation: send it out, and ask the referee to focus on whether the ν-Poincaré hypothesis can be verified in natural designs or replaced by a condition in primal coordinates. With that clarified, I would take the paper.","headline":"Solid diffusion-side theory with a genuinely new Markov-chain construction, but the advertised O(1/ε) chi-square rate is conditional on an external Poincaré inequality for the pushed-forward measure that the paper never verifies.","tokens_in":30816,"tokens_out":4921,"would_cite":true,"duration_ms":43482,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","60J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a two-step Gibbs sampler with exact stationary law $\\mu$ approximates the Mirror Langevin diffusion and contracts in $\\chi^2$ at rate $1-c_0\\epsilon$ per step, giving a mixing time consistent with the diffusion.","keywords":["Mirror Langevin diffusions","Hessian manifolds","Gibbs sampler","Poincare inequality","log-Sobolev inequality","strong data processing inequality","Sinkhorn algorithm","chi-square contraction"],"falsifier":"Construct a smooth, strongly convex $u$ with bounded Hessian and a smooth $V$ satisfying the paper's Assumption 1 for which the push-forward $\\nu=(\\nabla u)_\\#\\mu$ fails to satisfy a Poincar\\'e inequality (for example, a density with a sufficiently heavy tail), then compute the spectral gap or chi-square contraction coefficient of the two-step kernel $r_\\epsilon$; if the gap is not $\\Theta(\\epsilon)$ or the bound $(1-c_0\\epsilon)$ fails, the theorem's reachable scope is smaller than stated.","tokens_in":29779,"feed_emoji":"🎲","tokens_out":8577,"duration_ms":73335,"temperature":0.7,"pith_summary":"The paper's goal is to make the Mirror Langevin diffusion a practical sampling tool by giving checkable conditions under which it converges exponentially fast, and by providing a discrete Markov chain that inherits that speed while keeping the target distribution exactly invariant. The Lyapunov-function conditions in Theorems 1 and 2 only involve first derivatives of $V$ and the quadratic form of $\\nabla^2 u$, so they can be checked without computing the Ricci curvature of the Hessian manifold. The paper then introduces a two-step Gibbs sampler whose transition density is $r_\\epsilon(z|x)=\\int q_\\epsilon(y|x)\\hat q_\\epsilon(z|y)\\,dy$ with $q_\\epsilon(y|x)=N(\\nabla u(x),\\epsilon\\nabla^2 u(x))$; Theorem 4 shows the interpolated chain converges weakly to the MLD. Theorem 5 shows that, when the pushed-forward density $e^{-F}=(\\nabla u)_\\#\\mu$ satisfies a Poincar\\'e inequality and $F$ is $L$-smooth, the chain contracts in $\\chi^2$ distance at rate $1-c_0\\epsilon$ per step. Because the diffusion mixes in $O(1)$ continuous time, the chain's $O(1/\\epsilon)$ mixing time is exactly the discrete analog.","feed_headline":"Gibbs chain matches Mirror Langevin diffusion speed","feed_subtitle":"A two-step sampler keeps the exact target law and contracts in chi-square at rate 1-cε per step.","key_machinery":"The load-bearing object is the Hessian manifold with metric $\\nabla^2 u$ and its dual coordinate $x^*=\\nabla u(x)$, together with the two-step Gibbs kernel $r_\\epsilon(z|x)$ built from the Gaussian conditional $q_\\epsilon(y|x)=N(x^*, \\epsilon \\nabla^2 u(x))$ and its Bayes reverse $\\hat q_\\epsilon(z|y)$. The identity $F(x^*)=V(x)+\\log\\det\\nabla^2 u(x)$ ties the primal and dual densities and underlies the coordinate-invariance argument. The rate proof proceeds by comparing the discrete Dirichlet energy of $r_\\epsilon$ with the energy of the chain $(A, A+\\sigma_0\\sqrt{\\epsilon}Z)$ where $A\\sim e^{-F}$, whose maximal correlation is controlled by a strong data processing inequality; the comparison constant comes from the $L$-smoothness of $F$ and the uniform Hessian bounds of Assumption 1. A key technical observation is that the Laplace phase $\\Psi(w)=(w^*-y)^T(\\partial w/\\partial w^*)(w^*-y)$ has vanishing third derivatives at its minimizer $w_0=y^*$, which makes the Gaussian Laplace approximation accurate to order $\\epsilon^2$.","core_discovery":"The