arXiv subjects

Pierre Bizeul

Publications and source records attributed to Pierre Bizeul.

10 recordsLinked to original sources

arXiv2026

Optimal $MM^*$ bounds for convex bodies

Let $K\subset\R^n$ be a convex body in isotropic position. We prove the optimal mean-width estimate \[ M^*(K)\leq C\sqrt{n\log n}. \] The main new ingredient is a geometric inequality relating the Gaussian mean of the support function to its mean under the uniform measure on $K$, obtained through a heat-flow argument. Combined with the Gaussian-log-concave comparison of Eldan and Lehec and the newly available dimension-free bound on the third-moment parameter $\kappa_n$, this yields the result. The boundedness of $\kappa_n$ also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields \[ M(K)M^*(K)\leq C\log n, \] extending Pisier's $MM^*$ estimate to non-symmetric convex bodies and to the isotropic position.

math.MG↗

arXiv2026

Minimum Norm Interpolation via the Local Theory of Banach Spaces: The Role of $2$-Uniform Convexity

The minimum-norm interpolator (MNI) framework has recently attracted considerable attention as a tool for understanding generalization in overparameterized models, such as neural networks. In this work, we study the MNI under a $2$-uniform convexity assumption, which is weaker than requiring the norm to be induced by an inner product; in this setting, the MNI typically does not admit a closed-form solution. At a high level, we show that this condition yields an upper bound on the MNI bias in both linear and nonlinear models. We further show that this bound is sharp for overparameterized linear regression when the norm's unit ball is in isotropic or John's position and the covariates are i.i.d.\ sub-Gaussian, for example, when each covariate vector has i.i.d. Rademacher entries. Finally, under the same assumption on the covariates, we prove sharp generalization bounds for the $\ell_p$-MNI when $p \in \bigl(1 + C/\log d, 2\bigr]$. To the best of our knowledge, this is the \emph{first} work to establish sharp bounds for non-Gaussian covariates in linear models when the norm is not induced by an inner product. This work is deeply inspired by classical work on $K$-convexity and recent work on the geometry of $2$-uniformly convex and isotropic convex bodies.

math.FA↗

arXiv2025

Distances between non-symmetric convex bodies: optimal bounds up to polylog

In this paper we determine, up to polylogarithmic factors, the diameter of the Banach--Mazur compactum of $n$-dimensional convex bodies without symmetry assumptions. We prove that for any convex bodies $K_1,K_2\subset \mathbb{R}^n$, \begin{equation} d_{BM}(K_1,K_2)\le Cn\log^\alpha(n+1), \label{eq_1624} \end{equation} for universal constants $C,\alpha>0$, improving an earlier bound of Rudelson. We also study the partial-containment distance $d_{PC}$, in which the Banach-Mazur requirement to contain the other body in its entirety is relaxed to $99\%$-containment. We prove that this relaxation leads to a very different behavior: \begin{equation} d_{PC}(K_1,K_2) \leq C \log^{\alpha} (n+1) \label{eq_1625} \end{equation} for all convex bodies $K_1,K_2 \subseteq \mathbb{R}^n$. This demonstrates that in high dimensions, any convex body is not too far from an affine image of any other convex body, when we look at the bulk of their mass. \medskip In the centrally-symmetric case, the optimal upper bound for the Banach-Mazur distance is obtained in the John position. In contrast, our proofs rely on the isotropic position. The analytic core of our argument is a two-sided comparison, in the gauge order, between isotropic log-concave measures and Gaussian measures. This yields a new isotropic $M$-bound that complements E. Milman's $M^*$-bound. We also provide applications to linear symplectic geometry and to the first Dirichlet eigenvalue of the Laplacian.

math.MG↗

arXiv2025

Entropy and Learning of Lipschitz Functions under Log-Concave Measures

We study regression of $1$-Lipschitz functions under a log-concave measure $\mu$ on $\mathbb{R}^d$. We focus on the high-dimensional regime where the sample size $n$ is subexponential in $d$, in which distribution-free estimators are ineffective. We analyze two polynomial-based procedures: the projection estimator, which relies on knowledge of an orthogonal polynomial basis of $\mu$, and the least-squares estimator over low-degree polynomials, which requires no knowledge of $\mu$ whatsoever. Their risk is governed by the rate of polynomial approximation of Lipschitz functions in $L^2(\mu)$. When this rate matches the Gaussian one, we show that both estimators achieve minimax bounds over a wide range of parameters. A key ingredient is sharp entropy estimates for the class of $1$-Lipschitz functions in $L^2(\mu)$, which are new even in the Gaussian setting.

math.PR↗

arXiv2025

Polynomial Approximation in $ L^2 $ of the Double Exponential via Complex Analysis

