发表机构
LMU Munich; Munich Center for Machine Learning (MCML); DLR-German Aerospace Center; University of Tromsø（慕尼黑大学; 慕尼黑机器学习中心; 德国航空航天中心; 特罗姆瑟大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对高维结构化高斯目标预测建模的协方差矩阵近似问题，提出SCORE框架，将评分规则训练与谱空间协方差近似结合，在时间序列预测等任务上实现低计算成本下的性能提升。
AI 中文摘要
基于神经网络的高维结构化高斯目标预测建模，需要高效、数值稳定且具有表达能力的协方差矩阵近似方法。我们提出SCORE：一种可扩展框架，将评分规则训练与在谱空间中学习的高表达能力协方差近似相结合。对于d维数据，学习任务被分解为学习边际分布和学习结构化相关矩阵，这使得在具有线性存储和O(d log d)成本的情况下能够捕捉密集依赖关系。我们利用闭式高斯核评分进行训练，该评分即使对于退化协方差也保持定义，并在优化过程中具有有界梯度。我们表征了可逆变换下的核评分，并证明其在酉变换下具有精确不变性。在总体层面，我们的两级目标恢复了真实边际分布，并将目标相关投影到可表示类中；有限样本PAC界表明，两个阶段的误差呈加性进入。我们在多种具有常见假设高斯域的任务上评估了我们的模型：时间序列预测、单目深度估计和空间天气预测，结果显示在更低计算成本下性能得到提升。
英文摘要
Neural network-based predictive modeling with high-dimensional structured Gaussian targets requires an efficient and numerically stable, yet expressive approximation of the covariance matrix. We propose SCORE: a scalable framework, combining scoring rule training with an expressive covariance approximation learned in spectral space. For $d$-dimensional data, the learning task is decomposed into learning the marginal distributions and learning a structured correlation matrix, which enables dense dependencies with linear storage and $\mathcal{O}(d\log d)$ cost. We utilize the closed form Gaussian kernel score for training, which remains defined even for degenerate covariances and admits bounded gradients during optimization. We characterize kernel scores under invertible transforms and prove exact invariance under unitary transforms. At population level, our two-level objective recovers the true marginals and projects the target correlation onto the representable class; finite-sample PAC bounds show that the errors of the two stages enter additively. We evaluate our model on a variety of tasks with a commonly assumed Gaussian domain: Time-series forecasting, monocular depth estimation, and spatial weather prediction, showing improved performance at lower computational cost.
