From subgaussian marginals to logarithmic Sobolev inequalities

Source and verification

This article explains family 093 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Log-concavity and two kinds of concentration

A probability density $e^{-V}$ on ${\mathbb{R}}^n$ is log-concave when $V$ is convex, allowing $V=+\infty$ outside a convex support. Uniform measures on convex bodies are included.

Let $X$ have this law and mean zero. A uniform subgaussian bound on its linear marginals means that, for some $a>0$,

$$\underset{|\theta |=1}{\sup }\mathbb{E}\exp \left(\frac{⟨X,\theta {⟩}^2}{a^2}\right)\le 2.$$

Markov’s inequality immediately yields $\mathbb{P}(|⟨X,\theta ⟩|\ge t)\le 2e^{-t^2/a^2}$. This only concerns linear functions. A bound for every Lipschitz function is a stronger conclusion.

For a nonnegative integrable function $g$, define entropy by

$$\begin{array}{rl}{\mathrm{Ent}}_{\mu }(g)& ={\mathbb{E}}_{\mu }(g\log g)\\ & -{\mathbb{E}}_{\mu }g\log ({\mathbb{E}}_{\mu }g).\end{array}$$

A logarithmic Sobolev inequality controls this quantity by a gradient energy:

$${\mathrm{Ent}}_{\mu }(f^2)\le A\int |\nabla f{|}^{2}d\mu .$$

The coefficient convention matters: some sources write $2A$ on the right. Here $A$ is the entire coefficient.

🏷️ The claimed dimension-free implication

Statement in the manuscript

For a centered full-dimensional log-concave probability density satisfying the linear subgaussian bound above, the logarithmic Sobolev inequality holds with $A=C{a}^2$, where $C$ is absolute and independent of dimension. (OpenAI, 2026)

No isotropic normalization, smoothness, or uniformly positive Hessian is required in the final assertion. In particular, the theorem includes uniform measures on convex bodies with the stated marginal bound. The manuscript proves existence of a universal constant; it does not identify its optimal value.

The scale is forced by dilation. Replacing $X$ by $rX$ changes the subgaussian parameter to $ra$. Gradients of rescaled functions contribute $r^{-2}$, so the logarithmic Sobolev coefficient must grow like $r^2$.

🏷️ Two elementary consequences

Take $f=1+\epsilon g$ for bounded smooth $g$. Expanding the entropy at $\epsilon =0$ gives

$${\mathrm{Ent}}_{\mu }(f^2)=2{\epsilon }^2{\mathrm{Var}}_{\mu }(g)+o({\epsilon }^2).$$

The gradient term is ${\epsilon }^2\mathbb{E}|\nabla g{|}^2$. Dividing and passing to the limit yields the Poincaré inequality

$${\mathrm{Var}}_{\mu }(g)\le \frac{A}{2}{\mathbb{E}}_{\mu }|\nabla g{|}^2.$$

For a one-Lipschitz $F$, use $f=e^{\lambda F/2}$ and write $\psi (\lambda )=\log \mathbb{E}{e}^{\lambda F}$. After dividing the logarithmic Sobolev inequality by $\mathbb{E}{e}^{\lambda F}$,

$$\lambda {\psi }^{\prime }(\lambda )-\psi (\lambda )\le \frac{A{\lambda }^2}{4}.$$

Integrating the inequality for $(\psi (\lambda )/\lambda {)}^{\prime }$ gives

$$\log \mathbb{E}{e}^{\lambda (F-\mathbb{E}F)}\le \frac{A{\lambda }^2}{4}.$$

Chernoff optimization at $\lambda =2t/A$ then yields

$$\mathbb{P}(F-\mathbb{E}F\ge t)\le e^{-t^2/A}.$$

Truncation and approximation justify the exponential test when needed. The proposed theorem thus promotes one-dimensional Gaussian tails to concentration for nonlinear Lipschitz observables.

🏷️ Near-extremizers and Gaussian observations

The manuscript argues by contradiction. It normalizes the logarithmic Sobolev scale while forcing the marginal parameter to tend to zero, and uses regularization, truncation, and weak tilts to access smooth near-extremizers. One branch is governed by entropy extremizers; another is close to a Poincaré extremizer. (OpenAI, 2026)

A central diagnostic is prediction after adding Gaussian noise. If $Y$ has an auxiliary law and $w$ is its score field, the quantity

$$\mathbb{E}{\left|\mathbb{E}[w(Y)\mid Y+\sqrt{r}G]\right|}^2$$

measures how much of that field remains visible at noise level $r$. Here $G$ is an independent standard Gaussian. Conditional expectation is an orthogonal projection in $L^2$, so prediction energy cannot increase when further information is removed.

For the normalized near-extremizers, the source derives a precise limiting decay law. Repeated observations and independent copies convert it into constraints on tensors describing higher-order fluctuations. Discarding a mean direction is controlled explicitly rather than silently assuming the score has every desired orthogonality.

🏷️ Matrix rigidity and the information contradiction

The tensor data produce positive matrices with a trace normalization. Near equality in the preceding inequalities forces an approximate compatibility between matrices that need not initially commute. The argument retains these matrix errors instead of diagonalizing unrelated operators simultaneously.

A resulting limiting factor has lognormal form

$$\exp (\sqrt{8t}Z-4t),Z\sim N(0,1),$$

whose expectation equals one by the Gaussian moment-generating formula. This factor enters a comparison between information retained across many noise scales and the uniformly bounded information available from the marginal assumption.

