arXiv subjects

Jiaqian Liu

Publications and source records attributed to Jiaqian Liu.

11 recordsLinked to original sources

arXiv2026

Centro-sectional measures for log-concave functions

We introduce centro-sectional measures with parameters q,m for log-concave functions on Rn, defined in terms of the q-th moments of their Radon transforms with respect to the Haar measure on m-dimensional subspaces, where m=1,...,n-1, and establish the corresponding variational formulas. Our measures generalize the notion of dual curvature measure if q=1, and are related to the Sine transform if q=2. In line with the coarea formula for log-concave functions as BV functions, the variational formulas give rise to the Euclidean centro-sectional measures and the spherical centro-sectional measures. In the symmetric setting, we solve the associated even functional centro-sectional Minkowski problem, which asks which pairs of measures can arise as the centro-sectional measures of an even log-concave function.

math.AP↗

arXiv2026

The Lp centro-sectional Minkowski problem

As part of Lutwak's broadening of the Brunn-Minkowski theory, and extending the notion of affine quermassintegrals and dual curvature measure discussed by Milman, Yehudayoff and Huang, Lutwak, Yang and Zhang, centro-sectional measures with real parameter q have been recently introduced by Cai, Leng, Wu, Xi. In this paper, we introduce the Lp cross sectional Minkowski problem analogously to the Lp dual Minkowski problem formulated by Lutwak, Yang and Zhang. We solve the Lp centro-sectional Minkowski problem for p>1 and q>0, discuss the regularity and uniqueness of the solution, and prove Lp Brunn-Minkowski-type inequalities when $p$ is relatively large.

math.AP↗

arXiv2026

The Weighted $L^p$ Minkowski Problem

The Minkowski problem in convex geometry concerns showing that a given Borel measure on the unit sphere is, up to perhaps a constant, some type of surface area measure of a convex body. Two types of Minkowski problems in particular are an active area of research: $L^p$ Minkowski problems, introduced by Lutwak and (Lutwak, Yang, and Zhang), and weighted Minkowski problems, introduced by Livshyts. For the latter, the Gaussian Minkowski problem, whose primary investigators were (Huang, Xi and Zhao), is the most prevalent. In this work, we consider weighted surface area in the $L^p$ setting. We propose a framework going beyond the Gaussian setting by focusing on rotationally invariant measures, mirroring the recent development of the Gardner-Zvavitch inequality for rotationally invariant, log-concave measures. Our results include existence for all $p \in \mathbb R$ (with symmetry assumptions in certain instances). We also have uniqueness for $p \geq 1$ under a concavity assumption. Finally, we obtain results in the so-called "small mass regime" using degree theory, as instigated in the Gaussian case by (Huang, Xi and Zhao).

math.AP↗

arXiv2025

Grünbaum's inequality for Gaussian and convex probability measures

A celebrated result in convex geometry is Grünbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Grünbaum-type inequalities - with equality characterizations - for probability measures under certain concavity assumptions. As an application, we apply the renowned Ehrhard inequality and deduce an ``Ehrhard-Grünbaum'' inequality for the Gaussian measure on $\mathbb{R}^n$, which improves upon the bound derived from its log-concavity. For $s$-concave Radon measures, our framework provides a simpler proof of known results and, more importantly, yields the previously missing equality characterization. This is achieved by gaining new insight into the equality case of their Brunn-Minkowski-type inequality. Moreover, we show that these ``$s$-Grünbaum'' inequalities can hold only when $s > -1$. However, for convex measures on the real line, we prove Grünbaum-type inequalities involving their cumulative distribution function.

math.FA↗

arXiv2025

Com$^2$: A Causal-Guided Benchmark for Exploring Complex Commonsense Reasoning in Large Language Models

Large language models (LLMs) have mastered abundant simple and explicit commonsense knowledge through pre-training, enabling them to achieve human-like performance in simple commonsense reasoning. Nevertheless, LLMs struggle to reason with complex and implicit commonsense knowledge that is derived from simple ones (such as understanding the long-term effects of certain events), an aspect humans tend to focus on more. Existing works focus on complex tasks like math and code, while complex commonsense reasoning remains underexplored due to its uncertainty and lack of structure. To fill this gap and align with real-world concerns, we propose a benchmark Com$^2$ focusing on complex commonsense reasoning. We first incorporate causal event graphs to serve as structured complex commonsense. Then we adopt causal theory~(e.g., intervention) to modify the causal event graphs and obtain different scenarios that meet human concerns. Finally, an LLM is employed to synthesize examples with slow thinking, which is guided by the logical relationships in the modified causal graphs. Furthermore, we use detective stories to construct a more challenging subset. Experiments show that LLMs struggle in reasoning depth and breadth, while post-training and slow thinking can alleviate this. The code and data are available at https://github.com/Waste-Wood/Com2.

