发表机构
University of California, Los Angeles; Shanghai Jiao Tong University（加州大学洛杉矶分校; 上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出各向异性膨胀变换，通过显式轮廓实现吉布斯类中任意正则化器的最优适配，并给出闭式解、泛化界及实验验证，提升变分反问题的数据自适应正则化性能。
AI 中文摘要
数据驱动反问题中的一个核心问题是如何构造一个适应数据分布几何结构的正则化器。最优正则化的近期研究表明，在广泛的吉布斯类中，与分布 $P$ 最匹配的正则化器由单一、方向相关的径向汇总统计量 $\ ho_P$ 决定。这提示了一种通过变换数据来控制正则化器几何结构的方法。特别是，哪些变换以简单、显式的方式作用于 $\ ho_P$？它们能否改善由此产生的变分问题的优化性质，或将固定的基础正则化器适应于数据？为回答这些问题，我们引入了各向异性膨胀，这是一种保持方向的映射，通过球面上的正轮廓沿其欧几里得射线对每个点进行重新缩放。尽管其简单，该族在径向汇总统计量空间上传递作用：任何吉布斯类中的目标正则化器都可以通过单个显式轮廓从任何源分布到达。我们刻画了由此产生的分布上的轨道结构，以闭式形式推导出将固定基础正则化器最优适应于数据的轮廓，并表明其相对于各向同性重缩放的改进由詹森间隙控制，该间隙对若干自然族被证明为正。我们还建立了联合学习基础正则化器和各向异性轮廓的有限样本泛化界，当轮廓的对数由线性特征模型或稀疏ReLU网络参数化时，具有显式速率。基于此理论，我们将基础正则化器和各向异性轮廓参数化，并从样本中联合学习它们，为变分反问题产生数据自适应正则化器。在受控的二维族和MNIST去噪上的实验表明，学习轮廓进一步提高了性能。
英文摘要
A central question in data-driven inverse problems is how to construct a regularizer that adapts to the geometry of the data distribution. Recent work in optimal regularization shows that, within a broad Gibbs class, the regularizer best matched to a distribution $P$ is determined by a single, direction-dependent radial summary statistic $ρ_P$. This suggests a way to control regularizer geometry by transforming the data. In particular, which transformations act on $ρ_P$ in a simple, explicit way? Can they improve the optimization properties of the resulting variational problems or adapt a fixed base regularizer to data? To address these questions, we introduce anisotropic dilation, a direction-preserving map that rescales each point along its Euclidean ray by a positive profile on the sphere. Despite its simplicity, this family acts transitively on the space of radial summary statistics: any target regularizer in the Gibbs class can be reached from any source distribution by a single explicit profile. We characterize the resulting orbit structure on distributions, derive in closed form the profile that optimally adapts a fixed base regularizer to the data, and show that its improvement over isotropic rescaling is governed by a Jensen gap that is provably positive for several natural families. We also establish finite-sample generalization bounds for jointly learning the base regularizer and anisotropic profile, with explicit rates when the logarithm of the profile is parameterized by linear feature models or sparse ReLU networks. Building on this theory, we parameterize the base regularizer and anisotropic profile and learn them jointly from samples, yielding a data-adaptive regularizer for variational inverse problems. Experiments on a controlled two-dimensional family and MNIST denoising show that learning the profile further improves performance.
Comments58 pages, 3 figures, 3 tables
