A Sharp Jacobian-Deficit Inequality for Talagrand's Transportation Inequality under Strong Log-Concavity
Year: 2026
Type: preprint
Abstract: This paper establishes a sharp, dimension-free global inequality that quantifies the Talagrand deficit for centered, strongly log-concave probability measures relative to the standard Gaussian measure. The central object is Talagrand's transportation-entropy inequality, which asserts that for every probability measure μ on ℝⁿ and the standard Gaussian γ = N(0, I_n), one has W_2²(μ, γ) ≤ 2H(μ∥γ), where H(μ∥γ) = ∫ log(dμ/dγ) dμ is the relative entropy of μ with respect to γ and W_2 denotes the quadratic Wasserstein distance. This inequality is a cornerstone of concentration of measure, optimal transport, and functional inequalities; it is closely tied to the Gaussian logarithmic Sobolev inequality of Gross and admits a convexity interpretation through the Bakry–Émery Γ_2 criterion, while the HWI inequality of Otto and Villani and the Riemannian interpolation inequality of Cordero-Erausquin, McCann, and Schmuckenschläger provide the standard convexity routes to it and its refinements. The Talagrand deficit of μ is defined as the nonnegative quantity δ_T(μ) := 2H(μ∥γ) − W_2²(μ, γ) ≥ 0. By Talagrand's inequality, δ_T(μ) = 0 if and only if μ is a translate of γ, that is, μ = N(a, I_n) for some a ∈ ℝⁿ; under the centering assumption used throughout, this reduces to δ_T(μ) = 0 if and only if μ = γ. The stability problem for Talagrand's inequality asks for the rate at which δ_T(μ) → 0 forces μ → γ, typically in W_2 or in relative entropy. A stability exponent α > 0 is said to hold in a given class of measures if there is a constant C > 0 such that W_2(μ, γ) ≤ C δ_T(μ)^α for all measures in the class with δ_T(μ) small. The optimal exponent is the largest α for which such an inequality holds, and equivalently one may write δ_T(μ) ≥ c W_2(μ, γ)^β with β = 1/α, the optimal exponent corresponding to the smallest admissible β. In the classical stability literature for the Gaussian log-Sobolev and Talagrand inequalities several regimes are known. In the isotropic Gaussian regime, where μ = N(0, σ²I_n) and the Brenier map T from γ to μ is a scalar multiple of the identity, the optimal exponent 1/2 is known. In the general regime the best known exponents are far from optimal. The class of centered strongly log-concave measures μ = e^(−V) dx with ∇²V ≥ κ I_n for some κ ≥ 1 is the natural intermediate class in which one expects the optimal exponent 1/2 to hold. The purpose of the paper is to prove a sharp global form of the deficit–Wasserstein inequality in this class. The inequality proved is global: it holds for every centered measure in the class with κ ≥ 1. Its sharpness is understood uniformly over the union of the classes with κ > 1, in the limiting regime κ ↓ 1. This point is important. For each fixed κ > 1, the centered class is bounded away from γ in W_2, so the local statement "δ_T small implies W_2 small" is vacuous inside that fixed class. The content of the paper is the global inequality and its sharp constant in the limiting regime. The main result, Theorem 1.1, is a dimension-free inequality of the form δ_T(μ) ≥ c W_2(μ, γ)^β with exponent β = 2, i.e. with stability exponent α = 1/2, valid in the centered strongly log-concave regime κ ≥ 1. The proof is elementary and rests on three classical ingredients: the