arXiv subjects

Xinyuan Xie

Publications and source records attributed to Xinyuan Xie.

10 recordsLinked to original sources

arXiv2026

Sharp log-Sobolev inequalities and quartic stability on finite cyclic groups

Let $\mathbb Z_n$ be the cyclic group equipped with the uniform probability measure $π$, and let $A_{ψ_n}$ be the Laplacian with word length $$ ψ_n(k) = \min(k,n-k). $$ For every $n\ge4$, we prove the sharp log-Sobolev inequality $$ \text{Ent}_π(|f|^2) \le 2π(\bar{f}A_{ψ_n} f), \qquad f:\mathbb Z_n \to \mathbb{C}, $$ where $\text{Ent}_π$ is the relative entropy with respect to $π$. Equivalently, the Poisson semigroup $P_t=e^{-tA_{ψ_n}}$ satisfies the optimal hypercontractivity. The proof is inspired by the recent work of Frank and Ivanisvili [FI26] on a sharp log-Sobolev inequality for the nearest-neighbor simple random walk. Similar arguments yield a simple proof of Weissler's sharp log-Sobolev inequality for the Poisson semigroup on the circle \cite{Weissler1980}. The same inequalities were independently obtained by Yao~\cite{Yao2026} using a different method. We also prove quantitative stability estimates. For $n\ge 4$ and $f:\mathbb Z_n\to[0,\infty)$ with $π(f^2)=1$, $$ 2π(fA_{ψ_n}f)-\operatorname{Ent}_π(f^2) \ge \frac1{12}\|f-1\|_{L^2(π)}^4, $$ with coefficient $1/(12d)$ on products $(\mathbb Z_n)^d$. The quartic order and the $d^{-1}$ dependence are optimal.

math.CA↗

arXiv2026

The König constant is one

For each $N\geq1$, consider the normalized König bilinear form $B_{\mathrm K}:L_\infty(\mathbb R^N)\times L_\infty(\mathbb R^N)\to\mathbb R$ given by \[ B_{\mathrm K}(f,g):=\frac{1}{(\sqrt{2}π)^N} \iint_{\mathbb R^N\times\mathbb R^N} f(x)g(y)e^{-(\lVert x\rVert^2+\lVert y\rVert^2)/2} \sin\langle x,y\rangle\,\mathrm d x\,\mathrm d y, \] We define the König constant by \[ \mathfrak K_{\mathrm K}:=\sup_{N\geq1}\sup_{\substack{f,g:\mathbb R^N\to\{\pm1\}\\ f,g\ \mathrm{measurable}}}B_{\mathrm K}(f,g). \] The study of this bilinear form arose from efforts to determine the exact value of the Grothendieck constant. König~\cite{KONIG} conjectured that the sharp value should instead be given by the one-dimensional half-spaces $B_{\mathrm K}(\operatorname{sgn}(x_1),\operatorname{sgn}(x_1))=\frac{2}π\log(1+\sqrt{2})$. A positive answer to this conjecture, together with a classical upper bound of Krivine \cite{KRIVINE}, would determine the exact value of the Grothendieck constant. In a breakthrough~\cite{BMMN}, Braverman, Makarychev, Makarychev, and Naor disproved König's conjecture already in dimension two and used their counterexamples to obtain the first strict improvement over Krivine's bound. One question in \cite{BMMN} attempts to determine the Grothendieck constant through alternating Krivine rounding schemes arising from König's bilinear form in high dimension. More recently, Li et al.~\cite{LISK} constructed high-dimensional examples showing that $\mathfrak K_{\mathrm K}\ge 0.59357$. An elementary Fourier argument gives $\mathfrak K_{\mathrm K}\le 1$ and excludes equality for every finite-dimension. In this paper, we prove that $\mathfrak K_{\mathrm K}=1$ by constructing a family of Boolean pairs in high dimensions. In particular, this gives a negative answer to the high-dimensional aspect of the question in \cite{BMMN}.