central discovery is a transfer principle: the primal MLD with stationary density $\\mu=e^{-V}$ and the dual MLD with stationary density $\\nu=e^{-F}$, where $F(\\nabla u(x))=V(x)+\\log\\det\\nabla^2 u(x)$, are the same Langevin diffusion on the Hessian manifold $(\\mathbb{R}^d,\\nabla^2 u)$ written in two coordinate charts. This lets the paper prove functional inequalities for whichever side is easier, then transfer them to the other with the same constant. Using Lyapunov functions of the form $e^{\\alpha u}$ and $e^{\\alpha u^*}$, it obtains sufficient tail conditions for a Poincar\\'e or log-Sobolev inequality for the MLD without log-concavity of either $\\mu$ or $\\nu$. For the Markov chain, the paper proves that the Gibbs sampler's discrete Dirichlet energy is comparable to that of an isotropic Gaussian two-step chain, and that a strong data processing inequality bounds the chi-square contraction coefficient by $(1+\\sigma_0^2\\epsilon/c_F)^{-2}$, yielding a spectral gap of order $\\epsilon$.","pith_inferences":["The mirror map $u$ can be read as a user-chosen preconditioner: the theorems suggest a design rule of thumb, pick $u$ so the Brenier push-forward $\\nu=(\\nabla u)_\\#\\mu$ is as log-concave or Poincar\\'e-friendly as possible, since the dual condition is the bottleneck; this is not stated as an algorithm in the paper.","The comparison technique may apply beyond this chain: any Gibbs sampler whose forward conditional is close to an isotropic Gaussian in the duality coordinates, and whose reverse conditional has a uniform Gaussian lower bound, would inherit a $1-c\\epsilon$ chi-square contraction whenever the target side satisfies Poincar\\'e; this is an extension the paper leaves implicit.","The vanishing-third-derivative property of $\\Psi$ suggests the Gaussian approximation to the Schr\\\"odinger bridge is accurate to $O(\\epsilon^2)$, so one could build higher-order unbiased discretizations of the MLD or accelerated Sinkhorn schemes by adding $\\epsilon^2$ corrections; the paper does not pursue this."],"forward_implications":["For any target $\\mu$ for which a mirror map $u$ makes $e^{-F}$ Poincar\\'e with $F$ $L$-smooth, the two-step Gibbs chain is an unbiased sampler: every iterate has a density, the stationary law is exactly $\\mu$, and the $\\chi^2$ error contracts by $1-c_0\\epsilon$ per step.","Mixing time of the chain is $O(\\epsilon^{-1}\\log(1/\\delta))$ steps to come within $\\delta$ in $\\chi^2$, matching the continuous MLD's $O(\\log(1/\\delta))$ time under the $\\epsilon$ time rescaling.","The Euclidean choice $u(x)=\\|x\\|^2/2$ reduces to the proximal sampler with $N(x,\\epsilon I)$, so the paper supplies an unbiased, rate-guaranteed discretization of classical Langevin diffusion.","The Lyapunov conditions of Theorems 1 and 2 are checkable from first-order information on $V$ and the Hessian quadratic form of $u$, so they offer a concrete route to certify exponential mixing for targets that are not strongly log-concave.","Because primal and dual representations share the same Dirichlet energy, any Poincar\\'e or log-Sobolev constant proved for one side transfers to the other with the same constant."],"supporting_citations":[{"why":"Supplies the Lyapunov-to-functional-inequality theorems and the CD-implies-LSI framework used in Theorems 1-2.","marker":"[4]"},{"why":"Provides the defective log-Sobolev / super Poincar\\'e criteria used in the proof of Theorem 2.","marker":"[6]"},{"why":"Supplies the Markov-chain-to-diffusion weak convergence theorem (Theorem 8.7.1) used to prove Theorem 4.","marker":"[17]"},{"why":"Gives the strong data processing inequality bound on maximal correlation for the Gaussian pair used in Theorem 5 Step 4.","marker":"[22]"},{"why":"Links the chi-square contraction coefficient to the spectral gap and defines the discrete Dirichlet energy used in the comparison.","marker":"[34]"},{"why":"Introduces the Sinkhorn Markov chain whose Gaussian approximation motivates the two-step Gibbs sampler, and conjectures its diffusion limit.","marker":"[14]"},{"why":"Gives the earlier