Monge–Ampère identity for the Talagrand deficit, the Caffarelli contraction theorem, and the Gaussian Poincaré inequality. No auxiliary machinery such as log-Sobolev stability, Brascamp–Lieb stability, or entropy-power stability is used. The argument is short, direct, and yields the optimal constant in the limiting Gaussian regime. The centering assumption is essential for the uniform control of W_2 by δ_T; a non-centered version with an additional mean term is given in Proposition 2.2. A direct consequence of Theorem 1.1 makes the strengthened Talagrand structure explicit. Under the assumptions of Theorem 1.1, one has 2H(μ∥γ) ≥ W_2²(μ, γ) + ∫ ∥DT(x) − I_n∥_F² dγ(x), where DT denotes the Jacobian matrix of the Brenier map T, (DT)_ij = ∂_j T_i, and ∥·∥_F is the Frobenius norm. This inequality is not a new independent result: it is an equivalent rewriting of the Jacobian-defect bound, since by definition δ_T = 2H − W_2², so 2H = δ_T + W_2², and inserting the Jacobian-defect bound gives the stated form. It is stated explicitly because it makes manifest that the classical Talagrand inequality is strengthened by a positive Jacobian-defect term measuring the distortion of the Brenier map. The setup is as follows. The paper works on ℝⁿ with the standard Euclidean structure, writing γ = N(0, I_n) for the standard Gaussian, H(μ∥γ) = ∫ log(dμ/dγ) dμ for the relative entropy, W_2(μ, γ) for the quadratic Wasserstein distance, ⟨u, v⟩ for the Euclidean inner product, ∥u∥ for the norm, ∥A∥_F for the Frobenius norm of a matrix A, ∥A∥_op for the operator norm, I_n for the identity matrix, ∇²V for the Hessian of V with ∇²V ≥ κ I_n in the positive semidefinite order, and DT for the Jacobian matrix of a map T : ℝⁿ → ℝⁿ. Moreover ∥A∥_{S_3}³ := ∑_i |λ_i(A)|³ denotes the Schatten 3-norm cubed, where λ_i(A) are the eigenvalues of A. Let μ = e^(−V) dx be a probability measure with V ∈ C²(ℝⁿ). Three assumptions are imposed. (A1) Strong convexity: there exists κ ≥ 1 such that ∇²V(x) ≥ κ I_n for all x ∈ ℝⁿ. (A2) Centering at the origin: ∫ y dμ(y) = 0. (A3) Integrability and weak regularity: H(μ∥γ) < ∞, and the Brenier map T = ∇Φ from γ to μ satisfies T_#γ = μ. Under (A1), the Caffarelli contraction theorem gives that T is κ^(−1/2)-Lipschitz, hence T ∈ H¹(γ) and its Jacobian DT = ∇²Φ exists γ-a.e. with 0 ≤ DT(x) ≤ κ^(−1/2) I_n γ-a.e., and det DT(x) > 0 γ-a.e.; the Monge–Ampère equation holds in the weak (area-formula) sense γ(x) = μ(T(x)) det DT(x) γ-a.e. Assumption (A2) is essential for the uniform stability estimate. Without it, δ_T does not control W_2 uniformly. Under (A1)–(A3), the Brenier map T is the gradient of the convex Brenier potential Φ, T = ∇Φ, and DT = ∇²Φ ≥ 0 γ-a.e.; in particular DT is symmetric and positive semidefinite. A remark on the weak formulation explains that the deficit identity is first proved for C² Brenier maps with det DT > 0 everywhere, and then extended to the general case (A3) by a standard argument: since T is Lipschitz it belongs to H¹(γ) and DT exists γ-a.e.; by the area formula for Lipschitz maps the Monge–Ampère equation γ = μ(T) det DT holds γ-a.e.; the set where det DT = 0 has γ-measure zero because its image under T has μ-measure zero by the area formula and μ is absolutely continuous; integration by parts against the Gaussian is valid for Lipschitz T; and the integrability of log det DT follows from log det DT = log γ − log μ(T), where