Learning with latent group sparsity via heat flow dynamics on networks
利用网络热流动力学学习潜在组稀疏结构AI 解读 ↓
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利用网络热流动力学学习潜在组稀疏结构
- 研究问题
- 当解释变量具有未知的分组或聚类结构时，如何在无需事先识别组别和昂贵聚类预处理的情况下完成统计学习？
- 方法与思路
- 作者依据网络拉普拉斯几何构造惩罚，通过局部热流动力学计算，并提供从数据构建网络的方法及样本复杂度分析。
- 材料陈述的贡献
- 摘要称给出性能与样本复杂度定理，多种设定下热流运行时间仅随维度对数增长，并展示多个领域的真实数据应用。
- 阅读关注点
- 应核对数据构网误差是否纳入理论、组结构与网络社区的对应条件，以及对数运行时间结论的适用范围和实际计算成本。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实；摘要未列定理条件、界的具体形式或应用数值，不能判断普遍性能优势。
原文摘要
Group or cluster structure on explanatory variables in machine learning problems is a very general phenomenon, which has attracted broad interest from practitioners and theoreticians alike. In this work we contribute an approach to learning under such group structure, that does not require prior information on the group identities. Our paradigm is motivated by the Laplacian geometry of an underlying network with a related community structure, and proceeds by directly incorporating this into a penalty that is effectively computed via a heat flow-based local network dynamics. In fact, we demonstrate a procedure to construct such a network based on the available data. Notably, we dispense with computationally intensive pre-processing involving clustering of variables, spectral or otherwise. Our technique is underpinned by rigorous theorems that guarantee its effective performance and provide bounds on its sample complexity. In particular, in a wide range of settings, it provably suffices to run the heat flow dynamics for time that is only logarithmic in the problem dimensions. We explore in detail the interfaces of our approach with key statistical physics models in network science, such as the Gaussian Free Field and the Stochastic Block Model. We validate our approach by successful applications to real-world data from a wide array of application domains, including computer science, genetics, climatology and economics. Our work raises the possibility of applying similar diffusion-based techniques to classical learning tasks, exploiting the interplay between geometric, dynamical and stochastic structures underlying the data.