arXiv subjects

Ruijia Cao

Publications and source records attributed to Ruijia Cao.

4 recordsLinked to original sources

arXiv2026

On the Robustness of Distribution Support under Diffusion Guidance

Diffusion guidance is a powerful technique that enables controllable and high-fidelity sample generation with diffusion models. At a high level, it modifies the score function by incorporating a guidance term that steers the generative process toward a desired condition. Despite its empirical success, the theoretical properties of diffusion guidance remain largely unexplored, and it is not well understood why it consistently produces high-quality samples. In this work, we explain the effectiveness of diffusion guidance by establishing a robustness of support property. Specifically, we show that, given exact access to the score functions, guided diffusion processes almost always generate samples that remain close to the target support. This property is particularly desirable, as samples that lie off the support are often structurally implausible and may adversely affect downstream tasks. Our analysis covers both Denoising Diffusion Implicit Models (DDIM) and Denoising Diffusion Probabilistic Models (DDPM), and applies to a wide range of discretization schemes induced by exponential integrators. Our results provide a rigorous foundation for understanding why diffusion guidance produces physically meaningful and structurally plausible samples.

cs.LG↗

arXiv2024

Information geometric regularization of unidimensional pressureless Euler equations yields global strong solutions

Partial differential equations describing compressible fluids are prone to the formation of shock singularities, arising from faster upstream fluid particles catching up to slower, downstream ones. In geometric terms, this causes the deformation map to leave the manifold of diffeomorphisms. Information geometric regularization addresses this issue by changing the manifold geometry to make it geodesically complete. Empirical evidence suggests that this results in smooth solutions without adding artificial viscosity. This work makes a first step towards understanding this phenomenon rigorously, in the setting of the unidimensional pressureless Euler equations. It shows that their information geometric regularization has smooth global solutions. By establishing $\Gamma$-convergence of its variational description, it proves convergence of these solutions to entropy solutions of the nominal problem, in the limit of vanishing regularization parameter. A consequence of these results is that manifolds of unidimensional diffeomorphisms with information geometric regularization are geodesically complete.

math.AP↗

arXiv2023

Information geometric regularization of the barotropic Euler equation

Shock waves in gas dynamics feature jump discontinuities that hinder numerical simulations. Viscous regularizations are prone to excessive dissipation of fine-scale structures. In this work, we propose the first inviscid regularization of the multidimensional Euler equation based on ideas from semidefinite programming, information geometry, geometric hydrodynamics, and nonlinear elasticity. The Lagrangian flow maps of Euler solutions are a dynamical system on the manifold of diffeomorphisms. We observe that shock formation arises from the manifold's geodesic incompleteness. Our regularization embeds it into an ambient space equipped with the information geometry of the logarithmic barrier function. Thus, the diffeomorphism manifold inherits a geodesically complete geometry. The resulting regularized conservation law replaces shocks with smooth profiles without affecting oscillatory structures. One and two-dimensional numerical experiments show its practical potential to enable higher-order methods without explicit shock capturing. While we focus on the barotropic Euler equations for concreteness and simplicity of exposition, our regularization easily extends to more general Euler and Navier-Stokes-type equations. Our approach regularizes the Wasserstein geometry of the mass density with its information geometry. The former captures the natural trajectories of physical particles and the latter that of statistical estimators. Information geometric regularization accounts for the mass density's dual nature as a statistical/computational tool summarizing the motion of physical particles. Thus, our work is a starting point for information geometric mechanics that views solutions of continuum mechanical PDEs as parameters of statistical models for unresolved scales and uses their information geometry to evolve them in time.

math.NA↗

arXiv2023

Counting colorings of triangle-free graphs

By a theorem of Johansson, every triangle-free graph $G$ of maximum degree $Δ$ has chromatic number at most $(C+o(1))Δ/\log Δ$ for some universal constant $C > 0$. Using the entropy compression method, Molloy proved that one can in fact take $C = 1$. Here we show that for every $q \geq (1 + o(1))Δ/\log Δ$, the number $c(G,q)$ of proper $q$-colorings of $G$ satisfies $c(G, q) \,\geq\, \left(1 - \frac{1}{q}\right)^m ((1-o(1))q)^n$, where $n = |V(G)|$ and $m = |E(G)|$. Except for the $o(1)$ term, this lower bound is best possible as witnessed by random $Δ$-regular graphs. When $q = (1 + o(1)) Δ/\log Δ$, our result yields the inequality $c(G,q) \,\geq\, \exp\left((1 - o(1)) \frac{\log Δ}{2} n\right)$, which improves an earlier bound of Iliopoulos and yields the optimal value for the constant factor in the exponent. Furthermore, this result implies the optimal lower bound on the number of independent sets in $G$ due to Davies, Jenssen, Perkins, and Roberts. An important ingredient in our proof is the counting method that was recently developed by Rosenfeld. As a byproduct, we obtain an alternative proof of Molloy's bound $χ(G) \leq (1 + o(1))Δ/\log Δ$ using Rosenfeld's method in place of entropy compression (other proofs of Molloy's theorem using Rosenfeld's technique were given independently by Hurley and Pirot and Martinsson).

math.CO↗