标题： 拉普拉斯正则化分层模型中的联合图学习和模型拟合 显示英文标题

标题： Joint Graph Learning and Model Fitting in Laplacian Regularized Stratified Models

摘要： 拉普拉斯正则化分层模型（LRSM）是利用由称为层（strata）的分类特征定义的子问题显式或隐式网络结构的模型（例如，年龄、地区、时间、预测范围等），并从相邻层获取数据以增强每个子问题的参数学习。它们已被广泛应用于机器学习和信号处理问题，包括但不限于时间序列预测、表示学习、图聚类、最大间隔分类和一般的少样本学习。然而，现有的关于LRSM的研究要么假设已知图，要么仅限于特定应用。在本文中，我们首先展示了图权重在LRSM中的重要性和敏感性，并证明当节点间的参数规模和样本数量严重不平衡时，敏感性可以是任意大的。然后我们提出了一种通用方法，通过解决一个优化问题来联合学习图并拟合模型参数。我们从图连通性视角和端到端贝叶斯视角解释了所提出的公式，并提出了一种高效的算法来解决问题。尽管拉普拉斯正则化项通常在现有文献中需要全局强光滑性，但我们的优化算法也提供了收敛保证，这可能具有独立的兴趣。最后，我们通过各种现实世界的数值例子说明了我们的方法相比现有方法的效率。 显示英文摘要

摘要： Laplacian regularized stratified models (LRSM) are models that utilize the explicit or implicit network structure of the sub-problems as defined by the categorical features called strata (e.g., age, region, time, forecast horizon, etc.), and draw upon data from neighboring strata to enhance the parameter learning of each sub-problem. They have been widely applied in machine learning and signal processing problems, including but not limited to time series forecasting, representation learning, graph clustering, max-margin classification, and general few-shot learning. Nevertheless, existing works on LRSM have either assumed a known graph or are restricted to specific applications. In this paper, we start by showing the importance and sensitivity of graph weights in LRSM, and provably show that the sensitivity can be arbitrarily large when the parameter scales and sample sizes are heavily imbalanced across nodes. We then propose a generic approach to jointly learn the graph while fitting the model parameters by solving a single optimization problem. We interpret the proposed formulation from both a graph connectivity viewpoint and an end-to-end Bayesian perspective, and propose an efficient algorithm to solve the problem. Convergence guarantees of the proposed optimization algorithm is also provided despite the lack of global strongly smoothness of the Laplacian regularization term typically required in the existing literature, which may be of independent interest. Finally, we illustrate the efficiency of our approach compared to existing methods by various real-world numerical examples.