加权低秩逼近的重加权解
Reweighted Solutions for Weighted Low Rank Approximation
AI总结：
针对加权低秩逼近（WLRA）的NP困难性，提出一种利用权重矩阵对低秩解进行重加权的松弛解，在低秩权重下提供可证明的逼近保证，并首次给出匹配的通信复杂度界及特征选择相对误差保证。
摘要：
加权低秩逼近（WLRA）是一个重要但在计算上具有挑战性的原语，其应用范围涵盖统计分析、模型压缩和信号处理。为了应对该问题的NP困难性，先前的工作考虑使用启发式、双标准或固定参数可解的算法来解决此问题。在这项工作中，我们引入了一种针对WLRA的全新松弛解，其输出的矩阵不一定是低秩的，但可以使用极少的参数进行存储，并且在权重矩阵具有低秩时提供可证明的逼近保证。我们的核心思想是利用权重矩阵本身对低秩解进行重加权，这给出了一种极其简单的算法，在模型压缩应用和合成数据集上展现出卓越的实证性能。对于一个与此问题相关的自然分布式问题，我们的算法还给出了近乎最优的通信复杂度上界，并证明了与之匹配的通信下界。综合来看，我们的通信复杂度界表明，权重矩阵的秩可证明地参数化了WLRA的通信复杂度。我们还首次获得了针对带有加权目标的特征选择的相对误差保证。
英文摘要：
Weighted low rank approximation (WLRA) is an important yet computationally challenging primitive with applications ranging from statistical analysis, model compression, and signal processing. To cope with the NP-hardness of this problem, prior work considers heuristics, bicriteria, or fixed parameter tractable algorithms to solve this problem. In this work, we introduce a new relaxed solution to WLRA which outputs a matrix that is not necessarily low rank, but can be stored using very few parameters and gives provable approximation guarantees when the weight matrix has low rank. Our central idea is to use the weight matrix itself to reweight a low rank solution, which gives an extremely simple algorithm with remarkable empirical performance in applications to model compression and on synthetic datasets. Our algorithm also gives nearly optimal communication complexity bounds for a natural distributed problem associated with this problem, for which we show matching communication lower bounds. Together, our communication complexity bounds show that the rank of the weight matrix provably parameterizes the communication complexity of WLRA. We also obtain the first relative error guarantees for feature selection with a weighted objective.
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