There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the $\varphi^4$ model and weak lensing convergence maps with higher resolution than in previous works.

翻译：深度图像生成模型的显著成果与具有理论保证的经典算法之间存在日益扩大的差距。前者存在模式坍塌或记忆化问题，限制了其在科学数据中的应用；后者则需要如对数凹性等限制性假设以规避维度灾难。我们通过引入条件强对数凹（CSLC）模型部分弥合了这一差距——该模型将数据分布分解为一系列强对数凹条件概率分布的乘积形式。该分解通过适应数据分布的正交投影算子实现，使得尽管全局数据分布并非对数凹，仍能获得具有理论保证的高效参数估计与采样算法。我们证明，利用小波包正交投影算子，多个具有挑战性的多尺度过程满足条件对数凹性。数值实验展示了比先前工作更高分辨率的物理场结果，包括$\varphi^4$模型与弱引力透镜收敛图。