arXiv subjects

Luyao Fan

Publications and source records attributed to Luyao Fan.

8 recordsLinked to original sources

arXiv2026

An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture

We construct smooth, strictly log-concave counterexamples to the Gaussian completely monotone conjecture in every dimension. In one dimension, they form an explicit family $f_m$ whose signed $m$th entropy derivative at time zero is negative for every sufficiently large $m$; the inequality persists for all sufficiently small positive times. Tensorization with a broad Gaussian factor gives the higher-dimensional examples. The argument is analytic and self-contained. It reduces the sign to a two-frequency entropy calculation on the circle and transfers the resulting asymptotic to the real line through an exact heat-flow formula for Gaussian-windowed Fourier modes. The proof was developed by GPT-5.6 Sol Pro under the authors' guidance.

cs.IT↗

arXiv2026

The Randomized Query Complexity of Finding Minimal Elements in Bounded-Width Posets

We study the zero-error randomized query complexity of finding all minimal elements in an unknown $n$-element poset of width at most $w$. Previous work of Daskalakis, Karp, Mossel, Riesenfeld, and Verbin established a randomized upper bound with leading term $\frac{w+1}{2}n$, while the corresponding lower bound left a multiplicative gap in the leading constant that approaches a factor of 2 as $w$ grows. We prove the finite lower bound $ R^{\mathrm{LV}}_{n,w}\ge \frac{w+1}{2}n-\frac{w(w+3)}4 +w\left(1-\frac1w\right)^n +\frac{w(w-1)}4\left(1-\frac2w\right)^n. $ Consequently, for every fixed $w$, $ R^{\mathrm{LV}}_{n,w} = \left(\frac{w+1}{2}+o(1)\right)n. $ Thus the known randomized upper bound has the correct asymptotic leading constant for every fixed width. The argument is based on a pairwise accounting of incomparable queries under a random-chain hard distribution, using a component-flip involution and a unique ownership property for incomparable comparisons. Generative AI was used in the preparation of this manuscript.

cs.DS↗

arXiv2026

A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow

We construct a smooth, strictly positive, Gaussian-decaying density on $\mathbb{R}^2$ for which Fisher information along the heat flow is not log-convex. This disproves the Cheng--Geng log-convexity conjecture in dimension two and, by tensorization, in every dimension $d\ge2$. Consequently, the multidimensional forms of the Gaussian completely monotone conjecture, McKean's conjecture, and Toscani's entropy power conjecture also fail, complementing the one-dimensional counterexample of Gu and Sellke. Our construction is a small hexagonal perturbation on the triangular torus, transferred to $\mathbb{R}^2$ by a Gaussian envelope and supported by explicit two-dimensional numerics. We also initiate the study of the sharp constants $\theta_d^*$ by proving $\theta_1^*=1$, establishing monotonicity in the dimension, and identifying a dichotomy for the asymptotic constant $\theta_\infty^*$ governed by the sign of $\mathcal{D}$. The explicit two-dimensional counterexample was found by GPT-5.5 Pro.

cs.IT↗

arXiv2026

C$^2$FG: Control Classifier-Free Guidance via Score Discrepancy Analysis

Classifier-Free Guidance (CFG) is a cornerstone of modern conditional diffusion models, yet its reliance on the fixed or heuristic dynamic guidance weight is predominantly empirical and overlooks the inherent dynamics of the diffusion process. In this paper, we provide a rigorous theoretical analysis of the Classifier-Free Guidance. Specifically, we establish strict upper bounds on the score discrepancy between conditional and unconditional distributions at different timesteps based on the diffusion process. This finding explains the limitations of fixed-weight strategies and establishes a principled foundation for time-dependent guidance. Motivated by this insight, we introduce \textbf{Control Classifier-Free Guidance (C$^2$FG)}, a novel, training-free, and plug-in method that aligns the guidance strength with the diffusion dynamics via an exponential decay control function. Extensive experiments demonstrate that C$^2$FG is effective and broadly applicable across diverse generative tasks, while also exhibiting orthogonality to existing strategies.

cs.LG↗

arXiv2026

Rate-distortion Theory with Lower Semi-continuous Distortion on Noncompact Alphabets

In this paper, we study rate-distortion theory for general sources with an emphasis on the existence of optimal reconstruction distributions on noncompact alphabets. Classical attainability results typically rely on compactness of the reproduction alphabet together with continuity of the distortion function, which may fail in many noncompact settings. We identify two complementary existence mechanisms under lower semi-continuity on locally compact Polish alphabets. For bounded distortions, we prove that the rate-distortion infimum is attained via the one-point compactification argument. For unbounded coercive distortions, we establish existence via concentration-compactness. We also give several counterexamples showing that our attainability results are close to sharp. Our results provide a unified and transparent existence theorem for rate-distortion problems with lower semi-continuous distortions.

cs.IT↗

arXiv2025

A Revisit to Rate-distortion Theory via Optimal Weak Transport

This paper revisits the rate-distortion theory from the perspective of optimal weak transport, as recently introduced by Gozlan et al. While the conditions for optimality and the existence of solutions are well-understood in the case of discrete alphabets, the extension to abstract alphabets requires more intricate analysis. Within the framework of weak transport problems, we derive a parametric representation of the rate-distortion function, thereby connecting the rate-distortion function with the Schr\"odinger bridge problem, and establish necessary conditions for its optimality. As a byproduct of our analysis, we reproduce K. Rose's conclusions regarding the achievability of Shannon lower bound concisely, without reliance on variational calculus.

cs.IT↗

arXiv2025

Differential Properties of Information in Jump-diffusion Channels

We propose a channel modeling using jump-diffusion processes, and study the differential properties of entropy and mutual information. By utilizing the Kramers-Moyal and Kolmogorov-Feller equations, we express the mutual information between the input and the output in series and integral forms, presented by Fisher-type information and mismatched KL divergence. We extend de Bruijn's identity and the I-MMSE relation to encompass general Markov processes.

cs.IT↗

arXiv2025

Convexity of Mutual Information along the Fokker-Planck Flow

We study the convexity of mutual information as a function of time along the Fokker-Planck flow. The results are generalizations of that along heat flow and Ornstein-Ulenbeck flow, which were established by A. Wibisono and V. Jog. We prove the existence and uniqueness of the classical solutions to a class of Fokker-Planck equations and then we obtain the second derivative of mutual information along the Fokker-Planck equation. If the initial distribution is sufficiently strongly log-concave compared to the steady state, then mutual information always preserves convexity under suitable conditions. In particular, if there exists some time point at which the distribution is sufficiently strongly log-concave, then mutual information will preserve convexity after that time.

cs.IT↗