math.FA↗

arXiv2026

Sharp hypercontractivity for free group von Neumann algebras

In this paper, we settle the problem of optimal hypercontractivity for free group von Neumann algebras. Namely, for $n\ge 2$ and the free group $\mathbb{F}_n$ on $n$ generators, we prove that for any $1<p\le q<\infty$, the Poisson semigroup $P_t$ associated with the word-length function satisfies $$ \|P_t:L_p(\widehat{\mathbb{F}_n})\to L_q(\widehat{\mathbb{F}_n})\|\le 1 \qquad\text{ if and only if }\qquad t\ge \frac{1}{2}\log \frac{q-1}{p-1}. $$ The main idea is to apply a refined cubic majorant estimate from a recent work of Frank and Ivanisvili \cite{FrankIvanisvili2026} to the equivalent logarithmic Sobolev inequality, and use the Haagerup-type cancellation estimate \cite{Haagerup1979}. Similar ideas and techniques extend to free products \[ G=\left(*_{α\in A}\mathbb Z\right)*\left(*_{β\in B}\mathbb Z_2\right) \] and the free Gaussian von Neumann algebras. In the former setting, partial sharp estimates were previously obtained by Junge--Palazuelos--Parcet--Perrin--Ricard \cite{JungePalazuelosParcetPerrinRicard2015}; in the latter, our approach recovers Biane's free hypercontractivity theorem \cite{Biane1997}.

math.OA↗

arXiv2026

A Beckmann boundary form of Talagrand's conjecture on the discrete cube

We introduce the Beckmann boundary of a Boolean function \[ \mathsf{B}(f)=\inf_{\operatorname{div} V=Lf}\mathbb E\|V(x)\|_2. \] Here \[ L=\sum_iD_i,\qquad D_i f(x)=\frac{f(x)-f(x^{\oplus i})}{2}, \] and $\operatorname{div} V(x)=\sum_i (V_{i}(x)-V_{i}(x^{\oplus i}))$. This nonlocal quantity is no larger than the usual two-sided, one-sided, colored, optimized colored, or optimized fractional colored boundaries. Nevertheless, every nonconstant Boolean $f$ satisfies \[ \mathsf{B}(f)\gtrsim \operatorname{Var}(f) \sqrt{\log\!\left(1+\frac{1}{\sum_i\operatorname{Inf}_i(f)^2}\right)}. \] We also prove strong one-sided fractional spectral estimates. If $A\subset\{-1,1\}^n$ and \[ h_{A}(x)=\#\{i:x\in A,\ x^{\oplus i}\notin A\}, \] then, for $0<α<1$, \[ \sum_{S\ne\varnothing}|S|^α\widehat{\mathbf 1_{A}}(S)^2 \lesssim_α\mathbb Eω_α(h_{A}), \] where $ω_α(m)=\sqrt m$ for $α<1/2$, $ω_{1/2}(m)=\sqrt m\log(e+m)$, and $ω_α(m)=m^α$ for $α>1/2$. These profiles are sharp, up to $α$-dependent constants, for majority. We also show that the comparison is genuinely nonreversible: an explicit quotient-cube family makes the optimized fractional, and hence optimized colored, boundary exceed $\mathsf{B}$ by a factor $\gtrsim\sqrt{\log n}$. We further obtain a driftless Bernstein-multiplier inequality.

math.CA↗

arXiv2026

EvA: An Evidence-First Audio Understanding Paradigm for LALMs

Large Audio Language Models (LALMs) still struggle in complex acoustic scenes because they often fail to preserve task-relevant acoustic evidence before reasoning begins. We identify this error pattern as the evidence bottleneck: state-of-the-art systems show larger deficits in acoustic evidence extraction than in downstream reasoning, suggesting that upstream perception is often the limiting factor. To address this problem, we propose EvA (Evidence-First Audio), a dual-path architecture that enhances acoustic evidence preservation through hierarchical aggregation and non-compressive, time-aligned fusion. We also build EvA-Perception, a large-scale training set with about 54K event-ordered captions and 500K evidence-grounded QA pairs. Under a unified zero-shot protocol, EvA achieves the best open-source \emph{Perception} results on MMAU, MMAR, and MMSU, with the largest gains on perception-heavy splits. Human evaluation on open-ended captioning further shows improved fine-grained acoustic coverage and caption quality. These results support the evidence-first hypothesis: stronger audio understanding depends on preserving acoustic evidence before reasoning. Project can be found at https://satsuki2486441738.github.io/EvA/.

cs.SD↗

arXiv2026

Grokability in five inequalities

In this note, we report five mathematical discoveries made in collaboration with Grok, all of which have been subsequently verified by the authors. These include an improved lower bound on the maximal Gaussian perimeter of convex sets in $\mathbb{R}^n$, sharper $L_2$-$L_1$ moment comparison inequalities on the Hamming cube $\{-1,1\}^n$, a strengthened autoconvolution inequality, improved asymptotic bounds on the size of the largest $g$-Sidon sets in $\{1,\dots,n\}$, and an optimal balanced Szarek's inequality.

