Jiayang Zou

Jiayang Zou, Luyao Fan, Jiayang Gao et al.

We construct a smooth, strictly positive, Gaussian-decaying density on $\R^2$ 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. This complements the recent one-dimensional counterexample of Gu and Sellke. The counterexample is a small hexagonal perturbation on the triangular torus, transferred to $\R^2$ by a Gaussian envelope, and supported by explicit two-dimensional numerics. Finally, we initiate the study of the sharp constants $θ_d^*$ by proving $θ_1^*=1$, establishing monotonicity in the dimension, and recording an intrinsic simplex resonance family in every fixed dimension. The explicit two-dimensional counterexample was found by GPT-5.5 Pro.

Jiayang Zou, Luyao Fan, Jiayang Gao et al.

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.

Jiayang Gao, Tianyi Zheng, Jiayang Zou et al.

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.