arXiv subjects

Kai-Wen Yang

Publications and source records attributed to Kai-Wen Yang.

4 recordsLinked to original sources

arXiv2026

The Brunn--Minkowski inequality for the Gaussian measure

Let $γ_n$ be the standard Gaussian measure on $\mathbb{R}^n$, $n\ge2$, and let $α_γ(n)$ be the largest number for which \[ γ_n(λK+(1-λ)L)^{α_γ(n)} \ge λγ_n(K)^{α_γ(n)} +(1-λ)γ_n(L)^{α_γ(n)} \] holds for all convex bodies $K,L\subset\mathbb{R}^n$ containing the origin and all $λ\in[0,1]$. In this paper, we prove that \[ α_γ(n) =1-\frac{2}{n-1} \frac{Γ(\frac n2)^2}{Γ(\frac{n-1}{2})^2}. \] The core of the proof is a raywise radial--tangential localization of the Hessian energy of a solution of a Neumann problem, which reduces source selection of the Neumann problem to a one-dimensional optimization. Monotonicity in the segment length and Laguerre spectral analysis determine the sharp one-dimensional value, whereas the planar endpoint is treated separately.

math.MG↗

arXiv2026

A uniform bound in the dimensional Brunn--Minkowski inequality for even log-concave measures

For every $n\ge 2$, we prove that there exists an exponent $p_n$ such that, for every even log-concave probability measure $μ$ on $\mathbb R^n$, all nonempty symmetric convex sets $K,L\subseteq\mathbb R^n$, and all $λ\in[0,1]$, $$ μ(λK+(1-λ)L)^{p_n} \ge λμ(K)^{p_n}+(1-λ)μ(L)^{p_n}, $$ where $$ p_n\ge \frac{c}{n^2\ln n} $$ for some absolute constant $c>0$.

math.MG↗

arXiv2026

The Brunn-Minkowski inequality for the generalized Gaussian distribution

Let $μ_p$ be the generalized Gaussian distribution on $\mathbb{R}^n$ with density $e^{-\frac{|x|^p}{p}}$ multiplied by a constant depending on $p\ge 1$ and $n$, and $α_p(n)$ be the largest number such that the Brunn-Minkowski type inequality $$μ_p(λK+(1-λ) L)^{α_p(n)} \geq λμ_p(K)^{α_p(n)}+(1-λ) μ_p(L)^{α_p(n)}$$ holds for all convex bodies $K,L$ in $\mathbb{R}^n$ containing the origin and $λ\in[0,1]$. In this paper, the new lower and upper bounds for $α_p(n)$ are found, and their asymptotically optimality as $n\to +\infty$ is proved.

math.MG↗

arXiv2020

Clinical prediction system of complications among COVID-19 patients: a development and validation retrospective multicentre study

Existing prognostic tools mainly focus on predicting the risk of mortality among patients with coronavirus disease 2019. However, clinical evidence suggests that COVID-19 can result in non-mortal complications that affect patient prognosis. To support patient risk stratification, we aimed to develop a prognostic system that predicts complications common to COVID-19. In this retrospective study, we used data collected from 3,352 COVID-19 patient encounters admitted to 18 facilities between April 1 and April 30, 2020, in Abu Dhabi (AD), UAE. The hospitals were split based on geographical proximity to assess for our proposed system's learning generalizability, AD Middle region and AD Western & Eastern regions, A and B, respectively. Using data collected during the first 24 hours of admission, the machine learning-based prognostic system predicts the risk of developing any of seven complications during the hospital stay. The complications include secondary bacterial infection, AKI, ARDS, and elevated biomarkers linked to increased patient severity, including d-dimer, interleukin-6, aminotransferases, and troponin. During training, the system applies an exclusion criteria, hyperparameter tuning, and model selection for each complication-specific model. The system achieves good accuracy across all complications and both regions. In test set A (587 patient encounters), the system achieves 0.91 AUROC for AKI and >0.80 AUROC for most of the other complications. In test set B (225 patient encounters), the respective system achieves 0.90 AUROC for AKI, elevated troponin, and elevated interleukin-6, and >0.80 AUROC for most of the other complications. The best performing models, as selected by our system, were mainly gradient boosting models and logistic regression. Our results show that a data-driven approach using machine learning can predict the risk of such complications with high accuracy.

cs.CY↗