The contradiction is integrated over scales with a weighted measure. Under a change of variables such as $x=T{\beta }^2$, the corresponding ${\beta }^2$ weight must remain; losing it would incorrectly discard diffuse contributions. The source obtains an upper bound for the integrated information and a divergent lower bound from the near-extremal structure.

These prediction, compatibility, and scale-integral estimates constitute the research proof. The Poincaré and Herbst calculations above explain the consequences once the claimed logarithmic Sobolev inequality is available; they do not independently establish that inequality.

🏷️ Key steps and proof outlines

A normalized failure produces a smooth critical law

Assume the ratio between the logarithmic Sobolev coefficient and the squared marginal parameter is unbounded. The source chooses compact smooth tests with positive gradient energy, truncates the law, and adds Gaussian regularization. Dilation normalizes the analytic scale while the marginal parameter tends to zero. This makes the contradiction quantitative: the limiting object would retain a fixed entropy or variance defect although every linear direction becomes increasingly concentrated.

There are two critical alternatives. Above the Gaussian and Poincaré thresholds, an entropy extremizer can be obtained and differentiated. Near the Poincaré threshold, an almost first eigenfunction supplies the relevant structure. The proof keeps both alternatives because a maximizing sequence need not converge to an entropy extremizer in every regime. Regularization estimates must be uniform enough to pass back to the original nonsmooth law. (OpenAI, 2026)

Prediction energy links noisy observations to diffusion

For an auxiliary variable $Y$ and score field $w(Y)$, let $U_r=Y+\sqrt{r}G$ and $h_r(U_r)=\mathbb{E}[w(Y)\mid U_r]$. Conditional expectation gives the exact identity

$$\mathbb{E}⟨w(Y),h_r(U_r)⟩=\mathbb{E}|h_r(U_r){|}^2.$$

Thus the correlation with the original field equals the predictor’s own energy. Gaussian integration by parts expresses the Hermite coefficients of the observation in terms of derivatives of its smoothed conditional mean. Orthogonality of Gaussian chaoses then gives a sum of nonnegative contributions, each associated with one derivative order.

The critical law also carries a stationary diffusion with semigroup $e^{-uA}$. Its coordinate generator identity relates coordinates, the score field, and a controlled mean correction. Expanding along this diffusion connects the prediction curve to spectral information about $A$. Near equality forces saturation of several Bessel and energy inequalities, producing a hierarchy of tensors that behave like nearly first eigenfunctions. This is stronger than a monotonicity statement for prediction energy alone.

Matrix square roots preserve the quantitative error budget

The tensor hierarchy is packaged into rectangular matrices. Passing from a matrix $X$ to its reduced positive square root $(X{X}^{\ast }{)}^{1/2}$ is useful because it removes an irrelevant choice of coordinates on the column space. The source uses the Hilbert–Schmidt estimate

$$\Vert (X{X}^{\ast }{)}^{1/2}-(Y{Y}^{\ast }{)}^{1/2}{\Vert }_{\mathrm{HS}}^2\le 2\Vert X-Y{\Vert }_{\mathrm{HS}}^2.$$

A self-adjoint dilation reduces this to an absolute-value inequality for symmetric matrices. The estimate is dimension independent and remains valid when ranks change. It therefore transports small tensor errors to positive-matrix errors without dividing by a small eigenvalue.

Spectral blocks separate bounded columns from negligible tails. The proof then freezes a compatible matrix algebra on the retained blocks. Compatibility is obtained from near equality and controlled commutators; it is not assumed by diagonalizing unrelated positive matrices in one basis. This distinction is necessary for the later scalar observation model to represent the original matrix data.

Integrated information has incompatible upper and lower bounds

Covariance observations at many noise scales give an upper bound for the total information available under the marginal assumption. One scalar kernel in this calculation is bounded by an integral of the form

$${\int }_0^{\infty }\frac{dq}{(1+aq)(1+bq)}=\frac{\log (b/a)}{b-a},0<a<b.$$

Partial fractions prove the identity: integrate to a finite endpoint, subtract the two logarithms, and then take the limit. The source chooses parameters with $b-a=1$, giving logarithmic control uniform over the relevant spectral values.

The near-extremal matrix structure supplies a conflicting lower bound. Polynomial tests are first transferred to scalar observations on retained blocks. Their limiting diffusion law contains the factor $\exp (\sqrt{8t}Z-4t)$, whose mean is one but whose fluctuations increase with $t$. Those fluctuations force information to persist across too many scales.

The final contradiction integrates over an entire scale interval. Stability estimates are therefore uniform on that interval; convergence at each fixed scale would not suffice. When substituting $q=T{\beta }^2$, the spectral weight attached to $\beta$ is retained in the measure, including diffuse small-eigenvalue contributions. The lower scale integral diverges in the critical limit while the upper information budget remains bounded. Undoing normalization and regularization gives the proposed dimension-free logarithmic Sobolev estimate.

See Also

Logarithmic Brunn–Minkowski also derives a sharp variance estimate from log-concavity. Entropy and isotropic constants compares entropy directly with covariance through a different transport argument.

🏷️ Manuscripts and verification scope

• A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures — pinned manuscript and companion source.

No family scope page was present at lean/docs/093.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures.