arXiv subjects

Dongbin Li

Publications and source records attributed to Dongbin Li.

7 recordsLinked to original sources

arXiv2026

Gaussian Convolution, Internal Energies, and the Kneser--Poulsen Conjecture

We study the persistence under heat flow of internal energy comparisons induced by $1$-Lipschitz maps. We define a new hierarchy of internal energies based on iterative nonnegativity of the associated pressure law, and show that Gaussian marginalisation leads to hierarchical ascent. This mechanism, combined with a stronger dimension-free result for continuously contracting homotopies, allows us to prove that persistence holds for tiers in the hierarchy depending on the ambient dimension, including all convex energy densities in the two-dimensional case. We then connect these results to geometry, specifically to the volume of tubes, showing that several principal known cases of the Kneser--Poulsen conjecture follow from our results. Partly motivated by this connection, we conjecture that persistence holds for all convex energy densities in all dimensions. We also discuss a characterisation of isotropic Gaussians arising naturally in this framework.

math.PR↗

arXiv2026

An upper bound on the smallest singular value of dense random combinatorial matrices

Let $M$ be an $n\times n$ random matrix with entries in $\{0, 1\}$, where each row is independently and uniformly sampled from the set of all vectors in $\{0, 1\}^n$ containing exactly $d$ ones, with $d=pn$ for some fixed constant $p\in (0,1/2]$. A recent result of Tran states that the smallest singular value $s_n(M)$ is bounded below by $c_p n^{-1/2}$ with high probability. In this note, we establish a complementary upper bound for $s_n(M)$, proving that \[ \forall \varepsilon >0 \qquad \mathbb{P}\left(s_n(M)\le \frac{\sqrt{d}}{\varepsilon^2 n}\right)\ge 1-C_p\left(\varepsilon+\frac{1}{\sqrt{d}}\right), \]where $C_p$ is a positive constant depending only on $p$. This result confirms that the least singular value $s_n(M)$ of dense random combinatorial matrices is typically of the order $n^{-1/2}$.

math.PR↗

arXiv2026

Least singular values of shifted sparse random combinatorial matrices

Let $M_n$ be an $n\times n$ random matrix with entries in $\{0, 1\}$, where each row is independently and uniformly sampled from the set of all vectors in $\{0, 1\}^n$ containing exactly $d$ ones. we establish quantitative lower bounds on the smallest singular value of the shifted matrices $M_n-z \mathbf{I}_n$ whenever $|z| \leq \sqrt{d}\, \log\log d$ and $ C \log n \leq d \leq n/2$ for some absolute positive constant $C$. As an application, we show that the empirical spectral distribution of the appropriately rescaled matrix $M_n$ converges in probability to the circular law provided that $\log^{2+\varepsilon} n \leq d=o(n)$ for some fixed $\varepsilon \in (0,1)$.

math.PR↗

arXiv2025

Metric complexity is a Bryant--Tupper diversity

The metric complexity (sometimes called Leinster--Cobbold maximum diversity) of a compact metric space is a recently introduced isometry-invariant of compact metric spaces which generalizes the notion of cardinality, and can be thought of as a metric-sensitive analogue of maximum entropy. On the other hand, the notion of diversity introduced by Bryant and Tupper is an assignment of a real number to every finite subset of a fixed set, which generalizes the notion of a metric. We establish a connection between these concepts by showing that the former quantity naturally produces an example of the latter. Moreover, in contrast to several examples in the literature, the diversity that arises from metric complexity is Minkowski-superadditive for compact subsets of the real line.

math.MG↗

arXiv2025

Entropic and functional forms of the dimensional Brunn--Minkowski inequality in Gauss space

Given even strongly log-concave random vectors $X_{0}$ and $X_{1}$ in $\mathbb{R}^n$, we show that a natural joint distribution $(X_{0},X_{1})$ satisfies, \begin{equation} e^{ - \frac{1}{n}D ((1-t)X_{0} + t X_{1} \Vert Z)} \geq (1-t) e^{ - \frac{1}{n}D (X_{0} \Vert Z)} + t e^{ - \frac{1}{n}D ( X_{1} \Vert Z)}, \end{equation} where $Z$ is distributed according to the standard Gaussian measure $\gamma$ on $\mathbb{R}^n$, $t \in [0,1]$, and $D(\cdot \Vert Z)$ is the Gaussian relative entropy. This extends and provides a different viewpoint on the corresponding geometric inequality proved by Eskenazis and Moschidis, namely that \begin{equation} \gamma \left( (1-t) K_{0} + t K_{1} \right)^{\frac{1}{n}} \geq (1-t) \gamma (K_{0})^{\frac{1}{n}} + t \gamma (K_{1})^{\frac{1}{n}}, \end{equation} when $K_{0}, K_{1} \subseteq \mathbb{R}^n$ are origin-symmetric convex bodies. As an application, using Donsker--Varadhan duality, we obtain Gaussian Borell--Brascamp--Lieb inequalities applicable to even log-concave functions, which serve as functional forms of the Eskenazis--Moschidis inequality.

math.PR↗

arXiv2024

The Kneser--Poulsen phenomena for entropy

The Kneser--Poulsen conjecture asserts that the volume of a union of balls in Euclidean space cannot be increased by bringing their centres pairwise closer. We prove that its natural information-theoretic counterpart is true. This follows from a complete answer to a question asked in arXiv:2210.12842 about Gaussian convolutions, namely that the R\'enyi entropy comparisons between a probability measure and its contractive image are preserved when both undergo simultaneous heat flow. An inequality that unifies Costa's result on the concavity of entropy power with the entropic Kneser--Poulsen theorem is also presented.

math.MG↗

arXiv2022

Entropic exercises around the Kneser-Poulsen conjecture

We develop an information-theoretic approach to study the Kneser--Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether R\'enyi entropies of independent sums decrease when one of the summands is contracted by a $1$-Lipschitz map. We answer this question affirmatively in various cases.

math.MG↗