the first term is integrable and the second is integrable by H(μ∥γ) < ∞ and the second-moment bound. Theorem 1.1, the sharp global deficit inequality for centered measures, states the following. Let n ≥ 1, κ ≥ 1, and let μ = e^(−V) dx be a probability measure on ℝⁿ satisfying (A1)–(A3). Let T = ∇Φ be the Brenier map from γ = N(0, I_n) to μ, so T_#γ = μ, and define the Talagrand deficit δ_T(μ) := 2H(μ∥γ) − W_2²(μ, γ) ≥ 0. Then four conclusions hold. (i) Deficit controls the Jacobian defect: δ_T(μ) ≥ ∫ ∥DT(x) − I_n∥_F² dγ(x). (ii) Deficit controls the Wasserstein distance: δ_T(μ) ≥ W_2²(μ, γ). (iii) Deficit controls the covariance defect: letting C = Cov(μ) = ∫ y y^T dμ(y), one has ∥C − I_n∥_F ≤ 2√δ_T(μ) + δ_T(μ); moreover ∥C^(1/2) − I_n∥_F ≤ √δ_T(μ). In particular, if δ_T(μ) ≤ ε with 0 ≤ ε ≤ 1, then ∫ ∥DT(x) − I_n∥_F² dγ(x) ≤ ε, W_2(μ, γ) ≤ √ε, and ∥Cov(μ) − I_n∥_F ≤ 3√ε. (iv) Higher-order refinements: with B(x) := I_n − DT(x) and for every integer N ≥ 2, δ_T(μ) ≥ 2∑_{k=2}^N (1/k) ∫ Tr(B(x)^k) dγ(x); in particular, for N = 3, δ_T(μ) ≥ ∫ ∥DT(x) − I_n∥_F² dγ(x) + (2/3) ∫ ∥DT(x) − I_n∥_{S_3}³ dγ(x). All constants are independent of the dimension n and of κ. The exponent 1/2 and the constant 1 are sharp uniformly over the union of the centered classes with κ > 1, in the limit κ ↓ 1. Proposition 2.2 gives the non-centered version. Let μ = e^(−V) dx satisfy (A1) and (A3) only, and let m := ∫ y dμ(y) ∈ ℝⁿ. Then W_2²(μ, γ) ≤ ∥m∥² + ∫ ∥DT(x) − I_n∥_F² dγ(x) ≤ ∥m∥² + δ_T(μ), and hence W_2(μ, γ) ≤ ∥m∥ + √δ_T(μ). Theorem 1.1 is the case m = 0. Several remarks clarify the meaning of these results. Remark 2.3 concerns the local interpretation and the limit κ ↓ 1. For each fixed κ > 1, the centered strongly log-concave class is bounded away from γ in W_2. Indeed, if ∇²V ≥ κ I_n and μ is centered, then the Brascamp–Lieb covariance inequality gives Cov(μ) ≤ κ^(−1) I_n, and the classical Gelbrich lower bound for the quadratic Wasserstein distance between Gaussians gives W_2²(μ, γ) ≥ ∥Cov(μ)^(1/2) − I_n∥_F² ≥ n(1 − 1/√κ)² > 0. Therefore δ_T(μ) ≥ W_2²(μ, γ) ≥ n(1 − 1/√κ)² > 0. Thus, inside a fixed class with κ > 1, the condition δ_T(μ) → 0 cannot occur except at the degenerate boundary. For this reason the sharpness statement in Theorem 1.1 is global and is understood uniformly over the union of the classes with κ > 1, in the limit κ ↓ 1; the local terminology "stability exponent 1/2" refers to this limiting uniform statement, not to a non-vacuous local implication for a fixed κ > 1. Remark 2.4 addresses the necessity of centering. The centering assumption (A2) cannot be removed. For μ = N(m, σ²I_n) with σ < 1, one has κ = 1/σ² > 1 and δ_T(μ) = 2n(σ − 1 − log σ), which is independent of m, while W_2²(μ, γ) = n(σ − 1)² + ∥m∥². Thus, for σ close to 1 and ∥m∥ large, δ_T is small while W_2 is large. This shows that δ_T alone cannot control W_2 without a mean term, which is precisely the content of Proposition 2.2. Remark 2.5 concerns the sharp constant: the constant 1 in δ_T(μ) ≥ W_2²(μ, γ) is sharp uniformly over the union of the centered classes with κ > 1, in the limit κ ↓ 1; both inequalities in the chain are asymptotically sharp in the Gaussian example, since δ_T / ∫∥DT − I_n∥_F² dγ → 1 and ∫∥DT − I_n∥_F² dγ / W_2² → 1 as b → 0. Remark 2.6 treats th
Author: Mehennaoui, Sami
Institution: University of Skikda
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225571614