We study the polynomial approximation problem in $L^2(\mu_1)$ where $\mu_1(dx) = e^{-|x|}/2 dx$. We show that for any absolutely continuous function $f$, $$ \sum_{k=1}^{\infty} \log^2(e+k) \langle f, P_k \rangle^2 \ \leq C \left( \int_{\mathbb{R}} \log^2(e+\lvert x \rvert) f^2 \, d\mu_1 \ + \ \int_{\mathbb{R}} (f')^2 \, d\mu_1 \right) $$ for some universal constant $C>0$, where $(P_k)_{k \in N}$ are the orthonormal polynomials associated with $\mu_1$. This inequality is tight in the sense that $\log^2(e +k)$ on the left hand-side cannot be replaced by $a_k \log^2(e +k)$ with a sequence $a_k \longrightarrow \infty$. When the right hand-side is bounded this inequality implies a logarithmic rate of approximation for $f$, which was previously obtained by Lubinsky. We also obtain some rates of approximation for the product measure $\mu_1^{\otimes d}$ in $\mathbb{R}^d$ via a tensorization argument. Our proof relies on an explicit formula for the generating function of orthonormal polynomials associated with the weight $\frac{1}{2\cosh(\pi x/2)}$ and some complex analysis.

math.CA↗

arXiv2025

The slicing conjecture via small ball estimates

Bourgain's slicing conjecture was recently resolved by Joseph Lehec and Bo'az Klartag. We present an alternative proof by establishing small ball probability estimates for isotropic log-concave measures. Our approach relies on the stochastic localization process and Guan's bound, techniques also used by Klartag and Lehec. The link between small ball probabilities and the slicing conjecture was first observed by Dafnis and Paouris and is established through Milman's theory of M-ellipsoids.

math.FA↗

arXiv2023

On the Log-Sobolev Constant of Log-Concave Vectors

It is well known that if a random vector satisfies a log-Sobolev inequality, all of its marginals have subgaussian tails. In the spirit of the KLS conjecture, we investigate whether this implication can be reversed under a log-concavity assumption. In the general setting, we improve on a result of Bobkov, establishing the best dimension dependent bound on the log-Sobolev constant of subgaussian log-concave measures, and we investigate some special cases.

math.FA↗

arXiv2022

On measures strongly log-concave on a subspace

In this work we study the concentration properties of log-concave measures that are curved only on a subspace of directions. Proofs uses an adapted version of the stochastic localization process.

math.FA↗

arXiv2021

Entropy and Information jump for log-concave vectors

We extend the result of Ball and Nguyen on the jump of entropy under convolution for logconcave random vectors. We show that the result holds for any pair of vectors (not necessarily identically distributed) and that a similar inequality holds for the Fisher information, thus providing a quantitative Blachmann-Stam inequality

math.FA↗

arXiv2019

Positive solutions for large random linear systems

Consider a large linear system where $A_n$ is a $n\times n$ matrix with independent real standard Gaussian entries, $\boldsymbol{1}_n$ is a $n\times 1$ vector of ones and with unknown the $n\times 1$ vector $\boldsymbol{x}_n$ satisfying$$\boldsymbol{x}_n = \boldsymbol{1}_n +\frac 1{\alpha_n\sqrt{n}} A_n \boldsymbol{x}_n\, .$$We investigate the (componentwise) positivity of the solution $\boldsymbol{x}_n$ depending on the scaling factor $\alpha_n$ as the dimension $n$ goes to $\infty$. We prove that there is a sharp phase transition at the threshold $\alpha^*_n =\sqrt{2\log n}$: below the threshold ($\alpha_n\ll \sqrt{2\log n}$), $\boldsymbol{x}_n$ has negative components with probability tending to 1 while above ($\alpha_n\gg \sqrt{2\log n}$), all the vector's components are eventually positive with probability tending to 1. At the critical scaling $\alpha^*_n$, we provide a heuristics to evaluate the probability that $\boldsymbol{x}_n$ is positive.Such linear systems arise as solutions at equilibrium of large Lotka-Volterra systems of differential equations, widely used to describe large biological communities with interactions such as foodwebs for instance. In the domaine of positivity of the solution $\boldsymbol{x}_n$, that is when $\alpha_n\gg \sqrt{2\log n}$, we establish that the Lotka-Volterra system of differential equations whose solution at equilibrium is precisely $\boldsymbol{x}_n$ is stable in the sense that its jacobian $${\mathcal J}(\boldsymbol{x}_n) = \mathrm{diag}(\boldsymbol{x}_n)\left(-I_n + \frac {A_n}{\alpha_n\sqrt{n}}\right)$$ has all its eigenvalues with negative real part with probability tending to one. Our results shed a new light and complement the understanding of feasibility and stability issues for large biological communities with interaction.

math.PR↗