cs.CL↗

arXiv2024

The Generalized Gaussian Minkowski Problem

This article delves into the $L_p$ Minkowski problem within the framework of generalized Gaussian probability space. This type of probability space was initially introduced in information theory through the seminal works of Lutwak, Yang, and Zhang [49,50], as well as by Lutwak, Lv, Yang, and Zhang [45]. The primary focus of this article lies in examining the existence of this problem. While the variational method is employed to explore the necessary and sufficient conditions for the existence of the normalized Minkowski problem when $p \in \mathbb{R} \setminus \{0\}$, our main emphasis is on the existence of the generalized Gaussian Minkowski problem without the normalization requirement, particularly in the smooth category for $p \geq 1$.

math.AP↗

arXiv2023

Dual curvature measures for log-concave functions

We introduce dual curvature measures for log-concave functions, which in the case of characteristic functions recover the dual curvature measures for convex bodies introduced by Huang-Lutwak-Yang-Zhang in 2016. Variational formulas are shown. The associated Minkowski problem for these dual curvature measures is considered and sufficient conditions in the symmetric setting are demonstrated.

math.MG↗

arXiv2023

A class of generalized fully nonlinear curvature flows and its applications

In this paper, we concern a generalized fully nonlinear curvature flow involving $k$-th elementary symmetric function for principal curvature radii in Eulidean space $\rnnn$, $k$ is an integer and $1\leq k\leq n-1$. For $1\leq k< n-1$, based on some initial data and constrains on smooth positive function defined on the unit sphere $\sn$, we obtain the long time existence and convergence of the flow. Especially, the same result shall be derived for $k=n-1$ without any constraint on the smooth positive function.

math.AP↗

arXiv2023

A flow method to the Orlicz-Aleksandrov problem

In this paper, we obtain the existence result of smooth solutions to the Orlicz-Aleksandrov problem from the perspective of geometric flow. Furthermore, a special uniqueness result of solutions to this problem shall be discussed.

math.AP↗

arXiv2022

Decision-Making Under Uncertainty for Multi-stage Pipelines: Simulation Studies to Benchmark Screening Strategies

Multi-stage screening pipelines are ubiquitous throughout experimental and computational science. Much of the effort in developing screening pipelines focuses on improving generative methods or surrogate models in an attempt to make each screening step effective for a specific application. Little focus has been placed on characterizing generic screening pipeline performance with respect to the problem or problem parameters. Here, we develop models and algorithms to codify and simulate features and properties of the screening procedure in general. We outline and model common problem settings and potential opportunities to perform decision-making under uncertainty to optimize the execution of screening pipelines. We then illustrate the models and algorithms through several simulation studies. We finally show how such studies can provide a quantification of the screening pipeline performance with respect to problem parameters, specifically identifying the significance of stage-wise covariance structure. We show how such structure can lead to qualitatively different screening behaviors, and how screening can even perform worse than random in some cases.

math.OC↗

arXiv2021

On the Documentation of Refactoring Types

Commit messages are the atomic level of software documentation. They provide a natural language description of the code change and its purpose. Messages are critical for software maintenance and program comprehension. Unlike documenting feature updates and bug fixes, little is known about how developers document their refactoring activities. Developers can perform multiple refactoring operations, including moving methods, extracting classes, for various reasons. Yet, there is no systematic study that analyzes the extent to which the documentation of refactoring accurately describes the refactoring operations performed at the source code level. Therefore, this paper challenges the ability of refactoring documentation to adequately predict the refactoring types, performed at the commit level. Our analysis relies on the text mining of commit messages to extract the corresponding features that better represent each class. The extraction of text patterns, specific to each refactoring allows the design of a model that verifies the consistency of these patterns with their corresponding refactoring. Such verification process can be achieved via automatically predicting the method-level type of refactoring being applied, namely Extract Method, Inline Method, Move Method, Pull-up Method, Push-down Method, and Rename Method. We compared various classifiers, and a baseline keyword-based approach, in terms of their prediction performance, using a dataset of 5,004 commits. Our main findings show that the complexity of refactoring type prediction varies from one type to another. Rename method and Extract method were found to be the best documented refactoring activities, while Pull-up Method and Push-down Method were the hardest to be identified via textual descriptions. Such findings bring the attention of developers to the necessity of paying more attention to the documentation of these types.

cs.SE↗