基于非负低秩半定规划的统计最优K均值聚类
Statistically Optimal K-means Clustering via Nonnegative Low-rank Semidefinite Programming
AI总结：
针对K均值聚类中SDP松弛成本高、NMF缺乏理论保证的问题，提出基于Burer--Monteiro分解的类NMF非负低秩SDP算法，兼具可扩展性与统计最优性，误聚类误差显著更低。
摘要：
$K$-means聚类是一种广泛用于在大型数据集中识别模式的机器学习方法。近期，学界提出了用于求解$K$-means优化问题的半定规划（SDP）松弛方法，这类方法具备强统计最优性保证。然而，SDP求解器的实现成本极高，导致这些保证无法应用于实际数据集。相比之下，非负矩阵分解（NMF）是机器学习从业者广泛使用的简单聚类算法，但它缺乏坚实的统计基础与理论保证。本文提出一种类NMF算法，采用非凸的Burer--Monteiro分解方法求解SDP松弛$K$-means公式的非负低秩约束问题。所得算法与当前最优NMF算法同样简单且可扩展，同时具备与SDP相同的强统计最优性保证。实验中我们观察到，该算法在保持可扩展性的同时，相比现有最优方法的误聚类误差显著更低。
英文摘要：
$K$-means clustering is a widely used machine learning method for identifying patterns in large datasets. Recently, semidefinite programming (SDP) relaxations have been proposed for solving the $K$-means optimization problem, which enjoy strong statistical optimality guarantees. However, the prohibitive cost of implementing an SDP solver renders these guarantees inaccessible to practical datasets. In contrast, nonnegative matrix factorization (NMF) is a simple clustering algorithm widely used by machine learning practitioners, but it lacks a solid statistical underpinning and theoretical guarantees. In this paper, we consider an NMF-like algorithm that solves a nonnegative low-rank restriction of the SDP-relaxed $K$-means formulation using a nonconvex Burer--Monteiro factorization approach. The resulting algorithm is as simple and scalable as state-of-the-art NMF algorithms while also enjoying the same strong statistical optimality guarantees as the SDP. In our experiments, we observe that our algorithm achieves significantly smaller mis-clustering errors compared to the existing state-of-the-art while maintaining scalability.
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