math.PR↗

arXiv2026

Sharp isoperimetric inequalities on the Hamming cube II: The critical exponent

A sharp isoperimetric inequality for the Hamming cube is proved at the critical exponent $β=\frac12$. This follows up on previous work, where such bounds were established for $β$ near $\frac12$. As a consequence, this result settles a conjecture of Kahn and Park on cube partitions and yields a sharp $L^1$ Poincaré inequality for Boolean-valued functions. It also confirms a low-noise limit for balanced functions predicted by the Hellinger conjecture on noisy Boolean channels in information theory.

math.CA↗

arXiv2025

Counterexample to majority optimality in NICD with erasures

We asked GPT-5 Pro to look for counterexamples among a public list of open problems (the Simons ``Real Analysis in Computer Science'' collection). After several numerical experiments, it suggested a counterexample for the Non-Interactive Correlation Distillation (NICD) with erasures question: namely, a Boolean function on 5 bits that achieves a strictly larger value of $\mathbb{E}|f(z)|$ than the 5-bit majority function when the erasure parameter is $p=0.40.$ In this very short note we record the finding, state the problem precisely, give the explicit function, and verify the computation step by step by hand so that it can be checked without a computer. In addition, we show that for each fixed odd $n$ the majority is optimal (among unbiased Boolean functions) in a neighborhood of $p=0$. We view this as a little spark of an AI contribution in Theoretical Computer Science: while modern Large Language Models (LLMs) often assist with literature and numerics, here a concrete finite counterexample emerged.

math.PR↗

arXiv2025

FusionAudio-1.2M: Towards Fine-grained Audio Captioning with Multimodal Contextual Fusion

High-quality, large-scale audio captioning is crucial for advancing audio understanding, yet current automated methods often generate captions that lack fine-grained detail and contextual accuracy, primarily due to their reliance on limited unimodal or superficial multimodal information. Drawing inspiration from human auditory perception, which adeptly integrates cross-modal cues and performs sophisticated auditory scene analysis, we introduce a novel two-stage automated pipeline. This pipeline first employs specialized pretrained models to extract diverse contextual cues (e.g., speech, music, general sounds, and visual information from associated video). A large language model (LLM) then synthesizes these rich, multimodal inputs to generate detailed and context-aware audio captions. Key contributions of this work include: (1) the proposed scalable method for fine-grained audio caption generation; (2) FusionAudio, a new large-scale dataset comprising 1.2 million such detailed captions, combined with 6 million QA pairs; and (3) enhanced audio models developed using FusionAudio, specifically a CLAP-based audio encoder with superior audio-text alignment and instruction following. This paper paves the way for more nuanced and accurate automated understanding of complex audio environments. Code and data can be found in https://github.com/satsuki2486441738/FusionAudio.

cs.SD↗

arXiv2024

Jackson's inequality on the hypercube

We investigate the best constant $J(n,d)$ such that Jackson's inequality \[ \inf_{\mathrm{deg}(g) \leq d} \|f - g\|_{\infty} \leq J(n,d) \, s(f), \] holds for all functions $f$ on the hypercube $\{0,1\}^n$, where $s(f)$ denotes the sensitivity of $f$. We show that the quantity $J(n, 0.499n)$ is bounded below by an absolute positive constant, independent of $n$. This complements Wagner's theorem, which establishes that $J(n,d)\leq 1 $. As a first application we show that reverse Bernstein inequality fails in the tail space $L^{1}_{\geq 0.499n}$ improving over previously known counterexamples in $L^{1}_{\geq C \log \log (n)}$. As a second application, we show that there exists a function $f : \{0,1\}^n \to [-1,1]$ whose sensitivity $s(f)$ remains constant, independent of $n$, while the approximate degree grows linearly with $n$. This result implies that the sensitivity theorem $s(f) \geq Ω(\mathrm{deg}(f)^C)$ fails in the strongest sense for bounded real-valued functions even when $\mathrm{deg}(f)$ is relaxed to the approximate degree. We also show that in the regime $d = (1 - δ)n$, the bound \[ J(n,d) \leq C \min\{δ, \max\{δ^2, n^{-2/3}\}\} \] holds. Moreover, when restricted to symmetric real-valued functions, we obtain $J_{\mathrm{symmetric}}(n,d) \leq C/d$ and the decay $1/d$ is sharp. Finally, we present results for a subspace approximation problem: we show that there exists a subspace $E$ of dimension $2^{n-1}$ such that $\inf_{g \in E} \|f - g\|_{\infty} \leq s(f)/n$ holds for all $f$.

math.FA↗