CD(0,∞) criterion for Hessian manifolds and the curvature computations against which the new Lyapunov conditions are compared.","marker":"[23]"},{"why":"Defines the proximal sampler, the Euclidean specialization of the chain, and the restricted Gaussian oracle idea for sampling the reverse conditional.","marker":"[24]"},{"why":"Provides the Gaussian approximation to the Schr\\\"odinger bridge used to replace the unknown bridge by $N(x^*, \\epsilon \\nabla^2 u(x))$.","marker":"[32]"}],"fun_headline_variants":["Mirror Langevin duality gives exponential convergence rates","Gibbs chain matches MLD chi-square contraction rate","Transfer principle for Hessian Langevin diffusions and samplers","Lyapunov and dual charts yield functional inequalities for MLD"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rate guarantee depends on the pushed-forward density $\\nu=(\\nabla u)_\\#\\mu$ being well behaved in the sense that it satisfies a Poincar\\'e inequality; the paper does not derive this from its other assumptions, so the user must choose $u$ to make it true.","fun_headline_variants_meta":{"raw":{"variants":["Mirror Langevin duality gives exponential convergence rates","Gibbs chain matches MLD chi-square contraction rate","Transfer principle for Hessian Langevin diffusions and samplers","Lyapunov and dual charts yield functional inequalities for MLD"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000313,"raw_usage":{"total_tokens":1825,"prompt_tokens":1037,"completion_tokens":788,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":653,"completion_tokens_details":{"reasoning_tokens":721}},"tokens_in":653,"tokens_out":788,"duration_ms":7259,"temperature":1.0,"reasoning_tokens":721,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:29:52.397513+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a smooth, strongly convex $u$ with bounded Hessian and a smooth $V$ satisfying the paper's Assumption 1 for which the push-forward $\\nu=(\\nabla u)_\\#\\mu$ fails to satisfy a Poincar\\'e inequality (for example, a density with a sufficiently heavy tail), then compute the spectral gap or chi-square contraction coefficient of the two-step kernel $r_\\epsilon$; if the gap is not $\\Theta(\\epsilon)$ or the bound $(1-c_0\\epsilon)$ fails, the theorem's reachable scope is smaller than stated.","supporting_citations":[{"cited_title":"Bakry, I","cited_arxiv_id":null,"evidence_quote":"Supplies the Lyapunov-to-functional-inequality theorems and the CD-implies-LSI framework used in Theorems 1-2."},{"cited_title":"Cattiaux and A","cited_arxiv_id":null,"evidence_quote":"Provides the defective log-Sobolev / super Poincar\\'e criteria used in the proof of Theorem 2."},{"cited_title":"Durrett.Stochastic Calculus: A Practical Introduction","cited_arxiv_id":null,"evidence_quote":"Supplies the Markov-chain-to-diffusion weak convergence theorem (Theorem 8.7.1) used to prove Theorem 4."},{"cited_title":"Klartag and O","cited_arxiv_id":null,"evidence_quote":"Gives the strong data processing inequality bound on maximal correlation for the Gaussian pair used in Theorem 5 Step 4."},{"cited_title":"Raginsky","cited_arxiv_id":null,"evidence_quote":"Links the chi-square contraction coefficient to the spectral gap and defines the discrete Dirichlet energy used in the comparison."},{"cited_title":"Deb, Y.-H","cited_arxiv_id":null,"evidence_quote":"Introduces the Sinkhorn Markov chain whose Gaussian approximation motivates the two-step Gibbs sampler, and conjectures its diffusion limit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the earlier CD(0,∞) criterion for Hessian manifolds and the curvature computations against which the new Lyapunov conditions are compared."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the proximal sampler, the Euclidean specialization of the chain, and the restricted Gaussian oracle idea for sampling the reverse conditional."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Gaussian approximation to the Schr\\\"odinger bridge used to replace the unknown bridge by $N(x^*, \\epsilon \\nabla^2 u(x))$."}],"review_version":2}